![SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB®
,
THIRD EDITION
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Library of Congress Cataloging-in-Publication Data
Roberts, Michael J., Dr.
Signals and systems : analysis using transform methods and MATLAB /
Michael J. Roberts, professor, Department of Electrical and Computer
Engineering, University of Tennessee.
Third edition. | New York, NY : McGraw-Hill Education, [2018] |
Includes bibliographical references (p. 786–787) and index.
LCCN 2016043890 | ISBN 9780078028120 (alk. paper)
LCSH: Signal processing. | System analysis. | MATLAB.
LCC TK5102.9 .R63 2018 | DDC 621.382/2—dc23 LC record
available at https://lccn.loc.gov/2016043890
The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website
does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education
does not guarantee the accuracy of the information presented at these sites.
mheducation.com/highered](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-8-320.jpg)
![1.2 Types of Signals 3
The term system even encompasses things such as the stock market, government,
weather, the human body and the like. They all respond when excited. Some systems
are readily analyzed in detail, some can be analyzed approximately, but some are so
complicated or difficult to measure that we hardly know enough to understand them.
1.2 TYPES OF SIGNALS
There are several broad classifications of signals: continuous-time, discrete-time,
continuous-value, discrete-value, random and nonrandom. A continuous-time sig-
nal is defined at every instant of time over some time interval. Another common name
for some continuous-time signals is analog signal, in which the variation of the signal
with time is analogous (proportional) to some physical phenomenon. All analog sig-
nals are continuous-time signals but not all continuous-time signals are analog signals
(Figure 1.6 through Figure 1.8).
Sampling a signal is acquiring values from a continuous-time signal at discrete
points in time. The set of samples forms a discrete-time signal. A discrete-time signal
Figure 1.5
Modern farm tractor with enclosed cab
© Royalty-Free/Corbis
Figure 1.6
Examples of continuous-time and discrete-time signals
n
x[n]
Discrete-Time
Continuous-Value
Signal
t
x(t)
Continuous-Time
Continuous-Value
Signal](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-29-320.jpg)
![1.2 Types of Signals 5
Figure 1.9
Sampling, quantization and encoding of a signal to illustrate various signal types
t
kΔt (k+1)Δt (k+2)Δt
(k–1)Δt
kΔt (k+1)Δt (k+2)Δt
(k–1)Δt
xs[n]
n
k k+1 k+2
k–1
xsq[n]
n
k k+1 k+2
k–1
xsqe(t)
t
111 001 111 011
Continuous-Value
Continuous-Time
Signal
Continuous-Value
Discrete-Time
Signal
Discrete-Value
Discrete-Time
Signal
Discrete-Value
Continuous-Time
Signal
Sampling
Quantization
Encoding
x(t)
Figure 1.10
Asynchronous serial binary ASCII-encoded voltage signal for the
word SIGNAL
0 1 2 3 4 5 6 7
–1
0
1
2
3
4
5
6
Time, t (ms)
Voltage,
v(t)
(V)
Serial Binary Voltage Signal for the ASCII Message “SIGNAL”
S I G N A L
One common use of binary digital signals is to send text messages using the
American Standard Code for Information Interchange (ASCII). The letters of the al-
phabet, the digits 0–9, some punctuation characters and several nonprinting control
characters, for a total of 128 characters, are all encoded into a sequence of 7 binary
bits. The 7 bits are sent sequentially, preceded by a start bit and followed by 1 or 2
stop bits for synchronization purposes. Typically, in direct-wired connections between
digital equipment, the bits are represented by a higher voltage (2 to 5V) for a 1 and a
lower voltage level (around 0V) for a 0. In an asynchronous transmission using one
start and one stop bit, sending the message SIGNAL, the voltage versus time would
look as illustrated in Figure 1.10.](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-31-320.jpg)
![1.2 Types of Signals 7
and more easily manipulated than in the time domain. (In the frequency domain, signals
are described in terms of the frequencies they contain.) Without frequency-domain analy-
sis, design and analysis of many systems would be considerably more difficult.
Discrete-time signals are only defined at discrete points in time. Figure 1.14
illustrates some discrete-time signals.
Figure 1.12
A continuous-time signal described by a
mathematical function
...
...
–50
50
t = 10 ms
t
x(t)
Figure 1.13
A second continuous-time signal
...
...
5
t = 20 μs
t
x(t)
Figure 1.14
Some discrete-time signals
n
x[n]
n
x[n]
n
x[n]
n
x[n]
So far all the signals we have considered have been described by functions of time.
An important class of “signals” is functions of space instead of time: images. Most
of the theories of signals, the information they convey and how they are processed by
systems in this text will be based on signals that are a variation of a physical phenome-
non with time. But the theories and methods so developed also apply, with only minor
modifications, to the processing of images. Time signals are described by the variation
of a physical phenomenon as a function of a single independent variable, time. Spa-
tial signals, or images, are described by the variation of a physical phenomenon as a](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-33-320.jpg)
![Ch ap ter 1 Introduction
10
of change of water volume in the tank is the difference between the input volumetric
flow and the output volumetric flow and the volume of water is the cross-sectional area
of the tank times the height of the water. All these factors can be combined into one
differential equation for the water level h1 (t).
A 1 d
__
dt
( h 1 (t)) +
A 2 √
___________
2g[h 1 (t) − h 2 ] = f 1 (t) (1.2)
The water level in the tank is graphed in Figure 1.18 versus time for four volumetric
inflows under the assumption that the tank is initially empty.
Figure 1.17
Tank with orifice being filled from above
h (t)
1
h2
v (t)
2
f (t)
2
f (t)
1
A1
A2
Figure 1.18
Water level versus time for four different volumetric inflows with the
tank initially empty
0 1000 2000 3000 4000 5000 6000 7000 8000
0
0.5
1
1.5
2
2.5
3
3.5
Volumetric Inflow = 0.001 m3/s
Volumetric Inflow = 0.002 m3/s
Volumetric Inflow = 0.003 m3/s
Volumetric Inflow = 0.004 m3/s
Tank Cross-Sectional Area = 1 m2
Orifice Area = 0.0005 m2
Time, t (s)
Water
Level,
h
1
(t)
(m)
As the water flows in, the water level increases, and that increases the water out-
flow. The water level rises until the outflow equals the inflow. After that time the water
level stays constant. Notice that when the inflow is increased by a factor of two, the
final water level is increased by a factor of four. The final water level is proportional
to the square of the volumetric inflow. That fact makes the differential equation that
models the system nonlinear.](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-36-320.jpg)
![1.3 Examples of Systems 11
A DISCRETE-TIME SYSTEM
Discrete-time systems can be designed in multiple ways. The most common practical
example of a discrete-time system is a computer. A computer is controlled by a clock
that determines the timing of all operations. Many things happen in a computer at the
integrated circuit level between clock pulses, but a computer user is only interested in
what happens at the times of occurrence of clock pulses. From the user’s point of view,
the computer is a discrete-time system.
We can simulate the action of a discrete-time system with a computer program.
For example,
yn = 1 ; yn1 = 0 ;
while 1,
yn2 = yn1 ; yn1 = yn ; yn = 1.97*yn1 − yn2 ;
end
This computer program (written in MATLAB) simulates a discrete-time system
with an output signal y that is described by the difference equation
y[n] = 1.97y[n − 1] − y[n − 2] (1.3)
along with initial conditions y[0] = 1 and y[−1] = 0. The value of y at any time index
n is the sum of the previous value of y at discrete time n − 1 multiplied by 1.97, minus
the value of y previous to that at discrete time n − 2. The operation of this system can
be diagrammed as in Figure 1.19.
In Figure 1.19, the two squares containing the letter D are delays of one in discrete
time, and the arrowhead next to the number 1.97 represents an amplifier that multiplies
the signal entering it by 1.97 to produce the signal leaving it. The circle with the plus
sign in it is a summing junction. It adds the two signals entering it (one of which is
negated first) to produce the signal leaving it. The first 50 values of the signal produced
by this system are illustrated in Figure 1.20.
The system in Figure 1.19 could be built with dedicated hardware. Discrete-time
delay can be implemented with a shift register. Multiplication by a constant can be
done with an amplifier or with a digital hardware multiplier. Summation can also be
done with an operational amplifier or with a digital hardware adder.
Figure 1.20
Signal produced by the discrete-time system in Figure 1.19
n
50
y[n]
–6
6
Figure 1.19
Discrete-time system example
y[n]
y[n−2]
y[n−1]
1.97
+
–
D
D](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-37-320.jpg)
![Ch ap ter 1 Introduction
12
FEEDBACK SYSTEMS
Another important aspect of systems is the use of feedback to improve system perfor-
mance. In a feedback system, something in the system observes its response and may
modify the input signal to the system to improve the response. A familiar example is
a thermostat in a house that controls when the air conditioner turns on and off. The
thermostat has a temperature sensor. When the temperature inside the thermostat ex-
ceeds the level set by the homeowner, a switch inside the thermostat closes and turns
on the home air conditioner. When the temperature inside the thermostat drops a small
amount below the level set by the homeowner, the switch opens, turning off the air
conditioner. Part of the system (a temperature sensor) is sensing the thing the system is
trying to control (the air temperature) and feeds back a signal to the device that actually
does the controlling (the air conditioner). In this example, the feedback signal is simply
the closing or opening of a switch.
Feedback is a very useful and important concept and feedback systems are every-
where. Take something everyone is familiar with, the float valve in an ordinary flush
toilet. It senses the water level in the tank and, when the desired water level is reached,
it stops the flow of water into the tank. The floating ball is the sensor and the valve to
which it is connected is the feedback mechanism that controls the water level.
If all the water valves in all flush toilets were exactly the same and did not change
with time, and if the water pressure upstream of the valve were known and constant,
and if the valve were always used in exactly the same kind of water tank, it should be
possible to replace the float valve with a timer that shuts off the water flow when the
water reaches the desired level, because the water would always reach the desired level
at exactly the same elapsed time. But water valves do change with time and water pres-
sure does fluctuate and different toilets have different tank sizes and shapes. Therefore,
to operate properly under these varying conditions the tank-filling system must adapt
by sensing the water level and shutting off the valve when the water reaches the desired
level. The ability to adapt to changing conditions is the great advantage of feedback
methods.
There are countless examples of the use of feedback.
1. Pouring a glass of lemonade involves feedback. The person pouring watches
the lemonade level in the glass and stops pouring when the desired level is
reached.
2. Professors give tests to students to report to the students their performance levels.
This is feedback to let the student know how well she is doing in the class so she
can adjust her study habits to achieve her desired grade. It is also feedback to the
professor to let him know how well his students are learning.
3. Driving a car involves feedback. The driver senses the speed and direction of the
car, the proximity of other cars and the lane markings on the road and constantly
applies corrective actions with the accelerator, brake and steering wheel to
maintain a safe speed and position.
4. Without feedback, the F-117 stealth fighter would crash because it is
aerodynamically unstable. Redundant computers sense the velocity, altitude,
roll, pitch and yaw of the aircraft and constantly adjust the control surfaces to
maintain the desired flight path (Figure 1.21).
Feedback is used in both continuous-time systems and discrete-time systems. The
system in Figure 1.22 is a discrete-time feedback system. The response of the system
y[n] is “fed back” to the upper summing junction after being delayed twice and multi-
plied by some constants.](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-38-320.jpg)
![1.3 Examples of Systems 13
Let this system be initially at rest, meaning that all signals throughout the system
are zero before time index n = 0. To illustrate the effects of feedback let a = 1, let
b = −1.5, let c = 0.8 and let the input signal x[n] change from 0 to 1 at n = 0 and stay
at 1 for all time, n ≥ 0. We can see the response y [n] in Figure 1.23.
Now let c = 0.6 and leave a and b the same. Then we get the response in Figure 1.24.
Now let c = 0.5 and leave a and b the same. Then we get the response in Figure 1.25.
The response in Figure 1.25 increases forever. This last system is unstable because
a bounded input signal produces an unbounded response. So feedback can make a
system unstable.
Figure 1.21
The F-117A Nighthawk stealth fighter
© Vol. 87/Corbis
Figure 1.22
A discrete-time feedback system
x[n]
+
+
+
+
+
–
y[n]
D
D
b
a
c
Figure 1.23
Discrete-time system response with
b = −1.5 and c = 0.8
n
60
y[n]
6
a = 1, b = –1.5, c = 0.8
Figure 1.25
Discrete-time system response with
b = −1.5 and c = 0.5
n
60
y[n]
140
a = 1, b = –1.5, c = 0.5
Figure 1.24
Discrete-time system response with
b = −1.5 and c = 0.6
n
60
y[n]
12 a = 1, b = –1.5, c = 0.6
The system illustrated in Figure 1.26 is an example of a continuous-time feedback
system. It is described by the differential equation yʺ(t) + ay(t) = x(t). The homoge-
neous solution can be written in the form
y h (t) =
K h1 sin (
√
__
a t)+
K h2 cos (
√
__
a t). (1.4)
If the excitation x(t) is zero and the initial value y(
t 0 ) is nonzero or the initial deriva-
tive of y(t) is nonzero and the system is allowed to operate in this form after t =
t 0, y(t)
Figure 1.26
Continuous-time feedback system
x(t) y(t)
a
∫ ∫](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-39-320.jpg)
![Ch ap ter 1 Introduction
18
4. A chemical spectroscopy system in which white light excites the specimen and
the spectrum of transmitted light is the response.
5. A telephone network for which voices and data are the input signals and
reproductions of those voices and data at a distant location are the output signals.
6. Earth’s atmosphere, which is excited by energy from the sun and for which the
responses are ocean temperature, wind, clouds, humidity and so on. In other
words, the weather is the response.
7. A thermocouple excited by the temperature gradient along its length for which
the voltage developed between its ends is the response.
8. A trumpet excited by the vibration of the player’s lips and the positions of the
valves for which the response is the tone emanating from the bell.
The list is endless. Any physical entity can be thought of as a system, because if
we excite it with physical energy, it has a physical response.
1.5 USE OF MATLAB®
Throughout the text, examples will be presented showing how signal and system analy-
sis can be done using MATLAB. MATLAB is a high-level mathematical tool available
on many types of computers. It is very useful for signal processing and system analysis.
There is an introduction to MATLAB in Web Appendix A.
n
N[n] D[n]
N[n]
P[n]
Number of Cars
Crossing an Intersection
Between Red Lights
United States Population
2
4
6
8
n
Ball-Bearing Manufacturer’s
Quality Control Chart
for Diameter
1 cm
1.01 cm
0.99 cm
n
1800 1900 2000
300
Million
World War II
Great Depression
World War I
US Civil War
n
1950 2000
2500
Number of Annual Sunspots
Figure 1.33
Examples of signals that are functions of a discrete independent variable](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-44-320.jpg)
![Ch ap ter 2 Mathematical Description of Continuous-Time Signals
20
CH APTER G OA L S
1. To define some mathematical functions that can be used to describe signals
2. To develop methods of shifting, scaling and combining those functions to
represent real signals
3. To recognize certain symmetries and patterns to simplify signal and system analysis
2.2 FUNCTIONAL NOTATION
A function is a correspondence between the argument of the function, which lies in
its domain, and the value returned by the function, which lies in its range. The most
familiar functions are of the form
g(x)where the argument x is a real number and the
value returned g is also a real number. But the domain and/or range of a function can
be complex numbers or integers or a variety of other choices of allowed values.
In this text five types of functions will appear,
1. Domain—Real numbers, Range—Real numbers
2. Domain—Integers, Range—Real numbers
3. Domain—Integers, Range—Complex numbers
4. Domain—Real numbers, Range—Complex numbers
5. Domain—Complex numbers, Range—Complex numbers
For functions whose domain is either real numbers or complex numbers the argument
will be enclosed in parentheses (⋅). For functions whose domain is integers the argu-
ment will be enclosed in brackets [⋅]. These types of functions will be discussed in
more detail as they are introduced.
2.3 CONTINUOUS-TIME SIGNAL FUNCTIONS
If the independent variable of a function is time t and the domain of the function is the real
numbers, and if the function g(t) has a defined value at every value of t, the function is
called a continuous-time function. Figure 2.2 illustrates some continuous-time functions.
Figure 2.2
Examples of continuous-time functions
t
g(t)
t
g(t)
t
g(t)
t
g(t)
(a) (b)
(c) (d)
Points of Discontinuity of g(t)](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-46-320.jpg)
![2.3 Continuous-Time Signal Functions 21
Figure 2.2(d) illustrates a discontinuous function for which the limit of the func-
tion value as we approach the discontinuity from above is not the same as when we
approach it from below. If t =
t 0is a point of discontinuity of a function g(t) then
lim
ε→0
g(t 0+ ε) ≠
lim
ε→0
g(t 0 − ε).
All four functions, (a)–(d), are continuous-time functions because their values are de-
fined for all real values of t. Therefore the terms continuous and continuous-time mean
slightly different things. All continuous functions of time are continuous-time func-
tions, but not all continuous-time functions are continuous functions of time.
COMPLEX EXPONENTIALS AND SINUSOIDS
Real-valued sinusoids and exponential functions should already be familiar. In
g(t) = Acos(2πt/
T 0+ θ) = Acos(2πf 0t + θ) = Acos(ω 0t + θ)
and
g(t) = A
e (σ 0+jω 0)t
= A
e σ 0t
[cos(ω 0t) + j sin(ω 0t)]
A is the amplitude,
T 0is the fundamental period,
f 0is the fundamental cyclic frequency
and ω 0is the fundamental radian frequency of the sinusoid, t is time and
σ 0is the decay
rate of the exponential (which is the reciprocal of its time constant, τ) (Figure 2.3 and
Figure 2.4). All these parameters can be any real number.
Figure 2.3
A real sinusoid and a real exponential with
parameters indicated graphically
t
A
t
A
τ
g(t) = A cos(2πf0t+θ)
g(t) = Ae–t/τ
–θ/2πf0
T0
4
–4
... ...
t = 10 ms
t
–4sin(200πt) μA
10
–10
...
...
t = 2 μs
t
10cos(106πt) nC
2
t = 0.1 s
t
2e–10tm
5
–5 t = 1 s
t
5e–tsin(2πt) m
s2
Figure 2.4
Examples of signals described by real sines, cosines
and exponentials
In Figure 2.4 the units indicate what kind of physical signal is being described.
Very often in system analysis, when only one kind of signal is being followed through
a system, the units are omitted for the sake of brevity.
Exponentials (exp) and sinusoids (sin and cos) are intrinsic functions in
MATLAB. The arguments of the sin and cos functions are interpreted by MATLAB
as radians, not degrees.
[exp(1),sin(pi/2),cos(pi)]
ans =
2.7183 1.0000 −1.0000 (pi is the MATLAB symbol for π.)](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-47-320.jpg)
![Ch ap ter 2 Mathematical Description of Continuous-Time Signals
28
Using the definition of
Δ(t)we can rewrite the integral as
A =
1
__
a ∫
−a/2
a/2
g(t)dt.
The function
g(t)is continuous at
t = 0
. Therefore it can be expressed as a McLaurin
series of the form
g(t) = ∑
m=0
∞
g (m)
(0)
_____
m!
t m
= g(0) +
g′ (0)t +
g″ (0)
____
2!
t 2
+ ⋯ +
g (m)
(0)
_____
m!
t m
+ ⋯
Then the integral becomes
A =
1
__
a ∫
−a/2
a/2
[g(0) +
g′ (0)t +
g″ (0)
____
2!
t 2
+ ⋯ +
g (m)
(0)
_____
m!
t m
+ ⋯]
dt
All the odd powers of t contribute nothing to the integral because it is taken over sym-
metrical limits about
t = 0
. Carrying out the integral,
A =
1
__
a [ag(0) +
(
a 3
__
12
)
g″ (0)
____
2!
+
(
a 5
__
80
)
g (4)
(0)
_____
4!
+ ⋯]
Take the limit of this integral as a approaches zero.
lim
a→0
A = g(0)
.
In the limit as a approaches zero, the function
Δ(t)extracts the value of any continuous
finite function
g(t)at time
t = 0
, when the product of
Δ(t) and g(t)is integrated over
any range of time that includes time
t = 0
.
Now try a different definition of the function
Δ(t)
. Define it now as
Δ(t) =
{
(1/a)(1 −
|t|/a), |t| ≤ a
0,
|t| a
(See Figure 2.15.)
If we make the same argument as before we get the area
A = ∫
−∞
∞
Δ(t)g(t)dt = 1
__
a ∫
−a
a
(1 −
|t|
__
a )g(t)dt.
Δ(t)
t
1
a
2
a
2
– a
Figure 2.13
A unit-area rectangular pulse of
width a
t
Δ(t)
Δ(t)g(t)
1
a
2
a
2
– a
g(t)
Figure 2.14
Product of a unit-area rectangular
pulse centered at
t = 0and a func-
tion g(t)that is continuous and
finite at
t = 0](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-54-320.jpg)
![Figueroa’s expedition against Mindanao
In 1596 Capt. Esteban Rodriguez led an expedition into Mindanao, for its
conquest and pacification.
It is maintained that he proceeded up the Mindanao River as far as Bwayan,
the capital of the upper Mindanao Valley.
Don Esteban Rodriguez prepared men and ships, and what else was necessary for
the enterprise, and with some galleys, galleots, frigates, vireys,34 barangays,34
and lapis,35 set out with two hundred and fourteen Spaniards for the Island of
Mindanao, in February of the same year, of 1596. He took Capt. Juan de la Xara
as his master-of-camp, and some religious of the Society of Jesus to give
instruction, as well as many natives for the service of the camp and fleet.
He reached Mindanao River after a good voyage, where the first settlements,
named Tampakan and Lumakan, both hostile to the people of Bwayan, received
him peacefully and in a friendly manner, and joined his fleet. They were
altogether about six thousand men. Without delay they advanced about 8 leagues
farther up the river against Bwayan, the principal settlement of the island, where
its greatest chief had fortified himself on many sides. Arrived at the settlement,
the fleet cast anchor and immediately landed a large proportion of the troops with
their arms. But before reaching the houses and fort, and while going through some
thickets [cacatal]36 near the shore, they encountered some of the men of Bwayan,
who were coming to meet them with their kampilan,37 carazas38 and other
weapons, and who attacked them on various sides. The latter [i.e., the Spaniards
and their allies], on account of the swampiness of the place and the denseness of
the thickets [cacatal], could not act unitedly as the occasion demanded, although
the master-of-camp and the captains that led them exerted themselves to keep the
troops together and to encourage them to face the natives. Meanwhile Governor
Esteban Rodriguez de Figueroa was watching events from his flagship, but not
being able to endure the confusion of his men, seized his weapons and hastened
ashore with three or four companions and a servant who carried his helmet in
order that he might be less impeded in his movements. But as he was crossing a
part of the thickets [cacatal] where the fight was waging, a hostile Indian stepped
out unseen from one side and dealt the governor a blow on the head with his
kampilan that stretched him on the ground badly wounded.39 The governor’s](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-61-320.jpg)
![artillery to bear on the flagship (one of the galleys), but could not retard the
Spanish advance.
“I answered their fire with so great readiness,” said Ronquillo in his report, “that I
forced them to withdraw their artillery. But, as if they were goblins, they remained
here behind a bush or a tree, firing at us without being seen.” Reinforced by the
chief of the hill tribes, Lumakan, with 500 natives, Ronquillo resumed the
fighting after the delay of a few days. “Finally,” continued Ronquillo, “I planted
my battery of eight pieces somewhat over 100 paces from the fort. Although I
battered the fort hotly, I could not effect a breach through which to make an
assault. All the damage that I did them by day, they repaired by night. * * *
“I was very short of ammunition, for I had only 3,000 arquebus bullets left, and
very few cannon balls; and both would be spent in one day’s fighting, during
which, should we not gain the fort, we would be lost—and with no power to
defend ourselves while withdrawing our artillery and camp. * * *
“I reconnoitered the fort and its situation, for it is located at the entrance of a
lagoon, thus having only water at the back, and swampy and marshy ground at the
sides. It has a frontage of more than 1,000 paces, is furnished with very good
transversals, and is well supplied with artillery and arquebuses. Moreover it has a
ditch of water more than 4 brazas43 wide and 2 deep, and thus there was a space
of dry ground of only 15 paces where it was possible to attack; and this space was
bravely defended, and with the greatest force of the enemy. The inner parts were
water, where they sailed in vessels, while we had no footing at all.”
“Again, I reflected that those who had awaited us so long, had waited with the
determination to die in defense of the fort; and if they should see the contest
ending unfavorably for them, no one would prevent their flight. Further, if they
awaited the assault it would cost me the greater part of my remaining ammunition,
and my best men; while, if the enemy fled, nothing would be accomplished, but
on the contrary a long, tedious, and costly war would be entered upon. Hence,
with the opinion and advice of the captains, I negotiated for peace, and told them
that I would admit them to friendship under the following conditions:
“First, that first and foremost they must offer homage to his Majesty, and pay
something as recognition” (a gold chain). Second, “that all the natives who had
been taken from the Pintados Islands [Bisayan Islands] last year, must be
restored.” Third, “that they must break the peace and confederation made with the
people of Ternate, and must not admit the latter into their country.” Fourth, “that
they must be friends with Danganlibor and Lumakan, * * * and must not make](https://image.slidesharecdn.com/5645614-250523163026-26e4c924/85/Signals-And-Systems-Analysis-Using-Transform-Methods-And-Matlab-Third-Michael-J-Roberts-63-320.jpg)
Signals And Systems Analysis Using Transform Methods And Matlab Third Michael J Roberts
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- 7. Signals and Systems Analysis Using Transform Methods and MATLAB® Michael J. Roberts Professor Emeritus, Department of Electrical and Computer Engineering University of Tennessee Third Edition
- 8. SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB® , THIRD EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2018 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, and 2004. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 QVS 22 21 20 19 18 17 ISBN 978-0-07-802812-0 MHID 0-07-802812-4 Chief Product Officer, SVP Products & Markets: G. Scott Virkler Vice President, General Manager, Products & Markets: Marty Lange Vice President, Content Design & Delivery: Betsy Whalen Managing Director: Thomas Timp Brand Manager: Raghothaman Srinivasan/Thomas Scaife, Ph.D. Director, Product Development: Rose Koos Product Developer: Christine Bower Marketing Manager: Shannon O’Donnell Director of Digital Content: Chelsea Haupt, Ph.D. Director, Content Design & Delivery: Linda Avenarius Program Manager: Lora Neyens Content Project Managers: Jeni McAtee; Emily Windelborn; Sandy Schnee Buyer: Jennifer Pickel Content Licensing Specialists: Carrie Burger, photo; Lorraine Buczek, text Cover Image: © Lauree Feldman/Getty Images Compositor: MPS Limited Printer: Quad Versailles All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Roberts, Michael J., Dr. Signals and systems : analysis using transform methods and MATLAB / Michael J. Roberts, professor, Department of Electrical and Computer Engineering, University of Tennessee. Third edition. | New York, NY : McGraw-Hill Education, [2018] | Includes bibliographical references (p. 786–787) and index. LCCN 2016043890 | ISBN 9780078028120 (alk. paper) LCSH: Signal processing. | System analysis. | MATLAB. LCC TK5102.9 .R63 2018 | DDC 621.382/2—dc23 LC record available at https://lccn.loc.gov/2016043890 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. mheducation.com/highered
- 9. To my wife Barbara for giving me the time and space to complete this effort and to the memory of my parents, Bertie Ellen Pinkerton and Jesse Watts Roberts, for their early emphasis on the importance of education.
- 10. Preface, xii Chapter 1 Introduction, 1 1.1 Signals and Systems Defined, 1 1.2 Types of Signals, 3 1.3 Examples of Systems, 8 A Mechanical System, 9 A Fluid System, 9 A Discrete-Time System, 11 Feedback Systems, 12 1.4 A Familiar Signal and System Example, 14 1.5 Use of MATLAB® , 18 Chapter 2 Mathematical Description of Continuous-Time Signals, 19 2.1 Introduction and Goals, 19 2.2 Functional Notation, 20 2.3 Continuous-Time Signal Functions, 20 Complex Exponentials and Sinusoids, 21 Functions with Discontinuities, 23 The Signum Function, 24 The Unit-Step Function, 24 The Unit-Ramp Function, 26 The Unit Impulse, 27 The Impulse, the Unit Step, and Generalized Derivatives, 29 The Equivalence Property of the Impulse, 30 The Sampling Property of the Impulse, 31 The Scaling Property of the Impulse, 31 The Unit Periodic Impulse or Impulse Train, 32 A Coordinated Notation for Singularity Functions, 33 The Unit-Rectangle Function, 33 2.4 Combinations of Functions, 34 2.5 Shifting and Scaling, 36 Amplitude Scaling, 36 Time Shifting, 37 Time Scaling, 39 Simultaneous Shifting and Scaling, 43 2.6 Differentiation and Integration, 47 2.7 Even and Odd Signals, 49 Combinations of Even and Odd Signals, 51 Derivatives and Integrals of Even and Odd Signals, 53 2.8 Periodic Signals, 53 2.9 Signal Energy and Power, 56 Signal Energy, 56 Signal Power, 58 2.10 Summary of Important Points, 60 Exercises, 61 Exercises with Answers, 61 Signal Functions, 61 Shifting and Scaling, 62 Derivatives and Integrals of Functions, 66 Generalized Derivative, 67 Even and Odd Functions, 67 Periodic Signals, 69 Signal Energy and Power of Signals, 70 Exercises without Answers, 71 Signal Functions, 71 Scaling and Shifting, 71 Generalized Derivative, 76 Derivatives and Integrals of Functions, 76 Even and Odd Functions, 76 Periodic Functions, 77 Signal Energy and Power of Signals, 77 Chapter 3 Discrete-Time Signal Description, 79 3.1 Introduction and Goals, 79 3.2 Sampling and Discrete Time, 80 3.3 Sinusoids and Exponentials, 82 Sinusoids, 82 Exponentials, 85 3.4 Singularity Functions, 86 The Unit-Impulse Function, 86 The Unit-Sequence Function, 87 CONTENTS iv
- 11. Contents v The Signum Function, 87 The Unit-Ramp Function, 88 The Unit Periodic Impulse Function or Impulse Train, 88 3.5 Shifting and Scaling, 89 Amplitude Scaling, 89 Time Shifting, 89 Time Scaling, 89 Time Compression, 90 Time Expansion, 90 3.6 Differencing and Accumulation, 94 3.7 Even and Odd Signals, 98 Combinations of Even and Odd Signals, 100 Symmetrical Finite Summation of Even and Odd Signals, 100 3.8 Periodic Signals, 101 3.9 Signal Energy and Power, 102 Signal Energy, 102 Signal Power, 103 3.10 Summary of Important Points, 105 Exercises, 105 Exercises with Answers, 105 Functions, 105 Scaling and Shifting Functions, 107 Differencing and Accumulation, 109 Even and Odd Functions, 110 Periodic Functions, 111 Signal Energy and Power, 112 Exercises without Answers, 113 Signal Functions, 113 Shifting and Scaling Functions, 113 Differencing and Accumulation, 114 Even and Odd Functions, 114 Periodic Signals, 115 Signal Energy and Power, 116 Chapter 4 Description of Systems, 118 4.1 Introduction and Goals, 118 4.2 Continuous-Time Systems, 119 System Modeling, 119 Differential Equations, 120 Block Diagrams, 124 System Properties, 127 Introductory Example, 127 Homogeneity, 131 Time Invariance, 132 Additivity, 133 Linearity and Superposition, 134 LTI Systems, 134 Stability, 138 Causality, 139 Memory, 139 Static Nonlinearity, 140 Invertibility, 142 Dynamics of Second-Order Systems, 143 Complex Sinusoid Excitation, 145 4.3 Discrete-Time Systems, 145 System Modeling, 145 Block Diagrams, 145 Difference Equations, 146 System Properties, 152 4.4 Summary of Important Points, 155 Exercises, 156 Exercises with Answers, 156 System Models, 156 Block Diagrams, 157 System Properties, 158 Exercises without Answers, 160 System Models, 160 System Properties, 162 Chapter 5 Time-Domain System Analysis, 164 5.1 Introduction and Goals, 164 5.2 Continuous Time, 164 Impulse Response, 164 Continuous-Time Convolution, 169 Derivation, 169 Graphical and Analytical Examples of Convolution, 173 Convolution Properties, 178 System Connections, 181 Step Response and Impulse Response, 181 Stability and Impulse Response, 181 Complex Exponential Excitation and the Transfer Function, 182 Frequency Response, 184 5.3 Discrete Time, 186 Impulse Response, 186 Discrete-Time Convolution, 189
- 12. Contents vi Derivation, 189 Graphical and Analytical Examples of Convolution, 192 Convolution Properties, 196 Numerical Convolution, 196 Discrete-Time Numerical Convolution, 196 Continuous-Time Numerical Convolution, 198 Stability and Impulse Response, 200 System Connections, 200 Unit-Sequence Response and Impulse Response, 201 Complex Exponential Excitation and the Transfer Function, 203 Frequency Response, 204 5.4 Summary of Important Points, 207 Exercises, 207 Exercises with Answers, 207 Continuous Time, 207 Impulse Response, 207 Convolution, 209 Stability, 213 Frequency Response, 214 Discrete Time, 214 Impulse Response, 214 Convolution, 215 Stability, 219 Exercises without Answers, 221 Continuous Time, 221 Impulse Response, 221 Convolution, 222 Stability, 224 Discrete Time, 225 Impulse Response, 225 Convolution, 225 Stability, 228 Chapter 6 Continuous-Time Fourier Methods, 229 6.1 Introduction and Goals, 229 6.2 The Continuous-Time Fourier Series, 230 Conceptual Basis, 230 Orthogonality and the Harmonic Function, 234 The Compact Trigonometric Fourier Series, 237 Convergence, 239 Continuous Signals, 239 Discontinuous Signals, 240 Minimum Error of Fourier-Series Partial Sums, 242 The Fourier Series of Even and Odd Periodic Functions, 243 Fourier-Series Tables and Properties, 244 Numerical Computation of the Fourier Series, 248 6.3 The Continuous-Time Fourier Transform, 255 Extending the Fourier Series to Aperiodic Signals, 255 The Generalized Fourier Transform, 260 Fourier Transform Properties, 265 Numerical Computation of the Fourier Transform, 273 6.4 Summary of Important Points, 281 Exercises, 281 Exercises with Answers, 281 Fourier Series, 281 Orthogonality, 282 Forward and Inverse Fourier Transforms, 286 Relation of CTFS to CTFT, 293 Numerical CTFT, 294 System Response, 294 Exercises without Answers, 294 Fourier Series, 294 Forward and Inverse Fourier Transforms, 300 System Response, 305 Relation of CTFS to CTFT, 306 Chapter 7 Discrete-Time Fourier Methods, 307 7.1 Introduction and Goals, 307 7.2 The Discrete-Time Fourier Series and the Discrete Fourier Transform, 307 Linearity and Complex-Exponential Excitation, 307 Orthogonality and the Harmonic Function, 311 Discrete Fourier Transform Properties, 315 The Fast Fourier Transform, 321 7.3 The Discrete-Time Fourier Transform, 323 Extending the Discrete Fourier Transform to Aperiodic Signals, 323 Derivation and Definition, 324 The Generalized DTFT, 326 Convergence of the Discrete-Time Fourier Transform, 327 DTFT Properties, 327
- 13. Contents vii Numerical Computation of the Discrete-Time Fourier Transform, 334 7.4 Fourier Method Comparisons, 340 7.5 Summary of Important Points, 341 Exercises, 342 Exercises with Answers, 342 Orthogonality, 342 Discrete Fourier Transform, 342 Discrete-Time Fourier Transform Definition, 344 Forward and Inverse Discrete-Time Fourier Transforms, 345 Exercises without Answers, 348 Discrete Fourier Transform, 348 Forward and Inverse Discrete-Time Fourier Transforms, 352 Chapter 8 The Laplace Transform, 354 8.1 Introduction and Goals, 354 8.2 Development of the Laplace Transform, 355 Generalizing the Fourier Transform, 355 Complex Exponential Excitation and Response, 357 8.3 The Transfer Function, 358 8.4 Cascade-Connected Systems, 358 8.5 Direct Form II Realization, 359 8.6 The Inverse Laplace Transform, 360 8.7 Existence of the Laplace Transform, 360 Time-Limited Signals, 361 Right- and Left-Sided Signals, 361 8.8 Laplace-Transform Pairs, 362 8.9 Partial-Fraction Expansion, 367 8.10 Laplace-Transform Properties, 377 8.11 The Unilateral Laplace Transform, 379 Definition, 379 Properties Unique to the Unilateral Laplace Transform, 381 Solution of Differential Equations with Initial Conditions, 383 8.12 Pole-Zero Diagrams and Frequency Response, 385 8.13 MATLAB System Objects, 393 8.14 Summary of Important Points, 395 Exercises, 395 Exercises with Answers, 395 Laplace-Transform Definition, 395 Direct Form II System Realization, 396 Forward and Inverse Laplace Transforms, 396 Unilateral Laplace-Transform Integral, 399 Solving Differential Equations, 399 Exercises without Answers, 400 Region of Convergence, 400 Existence of the Laplace Transform, 400 Direct Form II System Realization, 400 Forward and Inverse Laplace Transforms, 401 Solution of Differential Equations, 403 Pole-Zero Diagrams and Frequency Response, 403 Chapter 9 The z Transform, 406 9.1 Introduction and Goals, 406 9.2 Generalizing the Discrete-Time Fourier Transform, 407 9.3 Complex Exponential Excitation and Response, 408 9.4 The Transfer Function, 408 9.5 Cascade-Connected Systems, 408 9.6 Direct Form II System Realization, 409 9.7 The Inverse z Transform, 410 9.8 Existence of the z Transform, 410 Time-Limited Signals, 410 Right- and Left-Sided Signals, 411 9.9 z-Transform Pairs, 413 9.10 z-Transform Properties, 416 9.11 Inverse z-Transform Methods, 417 Synthetic Division, 417 Partial-Fraction Expansion, 418 Examples of Forward and Inverse z Transforms, 418 9.12 The Unilateral z Transform, 423 Properties Unique to the Unilateral z Transform, 423 Solution of Difference Equations, 424 9.13 Pole-Zero Diagrams and Frequency Response, 425 9.14 MATLAB System Objects, 428 In MATLAB, 429 9.15 Transform Method Comparisons, 430 9.16 Summary of Important Points, 434
- 14. Contents viii Exercises, 435 Exercises with Answers, 435 Direct-Form II System Realization, 435 Existence of the z Transform, 435 Forward and Inverse z Transforms, 435 Unilateral z-Transform Properties, 438 Solution of Difference Equations, 438 Pole-Zero Diagrams and Frequency Response, 439 Exercises without Answers, 441 Direct Form II System Realization, 441 Existence of the z Transform, 441 Forward and Inverse z-Transforms, 441 Pole-Zero Diagrams and Frequency Response, 443 Chapter 10 Sampling and Signal Processing, 446 10.1 Introduction and Goals, 446 10.2 Continuous-Time Sampling, 447 Sampling Methods, 447 The Sampling Theorem, 449 Qualitative Concepts, 449 Sampling Theorem Derivation, 451 Aliasing, 454 Time-limited and Bandlimited Signals, 457 Interpolation, 458 Ideal Interpolation, 458 Practical Interpolation, 459 Zero-Order Hold, 460 First-Order Hold, 460 Sampling Bandpass Signals, 461 Sampling a Sinusoid, 464 Bandlimited Periodic Signals, 467 Signal Processing Using the DFT, 470 CTFT-DFT Relationship, 470 CTFT-DTFT Relationship, 471 Sampling and Periodic-Repetition Relationship, 474 Computing the CTFS Harmonic Function with the DFT, 478 Approximating the CTFT with the DFT, 478 Forward CTFT, 478 Inverse CTFT, 479 Approximating the DTFT with the DFT, 479 Approximating Continuous-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Discrete-Time Convolution with the DFT, 479 Aperiodic Convolution, 479 Periodic Convolution, 479 Summary of Signal Processing Using the DFT, 480 10.3 Discrete-Time Sampling, 481 Periodic-Impulse Sampling, 481 Interpolation, 483 10.4 Summary of Important Points, 486 Exercises, 487 Exercises with Answers, 487 Pulse Amplitude Modulation, 487 Sampling, 487 Impulse Sampling, 489 Nyquist Rates, 491 Time-Limited and Bandlimited Signals, 492 Interpolation, 493 Aliasing, 495 Bandlimited Periodic Signals, 495 CTFT-CTFS-DFT Relationships, 495 Windows, 497 DFT, 497 Exercises without Answers, 500 Sampling, 500 Impulse Sampling, 502 Nyquist Rates, 504 Aliasing, 505 Practical Sampling, 505 Bandlimited Periodic Signals, 505 DFT, 506 Discrete-Time Sampling, 508 Chapter 11 Frequency Response Analysis, 509 11.1 Introduction and Goals, 509 11.2 Frequency Response, 509 11.3 Continuous-Time Filters, 510 Examples of Filters, 510 Ideal Filters, 515 Distortion, 515 Filter Classifications, 516 Ideal Filter Frequency Responses, 516 Impulse Responses and Causality, 517 The Power Spectrum, 520 Noise Removal, 520 Bode Diagrams, 521
- 15. Contents ix The Decibel, 521 The One-Real-Pole System, 525 The One-Real-Zero System, 526 Integrators and Differentiators, 527 Frequency-Independent Gain, 527 Complex Pole and Zero Pairs, 530 Practical Filters, 532 Passive Filters, 532 The Lowpass Filter, 532 The Bandpass Filter, 535 Active Filters, 536 Operational Amplifiers, 537 The Integrator, 538 The Lowpass Filter, 538 11.4 Discrete-Time Filters, 546 Notation, 546 Ideal Filters, 547 Distortion, 547 Filter Classifications, 548 Frequency Responses, 548 Impulse Responses and Causality, 548 Filtering Images, 549 Practical Filters, 554 Comparison with Continuous-Time Filters, 554 Highpass, Bandpass, and Bandstop Filters, 556 The Moving Average Filter, 560 The Almost Ideal Lowpass Filter, 564 Advantages Compared to Continuous-Time Filters, 566 11.5 Summary of Important Points, 566 Exercises, 567 Exercises with Answers, 567 Continuous-Time Frequency Response, 567 Continuous-Time Ideal Filters, 567 Continuous-Time Causality, 567 Logarithmic Graphs, Bode Diagrams, and Decibels, 568 Continuous-Time Practical Passive Filters, 570 Continuous-Time Practical Active Filters, 574 Discrete-Time Frequency Response, 575 Discrete-Time Ideal Filters, 576 Discrete-Time Causality, 576 Discrete-Time Practical Filters, 577 Exercises without Answers, 579 Continuous-Time Frequency Response, 579 Continuous-Time Ideal Filters, 579 Continuous-Time Causality, 579 Bode Diagrams, 580 Continuous-Time Practical Passive Filters, 580 Continuous-Time Filters, 582 Continuous-Time Practical Active Filters, 582 Discrete-Time Causality, 586 Discrete-Time Filters, 587 Chapter 12 Laplace System Analysis, 592 12.1 Introduction and Goals, 592 12.2 System Representations, 592 12.3 System Stability, 596 12.4 System Connections, 599 Cascade and Parallel Connections, 599 The Feedback Connection, 599 Terminology and Basic Relationships, 599 Feedback Effects on Stability, 600 Beneficial Effects of Feedback, 601 Instability Caused by Feedback, 604 Stable Oscillation Using Feedback, 608 The Root-Locus Method, 612 Tracking Errors in Unity-Gain Feedback Systems, 618 12.5 System Analysis Using MATLAB, 621 12.6 System Responses to Standard Signals, 623 Unit-Step Response, 624 Sinusoid Response, 627 12.7 Standard Realizations of Systems, 630 Cascade Realization, 630 Parallel Realization, 632 12.8 Summary of Important Points, 632 Exercises, 633 Exercises with Answers, 633 Transfer Functions, 633 Stability, 634 Parallel, Cascade, and Feedback Connections, 635 Root Locus, 637 Tracking Errors in Unity-Gain Feedback Systems, 639 System Responses to Standard Signals, 640 System Realization, 641 Exercises without Answers, 642 Stability, 642 Transfer Functions, 642 Stability, 643
- 16. Contents x Parallel, Cascade, and Feedback Connections, 643 Root Locus, 646 Tracking Errors in Unity-Gain Feedback Systems, 647 Response to Standard Signals, 647 System Realization, 649 Chapter 13 z-Transform System Analysis, 650 13.1 Introduction and Goals, 650 13.2 System Models, 650 Difference Equations, 650 Block Diagrams, 651 13.3 System Stability, 651 13.4 System Connections, 652 13.5 System Responses to Standard Signals, 654 Unit-Sequence Response, 654 Response to a Causal Sinusoid, 657 13.6 Simulating Continuous-Time Systems with Discrete-Time Systems, 660 z-Transform-Laplace-Transform Relationships, 660 Impulse Invariance, 662 Sampled-Data Systems, 664 13.7 Standard Realizations of Systems, 670 Cascade Realization, 670 Parallel Realization, 670 13.8 Summary of Important Points, 671 Exercises, 672 Exercises with Answers, 672 Stability, 672 Parallel, Cascade, and Feedback Connections, 672 Response to Standard Signals, 673 Root Locus, 674 Laplace-Transform-z-Transform Relationship, 675 Sampled-Data Systems, 675 System Realization, 676 Exercises without Answers, 677 Stability, 677 Root Locus, 677 Parallel, Cascade, and Feedback Connections, 677 Response to Standard Signals, 677 Laplace-Transform-z-Transform Relationship, 679 Sampled-Data Systems, 679 System Realization, 679 General, 679 Chapter 14 Filter Analysis and Design, 680 14.1 Introduction and Goals, 680 14.2 Analog Filters, 680 Butterworth Filters, 681 Normalized Butterworth Filters, 681 Filter Transformations, 682 MATLAB Design Tools, 684 Chebyshev, Elliptic, and Bessel Filters, 686 14.3 Digital Filters, 689 Simulation of Analog Filters, 689 Filter Design Techniques, 689 IIR Filter Design, 689 Time-Domain Methods, 689 Impulse-Invariant Design, 689 Step-Invariant Design, 696 Finite-Difference Design, 698 Frequency-Domain Methods, 704 The Bilinear Method, 706 FIR Filter Design, 713 Truncated Ideal Impulse Response, 713 Optimal FIR Filter Design, 723 MATLAB Design Tools, 725 14.4 Summary of Important Points, 727 Exercises, 727 Exercises with Answers, 727 Continuous-Time Filters, 727 Finite-Difference Filter Design, 728 Matched-z Transform and Direct Substitution Filter Design, 729 Bilinear z-Transform Filter Design, 730 FIR Filter Design, 730 Digital Filter Design Method Comparison, 731 Exercises without Answers, 731 Analog Filter Design, 731 Impulse-Invariant and Step-Invariant Filter Design, 732 Finite-Difference Filter Design, 733 Matched z-Transform and Direct Substitution Filter Design, 733 Bilinear z-Transform Filter Design, 733 FIR Filter Design, 733 Digital Filter Design Method Comparison, 734
- 17. Contents xi Appendix I Useful Mathematical Relations, A-1 II Continuous-Time Fourier Series Pairs, A-4 III Discrete Fourier Transform Pairs, A-7 IV Continuous-Time Fourier Transform Pairs, A-10 V Discrete-Time Fourier Transform Pairs, A-17 VI Tables of Laplace Transform Pairs, A-22 VII z-Transform Pairs, A-24 Bibliography, B-1 Index, I-1
- 18. PREFACE MOTIVATION I wrote the first and second editions because I love the mathematical beauty of signal and system analysis. That has not changed. The motivation for the third edi- tion is to further refine the book structure in light of reviewers, comments, correct a few errors from the second edition and significantly rework the exercises. AUDIENCE This book is intended to cover a two-semester course sequence in the basics of signal and system analysis during the junior or senior year. It can also be used (as I have used it) as a book for a quick one-semester Master’s-level review of trans- form methods as applied to linear systems. CHANGES FROM THE SECOND EDITION 1. In response to reviewers, comments, two chapters from the second edition have been omitted: Communication Systems and State-Space Analysis. There seemed to be very little if any coverage of these topics in actual classes. 2. The second edition had 550 end-of-chapter exercises in 16 chapters. The third edition has 710 end-of-chapter exercises in 14 chapters. OVERVIEW Except for the omission of two chapters, the third edition structure is very similar to the second edition. The book begins with mathematical methods for describing signals and systems, in both continuous and discrete time. I introduce the idea of a transform with the continuous-time Fourier series, and from that base move to the Fourier trans- form as an extension of the Fourier series to aperiodic signals. Then I do the same for discrete-time signals. I introduce the Laplace transform both as a generalization of the continuous-time Fourier transform for unbounded signals and unstable systems and as a powerful tool in system analysis because of its very close association with the ei- genvalues and eigenfunctions of continuous-time linear systems. I take a similar path for discrete-time systems using the z transform. Then I address sampling, the relation between continuous and discrete time. The rest of the book is devoted to applications in frequency-response analysis, feedback systems, analog and digital filters. Through- out the book I present examples and introduce MATLAB functions and operations to implement the methods presented. A chapter-by-chapter summary follows. CHAPTER SUMMARIES CHAPTER 1 Chapter 1 is an introduction to the general concepts involved in signal and system analysis without any mathematical rigor. It is intended to motivate the student by xii
- 19. xiii Preface demonstrating the ubiquity of signals and systems in everyday life and the impor- tance of understanding them. CHAPTER 2 Chapter 2 is an exploration of methods of mathematically describing continuous- time signals of various kinds. It begins with familiar functions, sinusoids and exponentials and then extends the range of signal-describing functions to include continuous-time singularity functions (switching functions). Like most, if not all, signals and systems textbooks, I define the unit-step, the signum, the unit-impulse and the unit-ramp functions. In addition to these I define a unit rectangle and a unit periodic impulse function. The unit periodic impulse function, along with convolution, provides an especially compact way of mathematically describing arbitrary periodic signals. After introducing the new continuous-time signal functions, I cover the common types of signal transformations, amplitude scaling, time shifting, time scaling, differentiation and integration and apply them to the signal functions. Then I cover some characteristics of signals that make them invariant to certain transformations, evenness, oddness and periodicity, and some of the implications of these signal characteristics in signal analysis. The last section is on signal energy and power. CHAPTER 3 Chapter 3 follows a path similar to Chapter 2 except applied to discrete-time signals instead of continuous-time signals. I introduce the discrete-time sinu- soid and exponential and comment on the problems of determining period of a discrete-time sinusoid. This is the first exposure of the student to some of the implications of sampling. I define some discrete-time signal functions analo- gous to continuous-time singularity functions. Then I explore amplitude scaling, time shifting, time scaling, differencing and accumulation for discrete-time signal functions pointing out the unique implications and problems that occur, especially when time scaling discrete-time functions. The chapter ends with definitions and discussion of signal energy and power for discrete-time signals. CHAPTER 4 This chapter addresses the mathematical description of systems. First I cover the most common forms of classification of systems, homogeneity, additivity, linearity, time invariance, causality, memory, static nonlinearity and invertibility. By example I present various types of systems that have, or do not have, these properties and how to prove various properties from the mathematical description of the system. CHAPTER 5 This chapter introduces the concepts of impulse response and convolution as components in the systematic analysis of the response of linear, time-invariant systems. I present the mathematical properties of continuous-time convolution and a graphical method of understanding what the convolution integral says. I also show how the properties of convolution can be used to combine subsystems that are connected in cascade or parallel into one system and what the impulse response of the overall system must be. Then I introduce the idea of a transfer
- 20. xiv Preface function by finding the response of an LTI system to complex sinusoidal exci- tation. This section is followed by an analogous coverage of discrete-time impulse response and convolution. CHAPTER 6 This is the beginning of the student’s exposure to transform methods. I begin by graphically introducing the concept that any continuous-time periodic signal with engineering usefulness can be expressed by a linear combination of continuous-time sinusoids, real or complex. Then I formally derive the Fourier series using the concept of orthogonality to show where the signal description as a function of discrete harmonic number (the harmonic function) comes from. I mention the Dirichlet conditions to let the student know that the continuous-time Fourier series applies to all practical continuous-time signals, but not to all imaginable continuous-time signals. Then I explore the properties of the Fourier series. I have tried to make the Fourier series notation and properties as similar as possible and analogous to the Fourier transform, which comes later. The harmonic function forms a “Fourier series pair” with the time function. In the first edition I used a notation for har- monic function in which lower-case letters were used for time-domain quantities and upper-case letters for their harmonic functions. This unfortunately caused some confusion because continuous- and discrete-time harmonic functions looked the same. In this edition I have changed the harmonic function notation for continuous-time signals to make it easily distinguishable. I also have a section on the convergence of the Fourier series illustrating the Gibb’s phenomenon at function discontinuities. I encourage students to use tables and properties to find harmonic functions and this practice prepares them for a similar process in find- ing Fourier transforms and later Laplace and z transforms. The next major section of Chapter 6 extends the Fourier series to the Fourier transform. I introduce the concept by examining what happens to a continuous-time Fourier series as the period of the signal approaches infinity and then define and derive the continuous-time Fourier transform as a gener- alization of the continuous-time Fourier series. Following that I cover all the important properties of the continuous-time Fourier transform. I have taken an “ecumenical” approach to two different notational conventions that are commonly seen in books on signals and systems, control systems, digital signal processing, communication systems and other applications of Fourier methods such as image processing and Fourier optics: the use of either cyclic frequency, f or radian fre- quency, ω. I use both and emphasize that the two are simply related through a change of variable. I think this better prepares students for seeing both forms in other books in their college and professional careers. CHAPTER 7 This chapter introduces the discrete-time Fourier series (DTFS), the discrete Fou- rier transform (DFT) and the discrete-time Fourier transform (DTFT), deriving and defining them in a manner analogous to Chapter 6. The DTFS and the DFT are almost identical. I concentrate on the DFT because of its very wide use in digital signal processing. I emphasize the important differences caused by the differences between continuous- and discrete-time signals, especially the finite summation range of the DFT as opposed to the (generally) infinite summation range in the CTFS. I also point out the importance of the fact that the DFT relates
- 21. xv Preface a finite set of numbers to another finite set of numbers, making it amenable to direct numerical machine computation. I discuss the fast Fourier transform as a very efficient algorithm for computing the DFT. As in Chapter 6, I use both cyclic and radian frequency forms, emphasizing the relationships between them. I use F and Ω for discrete-time frequencies to distinguish them from f and ω, which were used in continuous time. Unfortunately, some authors reverse these symbols. My usage is more consistent with the majority of signals and systems texts. This is another example of the lack of standardization of notation in this area. The last major section is a comparison of the four Fourier methods. I emphasize particu- larly the duality between sampling in one domain and periodic repetition in the other domain. CHAPTER 8 This chapter introduces the Laplace transform. I approach the Laplace trans- form from two points of view, as a generalization of the Fourier transform to a larger class of signals and as result which naturally follows from the excitation of a linear, time-invariant system by a complex exponential signal. I begin by defining the bilateral Laplace transform and discussing significance of the re- gion of convergence. Then I define the unilateral Laplace transform. I derive all the important properties of the Laplace transform. I fully explore the method of partial-fraction expansion for finding inverse transforms and then show examples of solving differential equations with initial conditions using the uni- lateral form. CHAPTER 9 This chapter introduces the z transform. The development parallels the devel- opment of the Laplace transform except applied to discrete-time signals and systems. I initially define a bilateral transform and discuss the region of con- vergence. Then I define a unilateral transform. I derive all the important prop- erties and demonstrate the inverse transform using partial-fraction expansion and the solution of difference equations with initial conditions. I also show the relationship between the Laplace and z transforms, an important idea in the approximation of continuous-time systems by discrete-time systems in Chapter 14. CHAPTER 10 This is the first exploration of the correspondence between a continuous-time signal and a discrete-time signal formed by sampling it. The first section covers how sampling is usually done in real systems using a sample-and-hold and an A/D converter. The second section starts by asking the question of how many samples are enough to describe a continuous-time signal. Then the question is answered by deriving the sampling theorem. Then I discuss interpolation methods, theoret- ical and practical, the special properties of bandlimited periodic signals. I do a complete development of the relationship between the CTFT of a continuous-time signal and DFT of a finite-length set of samples taken from it. Then I show how the DFT can be used to approximate the CTFT of an energy signal or a periodic signal. The next major section explores the use of the DFT in numerically approx- imating various common signal-processing operations.
- 22. xvi CHAPTER 11 This chapter covers various aspects of the use of the CTFT and DTFT in fre- quency response analysis. The major topics are ideal filters, Bode diagrams, prac- tical passive and active continuous-time filters and basic discrete-time filters. CHAPTER 12 This chapter is on the application of the Laplace transform including block dia- gram representation of systems in the complex frequency domain, system stability, system interconnections, feedback systems including root locus, system responses to standard signals and lastly standard realizations of continuous-time systems. CHAPTER 13 This chapter is on the application of the z transform including block diagram representation of systems in the complex frequency domain, system stability, sys- tem interconnections, feedback systems including root-locus, system responses to standard signals, sampled-data systems and standard realizations of discrete-time systems. CHAPTER 14 This chapter covers the analysis and design of some of the most common types of practical analog and digital filters. The analog filter types are Butterworth, Chebyshev Types 1 and 2 and Elliptic (Cauer) filters. The section on digital filters covers the most common types of techniques for simulation of analog filters includ- ing, impulse- and step-invariant, finite difference, matched z transform, direct sub- stitution, bilinear z transform, truncated impulse response and Parks-McClellan numerical design. APPENDICES There are seven appendices on useful mathematical formulae, tables of the four Fourier transforms, Laplace transform tables and z transform tables. CONTINUITY The book is structured so as to facilitate skipping some topics without loss of continuity. Continuous-time and discrete-time topics are covered alternately and continuous-time analysis could be covered without reference to discrete time. Also, any or all of the last six chapters could be omitted in a shorter course. REVIEWS AND EDITING This book owes a lot to the reviewers, especially those who really took time and criticized and suggested improvements. I am indebted to them. I am also indebted to the many students who have endured my classes over the years. I believe that our relationship is more symbiotic than they realize. That is, they learn signal and system analysis from me and I learn how to teach signal and system analysis from them. I cannot count the number of times I have been asked a very perceptive question by a student that revealed not only that the students were not understand- ing a concept but that I did not understand it as well as I had previously thought. Preface
- 23. xvii WRITING STYLE Every author thinks he has found a better way to present material so that students can grasp it and I am no different. I have taught this material for many years and through the experience of grading tests have found what students generally do and do not grasp. I have spent countless hours in my office one-on-one with students explaining these concepts to them and, through that experience, I have found out what needs to be said. In my writing I have tried to simply speak directly to the reader in a straightforward conversational way, trying to avoid off-putting formality and, to the extent possible, anticipating the usual misconceptions and revealing the fallacies in them. Transform methods are not an obvious idea and, at first exposure, students can easily get bogged down in a bewildering morass of abstractions and lose sight of the goal, which is to analyze a system’s response to signals. I have tried (as every author does) to find the magic combination of ac- cessibility and mathematical rigor because both are important. I think my writing is clear and direct but you, the reader, will be the final judge of whether or not that is true. EXERCISES Each chapter has a group of exercises along with answers and a second group of exercises without answers. The first group is intended more or less as a set of “drill” exercises and the second group as a set of more challenging exercises. CONCLUDING REMARKS As I indicated in the preface to first and second editions, I welcome any and all criticism, corrections and suggestions. All comments, including ones I disagree with and ones which disagree with others, will have a constructive impact on the next edition because they point out a problem. If something does not seem right to you, it probably will bother others also and it is my task, as an author, to find a way to solve that problem. So I encourage you to be direct and clear in any re- marks about what you believe should be changed and not to hesitate to mention any errors you may find, from the most trivial to the most significant. Michael J. Roberts, Professor Emeritus Electrical and Computer Engineering University of Tennessee at Knoxville [email protected] Preface
- 24. Required=Results McGraw-Hill Connect® Learn Without Limits Connect is a teaching and learning platform that is proven to deliver better results for students and instructors. Connect empowers students by continually adapting to deliver precisely what they need, when they need it and how they need it, so your class time is more engaging and effective. Connect Insight® Connect Insight is Connect’s new one- of-a-kind visual analytics dashboard that provides at-a-glance information regarding student performance, which is immediately actionable. By presenting assignment, assessment and topical performance results together with a time metric that is easily visible for aggregate or individual results, Connect Insight gives the user the ability to take a just-in-time approach to teaching and learning, which was never before available. Connect Insight presents data that helps instructors improve class performance in a way that is efficient and effective. 73% of instructors who use Connect require it; instructor satisfaction increases by 28% when Connect is required. Analytics ©Getty Images/iStockphoto Using Connect improves passing rates by 12.7% and retention by 19.8%.
- 25. SmartBook® Proven to help students improve grades and study more efficiently, SmartBook contains the same content within the print book, but actively tailors that content to the needs of the individual. SmartBook’s adaptive technology provides precise, personalized instruction on what the student should do next, guiding the student to master and remember key concepts, targeting gaps in knowledge and offering customized feedback and driving the student toward comprehension and retention of the subject matter. Available on smartphones and tablets, SmartBook puts learning at the student’s fingertips—anywhere, anytime. Adaptive Over 5.7 billion questions have been answered, making McGraw-Hill Education products more intelligent, reliable and precise. THE ADAPTIVE READING EXPERIENCE DESIGNED TO TRANSFORM THE WAY STUDENTS READ More students earn A’s and B’s when they use McGraw-Hill Education Adaptive products. www.mheducation.com ©Getty Images/iStockphoto
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- 27. 1 C H A P T E R 1 Introduction 1.1 SIGNALS AND SYSTEMS DEFINED Any time-varying physical phenomenon that is intended to convey information is a signal. Examples of signals are the human voice, sign language, Morse code, traffic signals, voltages on telephone wires, electric fields emanating from radio or television transmitters, and variations of light intensity in an optical fiber on a telephone or com- puter network. Noise is like a signal in that it is a time-varying physical phenomenon, but usually it does not carry useful information and is considered undesirable. Signals are operated on by systems. When one or more excitations or input signals are applied at one or more system inputs, the system produces one or more responses or output signals at its outputs. Figure 1.1 is a block diagram of a single-input, single-output system. System Input Output Excitation or Input Signal Response or Output Signal Figure 1.1 Block diagram of a single-input, single-output system Transmitter Channel Receiver Information Signal Noisy Information Signal Noise Noise Noise Figure 1.2 A communication system In a communication system, a transmitter produces a signal and a receiver acquires it. A channel is the path a signal takes from a transmitter to a receiver. Noise is inevitably introduced into the transmitter, channel and receiver, often at multiple points (Figure 1.2). The transmitter, channel and receiver are all components or subsystems of the overall system. Scientific instruments are systems that measure a physical phenom- enon (temperature, pressure, speed, etc.) and convert it to a voltage or current, a sig- nal. Commercial building control systems (Figure 1.3), industrial plant control systems (Figure 1.4), modern farm machinery (Figure 1.5), avionics in airplanes, ignition and fuel pumping controls in automobiles, and so on are all systems that operate on signals.
- 28. Ch ap ter 1 Introduction 2 Figure 1.3 Modern office buildings © Vol. 43 PhotoDisc/Getty Figure 1.4 Typical industrial plant control room © Royalty-Free/Punchstock
- 29. 1.2 Types of Signals 3 The term system even encompasses things such as the stock market, government, weather, the human body and the like. They all respond when excited. Some systems are readily analyzed in detail, some can be analyzed approximately, but some are so complicated or difficult to measure that we hardly know enough to understand them. 1.2 TYPES OF SIGNALS There are several broad classifications of signals: continuous-time, discrete-time, continuous-value, discrete-value, random and nonrandom. A continuous-time sig- nal is defined at every instant of time over some time interval. Another common name for some continuous-time signals is analog signal, in which the variation of the signal with time is analogous (proportional) to some physical phenomenon. All analog sig- nals are continuous-time signals but not all continuous-time signals are analog signals (Figure 1.6 through Figure 1.8). Sampling a signal is acquiring values from a continuous-time signal at discrete points in time. The set of samples forms a discrete-time signal. A discrete-time signal Figure 1.5 Modern farm tractor with enclosed cab © Royalty-Free/Corbis Figure 1.6 Examples of continuous-time and discrete-time signals n x[n] Discrete-Time Continuous-Value Signal t x(t) Continuous-Time Continuous-Value Signal
- 30. Ch ap ter 1 Introduction 4 can also be created by an inherently discrete-time system that produces signal values only at discrete times (Figure 1.6). A continuous-value signal is one that may have any value within a continuum of allowed values. In a continuum any two values can be arbitrarily close together. The real numbers form a continuum with infinite extent. The real numbers between zero and one form a continuum with finite extent. Each is a set with infinitely many mem- bers (Figure 1.6 through Figure 1.8). A discrete-value signal can only have values taken from a discrete set. In a discrete set of values the magnitude of the difference between any two values is greater than some positive number. The set of integers is an example. Discrete-time signals are usually transmitted as digital signals, a sequence of values of a discrete-time signal in the form of digits in some encoded form. The term digital is also sometimes used loosely to refer to a discrete-value signal that has only two possible values. The digits in this type of digital signal are transmitted by signals that are continuous-time. In this case, the terms continuous-time and analog are not synonymous. A digital signal of this type is a continuous-time signal but not an analog signal because its variation of value with time is not directly analogous to a physical phenomenon (Figure 1.6 through Figure 1.8). A random signal cannot be predicted exactly and cannot be described by any math- ematical function. A deterministic signal can be mathematically described. A com- mon name for a random signal is noise (Figure 1.6 through Figure 1.8). In practical signal processing it is very common to acquire a signal for processing by a computer by sampling, quantizing and encoding it (Figure 1.9). The original signal is a continuous-value, continuous-time signal. Sampling acquires its values at discrete times and those values constitute a continuous-value, discrete-time signal. Quantization approximates each sample as the nearest member of a finite set of dis- crete values, producing a discrete-value, discrete-time signal. Each signal value in the set of discrete values at discrete times is converted to a sequence of rectangular pulses that encode it into a binary number, creating a discrete-value, continuous-time signal, commonly called a digital signal. The steps illustrated in Figure 1.9 are usually carried out by a single device called an analog-to-digital converter (ADC). Figure 1.8 Examples of noise and a noisy digital signal Noisy Digital Signal Continuous-Time Continuous-Value Random Signal t x(t) x(t) Noise t Figure 1.7 Examples of continuous-time, discrete-value signals Continuous-Time Discrete-Value Signal Continuous-Time Discrete-Value Signal t x(t) x(t) t Digital Signal
- 31. 1.2 Types of Signals 5 Figure 1.9 Sampling, quantization and encoding of a signal to illustrate various signal types t kΔt (k+1)Δt (k+2)Δt (k–1)Δt kΔt (k+1)Δt (k+2)Δt (k–1)Δt xs[n] n k k+1 k+2 k–1 xsq[n] n k k+1 k+2 k–1 xsqe(t) t 111 001 111 011 Continuous-Value Continuous-Time Signal Continuous-Value Discrete-Time Signal Discrete-Value Discrete-Time Signal Discrete-Value Continuous-Time Signal Sampling Quantization Encoding x(t) Figure 1.10 Asynchronous serial binary ASCII-encoded voltage signal for the word SIGNAL 0 1 2 3 4 5 6 7 –1 0 1 2 3 4 5 6 Time, t (ms) Voltage, v(t) (V) Serial Binary Voltage Signal for the ASCII Message “SIGNAL” S I G N A L One common use of binary digital signals is to send text messages using the American Standard Code for Information Interchange (ASCII). The letters of the al- phabet, the digits 0–9, some punctuation characters and several nonprinting control characters, for a total of 128 characters, are all encoded into a sequence of 7 binary bits. The 7 bits are sent sequentially, preceded by a start bit and followed by 1 or 2 stop bits for synchronization purposes. Typically, in direct-wired connections between digital equipment, the bits are represented by a higher voltage (2 to 5V) for a 1 and a lower voltage level (around 0V) for a 0. In an asynchronous transmission using one start and one stop bit, sending the message SIGNAL, the voltage versus time would look as illustrated in Figure 1.10.
- 32. Ch ap ter 1 Introduction 6 In 1987 ASCII was extended to Unicode. In Unicode the number of bits used to represent a character can be 8, 16, 24 or 32 and more than 120,000 characters are cur- rently encoded in modern and historic language characters and multiple symbol sets. Digital signals are important in signal analysis because of the spread of digital systems. Digital signals often have better immunity to noise than analog signals. In binary signal communication the bits can be detected very cleanly until the noise gets very large. The detection of bit values in a stream of bits is usually done by comparing the signal value at a predetermined bit time with a threshold. If it is above the thresh- old it is declared a 1 and if it is below the threshold it is declared a 0. In Figure 1.11, the x’s mark the signal value at the detection time, and when this technique is applied to the noisy digital signal, one of the bits is incorrectly detected. But when the signal is processed by a filter, all the bits are correctly detected. The filtered digital signal does not look very clean in comparison with the noiseless digital signal, but the bits can still be detected with a very low probability of error. This is the basic reason that digital signals can have better noise immunity than analog signals. An introduction to the analysis and design of filters is presented in Chapters 11 and 15. In this text we will consider both continuous-time and discrete-time signals, but we will (mostly) ignore the effects of signal quantization and consider all signals to be continuous-value. Also, we will not directly consider the analysis of random signals, although random signals will sometimes be used in illustrations. The first signals we will study are continuous-time signals. Some continuous-time signals can be described by continuous functions of time. A signal x(t) might be described by a function x(t) = 50sin(200πt) of continuous time t. This is an exact description of the signal at every instant of time. The signal can also be described graphically (Figure 1.12). Many continuous-time signals are not as easy to describe mathematically. Consider the signal in Figure 1.13. Waveforms like the one in Figure 1.13 occur in various types of instrumentation and communication systems. With the definition of some signal functions and an operation called convolution, this signal can be compactly described, analyzed and manipulated mathematically. Continuous-time signals that can be described by math- ematical functions can be transformed into another domain called the frequency domain through the continuous-time Fourier transform. In this context, transformation means transformation of a signal to the frequency domain. This is an important tool in signal analysis, which allows certain characteristics of the signal to be more clearly observed Figure 1.11 Use of a filter to reduce bit error rate in a digital signal x(t) –1 2 Noiseless Digital Signal t 2.6 –1 2 t 2.6 –1 2 t 2.6 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 0 1 1 Bit Error Bit Detection Threshold xf(t) x(t) + n(t) Noisy Digital Signal Filtered Digital Signal
- 33. 1.2 Types of Signals 7 and more easily manipulated than in the time domain. (In the frequency domain, signals are described in terms of the frequencies they contain.) Without frequency-domain analy- sis, design and analysis of many systems would be considerably more difficult. Discrete-time signals are only defined at discrete points in time. Figure 1.14 illustrates some discrete-time signals. Figure 1.12 A continuous-time signal described by a mathematical function ... ... –50 50 t = 10 ms t x(t) Figure 1.13 A second continuous-time signal ... ... 5 t = 20 μs t x(t) Figure 1.14 Some discrete-time signals n x[n] n x[n] n x[n] n x[n] So far all the signals we have considered have been described by functions of time. An important class of “signals” is functions of space instead of time: images. Most of the theories of signals, the information they convey and how they are processed by systems in this text will be based on signals that are a variation of a physical phenome- non with time. But the theories and methods so developed also apply, with only minor modifications, to the processing of images. Time signals are described by the variation of a physical phenomenon as a function of a single independent variable, time. Spa- tial signals, or images, are described by the variation of a physical phenomenon as a
- 34. Ch ap ter 1 Introduction 8 function of two orthogonal, independent, spatial variables, conventionally referred to as x and y. The physical phenomenon is most commonly light or something that affects the transmission or reflection of light, but the techniques of image processing are also applicable to anything that can be mathematically described by a function of two independent variables. Historically the practical application of image-processing techniques has lagged behind the application of signal-processing techniques because the amount of infor- mation that has to be processed to gather the information from an image is typically much larger than the amount of information required to get the information from a time signal. But now image processing is increasingly a practical technique in many situ- ations. Most image processing is done by computers. Some simple image-processing operations can be done directly with optics and those can, of course, be done at very high speeds (at the speed of light!). But direct optical image processing is very limited in its flexibility compared with digital image processing on computers. Figure 1.15 shows two images. On the left is an unprocessed X-ray image of a carry-on bag at an airport checkpoint. On the right is the same image after being pro- cessed by some image-filtering operations to reveal the presence of a weapon. This text will not go into image processing in any depth but will use some examples of image processing to illustrate concepts in signal processing. An understanding of how signals carry information and how systems process sig- nals is fundamental to multiple areas of engineering. Techniques for the analysis of sig- nals processed by systems are the subject of this text. This material can be considered as an applied mathematics text more than a text covering the building of useful devices, but an understanding of this material is very important for the successful design of useful devices. The material that follows builds from some fundamental definitions and concepts to a full range of analysis techniques for continuous-time and discrete-time signals in systems. 1.3 EXAMPLES OF SYSTEMS There are many different types of signals and systems. A few examples of systems are discussed next. The discussion is limited to the qualitative aspects of the system with some illustrations of the behavior of the system under certain conditions. These systems will be revisited in Chapter 4 and discussed in a more detailed and quantitative way in the material on system modeling. Figure 1.15 An example of image processing to reveal information (Original X-ray image and processed version provided by the Imaging, Robotics and Intelligent Systems (IRIS) Laboratory of the Department of Electrical and Computer Engineering at the University of Tennessee, Knoxville.)
- 35. 1.3 Examples of Systems 9 A MECHANICAL SYSTEM A man bungee jumps off a bridge over a river. Will he get wet? The answer depends on several factors: 1. The man’s height and weight 2. The height of the bridge above the water 3. The length and springiness of the bungee cord When the man jumps off the bridge he goes into free fall caused by the force due to gravitational attraction until the bungee cord extends to its full unstretched length. Then the system dynamics change because there is now another force on the man, the bungee cord’s resistance to stretching, and he is no longer in free fall. We can write and solve a differential equation of motion and determine how far down the man falls before the bungee cord pulls him back up. The differential equation of motion is a mathematical model of this mechanical system. If the man weighs 80 kg and is 1.8 m tall, and if the bridge is 200 m above the water level and the bungee cord is 30 m long (unstretched) with a spring constant of 11 N/m, the bungee cord is fully extended be- fore stretching at t = 2.47 s. The equation of motion, after the cord starts stretching, is x(t) = −16.85 sin(0.3708t) − 95.25 cos(0.3708t) + 101.3, t 2.47. (1.1) Figure 1.16 shows his position versus time for the first 15 seconds. From the graph it seems that the man just missed getting wet. Figure 1.16 Man’s vertical position versus time (bridge level is zero) 0 5 10 15 –200 –180 –160 –140 –120 –100 –80 –60 –40 –20 0 Time, t (s) Elevation (m) Bridge Level Water Level Free Fall Bungee Stretched A FLUID SYSTEM A fluid system can also be modeled by a differential equation. Consider a cylindrical water tank being fed by an input flow of water, with an orifice at the bottom through which flows the output (Figure 1.17). The flow out of the orifice depends on the height of the water in the tank. The vari- ation of the height of the water depends on the input flow and the output flow. The rate
- 36. Ch ap ter 1 Introduction 10 of change of water volume in the tank is the difference between the input volumetric flow and the output volumetric flow and the volume of water is the cross-sectional area of the tank times the height of the water. All these factors can be combined into one differential equation for the water level h1 (t). A 1 d __ dt ( h 1 (t)) + A 2 √ ___________ 2g[h 1 (t) − h 2 ] = f 1 (t) (1.2) The water level in the tank is graphed in Figure 1.18 versus time for four volumetric inflows under the assumption that the tank is initially empty. Figure 1.17 Tank with orifice being filled from above h (t) 1 h2 v (t) 2 f (t) 2 f (t) 1 A1 A2 Figure 1.18 Water level versus time for four different volumetric inflows with the tank initially empty 0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 3 3.5 Volumetric Inflow = 0.001 m3/s Volumetric Inflow = 0.002 m3/s Volumetric Inflow = 0.003 m3/s Volumetric Inflow = 0.004 m3/s Tank Cross-Sectional Area = 1 m2 Orifice Area = 0.0005 m2 Time, t (s) Water Level, h 1 (t) (m) As the water flows in, the water level increases, and that increases the water out- flow. The water level rises until the outflow equals the inflow. After that time the water level stays constant. Notice that when the inflow is increased by a factor of two, the final water level is increased by a factor of four. The final water level is proportional to the square of the volumetric inflow. That fact makes the differential equation that models the system nonlinear.
- 37. 1.3 Examples of Systems 11 A DISCRETE-TIME SYSTEM Discrete-time systems can be designed in multiple ways. The most common practical example of a discrete-time system is a computer. A computer is controlled by a clock that determines the timing of all operations. Many things happen in a computer at the integrated circuit level between clock pulses, but a computer user is only interested in what happens at the times of occurrence of clock pulses. From the user’s point of view, the computer is a discrete-time system. We can simulate the action of a discrete-time system with a computer program. For example, yn = 1 ; yn1 = 0 ; while 1, yn2 = yn1 ; yn1 = yn ; yn = 1.97*yn1 − yn2 ; end This computer program (written in MATLAB) simulates a discrete-time system with an output signal y that is described by the difference equation y[n] = 1.97y[n − 1] − y[n − 2] (1.3) along with initial conditions y[0] = 1 and y[−1] = 0. The value of y at any time index n is the sum of the previous value of y at discrete time n − 1 multiplied by 1.97, minus the value of y previous to that at discrete time n − 2. The operation of this system can be diagrammed as in Figure 1.19. In Figure 1.19, the two squares containing the letter D are delays of one in discrete time, and the arrowhead next to the number 1.97 represents an amplifier that multiplies the signal entering it by 1.97 to produce the signal leaving it. The circle with the plus sign in it is a summing junction. It adds the two signals entering it (one of which is negated first) to produce the signal leaving it. The first 50 values of the signal produced by this system are illustrated in Figure 1.20. The system in Figure 1.19 could be built with dedicated hardware. Discrete-time delay can be implemented with a shift register. Multiplication by a constant can be done with an amplifier or with a digital hardware multiplier. Summation can also be done with an operational amplifier or with a digital hardware adder. Figure 1.20 Signal produced by the discrete-time system in Figure 1.19 n 50 y[n] –6 6 Figure 1.19 Discrete-time system example y[n] y[n−2] y[n−1] 1.97 + – D D
- 38. Ch ap ter 1 Introduction 12 FEEDBACK SYSTEMS Another important aspect of systems is the use of feedback to improve system perfor- mance. In a feedback system, something in the system observes its response and may modify the input signal to the system to improve the response. A familiar example is a thermostat in a house that controls when the air conditioner turns on and off. The thermostat has a temperature sensor. When the temperature inside the thermostat ex- ceeds the level set by the homeowner, a switch inside the thermostat closes and turns on the home air conditioner. When the temperature inside the thermostat drops a small amount below the level set by the homeowner, the switch opens, turning off the air conditioner. Part of the system (a temperature sensor) is sensing the thing the system is trying to control (the air temperature) and feeds back a signal to the device that actually does the controlling (the air conditioner). In this example, the feedback signal is simply the closing or opening of a switch. Feedback is a very useful and important concept and feedback systems are every- where. Take something everyone is familiar with, the float valve in an ordinary flush toilet. It senses the water level in the tank and, when the desired water level is reached, it stops the flow of water into the tank. The floating ball is the sensor and the valve to which it is connected is the feedback mechanism that controls the water level. If all the water valves in all flush toilets were exactly the same and did not change with time, and if the water pressure upstream of the valve were known and constant, and if the valve were always used in exactly the same kind of water tank, it should be possible to replace the float valve with a timer that shuts off the water flow when the water reaches the desired level, because the water would always reach the desired level at exactly the same elapsed time. But water valves do change with time and water pres- sure does fluctuate and different toilets have different tank sizes and shapes. Therefore, to operate properly under these varying conditions the tank-filling system must adapt by sensing the water level and shutting off the valve when the water reaches the desired level. The ability to adapt to changing conditions is the great advantage of feedback methods. There are countless examples of the use of feedback. 1. Pouring a glass of lemonade involves feedback. The person pouring watches the lemonade level in the glass and stops pouring when the desired level is reached. 2. Professors give tests to students to report to the students their performance levels. This is feedback to let the student know how well she is doing in the class so she can adjust her study habits to achieve her desired grade. It is also feedback to the professor to let him know how well his students are learning. 3. Driving a car involves feedback. The driver senses the speed and direction of the car, the proximity of other cars and the lane markings on the road and constantly applies corrective actions with the accelerator, brake and steering wheel to maintain a safe speed and position. 4. Without feedback, the F-117 stealth fighter would crash because it is aerodynamically unstable. Redundant computers sense the velocity, altitude, roll, pitch and yaw of the aircraft and constantly adjust the control surfaces to maintain the desired flight path (Figure 1.21). Feedback is used in both continuous-time systems and discrete-time systems. The system in Figure 1.22 is a discrete-time feedback system. The response of the system y[n] is “fed back” to the upper summing junction after being delayed twice and multi- plied by some constants.
- 39. 1.3 Examples of Systems 13 Let this system be initially at rest, meaning that all signals throughout the system are zero before time index n = 0. To illustrate the effects of feedback let a = 1, let b = −1.5, let c = 0.8 and let the input signal x[n] change from 0 to 1 at n = 0 and stay at 1 for all time, n ≥ 0. We can see the response y [n] in Figure 1.23. Now let c = 0.6 and leave a and b the same. Then we get the response in Figure 1.24. Now let c = 0.5 and leave a and b the same. Then we get the response in Figure 1.25. The response in Figure 1.25 increases forever. This last system is unstable because a bounded input signal produces an unbounded response. So feedback can make a system unstable. Figure 1.21 The F-117A Nighthawk stealth fighter © Vol. 87/Corbis Figure 1.22 A discrete-time feedback system x[n] + + + + + – y[n] D D b a c Figure 1.23 Discrete-time system response with b = −1.5 and c = 0.8 n 60 y[n] 6 a = 1, b = –1.5, c = 0.8 Figure 1.25 Discrete-time system response with b = −1.5 and c = 0.5 n 60 y[n] 140 a = 1, b = –1.5, c = 0.5 Figure 1.24 Discrete-time system response with b = −1.5 and c = 0.6 n 60 y[n] 12 a = 1, b = –1.5, c = 0.6 The system illustrated in Figure 1.26 is an example of a continuous-time feedback system. It is described by the differential equation yʺ(t) + ay(t) = x(t). The homoge- neous solution can be written in the form y h (t) = K h1 sin ( √ __ a t)+ K h2 cos ( √ __ a t). (1.4) If the excitation x(t) is zero and the initial value y( t 0 ) is nonzero or the initial deriva- tive of y(t) is nonzero and the system is allowed to operate in this form after t = t 0, y(t) Figure 1.26 Continuous-time feedback system x(t) y(t) a ∫ ∫
- 40. Ch ap ter 1 Introduction 14 will oscillate sinusoidally forever. This system is an oscillator with a stable amplitude. So feedback can cause a system to oscillate. 1.4 A FAMILIAR SIGNAL AND SYSTEM EXAMPLE As an example of signals and systems, let’s look at a signal and system that everyone is familiar with, sound, and a system that produces and/or measures sound. Sound is what the ear senses. The human ear is sensitive to acoustic pressure waves typically between about 15 Hz and about 20 kHz with some sensitivity variation in that range. Below are some graphs of air-pressure variations that produce some common sounds. These sounds were recorded by a system consisting of a microphone that converts air-pressure variation into a continuous-time voltage signal, electronic circuitry that processes the continuous-time voltage signal, and an ADC that changes the continuous-time voltage signal to a digital signal in the form of a sequence of binary numbers that are then stored in computer memory (Figure 1.27). Figure 1.27 A sound recording system Microphone ADC Electronics Acoustic Pressure Variation Voltage Processed Voltage Binary Numbers Computer Memory Figure 1.28 The word “signal” spoken by an adult male voice 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 –1 –0.5 0.5 1 Time, t (s) Delta p(t) (Arbitrary Units) Adult Male Voice Saying the Word, “Signal” 0.07 0.074 0.078 –0.2 –0.1 0 0.1 0.2 Time, t (s) Delta p(t) 0.15 0.155 0.16 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 Time, t (s) Delta p(t) 0.3 0.305 0.31 –0.1 –0.05 0 0.05 Time, t (s) Delta p(t) Consider the pressure variation graphed in Figure 1.28. It is the continuous-time pressure signal that produces the sound of the word “signal” spoken by an adult male (the author).
- 41. 1.4 A Familiar Signal and System Example 15 Analysis of sounds is a large subject, but some things about the relationship between this graph of air-pressure variation and what a human hears as the word “signal” can be seen by looking at the graph. There are three identifiable “bursts” of signal, #1 from 0 to about 0.12 seconds, #2 from about 0.12 to about 0.19 seconds and #3 from about 0.22 to about 0.4 seconds. Burst #1 is the s in the word “signal.” Burst #2 is the i sound. The region between bursts #2 and #3 is the double consonant gn of the word “signal.” Burst #3 is the a sound terminated by the l consonant stop. An l is not quite as abrupt a stop as some other consonants, so the sound tends to “trail off” rather than stopping quickly. The variation of air pressure is generally faster for the s than for the i or the a. In signal analysis we would say that it has more “high-frequency content.” In the blowup of the s sound the air-pressure variation looks almost random. The i and a sounds are different in that they vary more slowly and are more “regular” or “predictable” (although not exactly predictable). The i and a are formed by vibrations of the vocal cords and therefore exhibit an approximately oscillatory behavior. This is described by saying that the i and a are tonal or voiced and the s is not. Tonal means having the basic quality of a single tone or pitch or frequency. This description is not mathematically precise but is useful qualitatively. Another way of looking at a signal is in the frequency domain, mentioned above, by examining the frequencies, or pitches, that are present in the signal. A common way of illustrating the variation of signal power with frequency is its power spectral density, a graph of the power density in the signal versus frequency. Figure 1.29 shows the three bursts (s, i and a) from the word “signal” and their associated power spectral densities (the G( f ) functions). Figure 1.29 Three sounds in the word “signal” and their associated power spectral densities t Delta p(t) f –22,000 22,000 t Delta p(t) f –22,000 22,000 t Delta p(t) f –22,000 22,000 G( f ) G( f ) G( f ) 0.16 s 0.1 s 0.12 s “s” Sound “i” Sound “a” Sound Power spectral density is just another mathematical tool for analyzing a signal. It does not contain any new information, but sometimes it can reveal things that are dif- ficult to see otherwise. In this case, the power spectral density of the s sound is widely distributed in frequency, whereas the power spectral densities of the i and a sounds are narrowly distributed in the lowest frequencies. There is more power in the s sound at
- 42. Ch ap ter 1 Introduction 16 higher frequencies than in the i and a sounds. The s sound has an “edge” or “hissing” quality caused by the high frequencies in the s sound. The signal in Figure 1.28 carries information. Consider what happens in conver- sation when one person says the word “signal” and another hears it (Figure 1.30). The speaker thinks first of the concept of a signal. His brain quickly converts the concept to the word “signal.” Then his brain sends nerve impulses to his vocal cords and di- aphragm to create the air movement and vibration and tongue and lip movements to produce the sound of the word “signal.” This sound then propagates through the air between the speaker and the listener. The sound strikes the listener’s eardrum and the vibrations are converted to nerve impulses, which the listener’s brain converts first to the sound, then the word, then the concept signal. Conversation is accomplished by a system of considerable sophistication. How does the listener’s brain know that the complicated pattern in Figure 1.28 is the word “signal”? The listener is not aware of the detailed air-pressure varia- tions but instead “hears sounds” that are caused by the air-pressure variation. The eardrum and brain convert the complicated air-pressure pattern into a few simple features. That conversion is similar to what we will do when we convert signals into the frequency domain. The process of recognizing a sound by reducing it to a small set of features reduces the amount of information the brain has to process. Signal processing and analysis in the technical sense do the same thing but in a more math- ematically precise way. Two very common problems in signal and system analysis are noise and interfer- ence. Noise is an undesirable random signal. Interference is an undesirable nonran- dom signal. Noise and interference both tend to obscure the information in a signal. Figure 1.31 shows examples of the signal from Figure 1.28 with different levels of noise added. As the noise power increases there is a gradual degradation in the intelligibility of the signal, and at some level of noise the signal becomes unintelligible. A measure of the quality of a received signal corrupted by noise is the ratio of the signal power to the noise power, commonly called signal-to-noise ratio and often abbreviated SNR. In each of the examples of Figure 1.31 the SNR is specified. Sounds are not the only signals, of course. Any physical phenomenon that is mea- sured or observed is a signal. Also, although the majority of signals we will consider in this text will be functions of time, a signal can be a function of some other independent Figure 1.30 Communication between two people involving signals and signal processing by systems “Signal” “Signal”
- 43. 1.4 A Familiar Signal and System Example 17 Signal-to-Noise Ratio = 23.7082 Original Signal Without Noise Signal-to-Noise Ratio = 3.7512 Signal-to-Noise Ratio = 0.95621 Figure 1.31 Sound of the word “signal” with different levels of noise added variable, like frequency, wavelength, distance and so on. Figure 1.32 and Figure 1.33 illustrate some other kinds of signals. Just as sounds are not the only signals, conversation between two people is not the only system. Examples of other systems include the following: 1. An automobile suspension for which the road surface excites the automobile and the position of the chassis relative to the road is the response. 2. A chemical mixing vat for which streams of chemicals are the input signals and the mixture of chemicals is the output signal. 3. A building environmental control system for which the exterior temperature is the input signal and the interior temperature is the response. Far-Field Intensity of Light Diffracted Through a Slit T(t) F(t) t Outside Air Temperature 24 hours t Neutron Flux in a Nuclear Reactor Core 1 ms S(λ) λ Optical Absorption Spectrum of a Chemical Mixture 400 nm 700 nm I(θ) θ 30° Two-Dimensional Image Correlation x y C(x,y) Figure 1.32 Examples of signals that are functions of one or more continuous independent variables
- 44. Ch ap ter 1 Introduction 18 4. A chemical spectroscopy system in which white light excites the specimen and the spectrum of transmitted light is the response. 5. A telephone network for which voices and data are the input signals and reproductions of those voices and data at a distant location are the output signals. 6. Earth’s atmosphere, which is excited by energy from the sun and for which the responses are ocean temperature, wind, clouds, humidity and so on. In other words, the weather is the response. 7. A thermocouple excited by the temperature gradient along its length for which the voltage developed between its ends is the response. 8. A trumpet excited by the vibration of the player’s lips and the positions of the valves for which the response is the tone emanating from the bell. The list is endless. Any physical entity can be thought of as a system, because if we excite it with physical energy, it has a physical response. 1.5 USE OF MATLAB® Throughout the text, examples will be presented showing how signal and system analy- sis can be done using MATLAB. MATLAB is a high-level mathematical tool available on many types of computers. It is very useful for signal processing and system analysis. There is an introduction to MATLAB in Web Appendix A. n N[n] D[n] N[n] P[n] Number of Cars Crossing an Intersection Between Red Lights United States Population 2 4 6 8 n Ball-Bearing Manufacturer’s Quality Control Chart for Diameter 1 cm 1.01 cm 0.99 cm n 1800 1900 2000 300 Million World War II Great Depression World War I US Civil War n 1950 2000 2500 Number of Annual Sunspots Figure 1.33 Examples of signals that are functions of a discrete independent variable
- 45. 19 2.1 INTRODUCTION AND GOALS Over the years, signal and system analysts have observed many signals and have real- ized that signals can be classified into groups with similar behavior. Figure 2.1 shows some examples of signals. t x(t) Amplitude-Modulated Carrier in a Communication System t x(t) Car Bumper Height after Car Strikes a Speed Bump t x(t) Light Intensity from a Q-Switched Laser t x(t) Step Response of an RC Lowpass Filter t x(t) Frequency-Shift-Keyed Binary Bit Stream t x(t) Manchester Encoded Baseband Binary Bit Stream Figure 2.1 Examples of signals In signal and system analysis, signals are described by mathematical functions. Some of the functions that describe real signals should already be familiar, exponentials and sinusoids. These occur frequently in signal and system analysis. One set of functions has been defined to describe the effects on signals of switching operations that often occur in systems. Some other functions arise in the development of certain system analysis tech- niques, which will be introduced in later chapters. These functions are all carefully chosen to be simply related to each other and to be easily changed by a well-chosen set of shifting and/or scaling operations. They are prototype functions, which have simple definitions and are easily remembered. The types of symmetries and patterns that most frequently occur in real signals will be defined and their effects on signal analysis explored. C H A P T E R 2 Mathematical Description of Continuous-Time Signals
- 46. Ch ap ter 2 Mathematical Description of Continuous-Time Signals 20 CH APTER G OA L S 1. To define some mathematical functions that can be used to describe signals 2. To develop methods of shifting, scaling and combining those functions to represent real signals 3. To recognize certain symmetries and patterns to simplify signal and system analysis 2.2 FUNCTIONAL NOTATION A function is a correspondence between the argument of the function, which lies in its domain, and the value returned by the function, which lies in its range. The most familiar functions are of the form g(x)where the argument x is a real number and the value returned g is also a real number. But the domain and/or range of a function can be complex numbers or integers or a variety of other choices of allowed values. In this text five types of functions will appear, 1. Domain—Real numbers, Range—Real numbers 2. Domain—Integers, Range—Real numbers 3. Domain—Integers, Range—Complex numbers 4. Domain—Real numbers, Range—Complex numbers 5. Domain—Complex numbers, Range—Complex numbers For functions whose domain is either real numbers or complex numbers the argument will be enclosed in parentheses (⋅). For functions whose domain is integers the argu- ment will be enclosed in brackets [⋅]. These types of functions will be discussed in more detail as they are introduced. 2.3 CONTINUOUS-TIME SIGNAL FUNCTIONS If the independent variable of a function is time t and the domain of the function is the real numbers, and if the function g(t) has a defined value at every value of t, the function is called a continuous-time function. Figure 2.2 illustrates some continuous-time functions. Figure 2.2 Examples of continuous-time functions t g(t) t g(t) t g(t) t g(t) (a) (b) (c) (d) Points of Discontinuity of g(t)
- 47. 2.3 Continuous-Time Signal Functions 21 Figure 2.2(d) illustrates a discontinuous function for which the limit of the func- tion value as we approach the discontinuity from above is not the same as when we approach it from below. If t = t 0is a point of discontinuity of a function g(t) then lim ε→0 g(t 0+ ε) ≠ lim ε→0 g(t 0 − ε). All four functions, (a)–(d), are continuous-time functions because their values are de- fined for all real values of t. Therefore the terms continuous and continuous-time mean slightly different things. All continuous functions of time are continuous-time func- tions, but not all continuous-time functions are continuous functions of time. COMPLEX EXPONENTIALS AND SINUSOIDS Real-valued sinusoids and exponential functions should already be familiar. In g(t) = Acos(2πt/ T 0+ θ) = Acos(2πf 0t + θ) = Acos(ω 0t + θ) and g(t) = A e (σ 0+jω 0)t = A e σ 0t [cos(ω 0t) + j sin(ω 0t)] A is the amplitude, T 0is the fundamental period, f 0is the fundamental cyclic frequency and ω 0is the fundamental radian frequency of the sinusoid, t is time and σ 0is the decay rate of the exponential (which is the reciprocal of its time constant, τ) (Figure 2.3 and Figure 2.4). All these parameters can be any real number. Figure 2.3 A real sinusoid and a real exponential with parameters indicated graphically t A t A τ g(t) = A cos(2πf0t+θ) g(t) = Ae–t/τ –θ/2πf0 T0 4 –4 ... ... t = 10 ms t –4sin(200πt) μA 10 –10 ... ... t = 2 μs t 10cos(106πt) nC 2 t = 0.1 s t 2e–10tm 5 –5 t = 1 s t 5e–tsin(2πt) m s2 Figure 2.4 Examples of signals described by real sines, cosines and exponentials In Figure 2.4 the units indicate what kind of physical signal is being described. Very often in system analysis, when only one kind of signal is being followed through a system, the units are omitted for the sake of brevity. Exponentials (exp) and sinusoids (sin and cos) are intrinsic functions in MATLAB. The arguments of the sin and cos functions are interpreted by MATLAB as radians, not degrees. [exp(1),sin(pi/2),cos(pi)] ans = 2.7183 1.0000 −1.0000 (pi is the MATLAB symbol for π.)
- 48. Ch ap ter 2 Mathematical Description of Continuous-Time Signals 22 Sinusoids and exponentials are very common in signal and system analysis because most continuous-time systems can be described, at least approximately, by linear, constant-coefficient, ordinary differential equations whose eigenfunctions are complex exponentials, complex powers of e, the base of the natural logarithms. Eigenfunction means “characteristic function” and the eigenfunctions have a particularly important relation to the differential equation. If the exponent of e is real, complex exponentials are the same as real exponentials. Through Euler’s identity e jx = cos(x) + j sin(x) and the relations cos(x) = (1/2)( e jx + e −jx ) and sin(x) = (1/j2)( e jx − e −jx ), complex expo- nentials and real-valued sinusoids are closely related. If, in a function of the form e jx , x is a real-valued independent variable, this special form of the complex exponential is called a complex sinusoid (Figure 2.5). t 2 Re(e j2πt) 1 Im(e j2πt) 1 –1 –1 t 2 Re(e–j2πt) 1 Im(e–j2πt) 1 –1 –1 t 2 Re 2 Im 1 –1 –2 t 2 Re 1 Im 2 –2 –1 Figure 2.5 The relation between real and complex sinusoids In signal and system analysis, sinusoids are expressed in either the cyclic fre- quency f form Acos(2πf 0t + θ) or the radian frequency ω form Acos(ω 0t + θ). The advantages of the f form are the following: 1. The fundamental period T 0 and the fundamental cyclic frequency f 0 are simply reciprocals of each other. 2. In communication system analysis, a spectrum analyzer is often used and its display scale is usually calibrated in Hz. Therefore f is the directly observed variable. 3. The definition of the Fourier transform (Chapter 6) and some transforms and transform relationships are simpler in the f form than in the ω form. The advantages of the ω form are the following: 1. Resonant frequencies of real systems, expressed directly in terms of physical parameters, are more simply expressed in the ω form than in the f form. The resonant frequency of an LC oscillator is ω0 2 = 1/LC = (2π f 0 ) 2 and the half-power corner frequency of an RC lowpass filter is ωc = 1/RC = 2π f c . 2. The Laplace transform (Chapter 8) is defined in a form that is more simply related to the ω form than to the f form.
- 49. 2.3 Continuous-Time Signal Functions 23 3. Some Fourier transforms are simpler in the ω form. 4. Use of ωin some expressions makes them more compact. For example, Acos( ω0 t + θ) is a little more compact than Acos(2π f 0 t + θ). Sinusoids and exponentials are important in signal and systems analysis because they arise naturally in the solutions of the differential equations that often describe sys- tem dynamics. As we will see in the study of the Fourier series and Fourier transform, even if signals are not sinusoids, most of them can be expressed as linear combinations of sinusoids. FUNCTIONS WITH DISCONTINUITIES Continuous-time sines, cosines and exponentials are all continuous and differentiable at every point in time. But many other types of important signals that occur in practical systems are not continuous or differentiable everywhere. A common operation in sys- tems is to switch a signal on or off at some time (Figure 2.6). Figure 2.6 Examples of signals that are switched on or off at some time 20 V –20 V t = 50 ns t 3 W t t = 10 s t = 2 ms t x(t) = x(t) = x(t) = 7 Pa t x(t) = 4 C 0 , t 0 3W , t 0 0 , t 10 s 4e0.1tC , t 10 s 7 Pa , t 2 ms 0 , t 2 ms 0 , t 0 20sin(4π×107t) V , t 0 The functional descriptions of the signals in Figure 2.6 are complete and accurate but are in a cumbersome form. Signals of this type can be better described mathemat- ically by multiplying a function that is continuous and differentiable for all time by another function that switches from zero to one or one to zero at some finite time. In signal and system analysis singularity functions, which are related to each other through integrals and derivatives, can be used to mathematically describe signals that have discontinuities or discontinuous derivatives. These functions, and functions that are closely related to them through some common system operations, are the subject of this section. In the consideration of singularity functions we will extend, modify and/or generalize some basic mathematical concepts and operations to allow us to efficiently analyze real signals and systems. We will extend the con- cept of what a derivative is, and we will also learn how to use an important mathe- matical entity, the impulse, which is a lot like a function but is not a function in the usual sense.
- 50. Ch ap ter 2 Mathematical Description of Continuous-Time Signals 24 t sgn(t) 1 –1 t sgn(t) 1 –1 Figure 2.7 The signum function The Signum Function For nonzero arguments, the value of the signum function has a magnitude of one and a sign that is the same as the sign of its argument: sgn(t) = { 1, t 0 0, t = 0 −1, t 0 } (2.1) (See Figure 2.7.) The graph on the left in Figure 2.7 is of the exact mathematical definition. The graph on the right is a more common way of representing the function for engineering purposes. No practical signal can change discontinuously, so if an approximation of the signum function were generated by a signal generator and viewed on an oscilloscope, it would look like the graph on the right. The signum function is intrinsic in MATLAB (and called the sign function). The Unit-Step Function The unit-step function is defined by u(t) = { 1, t 0 1/2, t = 0 0, t 0 (2.2) (See Figure 2.8.) It is called the unit step because the step is one unit high in the system of units used to describe the signal.1 1 Some authors define the unit step by u(t) = { 1, t ≥ 0 0, t 0 or u(t) = { 1, t 0 0, t 0 or u(t) = { 1, t 0 0, t ≤ 0 In the middle definition the value at t = 0 is undefined but finite. The unit steps defined by these definitions all have an identical effect on any real physical system. t u(t) 1 t u(t) 1 1 2 Figure 2.8 The unit-step function
- 51. 2.3 Continuous-Time Signal Functions 25 The unit step can mathematically represent a common action in real physical sys- tems, fast switching from one state to another. In the circuit of Figure 2.9 the switch moves from one position to the other at time t = 0. The voltage applied to the RC network is v RC(t) = V bu(t). The current flowing clockwise through the resistor and capacitor is i(t) = ( V b/R) e −t/RC u(t) and the voltage across the capacitor is v(t) = V b(1 − e −t/RC )u(t). There is an intrinsic function in MATLAB, called heaviside2 which returns a one for positive arguments, a zero for negative arguments and an NaN for zero arguments. The MATLAB constant NaN is “not a number” and indicates an undefined value. There are practical problems using this function in numerical computations because the re- turn of an undefined value can cause some programs to prematurely terminate or return useless results. We can create our own functions in MATLAB, which become functions we can call upon just like the intrinsic functions cos, sin, exp, etc. MATLAB functions are defined by creating an m file, a file whose name has the extension “.m”. We could create a function that finds the length of the hypotenuse of a right triangle given the lengths of the other two sides. % Function to compute the length of the hypotenuse of a % right triangle given the lengths of the other two sides % % a - The length of one side % b - The length of the other side % c - The length of the hypotenuse % % function c = hyp(a,b) % function c = hyp(a,b) c = sqrt(a^2 + b^2) ; The first nine lines in this example, which are preceded by %, are comment lines that are not executed but serve to document how the function is used. The first execut- able line must begin with the keyword function. The rest of the first line is in the form result = name(arg1, arg2,...) 2 Oliver Heaviside was a self-taught English electrical engineer who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations and reformulated and simplified Maxwell’s field equations. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of mathematics and science for years to come. It has been reported that a man once complained to Heaviside that his writings were very difficult to read. Heaviside’s response was that they were even more difficult to write! Vb R C t = 0 vRC(t) + – Figure 2.9 Circuit with a switch whose effect can be represented by a unit step
- 52. Ch ap ter 2 Mathematical Description of Continuous-Time Signals 26 where result will contain the returned value, which can be a scalar, a vector or a ma- trix (or even a cell array or a structure, which are beyond the scope of this text), name is the function name, and arg1, arg2,... are the parameters or arguments passed to the function. The arguments can also be scalars, vectors or matrices (or cell arrays or structures). The name of the file containing the function definition must be name.m. Below is a listing of a MATLAB function to implement the unit-step function in numerical computations. % Unit-step function defined as 0 for input argument values % less than zero, 1/2 for input argument values equal to zero, % and 1 for input argument values greater than zero. This % function uses the sign function to implement the unit-step % function. Therefore value at t = 0 is defined. This avoids % having undefined values during the execution of a program % that uses it. % % function y = us(x) % function y = us(x) y = (sign(x) + 1)/2 ; This function should be saved in a file named “us.m”. The Unit-Ramp Function Another type of signal that occurs in systems is one that is switched on at some time and changes linearly after that time or changes linearly before some time and is switched off at that time (Figure 2.10). Signals of this kind can be described with the use of the ramp function. The unit-ramp function (Figure 2.11) is the integral of the unit-step function. It is called the unit-ramp function because, for positive t, its slope is one amplitude unit per time unit. ramp(t) = { t, t 0 0, t ≤ 0 } = ∫ −∞ t u(λ)dλ = tu(t) (2.3) 20 t = 100 ms t 1V t = 6 s t t = 10 s t = 20 μs t x(t) x(t) x(t) t x(t) 4 mA –12 N cm s Figure 2.10 Functions that change linearly before or after some time, or are multiplied by functions that change linearly before or after some time t ramp(t) 1 1 Figure 2.11 The unit-ramp function
- 53. 2.3 Continuous-Time Signal Functions 27 The ramp is defined by ramp(t) = ∫−∞ t u(τ)dτ . In this equation, the symbol τ is the independent variable of the unit-step function and the variable of integration. But t is the independent variable of the ramp function. The equation says, “to find the value of the ramp function at any value of t, start with τ at negative infinity and move in τ up to τ = t, while accumulating the area under the unit-step function.” The total area accumulated from τ = −∞ to τ = t is the value of the ramp function at time t (Figure 2.12). For t less than zero, no area is accumulated. For t greater than zero, the area accumulated equals t because it is the area of a rectangle with width t and height one. τ u(τ) u(τ) u(τ) u(τ) 1 t Ramp(t) 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 –5–4–3–2–1 τ 1 1 2 3 4 5 –5–4–3–2–1 τ 1 1 2 3 4 5 –5–4–3–2–1 τ 1 1 2 3 4 5 –5–4–3–2–1 1 2 3 4 5 t = –1 t = 1 t = 3 t = 5 Figure 2.12 Integral relationship between the unit step and the unit ramp Some authors prefer to use the expression tu(t)instead of ramp(t) . Since they are equal, the use of either one is correct and just as legitimate as the other one. Below is a MATLAB m file for the ramp function. % Function to compute the ramp function defined as 0 for % values of the argument less than or equal to zero and % the value of the argument for arguments greater than zero. % Uses the unit-step function us(x). % % function y = ramp(x) % function y = ramp(x) y = x.*us(x) ; The Unit Impulse Before we define the unit impulse we will first explore an important idea. Consider a unit-area, rectangular pulse defined by Δ(t) = { 1/a, |t| ≤ a/2 0, |t| a/2 (See Figure 2.13.) Let this function multiply a function g(t)that is finite and continuous at t = 0and find the area A under the product of the two functions A = ∫−∞ ∞ Δ(t)g(t)dt (Figure 2.14).
- 54. Ch ap ter 2 Mathematical Description of Continuous-Time Signals 28 Using the definition of Δ(t)we can rewrite the integral as A = 1 __ a ∫ −a/2 a/2 g(t)dt. The function g(t)is continuous at t = 0 . Therefore it can be expressed as a McLaurin series of the form g(t) = ∑ m=0 ∞ g (m) (0) _____ m! t m = g(0) + g′ (0)t + g″ (0) ____ 2! t 2 + ⋯ + g (m) (0) _____ m! t m + ⋯ Then the integral becomes A = 1 __ a ∫ −a/2 a/2 [g(0) + g′ (0)t + g″ (0) ____ 2! t 2 + ⋯ + g (m) (0) _____ m! t m + ⋯] dt All the odd powers of t contribute nothing to the integral because it is taken over sym- metrical limits about t = 0 . Carrying out the integral, A = 1 __ a [ag(0) + ( a 3 __ 12 ) g″ (0) ____ 2! + ( a 5 __ 80 ) g (4) (0) _____ 4! + ⋯] Take the limit of this integral as a approaches zero. lim a→0 A = g(0) . In the limit as a approaches zero, the function Δ(t)extracts the value of any continuous finite function g(t)at time t = 0 , when the product of Δ(t) and g(t)is integrated over any range of time that includes time t = 0 . Now try a different definition of the function Δ(t) . Define it now as Δ(t) = { (1/a)(1 − |t|/a), |t| ≤ a 0, |t| a (See Figure 2.15.) If we make the same argument as before we get the area A = ∫ −∞ ∞ Δ(t)g(t)dt = 1 __ a ∫ −a a (1 − |t| __ a )g(t)dt. Δ(t) t 1 a 2 a 2 – a Figure 2.13 A unit-area rectangular pulse of width a t Δ(t) Δ(t)g(t) 1 a 2 a 2 – a g(t) Figure 2.14 Product of a unit-area rectangular pulse centered at t = 0and a func- tion g(t)that is continuous and finite at t = 0
- 55. Other documents randomly have different content
- 56. eighth, to uproot the “accursed doctrine” of Mohammed and to convert the Sulus to the Christian religion. The leader of the expedition was directed to carry out these instructions as carefully and as gently as possible; and there is no reason to think that he failed to comply with his orders to the letter. But no matter how careful and faithful Captain Rodriguez could have been, it was not difficult for the Sulus to understand the purpose of the expedition and the motives of the Spanish Government, and it does not stand to reason that such people would yield to vassalage and receive a direct insult to their religion without resentment and without a struggle. Governor Sandé knew the reputation of the Sulus, but he must have underestimated their strength and failed to provide garrisons for the occupation of the conquered territory and the protection of peaceful natives. In January, 1579, Governor Sandé sent an expedition to Mindanao, commanded by Capt. Gabriel de Ribera, under instructions similar to those given to Captain Rodriguez. Ribera had additional orders to visit Jolo and collect the tribute for that year, and special stress was laid on procuring from the Sultan of Sulu “two or three tame elephants.” Ribera accomplished nothing in Mindanao; the natives abandoned their villages and fled to the interior. On his return to Kawite or Caldera, he met a deputation from Jolo, which brought insignificant tribute and informed him of the existence of famine in Sulu and the extreme distress of the people. He returned their tribute, receiving in its place a cannon, which the Sulus had obtained from a wrecked Portuguese galley. Ribera then returned to Cebu, without producing any significant effect on conditions in Sulu. In April, 1580, Governor Sandé was relieved by Governor Gonzalo Ronquillo, who did not take the same interest in Borneo and Sulu. In the same year the kingdom of Portugal and its rich eastern colonies were annexed to the Spanish domain. No danger could then be expected from the direction of Borneo and Sulu, and the ambitious new Governor-General turned his attention to more desirable fields of conquest.
- 57. Piracy was not the primary cause of this invasion of Sulu. Public sentiment was not so strong against slavery in those days as it is now; for the Spaniards and other leading civilized nations were then diligently pursuing a profitable trade in it between the west coast of Africa and the West Indies and America. Piracy is always a crime among nations, but it can not be urged as the principal and leading cause of this war or as sufficient reason in itself for the early precipitation of such a deadly conflict between Sulu and Spain. Religion, on the other hand, was declared by Governor Sandé to be the “principal reason for going to their lands.” He ordered the Sulus not to admit any more preachers of Islam, but to allow the Spanish priests to preach Christianity to them. The Mohammedan preachers he directed to be arrested and brought to him, and the mosques to be burned or destroyed and not to be rebuilt. Part of the instructions the Adelantado24 Miguel Lopez de Legaspi received before embarking on his expedition to the Philippines read as follows: And you shall have especial care that, in all your negotiations with the natives of those regions some of the religious accompanying you be present, both in order to avail yourself of their good counsel and advice, and so that the natives may see and understand your high estimation of them; for seeing this, and the great reverence of the soldiers toward them, they themselves will hold the religious in great respect. This will be of great moment, so that, when the religious shall understand their language, or have interpreters through whom they may make them understand our holy Catholic faith, the Indians shall put entire faith in them; since you are aware that the chief thing sought after by his Majesty is the increase of our holy Catholic faith, and the salvation of the souls of those infidels.25 In 1566, a petition was sent from Cebu to the King of Spain, bearing the signatures of Martin de Goiti, Guido de Labezari, and the other leading officers under Legaspi, setting forth, among other requests, the following: That the Moros, “because they try to prevent our trade with the natives and preach to them the religion of Mohammed,” may be enslaved and lose their property. That slave traffic be allowed, “that the Spaniards may make use of them, as do the chiefs and natives of those regions, both in mines and other works that offer themselves.”26
- 58. In a letter addressed to Legaspi King Philip II said: We have also been petitioned in your behalf concerning the Moro Islands in that land, and how those men come to trade and carry on commerce, hindering the preaching of the holy gospel and disturbing you. We give you permission to make such Moros slaves and to seize their property. You are warned that you can make them slaves only if the said Moros are such by birth and choice, and if they come to preach their Mohammedan doctrine or to make war against you or against the Indians, who are our subjects and in our royal service. In a letter addressed to King Philip II Bishop Salazar writes, June 27, 1588, as follows: The second point is that, in the Island of Mindanao, which is subject to your Majesty, and for many years has paid you tribute, the law of Mohammed has been publicly proclaimed, for somewhat more than three years, by preachers from Bruney and Ternate who have come there—some of them even, it is believed, having come from Mecca. They have erected and are now building mosques, and the boys are being circumcised, and there is a school where they are taught the Quran. I was promptly informed of this, and urged the president to supply a remedy therefor at once, in order that that pestilential fire should not spread in these islands. I could not persuade them to go, and thus the hatred of Christianity is there; and we are striving no more to remedy this than if the matter did not concern us. Such are the calamities and miseries to which we have come, and the punishments which God inflicts upon us.27 In drawing a contract with Capt. Esteban Rodriguez de Figueroa, in 1591, for the pacification and conquest of Mindanao, the Governor and Captain- General Gomez Perez Dasmariñas makes the following declarations: His Majesty orders and charges me, by his royal instructions and decrees, as the most worthy and important thing in these islands, to strive for the propagation of our holy faith among the natives herein, their conversion to the knowledge of the true God, and their reduction to the obedience of His holy church and of the king, our sovereign. * * * Moreover, the Island of Mindanao is so fertile and well inhabited, and teeming with Indian settlements, wherein to plant the faith, * * * and is rich in gold mines and placers, and in wax, cinnamon, and other valuable drugs. And although the said island has been seen, discussed, and explored, * * * no effort has been made
- 59. to enter and reduce it, nor has it been pacified or furnished with instruction or justice—quite to the contrary being, at the present time, hostile and refusing obedience to his Majesty; and no tribute, or very little, is being collected. * * * Besides the above facts, by delaying the pacification of the said island greater wrongs, to the offense and displeasure of God and of his Majesty, are resulting daily; for I am informed that the king of that island has made all who were paying tribute to his Majesty tributary to himself by force of arms, and after putting many of them to death while doing it; so that now each Indian pays him one tae28 of gold. I am also told that he destroyed and broke into pieces, with many insults, a cross that he found, when told that it was adored by the Christians; and that in Magindanao, the capital and residence of the said king, are Bornean Indians who teach and preach publicly the false doctrine of Mohammed, and have mosques; besides these, there are also people from Ternate—gunners, armorers, and powder-makers, all engaged in their trades—who at divers times have killed many Spaniards when the latter were going to collect the tribute, * * * without our being able to mete out punishment, because of lack of troops. By reason of the facts above recited, and because all of the said wrongs and troubles will cease with the said pacification; and, when it is made, we are sure that the surrounding kingdoms of Bruney, Sulu, Java, and other provinces, will become obedient to his Majesty: therefore, in order that the said island may be pacified, subdued, and settled, and the gospel preached to the natives; and that justice may be established among them, and they be taught to live in a civilized manner, and to recognize God and His holy law, I have tried to entrust the said pacification to a person of such character that he may be entrusted with it.29 It is plain, therefore, that the sentiment of the times justified war on the Moros for the cause of religion alone, and that, though the primary object was conquest, no doubt the religious motives of the Spaniards were stronger than their desire to check piracy. But, of all the Christian nations, the Spaniards should have been most aware of the tenacity, determination, and courage with which the Mohammedans defend their faith, and the Sulus were no exception to the rule, for they had been born and reared in that religion for more than four generations. A wiser policy on the part of Governor Sandé would have either let the Moros of Sulu and Mindanao alone, or effected a complete reduction of the state of Sulu and immediate occupation of the coasts of Mindanao with strong forces; for it appears from all accounts that neither the Sulus nor the Magindanaos were as strongly organized then as they were a generation
- 60. later, and either alliance or war should have been easier then than afterwards. The Spaniards at that time were excellent warriors. Their conquests of the Bisayan Islands and Luzon were rapid and brilliant, but it appears that the system of government which they inaugurated there met with distinct failure the minute it was extended to the more organized communities and the greater forces they encountered in the south. The Sulus, on the other hand, fought in the defense of their national independence and religion, and never found life too dear to sacrifice in that cause. They resented the treatment of Spain, and in their rage and desire for revenge built stronger forts and fleets and became fiercer pirates. Rule of Batara Shah Tangah Pangiran must have died about 1579 and was followed by Sultan Batara Shah Tangah, who is in all probability the Paquian or Paguian Tindig of the Spanish writers. Tangah’s claim to the sultanate was strongly contested by his cousin, Abdasaolan30 who ruled over Basilan. The latter attacked Jolo with a strong force, but failed to reduce its forts. Tangah, however, felt insecure and went to Manila to request Governor Sandé’s aid and returned to Sulu with two Spanish armed boats (caracoas).31 Abdasaolan, whose power had in the meantime increased, prepared for defense and watched for the advance of the Sultan’s boat. Finding that the caracoas were at a considerable distance from the Sultan’s boat he manned two light salisipans32 with a strong force and dispatched them, with speed to intercept Tangah. The Sultan’s party was completely surprised, and in the fight that resulted Tangah was killed. On reaching Jolo the Spanish forces attacked the town. The Sulus fought valiantly, but their fort was reduced. The officers in command of the caracoas assembled the people and had Raja Bungsu, who was wounded in the fight, elected sultan to succeed Tangah. The full title of Bungsu was “The Sultan Muwallil Wasit Bungsu.”33
- 61. Figueroa’s expedition against Mindanao In 1596 Capt. Esteban Rodriguez led an expedition into Mindanao, for its conquest and pacification. It is maintained that he proceeded up the Mindanao River as far as Bwayan, the capital of the upper Mindanao Valley. Don Esteban Rodriguez prepared men and ships, and what else was necessary for the enterprise, and with some galleys, galleots, frigates, vireys,34 barangays,34 and lapis,35 set out with two hundred and fourteen Spaniards for the Island of Mindanao, in February of the same year, of 1596. He took Capt. Juan de la Xara as his master-of-camp, and some religious of the Society of Jesus to give instruction, as well as many natives for the service of the camp and fleet. He reached Mindanao River after a good voyage, where the first settlements, named Tampakan and Lumakan, both hostile to the people of Bwayan, received him peacefully and in a friendly manner, and joined his fleet. They were altogether about six thousand men. Without delay they advanced about 8 leagues farther up the river against Bwayan, the principal settlement of the island, where its greatest chief had fortified himself on many sides. Arrived at the settlement, the fleet cast anchor and immediately landed a large proportion of the troops with their arms. But before reaching the houses and fort, and while going through some thickets [cacatal]36 near the shore, they encountered some of the men of Bwayan, who were coming to meet them with their kampilan,37 carazas38 and other weapons, and who attacked them on various sides. The latter [i.e., the Spaniards and their allies], on account of the swampiness of the place and the denseness of the thickets [cacatal], could not act unitedly as the occasion demanded, although the master-of-camp and the captains that led them exerted themselves to keep the troops together and to encourage them to face the natives. Meanwhile Governor Esteban Rodriguez de Figueroa was watching events from his flagship, but not being able to endure the confusion of his men, seized his weapons and hastened ashore with three or four companions and a servant who carried his helmet in order that he might be less impeded in his movements. But as he was crossing a part of the thickets [cacatal] where the fight was waging, a hostile Indian stepped out unseen from one side and dealt the governor a blow on the head with his kampilan that stretched him on the ground badly wounded.39 The governor’s
- 62. followers cut the Mindanao to pieces and carried the governor back to the camp. Shortly after the master-of-camp, Juan de la Xara, withdrew his troops to the fleet, leaving behind several Spaniards who had fallen in the encounter. The governor did not regain consciousness, for the wound was very severe, and died next day. The fleet after that loss and failure left that place, and descended the river to Tampakan, where it anchored among the friendly inhabitants and their settlements. The master-of-camp, Juan de la Xara, had himself chosen by the fleet as successor in the government and enterprise. He built a fort with arigues40 and palms near Tampakan, and founded a Spanish settlement to which he gave the name of Murcia. He began to make what arrangements he deemed best, in order to establish himself and run things independently of, and without acknowledging the governor of Manila, without whose intervention and assistance this enterprise could not be continued.41 Bwayan was 30 miles up the river and 25 miles above Magindanao or Kotabato where Bwisan, the Sultan of Magindanao, was strongly fortified. It is difficult to believe that Rodriguez could advance so far even with a small scouting party. A careful review of the Spanish reports referring to these early campaigns in Mindanao indicates that Bwayan has been erroneously used in place of Magindanao, the ancient capital of the sultanate of Magindanao. Bent on the conquest of Mindanao, Governor Tello prepared another expedition under Gen. Juan Ronquillo42 and dispatched it by the way of Cebu. At Caldera, it was joined by the fleet of Mindanao and the whole force proceeded east in the direction of the Mindanao River, on the 6th of February, 1597. Captain Chaves arrived with his frigates at the river on the 8th of January. In a battle fought at Simway to capture Moro vessels going to seek aid from Ternate he had a leg cut off and received a shot in the helmet above the ear. Ronquillo arrived at the mouth of the river on February 21, and on the 17th of April he engaged a Moro fleet with 40 arquebusiers and defeated them, killing a number of their brave men and some Ternatans without losing any of his men except 5 Bisayans. Leaving a guard of 34 men under Chaves at the fort of Tampakan he advanced up the river with a force of 230 sailors and gunners. The enemy retired behind some parapets as soon as the artillery opened upon them, and brought some
- 63. artillery to bear on the flagship (one of the galleys), but could not retard the Spanish advance. “I answered their fire with so great readiness,” said Ronquillo in his report, “that I forced them to withdraw their artillery. But, as if they were goblins, they remained here behind a bush or a tree, firing at us without being seen.” Reinforced by the chief of the hill tribes, Lumakan, with 500 natives, Ronquillo resumed the fighting after the delay of a few days. “Finally,” continued Ronquillo, “I planted my battery of eight pieces somewhat over 100 paces from the fort. Although I battered the fort hotly, I could not effect a breach through which to make an assault. All the damage that I did them by day, they repaired by night. * * * “I was very short of ammunition, for I had only 3,000 arquebus bullets left, and very few cannon balls; and both would be spent in one day’s fighting, during which, should we not gain the fort, we would be lost—and with no power to defend ourselves while withdrawing our artillery and camp. * * * “I reconnoitered the fort and its situation, for it is located at the entrance of a lagoon, thus having only water at the back, and swampy and marshy ground at the sides. It has a frontage of more than 1,000 paces, is furnished with very good transversals, and is well supplied with artillery and arquebuses. Moreover it has a ditch of water more than 4 brazas43 wide and 2 deep, and thus there was a space of dry ground of only 15 paces where it was possible to attack; and this space was bravely defended, and with the greatest force of the enemy. The inner parts were water, where they sailed in vessels, while we had no footing at all.” “Again, I reflected that those who had awaited us so long, had waited with the determination to die in defense of the fort; and if they should see the contest ending unfavorably for them, no one would prevent their flight. Further, if they awaited the assault it would cost me the greater part of my remaining ammunition, and my best men; while, if the enemy fled, nothing would be accomplished, but on the contrary a long, tedious, and costly war would be entered upon. Hence, with the opinion and advice of the captains, I negotiated for peace, and told them that I would admit them to friendship under the following conditions: “First, that first and foremost they must offer homage to his Majesty, and pay something as recognition” (a gold chain). Second, “that all the natives who had been taken from the Pintados Islands [Bisayan Islands] last year, must be restored.” Third, “that they must break the peace and confederation made with the people of Ternate, and must not admit the latter into their country.” Fourth, “that they must be friends with Danganlibor and Lumakan, * * * and must not make
- 64. war on their vassals.” Fifth, “that all the chiefs must go to live in their old villages.”44 Ronquillo later reported the place indefensible and was authorized to retire to Caldera. Ronquillo must have advanced as far as the settlement of Kalangnan or possibly Magindanao (Kotabato), the capital of Sultan Bwisan. The report he rendered relative to the country, its people and chiefs, is very interesting and an excerpt of the same is herewith quoted because of its bearing on conditions throughout Moroland: The leading chiefs collect tribute from their vassals. * * * These Indians are not like those in Luzon, but are accustomed to power and sovereignty. Some collect five or six thousand tributes. * * * Hitherto it has not been possible to tell your lordship anything certain of this country except that it will be of but little advantage to his Majesty, but a source of great expense. It has far fewer inhabitants than was reported, and all are very poor, so that their breakfast consists only in cleaning their arms, and their work in using them, and not in cultivating the land, which is low and swampy in this river. There is no chief who can raise 20 taes of gold. Rice is very scarce; in the hills is found a small amount, which is used for food by the chiefs only. There are some swine, and a few fowls that are very cunning, and less fruit.45 These early expeditions of the Spaniards against the Moros undoubtedly aroused in the latter a great desire for vengeance. The forces the Spaniards sent to conquer Mindanao and Sulu were very small. Such forces would have been strong enough to reduce any island of the Bisayan group, or even Luzon, but against the Moros they proved insufficient and inadequate. They however succeeded in provoking bitter hostilities and marked the beginning of a long period of terror and bloodshed. Moro raids46
- 65. In 1599 combined Moro fleets invaded and plundered the coasts of the Bisayan Islands, Cebu, Negros, and Panay. Captain Paches, who was in command of the fort of Caldera, attacked the northern coast of the Island of Sulu. After landing at some point, it was observed by the Sulus that his fuses were wet and that his guns could not fire well. They then rushed his position, killed him, and dispersed his forces. The following year saw the return of a larger and still more dreadful expedition. The people of Panay abandoned their towns and fled into the mountains under the belief that these terrible attacks had been inspired by the Spaniards. To check these pirates, Juan Gallinato, with a force of 200 Spaniards, was sent against Sulu, but like so many expeditions that followed his, he accomplished nothing. * * * “From this time until the present day” (about the year 1800), wrote Zuñiga, “these Moros have not ceased to infest our colonies; innumerable are the Indians they have captured, the towns they have looted, the rancherias they have destroyed, and the vessels they have taken. It seems as if God has preserved them for vengeance on the Spaniards that they have not been able to subject them in two hundred years, in spite of the expeditions sent against them, the armaments sent almost every year to pursue them. In a very little while we conquered all the islands of the Philippines, but the little Island of Sulu, a part of Mindanao, and the other islands nearby, we have not been able to subjugate to this day.”47 Gallinatos’s expedition occurred in 1602.48 After three months of protracted fighting at Jolo, he was unable to reduce the fortifications of the town and retired to Panay. In 1616 a large Sulu fleet destroyed Pantao in the Camarines and the shipyards of Cavite and exacted large sums for the ransom of Spanish prisoners. Moro fleets in 1625 sacked Katbalogan in Samar. In 1628 Governor Tavora sent an expedition to Sulu under Cristobol de Lugo. Cristobol disembarked half of his infantry, sacked the town of Jolo, set part of it on fire and sailed back to Cebu. In 1629 the Moros raided Samar and Leyte. In 1630 an armada composed of 70 vessels and having 350 Spanish and 2,000 native soldiers, under
- 66. Lorenzo de Olaso Ochotegui, arrived at Jolo. Olaso misdirected his forces and, advancing too near to the wall of the fort, was wounded in his side and fell. He was rescued by the officers who followed him, but the troops were demoralized and retired. The expedition, however, landed at various points on the coast and burned and pillaged small settlements.49 In the same year P. Gutierrez came to Mindanao on a mission to Corralat.50 On his return he met Tuan Baluka, wife of Raja Bungsu, at Zamboanga. Baluka urged P. Gutierrez to delay his departure from Zamboanga and warned him of the danger of meeting the Sulu expedition under Datu Ache. He, however, continued on his way and was overtaken by Datu Ache’s force, but on account of the message and flag he delivered to Ache from Tuan Baluka, he was allowed to proceed safely. For some time the Jesuits had been urging upon the Philippine Government the occupation of the southern coast of Mindanao. This meant an advance into the enemy’s camp and a bloody struggle for supremacy in the southern seas. The consequences of such a step were foreseen by the Government and very few governors would have dared undertake such a grave responsibility. In 1635, Governor Juan Cerezo de Salamanca was petitioned by the Jesuits to establish an advance post of the Spanish forces at Zamboanga for the protection of missionaries and the Christians who had to navigate in the southern seas. Salamanca granted their request and sent Capt. Juan de Chaves, who disembarked at Zamboanga on the 6th of April, 1635. The force under Captain Chaves consisted of 300 Spanish and 1,000 native soldiers. In June they began the construction of a stone fort on a plan designed by the Jesuit missionary P. Melchor de Vera, who was an expert engineer. The advantages to be derived from the position of this garrison were demonstrated before the year was over. As a piratical fleet was returning from Cuyo, Mindoro, and the Kalamian Islands, the favorable opportunity was watched for, and as the two divisions of the fleet separated, the Spanish forces pursued Corralat’s pirates and dealt them a deadly blow in the neighborhood of Point Flechas, killing about 300 Moros and saving 120 Christian captives.51
- 67. First Spanish conquest and occupation of Sulu, 1635–1646 Gen. Sebastian Hurtado de Corcuera relieved Salamanca before the end of the year 1635 and continued the same policy with additional vigor and great ability. He quickly resolved upon attacking the Moros in their own strongholds, and thought that by crushing their power at home he would be able to put an end to their piratical raids. He arrived at Zamboanga February 22, 1636, proceeded first to Mindanao, fought Corralat and destroyed some of his forts and sailed back to Manila.52 Corcuera returned to Zamboanga in December, 1637, and prepared for an expedition against Sulu. On January 1, 1638, he embarked for Sulu with 600 Spanish soldiers, 1,000 native troops, and many volunteers and adventurers. He had 80 vessels all told and arrived at Jolo on the 4th.53 Anticipating an invasion, Sultan Bungsu had strengthened his garrisons and called for aid and reënforcements from Basilan, Tapul, and Tawi-tawi. On his arrival Corcuera found the town well fortified and the enemy strongly intrenched. The Moros were well disciplined and had a well organized guard. The forts occupied strategic points and were strongly defended; the trenches were well laid, and the Moros shot well and fought fearlessly. Corcuera besieged the town with all his forces and attacked it repeatedly and valiantly using powerful artillery, but he could not reduce it. Several efforts to tunnel the walls or effect a breach in them by mines were frustrated by the vigilance and intrepidity of the Sulus. The siege lasted three months and a half, at the end of which time the Sulus evacuated the town and retired to the neighboring hills, where they intended to make the next stand. Corcuera, taking possession of the town, reconstructed its forts and established three posts, one on the hill, one at the river, and one on the sandbank in front of the town. The garrison he established there consisted
- 68. of 200 Spanish soldiers and an equal number of Pampangans, under the command of Capt. Ginés Ros and Gaspar de Morales. In May Corcuera returned to Manila with all the triumph of a conqueror, leaving Gen. Pedro Almonte, the senior officer next to himself in command of the expedition, as governor of Zamboanga and Ternate and chief of the forces in the south. Soon after the establishment of the Jolo garrison, the Sulus under Datu Ache attacked the soldiers in the quarry and killed a few Spaniards and captured 40 Chinese and Negroes (galley slaves). This and other depredations committed by the Sulus from time to time, some of which were provoked by the ill behavior of the Spanish officers and troops, forced Almonte in June, 1639, to come over to Sulu and take the field a second time. With 3 captains and 1,200 Spanish and native soldiers, he marched over the island, attacked the Sulus in their homes, burned their houses and killed every man he could reach. It is said that he hung 500 heads on the trees, liberated 112 Christian captives, and captured quantities of arms. When he asked the Gimbaha Sulus (at one of the settlements of Parang) to submit to the sovereignty of Spain, they refused to recognize his authority, challenged his forces, and fought him desperately. They wore helmets and armor and used spears and swords. On one occasion, Captain Cepeda engaged them in battle and returned with 300 captives, leaving on the field 400 dead, a fearful lesson to those who survived. Cepeda lost 7 Spaniards and 20 natives only, but he had a large number wounded. Not satisfied with the havoc he wrought on the Island of Sulu, and desiring to follow and catch the fugitive sultan, Almonte invaded the other large islands and followed the sultan and the datus all over the Archipelago. At Tawi-tawi, however, he met with a reverse, and the captain who led the expedition returned with considerable loss. Soon after Almonte’s departure, the Sulus who had fled returned and lost no time or opportunity in harassing the garrison. Several piratical excursions invaded the Bisayas and Camarines. Soon Dutch vessels, invited by Sulu emissaries sent to Java, appeared in the vicinity of Zamboanga and Jolo and threatened the Spanish garrison and incited the Moros to resist the Spaniards and attack their forces. Anticipating trouble with the Dutch, and
- 69. foreseeing the danger of maintaining a garrison at Jolo under the circumstances, the Spaniards planned to evacuate the town. Accordingly on the 14th of April, 1646, they left Jolo. Before withdrawing their troops, they managed to make a treaty with the Sulus, which took the form of an alliance both offensive and defensive. The purpose of the treaty was declared to be the maintenance of peace between both parties and mutual aid against foreign enemies. In case of assistance against a foreign nation, the expenses of the war were to be defrayed by the party requesting aid. The Spanish Government recognized the supreme authority of the Sultan of Sulu from Tawi-tawi to Tutup and Pagahak, reserving sovereignty rights for the King of Spain over Tapul, Siasi, Balangingi, and Pangutaran only. In return for the evacuation of Jolo, and as a sign of brotherhood, the Sultan of Sulu promised to send yearly to Zamboanga three boats, 8 fathoms long, full of rice, and to allow the Jesuit priests to come to Jolo unmolested. Other provisions were inserted in the treaty for the exchange and redemption of slaves, criminals, or others who happened to run away from Zamboanga to Sulu and vice versa. This treaty did not remain in force for any great length of time, for we hear again in 1647 that the Sulus invaded the Bisayas and harassed the vicinity of Zamboanga. Sulu supremacy in the Archipelago, 1647–1850 Successors of Bungsu Bungsu had a very long reign marked with reverses and misfortunes. He died before 1640, and was succeeded by Sultan Nasirud Din II and Sultan Salahud Din Karamat. The latter was known to the Spanish writers as
- 70. Baktial, which was his Sulu name before the sultanate. During the reign of Karamat the Philippines were threatened by a Chinese invasion from the north and by war with Holland, and the government, under the circumstances, decided to abandon Zamboanga and the Moluccas. This purpose they carried out in 1663. In the days of Karamat the Sulus became very active and made many raids in various directions. The decline of Spain’s political power and her inactivity in the century that followed the evacuation of Zamboanga caused obscurity in the Spanish records of the history of Sulu and Mindanao. The events of this century are, with few exceptions, lacking in significance and interest.54 The sultans who followed Karamat are, in the order of their succession, Shahabud Din, Mustafa Shafiʿud Din, Badarud Din I, Nasarud Din, and Alimud Din I, better known as Amirul Mu’minin (Ferdinand I of Sulu). The first three were brothers, the sons of Karamat, while the last two were the sons of Badarud Din. In 1718 Governor Bustamante reoccupied Zamboanga for the purpose of waging war against piracy. “The citadel (Fuerza del Pilar) was rebuilt on an elaborate plan under the direction of the engineer, Juan Sicarra. Besides the usual barracks, storehouses, and arsenals, there were, within the walls, a church, a hospital, and quarters for the Pampangan soldiers. Sixty-one cannon were mounted upon the defenses.” In 1725, a Chinese named Ki Kuan was sent to Manila to arrange for peace and returned with two Spanish commissioners, who made a treaty with the sultan of Sulu providing for trade between Manila and Jolo, the return or ransom of captives, and the ceding to Spain of the Island of Basilan. Notwithstanding this treaty Moro raids continued either by toleration of the sultan and datus or at their instigation. In 1730 a brother of the sultan commanded an expedition of 31 vessels, which attacked the fort of Taytay and ravaged the coast of Palawan. Another expedition spent nearly a whole year cruising and destroying among the Bisayas.
- 71. In retaliation a large Spanish fleet united at Zamboanga and, under Ignacio de Irebri and Manuel del Rosal, invaded the shores of Sulu and ravaged and burned some settlements. At Bwal they found the settlement well protected and extensively fortified, so they contented themselves with destroying some plantations and burning outlying houses. At Tapul considerable damage was inflicted. A force of 600 disembarked, dispersed the Sulus, burned their settlements, destroyed many farms, the salt works, and many boats, and returned to Zamboanga. In 1732 similar raids were made and hostilities continued until 1737. Reign of Sultan Alimud Din I One of the earliest events in the reign of Alimud Din I was his ratification of the treaty of 1737. The sultan was represented in Manila by Datu Mohammed Ismael and Datu Jaʿfar, who signed the document. The treaty was drawn in January, 1737, by Governor-General Fernando Valdés y Tamon and contained five articles. The first article declared the determination of both parties to preserve permanent peace between the two states, all differences or grievances to be settled amicably, and hostilities between subjects or vassals to be strictly prohibited and punished; the second provided for alliance and mutual aid against any foreign foe. European nations were, however, excluded from the provisions of this article; the third provided for free trade between the two states, restricted by the use of passports to be issued by superior authority; the fourth provided that each state should be held responsible for all infractions of the peace committed by its subjects and should be bound to punish the same and make proper amends to the proper party; the fifth provided for the exchange of captives and return of all church images and ornaments in the possession of the Sulus. To all appearances Alimud Din I was a man of peace and a reformer. He kept his part of the treaty faithfully and piracy was actually suppressed during the whole period in which he held the reins of government. He
- 72. revised the Sulu code of laws and system of justice. He caused to be translated into Sulu parts of the Quran and several Arabic texts on law and religion. He strongly urged the people to observe faithfully their religion and the ordained five daily prayers. He even went so far as to prescribe punishment for failure to observe this rule. He wanted all pandita to learn Arabic and prepared Arabic-Sulu vocabularies as a preliminary step to making the Arabic the official language of the state. He coined money, organized a small army, and tried to establish a navy. His name is foremost in the memory of the Sulus, partly because of his able administration and partly on account of the fact that he is the grandfather of all the present principal datus of the Sulus. In September, 1746, a special commission from Manila carried to Alimud Din a letter written by King Philip V in 1744, requesting the admission of Jesuit missionaries to Jolo with permission to preach the Christian religion to the Sulus. The sultan entertained the commission very hospitably and gave in their honor a royal reception and a review of the troops. A council was held in which the sultan conferred with the leading datus of Sulu and granted the request of King Philip V. He further authorized the building of a church and recommended the erection of a fort at some convenient locality for the safe protection of the missionaries. In return for this favor he requested that the Spanish Government give him, as an aid in building a navy, the sum of ₱6,000, 12 piculs55 of gunpowder, 12 piculs of nails, and 1 picul of steel. This, he represented, was needed to enable him to suppress piracy and to check the depredations of his enemies in Borneo. This request the Spanish Government granted, and Jesuit missionaries entered Jolo, translated the catechism into Sulu, and distributed it freely among the people. The liberties exercised by the Jesuits in their endeavor to proselyte the Sulus and the strong friendship the sultan manifested toward them created great dissatisfaction among the people, and an opposition party was formed, under the leadership of Prince Bantilan, for the purpose of expelling the missionaries and deposing Alimud Din. Bantilan was the son of Sultan Shahabud Din and had as much right to the sultanate of Sulu as any son of Sultan Badarud Din. After the death of the latter the sultanate should have
- 73. reverted to the line of Shahabud Din; but it happens very often that the sons of the last sultan are either older than those of the former or meet with more favor and are, as a rule, supported by the majority of the council of datus; thus the regular order of descent changes in favor of the stronger person. Probably Bantilan was preceded by both Nasarud Din and Alimud Din for some such reason as the above. This he resented at heart, but suppressed his resentment until this favorable opportunity offered itself. He then headed the opposition to the sultan and the missionaries and won the majority of the datus and panditas to his side. Hostilities soon increased and civil war was imminent. In an effort to assassinate the sultan, Bantilan thrust a spear at Alimud Din and inflicted a severe wound in his side or thigh. During the disturbances and confusion which followed it became dangerous for the missionaries to remain at Jolo. One of the ministers of the sultan provided them with a salisipan in which they escaped without harm and withdrew to Zamboanga. This occurred late in 1748. Overpowered, disheartened, and grieved, Alimud Din left Jolo with his family and numerous escort and came to Zamboanga, seeking the aid of Spain against Bantilan. The latter proclaimed himself sultan with the title of Muʿizzud Din,56 strengthened the defenses of his capital, and waged war on all the datus who had supported Alimud Din. His power soon became supreme, and he reigned with a strong hand. At Zamboanga Alimud Din is said to have given the officers many presents and offered the Governor Zacharias 40 male Papuan slaves, who were well dressed. Zacharias, unreasonably prejudiced and distrustful, suspected some ill design and refused the present. Not receiving sufficient attention and consideration at Zamboanga, Alimud Din asked leave to go to Manila. This granted, he sailed and arrived at Cavite January 2, 1749. At Manila “he was received with all the pomp and honor due to a prince of high rank. A house for his entertainment and his retinue of seventy persons was prepared in Binondo. A public entrance was arranged which took place some fifteen days after he reached the city. Triumphal arches were erected across the streets, which were lined with more than 2,000 native militia under arms. The sultan was publicly received in the hall of the Audiencia, where the governor promised to lay his case before the King of Spain. The sultan was showered with presents, which included chains of gold, fine garments,
- 74. precious gems, and gold canes, while the Government sustained the expense of his household.”57 Following this reception, steps were taken for his conversion. His spiritual advisers cited to him the example of the Emperor Constantine whose conversion enabled him to effect triumphant conquests over his enemies. Under these representations Alimud Din expressed his desire for baptism. The governor-general, who at this time was a priest, the bishop of Nueva Segovia, was very anxious that the rite should take place; but this was opposed by his spiritual superior, the archbishop of Manila, who, with some others, entertained doubts as to the sincerity of the Sultan’s profession. “In order to accomplish his baptism, the governor sent him to his own diocese, where at Paniki, on the 29th of April, 1750, the ceremony took place with great solemnity. On the return of the party to Manila, the sultan was received with great pomp, and in his honor were held games, theatrical representations, fireworks, and bull fights. This was the high-water mark of the sultan’s popularity.58 At his baptism the sultan received the name of Ferdinand, and Spanish authors often referred to him as “Don Fernando de Alimud Din I, Catholic Sultan of Joló.” It is further stated that two datus and five of his principal followers were baptized. The crown prince, Raja Muda Mohammed Israel and his sister Fatimah attended school in Manila and learned Spanish manners and customs. A year and a half passed and no action was taken by the authorities to restore Alimud Din. In the meantime Bantilan’s fleets were busy ravaging and pillaging the Bisayas. In July, 1750, a new governor, the marquis of Obando (Francisco José de Obando) arrived in Manila. After some deliberation he resolved to reinstate Alimud Din and punish Bantilan and his pirates.59 Accordingly, on May 19, 1751, the sultan and his retinue were sent on board the Spanish frigate San Fernando and were convoyed by a squadron composed of seven war vessels under the command of Field Marshal Ramon de Abad. Falling in with bad weather off the shore of Mindoro, the San Fernando was disabled and made for Kalapan. The
- 75. squadron, however, continued its voyage uninterrupted to Jolo, arriving there on the 26th of June. After some desultory fighting, Abad arrived at an understanding with the Sulus and arranged for Datu Asin to come to Zamboanga with sufficient boats to escort the sultan back to Jolo. The sultan in the meantime stopped at Iloilo where he changed boats. Meeting with contrary winds he was carried off his course to Dapitan, and from there he set sail again for Zamboanga, which he reached on July 12. Before Ferdinand I left Manila, he had addressed a letter to the sultan of Mindanao, at the instance of the Spanish Governor-General. The original was written by Ferdinand I in Moro; a version in Spanish was dictated by him, and both were signed by him. These documents reached the governor of Zamboanga, but he had the original in Moro retranslated and found that it did not at all agree with the sultan’s Spanish rendering. The translation of the Moro text runs thus: “I shall be glad to know that the Sultan Mohammed Amirud Din and all his chiefs, male and female, are well. I do not write a lengthy letter, as I intended, because I simply wish to give you to understand, in case the sultan or his chiefs and others should feel aggrieved at my writing this letter in this manner, that I do so under pressure, being under foreign dominion, and I am compelled to obey whatever they tell me to do, and I have to say what they tell me to say. Thus the governor has ordered me to write to you in our style and language; therefore, do not understand that I am writing you on my own behalf, but because I am ordered to do so, and I have nothing more to add. Written in the year 1164 in the month Rabiʿ-ul Akir. Ferdinand I, King of Sulu, who seals with his own seal.” This letter was pronounced treasonable. Impressed with, or feigning this idea, Governor Zacharias saw real or imaginary indications of a design on the part of the sultan to throw off the foreign yoke at the first opportunity.60 After the landing of Datu Asin and his followers at Zamboanga, the governor found out by his spies that they had many arms and quantities of ammunition in their boats which lay in the roadstead opposite the town and fort. Suspicious and distrustful from the beginning, Zacharias interpreted these facts as positive proof of an intention on the part of the sultan and Datu Asin treacherously to attack the town when an opportunity offered itself. He then at once confiscated part of the arms, ordered the boats to leave the port, imprisoned the sultan and Datu Asin and all their retinue, and
- 76. communicated his suspicions and the action taken to Manila. Among the prisoners were the sons and daughters of the sultan, several datus and dignitaries and panditas, and many male and female followers and servants. In all 217 persons entered the prisons of the fort, most of whom were later transferred to Manila and confined in Fort Santiago. Zacharias’s interpretation of the action of the sultan and Datu Asin was simply absurd and his behavior reflected considerable discredit on his ability as an officer and administrator. It was further most regrettable that his views were accepted as true by higher authority in Manila where no clemency or redress was extended to the unfortunate sultan and datus. By a degree of the Governor-General, the following accusations were set forth against the sultan and Datu Asin, viz: 1. That Prince Asin had not surrendered captives; 2. That whilst the sultan was in Manila, new captives were made by the party who expelled him from the throne; 3. That the number of arms brought to Zamboanga by Sulu chiefs was excessive; 4. That the letter to Sultan Mohammed Amirud Din insinuated help wanted against the Spaniards; 5. That several Mohammedan, but no Christian books, were found in the sultan’s baggage; 6. That during the journey to Zamboanga he had refused to pray in Christian form; 7. That he had only attended mass twice; 8. That he had celebrated Mohammedan rites, sacrificing a goat, and had given evidence in a hundred ways of being a Mohammedan; 9. That his conversation generally denoted a want of attachment to the Spaniards, and a contempt for their treatment of him in Manila,61 and, 10. That he still cohabited with his concubines. The greatest stress was laid on the recovery of the captive Christians, and the governor added, that although the mission of the fleet was to restore the sultan to the throne (which, by the way, he does not appear to have attempted), the principal object was the rescue of Christian slaves. He therefore proposed that the liberty of the imprisoned nobles and chiefs should be bartered at the rate of 500 Christian slaves for each one of the chiefs and nobles, and the balance of the captives for Prince Asin and the clergy.62 It is not therefore surprising to hear of the extraordinarily revengeful activity which the Sulus exhibited during the period of humiliation to which their sultan and nobles were subjected in Manila.
- 77. Bantilan was a man of strong personality, a warrior, and a leader. The expeditions which he organized against his enemies were unusually strong and left havoc everywhere. The towns he pillaged and the captives he carried away alarmed the Spanish Government to a high degree. A high council of war was convened in Manila in 1752, which declared for an unmerciful campaign and a war of extermination to be conducted with the utmost conceivable cruelty. Volunteers and Bisayan corsairs were called to aid the regular troops. Unlimited authority was granted them to annihilate the foe, burn his villages, destroy his crops, and desolate his lands. The corsairs were exempted from all taxes. They were allowed to keep or sell all female captives and all males under 12 and over 30 years of age. Old men and crippled persons were to be killed. Male captives between 12 and 30 years of age were to be turned in to the government; the captors to receive in compensation from ₱4 to ₱6 per man. Nursing children were ordered to be baptized. At first the corsairs were required to turn in to the government one-fifth of all valuables looted, but this was soon afterwards revoked and all corsairs who equipped themselves retained all their booty. As part of the general campaign, Field Marshal Abad made another attack on Jolo with a force amounting to 1,900 men. The fleet cannonaded the forts for seventy-two continuous hours. A division of the troops landed and engaged the Sulus, but after suffering considerable loss retreated disastrously. The raids of the Spaniards and Bisayans helped to increase the vigilance of the Sulus and excited them to extreme cruelty and an abnormal degree of revenge.63 The year 1753 is stated to have been the bloodiest in the history of Moro piracy. No part of the Bisayas escaped ravaging in this year, while the Camarines, Batangas, and Albay suffered equally with the rest. The conduct of the pirates was more than ordinarily cruel. Priests were slain, towns wholly destroyed, and thousands of captives carried south into Moro slavery. The condition of the Islands at the end of this year was probably the most deplorable in their history.64 In the meantime Prince Asin died of grief in his prison.
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