Signals and classification

SIGNALS AND SYSTEM
SURAJ MISHRA
SUMIT SINGH
AMIT GUPTA
PRATYUSH SINGH
(E.C 2ND YEAR ,MCSCET)
1

Topics





Introduction
Classification of Signals
Some Useful Signal Operations
Some useful signal models

2

Introduction
 The

concepts of signals and systems
arise in a wide variety of areas:
 communications,
 circuit design,
 biomedical engineering,
 power systems,
 speech processing,
 etc.

3

What is a Signal?
SIGNAL





A set of information or data.
Function of one or more
independent variables.
Contains information about the
behavior or nature of some
phenomenon.
4

Examples of Signals
 BRAIN

WAVE

5

Examples of Signals
 Stock

Market data as signal (time series)

6

What is a System?
SYSTEM
Signals

may be processed further
by systems, which may modify
them or extract additional from
them.
A
system is an entity that
processes a set of signals
(inputs) to yield another set of
signals (outputs).

7

What is a System? (2)
A

system may be made up of
physical components, as in
electrical or mechanical systems
(hardware realization).
A system may be an algorithm
that computes an outputs from
an inputs signal (software
realization).
8

Examples of signals and systems




Voltage (x1) and current (x2) as functions of
time in an electrical circuit are examples of
signals.
A circuit is itself an example of a system (T),
which responds to applied voltages and
currents.

9

Signal Models: Unit Step Function
 Continuous-Time

unit step function, u(t):

 u(t)

is used to start a signal, f(t) at t=0
 f(t) has a value of ZERO for t <0
11

Signal Models: Unit Impulse Function
A

possible approximation
to
a
unit
impulse:
An overall area that has
been
maintained
at
unity.

Graphically, it is
represented by an arrow
"pointing to infinity" at
t=0 with its length equal
to its area.

 Multiplication

of
a
function by an Impulse?
 bδ(t) = 0; for all t≠0
is an impulse function
which the area is b.

12

Signal Models: Unit Impulse Function
(3)
 May

use functions other than a rectangular
pulse. Here are three example functions:
 Note that the area under the pulse function
must be unity.

13

Signal Models: Unit Ramp Function
 Unit


ramp function is defined by:

r(t) = t∗u(t)

 Where

can it be used?

14

Signal Models: Exponential Function
est
 Most

important function in SNS where s is
complex in general, s = σ+jϖ
 Therefore,
est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt)
(Euler’s formula: ejϖt = cosϖt + jsinϖt)
s∗ = σ-jϖ,
 es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt)
 If

 From

the above, e cosϖt = ½(e +e )
σt

st

-st

15

Signal Models: Exponential Function
est (2)





Variable s is complex frequency.
est = e(σ+jϖ)t = eσtejϖt = eσt(cosϖt + jsinϖt)
es∗ t = e(σ-jϖ)t = eσte-jϖt = eσt(cosϖt - jsinϖt)
eσtcosϖt = ½(est +e-st )
There are special cases of est :
1.
2.
3.
4.

A constant k = ke0t (s=0  σ=0,ϖ=0)
A monotonic exponential eσt (ϖ=0, s=σ)
A sinusoid cosϖt (σ=0, s=±jϖ)
An exponentially varying sinusoid eσtcosϖt
(s= σ ±jϖ)

16

Signals Classification
 Signals






may be classified into:

1. Continuous-time and Discrete-time signals
2. Deterministic and Stochastic Signal
3. Periodic and Aperiodic signals
4. Even and Odd signals
5. Energy and Power signals

17

Continuous v/S Discrete Signals
 Continuous-time

A signal that is
specified for every
value of time t.

 Discrete-time

A signal that is
specified only at
discrete values
of time t.
18

Deterministic v/s Stochastic
Signal
 Signals

that can be written in any
mathematical expression are called
deterministic signal.
 (sine,cosine..etc)
 Signals that cann’t be written in mathematical
expression are called stochastic signals.
 (impulse,noise..etc)

19

Periodic v/s Aperiodic Signals
 Signals

that repeat itself at a proper interval
of time are called periodic signals.
 Continuous-time signals are said to be
periodic.
 Signals that will never repeat themselves,and
get over in limited time are called aperiodic or
non-periodic signals.

20

Even v/s Odd Signals
A

signal x(t) or x[n] is referred to as an even
signal if



CT:
DT:

A

signal x(t) or x[n] is referred to as an odd
signal if



CT:
DT:
22

Even and Odd Functions: Properties
 Property:

 Area:


Even signal:



Odd signal:
23

Even and Odd Components of a
Signal (1)
 Every

signal f(t) can be expressed as a sum
of even and odd components because

 Example,

f(t) = e-atu(t)

24

Energy v/s Power Signals


Signal with finite energy (zero power)



Signal with finite power (infinite energy)



Signals that satisfy neither property are referred
as neither energy nor power signals

25

Size of a Signal, Energy (Joules)
 Measured

by signal energy Ex:

 Generalize


CT:

 Energy

for a complex valued signal to:
DT:

must be finite, which means
26

Size of a Signal, Power (Watts)
 If

amplitude of x(t) does not → 0 when t → ∞,
need to measure power Px instead:

 Again,

generalize for a complex valued signal

to:


CT:



DT:
27

OPERATIONS ON SIGNALS
 It

includes the transformation of independent
variables.
 It is performed in both continuous and
discrete time signals.
 Operations that are performed are-

28

1.ADDITION &SUBSTRACTION





Let two signals x(t) and y(t) are given,
Their addition will be,
z(t) = x(t) + y(t)

Their substraction will be,
z(t) = x(t) – y(t)
29

2.MULTIPLICATION OF
SIGNAL BY A CONSTANT


If a constant ‘A’ is given with a signal x(t)
z(t) = A.x(t)



If A>1,it is an amplified signal.
If A<1,it is an attenuated signal.



30

3.MULTIPLICATION OF TWO
SIGNALS


If two signals x(t) and y(t) are given,than their
multiplication will be
z(t) = x(t).y(t)

31

4.SHIFTING IN TIME


Let a signal x(t),than the signal x(t-T)
represented a delayed version of x(t),which is
delayed by T sec.

32

Signal Operations: Time Shifting
 Shifting

of a signal in time
  adding or subtracting the amount of the
shift to the time variable in the function.
 x(t)  x(t–t )
o




to > 0 (to is positive value),
signal is shifted to the right (delay).
to < 0 (to is negative value),
signal is shifted to the left (advance).

 x(t–2)?

x(t) is delayed by 2 seconds.
 x(t+2)? x(t) is advanced by 2 seconds.

33

Signal Operations: Time Shifting (2)
 Subtracting

a fixed amount from the time
variable will shift the signal to the right that
amount.

 Adding

to the time variable will shift the signal
to the left.

34

Signal Operations: Time Shifting
 Shifting

of a signal in time

35

5.COMPRESSION/EXPANSION
OF SIGNALS





This is also known as ‘Time Scaling’ process.
Let a signal x(t) is given,we will examine as
x(at)
where a =real number
and how it is related to x(t) ?

36

Signal Operations: Time Inversion
 Reversal

of the time axis, or folding/flipping
the signal (mirror image) over the y-axis.

38

THANKS....................... FOR
YOUR

ATTENTION !
39

Signals and classification

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