Design of ternary sequence using msaa

Journal of Advanced Computing and Communication Technologies (ISSN: 2347 - 2804)
Volume No. 3 Issue No.1, February 2015
1
Design of Ternary Sequence Using MSAA
By
1
Mohammed Khaleel Anwar,2
M Vijay Krishna
1
College of Engineering, Shaqra University, Al Dawadmi, Kingdom Of Saudi Arabia
2
MVSR Engineering College, Hyderabad, India
1
kmohammad@su.edu.sa,2
vijay_mendu@yahoo.com
ABSTRACT
Pulse Compression Sequence (PCS) are widely used in radar to
increase the range resolution. Binary sequence has the limitation
that the compression ratio is small. Ternary code is suggested
as an alternative. The design of ternary sequence with good
Discriminating Factor (DF) and merit factor can be considered
as a nonlinear multivariable optimization problem which is
difficult to solve. In this paper, we proposed a new method for
designing ternary sequence by using Modified Simulated
Annealing Algorithm (MSAA). The general features such as
global convergence and robustness of the statistical algorithm
are revealed.
Keywords
Pulse Compression, Ternary Code, Multivariable Optimization
and Discriminating Factor.
INTRODUCTION
Pulse compressionpermitsmeasuring instrumentto realizethe
typicaltransmitted power ofa comparativelylong
pulse,whereasgettingthevaryresolution of short pulse.
Inmeasuring instrumentwherevertheresquare measurelimitations
onthe heightpower, pulse compressionis that thesolelysuggests
thatto getthe resolution and accuracyrelated toa
pointypulsehowever ata similartimegettingthe detection
capability ofa protractedpulse. There searchers developed
severalpulse compressionmeasuring
instrumentsignalsassistedbytrendysignalprocess systems.
Consequently, signals in severalshapesaregivenlike partcoded
signalslikeBarker codes, nested Barker codes and frequency
coded signalslike easypulse, Linearmodulation (LFM),
Hyperbolicmodulation(HFM) and Costas undulation.Everyof
thosesignals have its ownblessing sand downsides. Inmeasuring
instrumentstate of affairs, noundulationis optimum for target
resolutiongenerally. The applications likeradars,
communications and system identificationsquare measurein
generating the sequences withsensibleautocorrelation properties.
Polyphase sequence has been suggested by Levanon N [3]
which uses in applications like signal processing of sonar and
also radar significantly. A very important criterion within the
field of signal processing of sonar, system identification and
radar has been given by Barker [5].
A problem of optimization for designing of signals for the
application of radar sequences like binary, ternary, polyphase
and quaternary has viewed and suggested as optimization
problem by Griep Karl R John et al [7]. For obtaining good
discriminating factor as well as merit factor values work has
been carried out extensively.By using shift registers Ipatov [1-2]
has designed a large scale of ternary sequences. Ternary
sequences of length 2n-1 have been constructed by Shedd and D
Sarwatte[4] and Moharir[10] has perfectly given some
conditions for the existence of ternary sequences. For generation
of ternary codes with good discriminating factor has been given
in architecture of VLSI by N Balaji et al [6]. Hoholdt, Tom et al
in [8] constructed ternary sequence with periodic
autocorrelation.
J.J Blakley,1998 [9] implemented programmable hardware
architecture ternary de Brujin which generates ternary
sequences. By considering Hamming scan algorithm Pasha
I.A, P. S. Moharir and N. SudarshanRao [11] has view that the
generation of ternary sequencesas a problem of optimization.
The technique for the generation of two different lengths of
ternary preamble sequences has been proposed by Yuen-
Sam[12].Naga Jyothi.A et al[18] has proposed an efficient VLSI
architecture for generation and implementation of the ternary
sequences using Finite State Machines (FSM). K SubbaRao and
S P Singh [15,16] combined Hamming scan algorithm with
Simulated Annealing algorithm and proposed Modified
Simulated. Annealing Algorithm (MSAA) to design binary and
thirty –two phase sequences .In this paper,MSAA is used for
generating ternary codes with good discriminating factor values.
2. TERNARY SEQUENCE
The ternary sequences are also known as non-binary sequences
and have the elements of unequal magnitude. Hence they do not
have the ideal energy efficiency i.e. their energy efficiency is
less than unity. The alphabet of a ternary sequence is [-1, 0, +1].
The ternary alphabet has zero as an element, which implies no
transmission during some time slots. It is considered difficult to
have on±off switching at high power in comparison to phase
shifting. Binary sequence has a disadvantage that they do not
have high merit factor.
Ternary sequence (TSs) eliminates the drawbacks of the binary
and polyphase sequences. Generally TSs has good merit factors
at all length. Ternary alphabet shares one common property of
binary and poly phase sequence of peakiness.

2
∑
−
=
= 1
1
|)(|2
)0(
N
k
kr
r
p (1)
For a sequence of length N=20, the number of search would be
only 3^20 as against M^20 where M is generally greater than 4.
Hence, the search of ternary sequence is relatively easier than
polyphase sequence. The major demerit of TSs is due to
inclusion of zero in the alphabet, which is corresponds to a
pause in transmission.
The main criteria of goodness of pulse
compression sequences or codes are the discriminating factor
(DF) and merit factor (MF).The factors DF and MF must be as
large as possible for a good sequence or code.
Let S= [x0, x1, x2…xN-1] be a real sequence of length N. Its
aperiodic auto correlation is then defined as
γ(k)= ki
k1N
0i
i xx +
−−
=
∑ (2)
where k=0, 1, 2…N-1.Ideally, the range resolution radar signal
should have large auto-correlation for zero shift and zero auto-
correlation for non-zero shift.
3. DISCRIMINATING FACTOR:
The discrimination (DF) is defined as ratio of main peak in
autocorrelation to the absolute maximum amplitude among the
side lobes.
DF=
0k
|γ(k)|Max
γ(0)
≠
(3)
4. PROPOSED METHODOLOGY
4.1 Simulated annealing algorithm
In this section, the Simulated annealing (SA)
algorithm has been used for designing Ternary sequences with
good autocorrelation properties. In this method, each point “s”
of the search space is analogous to a state of some physical
system, and the function E(s) to be minimized is analogous to
the internal energy of the system in that state. The goal is to
bring the system, from an arbitrary initial state, to a state with
the minimum possible energy.
The implementation procedure is as follows:
1. An arbitrary code matrix X (0) is chosen as initial sequence
set for optimization.
X(0)=
















1-N1,-K1,1-K1,0-K
1-N1,1,11,0
1-N0,0,10,0
x..xx
.....
.....
x..xx
x..xx
(4)
where X (i , j) Є (-1,0,1)0 ≤ i ≤ K-1 and 0 ≤ j ≤ N-1
2. T(0) is chosen as initial temperature for annealing. Set the
value of i, iterations tobe performed at each temperature and the
value of ε, lowest possible temperature..
3. In addition the initial energy function value is calculated and
designated as E(0).
4. Make a perturbation to the code matrix X(0) by randomly
selecting an element X(i , j) from X(0) and changing it to - X(i , j)
hence a new code matrix X(1) is generated and the new energy
function value is designated as E(1).
5. If the energy is decreased i.e., E(1)<E(0) the new code matrix
is accepted.
6. If the energy is increased i.e., E(1)>E(0) the new code matrix
is accepted with Probability exp (-ΔE/T).
7. In the same way code matrix perturbation is repeated until the
required iterations are performed at each temperature.
8. Then the temperature is reduced and new equilibrium is
setup.
9. Repeat this cooling process until energy function reaches
global minimum or the System is frozen (temperature is reduced
to the lowest possible temperature.)
4.2 Hamming Scan Algorithm
The Hamming scan is employed for obtaining the pulse
compression sequences of larger length with good
autocorrelation and cross correlation properties. The basic
difference between Genetic algorithm and Hamming scan
algorithm is that Genetic algorithm uses random but possibly
multiple mutations.
Mutation is a term metaphorically used for a change
in an element in the sequence. For example, in the case of
binary sequence, a mutation of ternary element implies -1→ +1,
-1 → 0, +1→ 0, +1→ -1, 0 → -1, and0→ 1. Thus, a single
mutation in a sequence results in hamming distance of one from
the original sequence.
The Hamming scan algorithm mutates all the elements in a
given sequence one by one and looks at all the first order
hamming neighbors of the given sequence. Thus, Hamming
scan performs recursively local search among all the Hamming-
1 neighbors of the sequence and selects the one whose objective
function value is minimum.
4.3 Modified Simulated Annealing Algorithm

3
Modified Simulated Annealing Algorithm is a combination of
both Simulated Annealing and Hamming scan algorithm. It
excerpts the good methodologies of these algorithms like fast
convergence rate of Hamming scan algorithm and Global
minima trapping capability of Simulated annealing algorithm to
increase the probability of converging to the global minimum
point.
The new modified simulated annealing algorithm overcomes
these drawbacks as it makes use of simulated annealing to
randomly generate a sequence and then it invokes the Hamming
scan to converge to the local minima corresponding to that
point. Thus the selection of Simulated Annealing and mutations
of Hamming scan work well for this algorithm.
4.4 Working of Modified Simulated Annealing
Algorithm
The figure 1 gives us a complete picture about
working of the New Modified Simulated Annealing algorithm.
The X-axis contains all the possible sequences and the Y-axis
represents the Cost function.
Let us consider that we initially start from a point
say ‘A’. Now at this point we invoke the Simulated Annealing
for selection procedure that is we randomly select a point. Let
the new point chosen by this Algorithm be ‘B’. As the selection
process is complete it is now time to invoke Hamming scan.
Hamming scan ensures that the local minimum is reached which
is at point C.This local minimum point is stored. Then again
Simulated Annealing is invoked to get the point ‘D’ even
though it is of higher cost because simulated annealing
algorithm is a stochastic algorithm which accepts the higher cost
function if it lies within certain range of the present cost in the
hope that the selected point is in the valley of global minima
then the Hamming scan algorithm is invoked for optimization to
reach another Local minimum point say ‘E’ which might be
global minima because one can find the global minima only
after viewing the results of simulation.
Figure 1. Working of Modified simulated annealing algorithm
At each temperature ‘T ’of simulated annealing algorithm,
we continue the above process to find out all the local
minimum points since the global minimum point is also
one of the local minimum points. Thus this algorithm
proves to be much more efficient in converging to a global
minimum point.
5.RESULTS AND DISCUSSION
Various pairs of ternary sequences of different lengths
having good autocorrelation properties which have been
obtained using optimization techniques are given in this
section.
Ternary sequences are designed using the MSAA. The
length of the sequence N, is varied from 5 to 250. The cost
function for the optimization is based on
DF=
0k
|γ(k)|Max
γ(0)
≠
(5)
Below Table shows the synthesized results in Matlab. In
column 2 and 5 show sequence length N, column 3 and
6show Discriminating factor (DF). From sequences of
length from 5 to 250, the correlation properties are good. It
may be observed that as the length N, increases, the DF
also increases, which is the conformity with other finding.

4
S.No Sequence DF S.No Sequence DF
(1) Length N (2) (3) (4) Length N (5) (6)
1 5 5 37 107 14.3333
2 10 9 38 110 13.8333
3 12 12 39 120 13. 1667
4 13 13.0000 40 130 13.2500
5 15 14.0000 41 140 13.5000
6 17 16.0000 42 150 13.8571
7 18 15.0000 43 160 14.0000
8 20 16.0000 44 175 14.7500
9 25 18.0000 45 180 15.5714
10 35 14.0000 46 190 15.8000
11 40 14.5000 47 195 15.8571
12 45 15.0000 48 200 16.0000
13 47 12.6667 49 210 16.1667
14 48 13.3333 50 215 17.0000
15 50 12.6667 51 220 17.3000
16 52 14.0000 52 225 18.0000
17 53 14.3333 53 230 18.6667
18 55 13.0000 54 235 19.0000
19 57 14.6667 55 240 19.5000
20 59 12.5000 56 245 19.5000
36 105 13.8333 57 250 20.5000
Table 1:
Figure 2: Synthesized generated ternary sequence
From figure 2, it is observe that as the length of the sequence increases the DF is also increased.
6.CONCLUSION
A novel methodology based on Modified simulated
annealing algorithm was proposed in this paperto generate
ternary codes for various lengths with good discriminating
factor value.Length of the sequence and the number of
iterations to be performed at a constant temperature is
defined. Then the cost function value was also calculated
for the generated sequence. Out proposed algorithm
overcomes these disadvantages and proved to be more

5
effective in synthesizing ternarycodes having good auto
correlation and cross correlation properties. Results
obtained by using this algorithm are better than the results
existing in the literature.
7. REFERENCE
[1] Ipatov, V. P. "Ternary sequences with ideal periodic
autocorrelation properties."Radio Engineering and
Electronic Physics 24 (1979): 75-79.
[2] Ipatov, V. P., V. D. Platonov, and I. M. Samilov. "A
new class of ternary sequences with ideal periodic
autocorrelation properties." Soviet Math.(IzvestiyaVuz)
English Translation 27 (1983): 57-61.
[3] Levanon. N, Eli Mozeson, “Radar Signals”, Wiley,
New York, 2004
[4]Shedd, D., and D. Sarwate."Construction of sequences
with good correlation properties (Corresp.)."Information
Theory, IEEE Transactions on 25.1 (1979): 94-97.
[5] R. H. Barker, “Group synchronizing of binary digital
systems, in Communication theory”, Butterworth, London,
1953, pp. 273-287.
[6] Balaji, N., K. SubbaRao, and M. SrinivasaRao. "FPGA
implementation of ternary pulse compression sequences
with superior merit factors." NAUN international Journal
of Circuits, systems and signal processing 2.3 (2009): 47-
54.
[7] Griep, Karl R., James A. Ritcey, and John J.
Burlingame. "Poly-phase codes and optimal filters for
multiple user ranging."Aerospace and Electronic Systems,
IEEE Transactions on 31.2 (1995): 752-767
[8] Hoholdt, Tom, and JørnJustesen. "Ternary sequences
with perfect periodic autocorrelation (Corresp.)."
Information Theory, IEEE Transactions on 29.4 (1983):
597-600.
[9] Blakley, J. J. "Architecture for hardware
implementation of programmable ternary de Bruijn
sequence generators." Electronics Letters 34.25 (1998):
2389-2390
[10] Moharir, P. "Generalized PN sequences (Corresp.)."
Information Theory, IEEE Transactions on 23.6 (1977):
782-784
[11] Pasha, I. A., P. S. Moharir, and N. SudarshanRao. "Bi-
alphabetic pulse compression radar signal design."
Sadhana25.5 (2000): 481-488.
[12] Lei, Zhongding, Francois Chin, and Yuen-Sam Kwok.
"UWB ranging with energy detectors using ternary
preamble sequences." Wireless Communications and
Networking Conference, 2006. WCNC 2006.IEEE.Vol.
2.IEEE, 2006.
[13] Moharir, P.S., “Signal Design” Journal of IETE,
Vol.41, Oct. 1976, pp. 381-398
[14] Moharir, P. S., R. Singh, and V. M. Maru. "SKH
algorithm for signal design."Electronics letters 32.18
(1996): 1648.
[15] S P Singh and K SubbaRao, “ Binary Sequence
Design” Journel of Technology and Engineering Sciences.
Vol 1,2009.
[16] S P Singh and K SubbaRao"Thirty-Two phase
sequences with good Autocorrelation properties"
Sadhana Vol. 35, Part 1, February 2010, pp.63-73. Indian
Academy of Sciences India.
[17] Naga Jyothi.A., and K.RajaRajeswari.,
“Implementation and Generation of Barker and nested
Barker codes” ARCNET-2013, NSTL Visakhapatnam
[18] Naga JyothiAggala and Raja Rajeswari K, “ Design
and Implementation of the Ternary Sequences with Good
Merit Factor Values” IJCA 92(9):5-7, April 2014.

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