Polyphase Sequences Design-Using MSAA

S.P SINGH K SUBBA RAO

ECE Dept, MGIT ECE Dept, College Of Engg, (O.U) Hyderabad -75, India Hyderabad -7, India

singh_spgahlot@rediffmail.com kakarlasubbarao.@yahoo.com

Abstract

Sequences having the minimum peak aperiodic autocorrelation sidelobe level one (1) are called Barker Sequences. Such sequences have been used in numerous real-world applications such as channel estimation, radar and spread spectrum communication etc. Unfortunately, the longest known biphase and quadriphase Barker sequences are of lengths 13 and 15 respectively. In this paper Modified Simulated Annealing Algorithm (MSAA) is used to design thirty-two phase sequences, which have good autocorrelation properties. Some of the synthesized results are presented here. The properties of the sequences up to length 24 have Barker properties which were not reported in literature earlier. The sequences of lengths from 4 to 289 have autocorrelation properties better than Frank codes. The synthesized 32-phase sequence sets are promising for practical application to radar and spread spectrum communication systems. Keywords Barker codes, hamming scan, polyphase codes, radar signal design, simulated annealing.

1. Introduction

Sequences with low aperiodic autocorrelation sidelobe levels are useful for channel estimation, radar, and spread spectrum communication applications. Sequences achieving the minimum peak aperiodic autocorrelation sidelobe level one (i.e. the Barker condition) are called Barker Sequences. The aperiodic autocorrelation function (ACF) of sequence S of length N is given

≤≤+

≤≤=

∑

∑+

=

∗

=

∗+

1-kN

0 n k-nn

1-k - N

0 n knn

0k1N- ; s s

1- Nk0 ; ssk) (A

… (1) If all the sidelobes of the ACF of any polyphase sequence are bounded by |A (k)| ≤ 1, 1 ≤ | k | ≤ N-1 … (2) then the sequence is called a generalized Barker sequence In 1953 [1] Barker introduced binary sequences for lengths N = 2,3,4,5,7,11, and 13, fulfilling the condition in (2). The binary Barker can be regarded as a special case of polyphase Barker sequences. If the sequence elements are taken from an alphabet of size M, consisting of the Mth roots of unity.

1Mm0 )iexp(:

Mm.i2expS mm −≤≤φ=

π=

.. (3) the sequence is alternatively named as M-phase Barker sequence.

In 1965, Golomb and Scholtz [2] first investigated generalized Barker sequences and presented six phase Barker sequence of lengths N ≤ 13. In [3], 60-phase Barker sequences upto length 18 were reported. Polyphase Barker sequences of lengths up to 31 were given in [4-6],

International Conference on Computational Intelligence and Multimedia Applications 2007

0-7695-3050-8/07 $25.00 © 2007 IEEEDOI 10.1109/ICCIMA.2007.89

357

International Conference on Computational Intelligence and Multimedia Applications 2007

0-7695-3050-8/07 $25.00 © 2007 IEEEDOI 10.1109/ICCIMA.2007.89

357

wherein, at length 30, an alphabet size of 7200 had to be used. In [7] , polyphase Barker sequences of lengths 32-36 were presented, wherein, at length 35, an alphabet size of 11520 was necessary. In [8], polyphase Barker sequences of lengths 37-45 were presented, wherein, an alphabet size of 120. Recently, in [9] polyphase Barker sequences of lengths 46-63 were presented, wherein, an alphabet size of 2000 had to be used. However, polyphase Barker sequences for larger lengths require larger alphabets and the possibility for exhaustive search diminishes. The synthesis of polyphase codes with good correlation properties is a nonlinear multivariable optimization problem, which is usually difficult to tackle. The Simulated Annealing (SA) technique, introduced by Kirkpatrick et al [10] proved to be an efficient and powerful tool to find optimal or near optimal solutions for complex multivariable nonlinear functions. The concept of Hamming scan algorithm has been employed for obtaining the pulse compression sequences at larger lengths with good correlation properties [11,12]. This algorithm has fast convergence rate but has demerit viz., the tendency to get stuck with local minima. The MSAA has global minimum estimation capability of SA algorithm and fast convergence rate of Hamming scan algorithm.

Binary code is one of the most commonly used radar pulse compression signals due to the easy signal generation and processing [13]. However, polyphase signal has larger main lobe-to- peak sidelobe ratio over binary signal of the same code length. In addition, polyphase waveforms have a more complicated signal structure and thus, are more difficult to detect and analyze by an enemy’s electronic support measures (ESMs). With the maturity of digital signal processing, the generation and processing of polyphase signals has become easy and less costly. Therefore, polyphase code is increasingly becoming a favorable alternative to the traditional binary code for radar signals and will be used as the basic code form for radar and signal design. In this paper, MSAA have been used for the design of thirty- two phase coded sequences having good correlation properties

2. Thirty- Two Phase Sequences Design

A thirty-two phase coded sequence of length N bits is represented by a complex number sequence

{ } N ..., 2, 1, n ,e n)(s n)(j m == φ …. (4)

where φm(n) is the phase of nth bit in the sequence and lies between 0 and 2π. If the number of the distinct phases available to be chosen for each bit in a code sequence is M, the phase for the bit can only be selected from the following admissible values:

π−ππ∈φ

M2)1M(,....,

M22 ,

M2 ,0)n(m

…. (5)

Considering a polyphase coded sequence S with code length N, and distinct phase number M, one can concisely represent the phase values of S with the following 1 by N phase matrix: S = [φm(1) , φm(2),….., φm(N)] … (6) where all the elements in the matrix can only be chosen from the phase set in (5).

A more practical approach to design polyphase sequences with properties in (2) is to numerically search the best polyphase sequences by minimizing a cost function that measures the degree to which a specific result meets the design requirements. For the design of thirty-two phase coded sequences, the cost function is based on the sum of squares of autocorrelation sidelobe peaks. Hence, from (1) the cost function can be written as,

=

≠ )k(A maxE

0k

… (7)

the minimization of cost function in (7) generates thirty-two phase sequences that are automatically constrained by (2).

358358

3. Discriminating Factor (DF)

The discriminating factor (DF) is defined by Golay as follows.

)k(A max)0(A

0k ≠

=DF … (8)

The denominator is the peak sidelobe value and is related to the L∞ norm of the sidelobes. 4. Simulated Annealing Algorithm

The SA technique, introduced by Kirkpatrick et al [10] proved efficient and powerful tool to find optimal or near optimal solutions for complex multivariable nonlinear functions. The major advantage of the SA algorithm over the traditional “greedy” optimization algorithms is the ability to avoid becoming trapped in local optima during the search process. The algorithm employs a random variable search that not only accepts the changes that decrease the cost function but also accepts some changes that increase it with a probability

<∆>∆∆

=ρ0E ,10E ),E/Texp(- i … (9)

where ∆E is the change in cost due to mutation. Ti is the control parameter, which by analogy is known as the system “temperature.” Normally, the temperature Ti is slowly decreased from a large value to a very small one during the annealing process. The SA algorithm can find the global optimum of a nonlinear multivariable function by carefully controlling the rate of decrease of the system temperature Ti. 5. Hamming Scan Algorithm (HSA)

The HSA is a traditional greedy optimization algorithm, which searches in the neighborhood of the point in all directions to reduce the cost function and has fast convergence rate. The basic difference between Genetic Algorithm (GA) and HSA is that GA uses random but possibly multiple mutations. The Mutation is a term metaphorically used for a change in an element in the sequence. For example if a phase value of a polyphase sequence is φm (1 ≤ m ≤ M), it is replaced with phase φi, i =1, 2, M, i ≠ m, and the cost for each φi, is evaluated. If the cost is reduced due to a change in phase value, the new phase value is accepted; otherwise, the original phase value is retained. This process is recursively applied to the entire sequence. The HSA mutates all the elements in the given sequence one by one and looks at all the first order-Hamming neighbors of the 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. 6. MSAA for Polyphase Sequences Design

The MSAA is a combination of both SA and Hamming scan algorithms. It combines the good methodologies of the two algorithms like global minimum converging property of SA algorithm and fast convergence rate of Hamming scan algorithm. The demerit of Hamming scan algorithm is that it gets stuck in the local minimum point because it has no way to distinguish between local minimum point and a global minimum point. Hence it is sub-optimal [11,12]. The drawback in SA is that it has a slow convergence rate because even though it may get closer to the global minimum point, it may skip it because of the methodology it employs,

359359

generating the sequences randomly and accepting them with probability based on annealing schedule. The MSAA overcomes these drawbacks [14].

Fig. 1 Flow chart of Modified Simulated Annealing Algorithm

The computational cost for searching the best polyphase, code of length N, and distinct phase number M through an exhaustive search, i.e., minimizing (7), is of the order of MN grows exponentially with the code length and the set size. Therefore, the numerical optimization of polyphase codes is an NP-complete problem. During the optimization process of polyphase code sets, the random search is carried out through code phase value “mutation”, i.e., randomly selecting an entry in the (6) and replacing it with different admissible value. With each phase “mutation”, the cost function before and after the phase change are evaluated, and the phase change is accepted with a probability calculated using (9). More specifically, the phase values of a polyphase code set S is “mutated” as follows: First, a polyphase sequence set S as given in (6) is randomly chosen; then the selected phase value is replaced with a phase value randomly chosen from the other M-1 possible distinct phase values that are from{ }.)M(, ),......2(),1( mmm φφφ Now the cost function for the new polyphase code set is evaluated according to (7). If the new cost value is reduced, then accept the new set. Otherwise, accept it with a probability given in (9). The probability density function for all random selections is a uniform function among all possible values. The next step of the algorithm is to invoke the Hamming scan as shown in Fig.1. To successfully implement the MSAA, one needs

360360

to determine few key parameters or criteria for the annealing process. These are the starting temperature, the rate of decrement of temperature, i.e., cooling schedule, the determination of equilibrium condition at each temperature, and the annealing stopping criterion. In this work, the starting temperature T0 is decided based on the standard deviation σ of the initial cost distribution by setting. T0 = 15 σ …. (10) From the initial temperature T0, the system temperature is systematically reduced according to Ti+1 = αTi (0 < α < 1) ….. (11) where α is constant and chosen to be 0.95 in this design. At a temperature Ti (i > 0 ), the alphabets the sequence are constantly “mutated” and accepted with a probability according to (9), until the cost function distribution reaches equilibrium state. Then, the temperature is reduced to Ti+1 according to (11), and the code “mutations” are repeated until a new equilibrium state is reached at the updated temperature.

The annealing process is stopped if no “mutated” phase is accepted during three consecutive temperature reductions or T< ε, where ε is the minimum stopping temperature. The value of ε is chosen as 0.01. 7. Design Results

Thirty–two phase sequences are designed using the MSAA. The length of the sequence, N, is varied from 2 to 500. The cost function for the optimization is based on (7). In this paper all the autocorrelation sidelobe peaks (ASP) are single realizations obtained using Pentium - IV, processor. We have found Barker codes up to length 24 with maximum alphabet size of only 32, which have not reported in literature. Table I shows the synthesized results. In table I, column 1, 4 &7 show sequence length, N, column 2, 5 & 8 show autocorrelation sidelobe peaks (ASP), columns 3, 6 & 9 show the Discrimination factor (DF). From sequences of length from 25 to 500, 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 findings. Fig 2 shows comparison between Frank code and 32 phase-synthesized sequences. As shown in the figure-synthesized sequences from length 4 to 289 have better autocorrelation properties than Frank codes [15].

Table : 1 Correlation Properties ( M = 32 & N, varied From 2 to 500)

Sequence Length

(1)

ASP

(2)

DF

(3)

Sequence Length

(4)

ASP

(5)

DF

(6)

Sequence Length

(7)

ASP

(8)

DF

(9) 2 1.0 2.0 16 1.0 15.8 58 1.9 29 3 1.0 3.0 17 1.0 17.0 64 2.1 29.8 4 1.0 4.0 18 1.0 17.3 81 2.4 33.0 5 1.0 5.0 19 1.0 19.0 100 2.6 38.1 6 1.0 6.0 20 1.0 19.5 121 3.3 36.2 7 1.0 7.0 21 1.0 20.5 144 3.5 40.9 8 1.0 8.0 23 1.0 22.1 169 3.8 44.1 9 1.0 9.0 24 1.0 22.6 196 4.4 44.3 10 1.0 10.0 25 1.1 22.7 225 4.6 47.9 11 1.0 11.0 29 1.2 23.7 256 5.0 50.8 12 1.0 12.0 36 1.5 23.1 289 5.0 57.0 13 1.0 13.0 40 1.7 21.7 324 5.9 54.4 14 1.0 14.0 49 2.0 24.3 400 7.4 54.2 15 1.0 15.0 54 1.7 31.4 500 7.7 64.9

361361

Comparison between 32-phase sequences and Frank codes

11.5

22.5

33.5

44.5

55.5

16 25 36 49 64 81 100 121 144 169 196 225 256 289

sequences length(N)

AS

P

32-phase Frank codes

Fig. 4 Comparison of autocorrelation sidelobe peaks of Frank code and thirty-two phase synthesized sequences 8. Conclusions

An effective Modified Simulated Annealing Algorithm has been developed for the design of thirty –two phase-coded sequences used in radar systems and spread spectrum communications for significantly improving performance of the system. This new approach combines the SA algorithm and the Hamming Scan algorithm and provides a powerful tool for the design of polyphase sequences. With the proposed optimization algorithm, a thirty-two phase sequences are designed with different code lengths, N. Some of the synthesized results are presented which have good correlation properties. We have found Barker codes up to length 24 with maximum alphabet size of only 32. The sequences of length from 25 to 500 also have good correlation properties. References

[1] Barker, R.H: “Group synchronizing of binary digital system” in Jackson, W, (Ed): Communication theory (

Butterworths, London, 1953), pp. 273-287. [2] Golomb, S. W, and Scholtz, R. A, “Generalized Barker sequences”, IEEE Trans, Inf. Theory, 1965, IT-11,(4),

pp. 533-537. [3] Zhang., N, and Golomb, S. W, “ Sixty phase Generalized Barker sequences”, IEEE Trans, Inf. Theory, 1989,

IT-35,(4), pp. 911-912. [4] Bomer, L and Antweiler,M., “ Polyphase Barker sequences”. Electronics letters Vol. 34, No 16, 1989 25, pp

1577-1579. [5] Friese, M, and Zottmann, H: “ Polyphase Barker sequences upto length 31 ”, Electronics letters, 1994,30,(23),

pp 1930-1931. [6] Zhang., N, and Golomb, S. W, “ 7200 - phase Generalized Barker sequences”, IEEE Trans, Inf. Theory, 1996,

IT-35,(4), pp. 1236-1238 [7] Friese, M, “ Polyphase Barker sequences upto length 36 ”, IEEE Trans, Inf. Theory, 1996, IT-42,(4), pp.1248 [8] A.R Brenner., “ Polyphase Barker Sequences upto length 45 with small alphabets” Electronics letters Vol. 34,

No 16, Aug 1998, pp 1576-1577. [9] Peter Browein and Ron Fergusion “ Polyphase sequence with low autocorrelation”, IEEE Trans, Inf. Theory,

2005, IT -51, No 4, pp1564-1567. [10] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science, Vol. 220 May

1983, pp. 671–680,. [11] Moharir.P.S, Singh.R.and Maru. V.M., ”S-K-H algorithm for signal design”, Electronics letters, Vol 32, no 18,

Aug 1996, pp.1642-1649. [12] Moharir.P.S and Maru. V.M and Singh.R., “Bi-parental Product algorithm for coded waveform design in radar”,

Sadhana, Vol.22, no.5, Oct. 1997, pp 589-599. [13] E. C. Farnett and G. H. Stevens, “Pulse compression radar,” Radar Handbook, Second ed. New York: McGraw-

Hill, 1990, [14] S.P Singh., and K. Subba Rao., “Modified simulated Annealing Algorithm for Poly Phase Code Design” Proc

of IEEE ISIE -06, Canada, 09-13 July 2006, pp 2961-2971. [15] R. L. Frank, “Polyphase codes with good non-periodic correlation properties,” IEEE Trans. Inform. Theory, Vol.

IT-9, Jan. 1963, pp. 43–45.

362362