Optimizing the sidelobe properties of radar waveformsKavish Gandhi1,2, Ryan Volz2

1Massachusetts Institute of Technology, 2MIT Haystack Observatory

Introduction

• This project studies the optimization radar waveforms for signal processing.• Typically, to maximize the signal-to-noise ratio (SNR), a matched filter is applied

to the signal received. This can created unwanted sidelobes.• Fundamental problem: how to minimize size or integrated power in sidelobes?• Possible solution: design a waveform that minimizes these under the matched filter

Given a code c, define the autocorrelation function  and the ambiguity function ‰

„(c)d =⁄ Œ≠Œ c(t)cú(t ≠ d)dt.

‰(c)n,d =⁄ Œ≠Œ c(t)cú(t ≠ d)e2fiintdt.

A sidelobe of „(c) or ‰(c) is any value that is not the central peak and the peak

sidelobe value (PSL) is the maximal such sidelobe.

Figure: Example of the autocorrelation of a binary code.

Problem Statement (Autocorrelation variant)

Given an integer L and a function f : CL æ R, find a code c = {ci}, ci œ C oflength L that minimizes f („(c)).

Types of CodesIn this project, we considered only coding using bauds, in which the waveform is splitinto equal segments, encoded by the phase of that segment.• Binary codes:

c = {ck}, ck œ {≠1, 1}

• Polyphase codes with fixed generator m:c = {ck}, ck œ {e2fiij/m : j œ [0, m ≠ 1]}

.• Arbitrary polyphase codes:

c = {ck}, ck = ei„k, „k œ R.

Binary codes are the simplest to generate, but the more flexible the code, the easier itis to minimize its sidelobes. Furthermore, these are all amplitude one codes; can alsoconsider amplitude-modulated codes, but these are usually very di�cult to generate.

Exhaustive Search

• Simplest way to generate optimal code: search through all possible codes• Naively, this takes exponential time• Number of ways to optimize this process

1 Exploit PSL-preserving operations: reversal, negation, alternate negation2 Branch-and-bound algorithms

• Wrote my own e�cient algorithm in Python, modeled o� ofexisting branch-and-bound and DFS methods [1].

Exhaustive Search (cont.)

Figure: Comparison of naive and e�cient exhaustive algorithms for binary codes

Heuristic Optimization Algorithms

• For lengths > 40, exhaustive search takes too long• Enter optimization: ensemble of techniques to estimate the global extrema of a

function• Many optimization techniques require di�erentiation, however, or heavily depend

in performance on a continuous search spaceSolution: Heuristic OptimizationThe main idea of heuristic optimization is to start with a code or set of codes, anduse an update heuristic to move around in the search space starting with these codes.Ideally, the heuristic allows you to move towards minima without searching the entirespace nor getting stuck in local minima, thus reaching a global minima faster than anexhaustive approach.

1 Population-based approaches:• Evolutionary algorithm• Particle swarm• Genetic algorithms

2 Single code approach:• Simulated annealing• Hill Climbing• Tabu Search• Great deluge• Threshold accepting

3 Most powerful algorithms: globalheuristic with local search subroutine

Figure: Flow chart illustrating genetic algorithms.Credit: http://www.jade-cheng.com/au/

coalhmm/optimization/

Figure: Depiction of simulated annealing; the algorithm starts at higher temperatures, allowing morefreedom for a code to escape local minima, and slowly cools as the code moves closer to a globalminima. The initial temperature and cooling schedule significantly a�ect the e�cacy of this algorithm.Credit: Ledesma, Ruiz, and Garcia, “Simulated Annealing: Advances, Applications, and Hybridizations”

Binary Codes

• Extensive literature applying optimization algorithms to minimizing PSL or ISL ofbinary code [2].

• To compare existing techniques, we wrote six binary optimization algorithms basedon the defined heuristics

• Our algorithms were able to, even with short time constraints, findcodes close to optimal for lengths 2-55.

• Best algorithms: basic hill climbing, evolutionary algorithms, memetic tabu search.• We also compared the relative success of di�erent crossover and parent selection

options in genetic algorithms, as well as temperature schedules for annealingalgorithms

• Finally, we extended our algorithms to also minimize overpolyphase codes with a fixed generator m.

Binary Codes (cont.)

Figure: Comparison of our binary optimization algorithms: displaying cumulative distance from theoptimal code on binary codes from 2-55, given 5 minutes to run on each input. Note that memetic,evolutionary, hill climbing, and anneal all have variants with similar performance.

Arbitrary Polyphase Codes

• We also considered arbitrary polyphase codes, codes of the formc = {ck}, ck = ei„k, „k œ R

• There is some literature applying optimization techniques to this problem; wetested an existing technique [3], the great deluge algorithm, against a simulatedannealing and particle swarm approach.

• All work better than a random approach.

Figure: Comparison of our polyphase optimization algorithms, displaying the peak sidelobe values foundfor lengths 2-39 given five minutes of runtime. Note that all of our algorithms outperform randomselection, with the great deluge approach outperforming it significantly.

Minimizing over the Ambiguity Function

• This project‘s main novel contribution is studying the ambiguity function variantof our fundamental problem

• Our algorithms can be extended naturally to minimize any function of theambiguity function of a code, in any Doppler width N .

• Used these algorithms to generate a number of novel, near-optimalbinary, polyphase, and arbitrary polyphase codes for Dopplerwidths 2-6.

• These codes, for Barker lengths, significantly outperform these codes in Dopplerspace, reflecting the poor Doppler tolerance of Barker codes.

Figure: Peak sidelobe values of ambiguity functions found by our algorithms for triphase codes, N=1-5.

Alternating Codes

• Finally, we applied our algorithms to the generation of alternating codes• Already exists a way to generate alternating codes with lengths a power of 2.• We were able to reimplement our algorithms to find

(near)-alternating binary and polyphase codes with lengths otherthan a power of 2.

• At the moment, our best results are only for lengths 8-20, and for longer lengthswe do not improve on single code peak sidelobe values.

Figure: A quadruple of alternating codes, each of length 12, found by our hill climbing algorithm.

ConclusionIn this project, we have developed a number of optimization algorithms that can

both reproduce optimal peak sidelobe results from the literature concerning binary,polyphase, and arbitrary polyphase codes and find novel near-optimal codes in dopplerspace. We have also compared the relative success of di�ering optimization algorithms,as well as compared local search routines, temperature schedules, and crossover types.Finally, we have shown that these algorithms are versatile, and can be applied to manyoptimization problems, including the generation of alternating codes.

These results have a number of important consequences; optimally designed wave-forms improve speed and accuracy of the reconstruction of radar signals, and specifi-cally our novel waveforms that minimize Doppler sidelobes will aid in detecting objectsin cluttered environments in which velocity is a distinguishing factor. Our algorithmsare also flexible, allowing for extensions to di�erent codes, such as the less-investigatedpolyphase or amplitude-modulated variants, as well as potentially being useful for awhole suite of problems completely separate from radar.

Future Work

• I optimized the parameters of most of these algorithms, but a number of additionalimprovements could be made.• Parallelization• Addition of C or C++ wrapper• Adaptive parameter selection, or tests to optimize for the best parameters• Ensemble algorithms, or additions of relevant local search routine to the global algorithms• Use of characteristics of the autocorrelation function to better inform search, crossover.

• Study of general waveforms: briefly studied linear chirps and optimizedunsuccessfully over their parameters.• Particle swarm and associated algorithms (gravitational search) seem well suited to this problem

• Alternating codes: take advantage of symmetry to improve current algorithm.• Perfect codes: limiting amplitude modulation

References

[1] Coxson, G. and Russo, J. “E�cient exhaustive search for optimal peak-sidelobebinary codes.” Proceedings of the IEEE 2004 National Radar Conference,Philadelphia PA, April 2004.

[2] Deng, X. and Fan, P. “New binary sequences with good aperiodic autocorrelationsobtained by evolutionary algorithm.” IEEE Communication Letters, vol. 3, no.10, pp. 288-90, October, 1999.

[3] Friese, M. and Zottman, H. “Polyphase sequences up to length 31.” Electronic

Letters, vol. 34, no. 16, pp. 1576-1577, 1998

AcknowledgementsI would like to acknowledge Ryan Volz, my mentor, for his endless help throughout theproject. I would also like to thank the MIT Haystack Observatory for this wonderfulopportunity, as well as all of the sta� and students that I worked and talked with here.