Compressive Radar Imaging Using White StochasticWaveforms

Mahesh C. Shastry∗, Ram M. Narayanan∗ and Muralidhar Rangaswamy†∗Electrical Engineering Department, The Pennsylvania State University, University Park, Pennsylvania 16802,

Email: {mcs312, rmn12}@psu.edu†AFRL/RYRT Building 620, 2241 Avionics Circle, WPAFB, OH 45433,

Email: muralidhar.rangaswamy@wpafb.af.mil

Abstract—In this paper, we apply the principles of compressivesampling to ultra-wideband (UWB) stochastic waveform radar.The theory of compressive sampling says that it is possibleto recover a signal that is parsimonious when representedin a particular basis, by acquiring few projections on to anappropriate basis set. Drawing on literature in compressivesampling, we develop the theory behind stochastic waveform-based compressive imaging. We show that using stochasticwaveforms for radar imaging, it is possible to estimate targetparameters and detect targets by sampling at a rate that isconsiderably slower than the Nyquist rate and recovering usingcompressive sensing algorithms. Thus, it is theoretically possibleto increase the bandwidth (and hence the spatial resolution)of an ultra-wideband radar system using stochastic waveforms,without significant additions to the data acquisition system.Further, there is virtually no degradation in the performanceof a UWB stochastic waveform radar system that employscompressive sampling. We present numerical simulations to showthat the performance guarantees provided by theoretical resultsare achieved in realistic scenarios.

I. INTRODUCTION

Radar systems can be classified to be of two types [1], (1)those employing an analog system for detecting the targetsusing an analog matched filter, and (2) systems that sampleand quantize the reflected radar signal and process it as digitalsignals. With the advent of fast analog to digital converters(ADC), radar systems of the latter type are gaining popularityfor high resolution radar imaging. While digital radar systemsprovide operational flexibility, their resolution has been limitedby the sampling rate of the fastest available ADCs. Theperformance of digital radar systems is further degraded bythe trade-off that exists between the rate of sampling and thenumber of quantization levels of an ADC.

Compressive sensing is a new paradigm in signal process-ing with far-reaching implications. In its most useful form,it presents a new approach to digital signal processing byenabling signal recovery from under-sampled measurements.It has been shown that exact recovery of sparse signals ispossible by acquiring as few as O(S log (N)) (N is the lengthof the vector that is also S-sparse in some representation)projections of a signal into an appropriate measurement ba-sis. While it is possible to recover signals that are sparsein a particular basis of representation, from under-sampledmeasurements by searching for the vector that, at the sametime, is most consistent with the measurements and also

minimizes the number of non-zero values, as given by the l0norm, the problem of finding such a vector is intractable. Themost important results concerning compressive sensing werederived in [7] and [8]. It was shown that the minimizationof the l0 norm is equivalent, under certain easily verifiableconditions (the restricted isometry property (RIP)), to thetractable convex optimization problem of minimizing the l1norm. The original papers on compressive sensing provedthat random measurement bases drawn from sub-Gaussianand Gaussian distributions almost always satisfied RIP fora large class of representation vectors. Thus, compressivesensing has the potential to significantly reduce the samplingrequirements while maintaining and in some cases, improvingthe performance of the systems.

Random noise radar [5] involves transmitting waveformsthat are generated as ultra-wideband stochastic processes. Theultra-wideband property implies that high range resolutionscan be obtained. Random noise waveforms have the ability toachieve approximately thumbtack ambiguity functions due tothe statistical independence of the waveform at different timeintervals. Further, the stochastic and non-repetitive nature ofcontinuous time random waveforms means that they cannotbe intercepted or detected by an adversary. Early stochasticwaveform radars used analog processing to detect targets [5].However, with advances in ADC technology, digital systemshave become popular in recent times [6] for random waveformradar. In digital stochastic waveform radar systems, a high-rateADC is employed to acquire samples of the reflected signal,which are then stored and processed as quantized discretevectors to extract information about the targets.

The randomness of the transmit waveform makes stochasticwaveforms suitable for compressive sensing systems. We un-dertake a systematic study of the performance of compressivestochastic waveform radar systems. Applying the state ofthe art in compressive sensing to radar systems, we: (1)show theoretical results regarding target estimation, (2) studythe robustness of compressive radar systems to uncorrelatedadditive noise; (3) study the performance of compressiveestimators, (4) simulate receiver operating characteristics ofcompressive radar using phase transition diagrams [10] andreceiver operating curves. We demonstrate the suitability ofstochastic waveforms (noise waveforms) for building practicalradar systems based on compressive sensing. The paper is

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organized as follows. In section 2, we present the theory ofcompressive sensing with circulant/Toeplitz matrices [9] andoutline its implications to white stochastic waveform radar.The estimation performance of compressively sampled randomwaveforms was shown in [4] to be better than the best-caseleast-squares estimators in the context of channel sensing.We state this result to emphasize the advantages of com-pressive sensing. Then, we outline the matched-pursuit-basedcompressive detection scheme presented in [11] and apply itto radar systems. In section 3, we present numerical resultson the performance of compressive-stochastic-waveform radarsystems. We detail the application of Donoho-Tanner phasetransition diagrams [10] for characterizing the performance ofradar systems, and show that even with stochastic waveformsand under-sampled reflected radar signals, it is possible torecover sufficiently sparse signals with adequate accuracy. Us-ing Monte-Carlo simulations, we study the receiver operatingcharacteristics of the proposed compressive radar system. Weconclude the paper in section 4 with comments on the scopefor future work.

II. BASICS OF COMPRESSIVE STOCHASTIC WAVEFORMRADAR

A. Compressive Radar

We consider one-dimensional radar imaging in our studyof compressive radar. The problem of radar imaging involvessensing and processing a transmitted signal that has beenlinearly filtered by a transfer function representing the targetscene. In the most general case, the reflected signal y(t) canbe written as a convolution of the transmitted signal x(t) withthe target scene impulse response denoted by s(t). In thecontinuous domain, this can be written as,

y(t) =

∫ +∞

−∞x(t)s(t− τ)dτ + η(t). (1)

When discretized, this becomes,

y[n] =N∑k=1

x[k]s[n− k] + η[n], (2)

y = Xs+ η, (3)

with s, y ∈ RN . Furthermore we assume that the vectors, representing the target scene is sparse, i.e, supp(s) =S << N , where supp(.) is the count of the number ofnon-zero entries. The matrix X ∈ RN×N is Toeplitz (orcirculant). The objective of the radar imaging problem isto accurately recover the target characteristics s from thereflected waveform y. Conventionally, the problem can becast as the minimization of the l2 norm of the estimationerror, i.e., s∗ = arg mins∈RN ||y − Xs||l2 . The solution tothis problem is well known and is given by the estimator,s∗ = (XTX)−1XT y. However, in the absence of any modelorder selection, the vector y needs to be at least as longas s, in other words, the system should be fully, or over-determined. Compressive sensing answers the question- is itpossible to blindly acquire fewer measurements, without using

any adaptive, model order selection? In terms of radar imaging,this question can be restated as whether it is possible to acquirefewer measurements while still achieving the resolution limitsimposed by the bandwidth. Compressive sensing proposes anestimator based on measuring the vector y using a collection ofM < N measurement basis in RN . Rather than the traditionalleast squares approach, the solution to the estimation problemis,

s∗ = arg mins∈RN

||s||l1 subject to ||z −ΨXs||l2 ≤ ξ. (4)

In its most general form, the reconstruction theorem of com-pressive sensing states that this problem yields an accuratesolution when the matrix A = ΨX satisfies the restrictedisometry property (RIP) with as few as O(S logN) measure-ments, i.e., for some δS , and u ∈ RN , (1 − δS)||u||l2 ≤||Au||l2 ≤ (1+δS)||u||l2 . RIP and incoherence properties havebeen shown to hold for different pairs of matrices, such as therandom matrices Ψ and arbitrary X . Examples of incoherentpairs include spike bases and Fourier bases. The most generalprocedure proven to recover exactly is when each elementof Ψ is drawn randomly from Gaussian and sub-Gaussiandistributed random variables. Implementing this would needthe acquisition of M random measurements of a time domainsignal. The measurement hardware should then consist of Mchannels of correlation with random measurements. Designingsuch hardware would be problematic for the sampling ratestypically required for ultra-wideband radar systems. The sim-ple system we envision for compressive radar systems shouldconsist only of sampling the reflected signal at a rate lowerthan the Nyquist rate. Stochastic waveforms are a naturalcandidate, as we outline in the following sub-section.

B. Stochastic Waveforms and RIP

Proving the uniqueness of the solution to Equation ( 4)requires that X satisfies RIP. We outline that a circulant orToeplitz matrix generated from a random, Gaussian, vectorsatisfies RIP. Particular forms of such matrices have beenstudied in [3], [4], [9]. In [3], circulant/Toeplitz matrices wereconstructed such that they represent the operation of randomdemodulation and were shown to satisfy RIP. The results of[3] however, do not generalize to random circulant matriceswhich are generated from a sequence of independent randomvariables as is needed in stochastic waveform radar imaging.Random circulant matrices with the first row made up of asequence of independent random variables were studied by [4]in the context of channel estimation. Using a method primarilybased on the Gershgorin disk theorem, the authors statedthat RIP and subsequently, recovery using an l1 minimizationprocedure, is guaranteed when M is at least O(S2 logN).Numerical simulations in the present paper (phase transitiondiagrams in Section 3.2) and elsewhere suggest that thisbound is too conservative, as we see in numerical simulationsthat with as few as O(S logN) samples, reconstruction isaccurate. The theoretical result was improved upon in [9], forthe particular case where the first row of X is a sequence

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of Bernoulli random variables taking values ±1 with equalprobability. The main result of [9] presents a stronger boundfor RIP than [4], with the minimum number of samples neededbeing O(S logαN) (α ≥ 1) using a signal modeled such thatthe signs of the nonzero entries are Bernoulli random. Theresults of [9] also hold for circulant and Toeplitz matricesconstructed from Gaussian random variables. We restate asTheorem 1, the main theorem of [9] for the case of a circulantmatrix that is constructed with the first row as entries drawn atrandom from a Gaussian distribution of variance 1/

√N and

mean 0.Theorem 1: Let s ∈ RN and X ∈ RN×N , Ψ ∈ RM×N

be as defined earlier, and z = ΨXs. There exist constants Cand α, such that recovery of the vector s is guaranteed withprobability exceeding 1−ε by solving the problem in Equation4 if, M ≥ CS logα N

ε .

C. Comparison with Best Model Selection

Conventionally, model order selection methods have beenused to reduce the dimensionality of problems in radar signalprocessing. When viewed as a method of estimation, thel1 minimization problem can be interpreted as selecting themodel order of the estimator. It has been shown [2] that theperformance of compressive sensing is within a log factor ofthe so called ’oracle’ estimator- an estimator with completeknowledge of the location of non-zero members of s. Thissuggests that even the unrealistic estimator that works withfull knowledge of the indices of the non-zero elements, wouldstill only provide performance that is better than compressivesensing by a log factor [2], [4].

III. NUMERICAL SIMULATIONS

In order to understand the implications of the theoreticalresults underlying compressive radar imaging using stochasticwaveforms, it is necessary to look at numerical simulationsinvolving basic detection and estimation. In our simulations,we used discrete sequences of arbitrary lengths to simulate theperformance of the compressive detector, and compressive es-timator. Numerically, the difference between the compressiveradar imaging system proposed above and other applications ofcompressive sensing arises due to the fact that the matrix thatmaps the target scene to the under-sampled reflected signal isa random Toeplitz (or circulant) matrix. To a limited extent,such a measurement system has been studied for channelestimation [4] and proposed for use in Fourier optics andsynthetic aperture radar imaging [3]. Adopting the theoryof compressive sensing for radar systems begs a thoroughnumerical study of the resolution limitations compressiverecovery, detection performance of a compressive receiver, androbustness to noise.

A. Receiver Operating Characteristics for Compressive Esti-mation

We study the detection performance of compressive radarin the context of detecting the presence or absence of asingle target, using the greedy ‘incoherent detection algorithm’

described in [11]. The two hypothesis considered are asin conventional binary hypothesis testing, H0 : z = Φη(target absent) and H1 : z = ΦXs + Φη (target present).The algorithm is essentially the traditional matching pursuitalgorithm, with the modification that the pursuit stops whenthe l∞ norm, defined as the maximum element of the estimatedvector is found to be above a fixed threshold. The algorithmhas only been studied empirically, without a thorough theo-retical analysis. In this paper, we present receiver operatingcharacteristics of this algorithm in the present context. Thematching pursuit algorithm, which the incoherent detectionalgorithm is based on is guaranteed to converge to a solutionwhen RIP is satisfied, as is the case here. Thus, a convergenceto the solution would be seen in the presence of a target, withthe subsequently recovered vector lying above the detectionthreshold. Figure 1 shows the curves for the probability oferror and false alarm for an arbitrarily fixed threshold. In theabsence of theoretical results, we study the application of thisalgorithm using Monte-Carlo simulations for computing theprobabilities of false alarm and detection. The performance ofthe above compressive detector in the context of radar systemsis seen in Figure 2 to be reasonably good at even low signalto noise ratios.

B. Phase Transition Diagrams

Conventional radar systems employ receiver operating char-acteristics (RoC) curves to calibrate the performance of radar

0 5 10 15 200

0.2

0.4

0.6

0.8red curve - false alarm, blue curve - error

Pro

babi

lity

Fraction of samples

Fig. 1: The probability of error (red curve) and false alarm(blue curve) plotted as a function of the rate of fraction ofsamples acquired.

0 0.2 0.4 0.6 0.8 10

0.5

1

Probability of false alarm

Pro

babi

lity

of d

etec

tion

ROC Curve

0 dB SNR-2 dB SNR-4 dB SNR-6 dB SNR

Fig. 2: The receiver operating curves for different values ofsignal to noise ratio.

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detection systems. However, the performance of compressiveradar systems cannot be adequately described using RoCcurves due to the additional parameters of sparsity and rate ofunder-sampling affecting receiver performance. In this section,we study phase transition diagrams [10] in the context ofcompressive sensing to verify the performance of circulantrandom matrices with random under sampling. Phase transitiondiagrams present a numerical study of the l0-l1 equivalencethat forms the basis of compressive sensing. They are anindirect way of numerically verifying RIP. In our simulations,we used a 256 length transmit vector and discretized the (δ, ρ)phase space into a 64 × 64 grid. Over several realizations,we computed the average miss rate, which is the averagepercentage of non-zero samples (l0 norm) that are recoveredwith a sample-wise error that is greater than a thresholdthat is related to the variance of the additive noise. Therecovery was performed using the spectral projected gradientalgorithm [12], [13]. The suitability of stochastic waveformsfor compressive radar systems is indicated by the fact that thephase transition diagrams are close to the phase transition dia-grams that arise from the general compressive sensing settingof acquisition using random measurements [10]. The phasetransition diagram of Figure 3 was generated by acquiringsamples of the reflected signal at random intervals. Similar,desirable phase transitions are seen when the reflected signalis uniformly under-sampled as shown in Figure 4, suggestingthat a further simplification of the hardware will not degradethe performance significantly. The boundary line between

0 0.5 10

0.2

0.4

0.6

0.8

1

δ = M/N

ρ =

S/M

0

0.2

0.4

0.6

0.8

Fig. 3: Phase transition diagram for a Gaussian randomcirculant matrix with signal to noise ratio of 20 dB and randomunder-sampling.

the red and blue areas is similar to the case of randommeasurement matrices [10] and indicates that RIP holds withM = O(S logN). In the context of compressive sensing,phase transition diagrams are a representation of the successof compressive recovery in terms of the probability of successplotted in the phase space of the pair (δ, ρ) where, δ = M/Ncorresponds to the number of samples acquired and ρ = S/M ,is a ratio of the sparsity of the signal and the number ofsamples acquired. The performance of compressive estimation

0 0.2 0.4 0.6 0.8 10

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δ = M/N

ρ =

S/M

0

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0.9

Fig. 4: Phase transition diagram for a Gaussian random cir-culant matrix with signal to noise ratio of 20 dB and uniformunder-sampling.

is seen to deteriorate for low SNRs in Figure 5(a) (10 dB SNR)and Figure 5(b) (0 dB SNR). At 0 dB SNR, recovery is only

0 0.2 0.4 0.6 0.8 10

0.1

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δ = M/N

ρ =

S/M

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(a) SNR 10 dB

0 0.2 0.4 0.6 0.8 10

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δ = M/N

ρ =

S/M

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(b) SNR 0 dB

Fig. 5: Phase transitions at low SNRs.

possible when a large number of samples are acquired and thattoo only for very sparse target scenes. This suggests that there

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is scope for developing compressive recovery algorithms thatare more robust to noise. Further, it is seen that lower the SNR,the further the deviation from RIP and l0-l1 equivalence. Froma radar systems perspective, phase transition diagrams are away of calibrating compressive radar systems. Compressiveradar imaging is only non-blind in the sense that the operatorof the system needs to have an approximate idea of the sparsityof the target scene. The uncertainty in the knowledge of theapproximate sparsity of the target scene can be quantified interms of the phase transition diagrams. For instance, it can beinferred from Figure 3 that if a radar system is intended tooperate in an environment with 20 dB SNR, and with 2% ofthe range cells populated by point scatterers, almost all thetargets can be recovered by acquiring as few as 10% of thetotal samples.

IV. CONCLUSION

In this paper, through theoretical arguments and numericalsimulations, we showed the suitability of stochastic waveformsfor compressive radar systems. It is possible to recover targetinformation and perform detection by sampling the reflectedsignal at rates far lower than the Nyquist rate. The restrictedisometry property of circulant matrices generated from aGaussian random vector was shown to compare well withthat of the fully random matrix by looking at phase transitiondiagrams, confirming theoretical results. We also outlined howphase transition diagrams can be used to build and operatecompressive radar systems.

We are currently working on extending this paper by devel-oping the complete proof for the RIP for Gaussian circulantmatrices with M as close to the experimental asymptoticbound of O(S logN) and by studying the performance ofdifferent algorithms for estimation and detection.

ACKNOWLEDGMENT

The research is supported by Air Force Office of ScientificResearch (AFOSR) contract number FA9550-09-1-0605. Theauthors would like to thank Dr. Jon Sjogren of AFOSR forhis valuable comments.

REFERENCES

[1] R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in Proceed-ings of 2007 IEEE Radar Conference, Boston, MA 2007, pp. 128-133.

[2] E. J. Candes, “Compressive sampling,” in Proceedings of the Interna-tional Congress of Mathematicians, Madrid, Spain, 2006, pp. 1433-1452.

[3] J. Romberg, “Compressive sensing by random convolution,” SIAMJournal of Imaging Science, vol. 2, no. 4, pp. 1098–1128, Nov. 2009.

[4] J. Haupt, W.U. Bajwa, G. Raz, and R. Nowak, “Toeplitz compressedsensing matrices with applications to sparse channel estimation,” toappear in IEEE Transactions on Information Theory, 2010.

[5] R. M. Narayanan, Y. Xu, P. D. Hoffmeyer, and J. O. Curtis, “Design,performance, and applications of a coherent ultrawideband random noiseradar,” Optical Engineering, vol. 37, no. 6, pp. 1855-1869, June 1998.

[6] R. M. Narayanan, “Through-wall radar imaging using UWB noisewaveforms,” Journal of the Franklin Institute, vol. 345, no. 6, pp. 659–678, September 2008.

[7] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exactsignal reconstruction from highly incomplete frequency information,”IEEE Transations on Information Theory, vol. 52, pp. 489-509, 2006.

[8] D. Donoho, “Compressed sensing,” IEEE Transactions on InformationTheory, vol. 52, pp. 1289-1306, 2006.

[9] H. Rauhut, “Compressive sensing and structured random matrices,” inTheoretical Foundations and Numerical Methods for Sparse Recovery,M. Fornasier, Ed., Berlin: de Gruyter, 2010, pp. 1–93.

[10] D. Donoho and J. Tanner, “Observed universality of phase transitions inhigh-dimensional geometry, with implications for modern data analysisand signal processing,” Philosophical Transactions of The Royal SocietyA, vol. 367, no. 1906, pp. 4273-4293, 2009.

[11] M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk,“Sparse Signal Detection from Incoherent Projections,” in Proceedingsof the International Conference on Acoustics, Speech, and Signal Pro-cessing (ICASSP), Toulouse, France, 2006, pp. III-305–III-308.

[12] E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier forbasis pursuit solutions,” SIAM Journal on Scientific Computing, vol. 31,no. 2, pp. 890–912, November 2008.

[13] E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scalesparse reconstruction,” http://www.cs.ubc.ca/labs/scl/spgl1, June 2007.

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