Sparsity-Based Signal Processing for Noise Radar ... - IEEE Xplore

wave noise radar validate our analytical results and the theoretical guarantees of compressive sensing.

Sparsity-Based Signal Processing for Noise Radar Imaging

MAHESH C. SHASTRY, Member, IEEE 3M Corporate Research Labs St. Paul, MN, USA RAM M. NARAYANAN, Fellow, IEEE The Pennsylvania State University University Park, PA, USA MURALIDHAR RANGASWAMY, Fellow, IEEE Air Force Research Laboratory Sensors Directorate Wright Patterson Air Force Base, OH, USA

Noise radar systems transmitting incoherent signal sequences have been proposed as powerful candidates for implementing compressively sampled detection and imaging systems. This paper presents an analysis of compressively sampled noise radar systems by formulating ultrawideband (UWB) compressive noise radar imaging as a problem of inverting ill-posed linear systems with circulant system matrices. The nonlinear nature of compressive signal recovery presents challenges in characterizing the performance of radar imaging systems. The suitability of noise waveforms for compressive radar is demonstrated using phase transition diagrams and transform point spread functions (TPSFs). The numerical simulations are designed to provide a compelling validation of the system. Nonidealities occurring in practical compressive noise radar systems are addressed by studying the properties of the transmit waveform. The results suggest that waveforms and system matrices that arise in practical noise radar systems are suitable for compressive signal recovery. Field imaging experiments on various target scenarios using a UWB millimeter

Manuscript received November 13, 2013; revised July 7, 2014, July 16, 2014; released for publication October 1, 2014. DOI. No. 10.1109/TAES.2014.130733. Refereeing of this contribution was handled by L. Kaplan. This work was supported by AFOSR through Contract FA9550-09-1-0605. Authors’ addresses: M. C. Shastry, 3M Corporate Research Labs, St. Paul, MN 55144, USA; R. Narayanan, The Pennsylvania State University, Dept. of Electrical Engineering, 202 EE East Building, University Park, PA 16802-2705, USA. E-mail: ([email protected]). M. Rangaswamy, Air Force Research Laboratory Sensors Directorate, Radar Signal Processing, AFRL/RYRT, Building 620, 2241 Avionics Circle, Wright Patterson Air Force Base, OH 45433-7132, USA, and the Electrical and Computer Engineering Department, Purdue University, West Lafayette, IN 47907, USA. C 2015 IEEE 0018-9251/15/$26.00

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LIST OF SYMBOLS x(t) = Transmit waveform s(t) = Target scene impulse response y(t) = Received waveform n(t) = Additive noise x ∈ RN = Discretized transmit waveform s ∈ RN = Discretized target impulse response y ∈ RN = Discretized received waveform n ∈ RN = Discretized additive noise z ∈ RM = Undersampled received waveform n ∈ RM = Undersampled additive noise X ∈ RN×N = Circulant system matrix generated from x R ∈ RM×N = Undersampling operator s∗ ∈ RN = Recovered target scene σ = Basis pursuit de-noising parameter δ = Fraction of samples acquired ρ = Fraction of nonzero elements in s p = Energy of error residue χ = Energy of off-diagonal elements in the transform point spread function matrix s∗CR = Target scene recovered using cross-correlation based imaging s∗LS = Target scene recovered using least squares s∗CS = Target scene recovered using compressive signal recovery G = Gram matrix corresponding to X I. INTRODUCTION

Radar range imaging in the far field can be viewed as a linear inverse problem. Let x(t) be the transmit waveform, s(t) denote the impulse response modeling the target scene, and n(t) the additive noise in the system. For a monostatic radar with colocated transmit and receive antennas, the received radar signal y(t) can be modeled as a filtered version of the transmit waveform, with the filter coefficients representing the target model. In the time-domain, the received signal can be expressed as ∞ y(t)= x(t − τ )s(τ )dτ + n(t). (1) −∞

The discrete form of the above equation is a system of linear equations, ym = xm−k sk + nm , (2) k

that is, y = Xs + n

(3)

where y, s ∈ RN and X ∈ RN×N . When we consider this linear time-invariant system over a finite time-interval, the matrix X can be approximated to have a circulant structure.

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Based on whether signal processing is performed in the analog or digital domain, there are two types of radar ranging systems: 1) analog radar systems which use analog hardware to perform matched-filter based imaging, and 2) digital radar systems that sample and quantize the reflected radar signal and process it digitally. An important objective in designing radar systems is to maximize the imaging resolution. It is well known that finer range-resolution requires high-bandwidth waveforms, since the resolution cell extent is inversely proportional to the transmit bandwidth. For example, a radar system transmitting an instantaneous bandwidth of 500 MHz achieves a resolution of about 30 cm. At such high bandwidths, we are faced with signals with high-throughput which in turn require suitable algorithms to manage the large amounts of data generated. Further, to move up to higher bandwidths a limiting factor is the cost of analog-to-digital converters (ADCs). The theory of compressive sensing provides the framework for signal recovery from samples acquired at considerably lower sampling rates by using and exploiting the information rate. The question of whether compressive sensing is a practical approach to radar imaging remains open. We address this issue by exploring the application of noise signals as transmit waveforms. We also present experimental results that validate the feasibility of using compressive sensing for radar range imaging applications. Carefully designed simulations allow us to characterize the performance of such systems in the presence of hardware system-related nonidealities. Since continuous-wave noise radar systems are simpler to design and implement compared with pulsed radar, owing to the lack of synchronization requirements, they emerge as attractive alternatives for implementing compressively sampled radar imaging systems. The term noise radar traditionally refers to systems that transmit waveforms that are typically thermally generated random noise, or could also be digitally generated pseudorandom noise. Such a system was first proposed in 1959 by Horton [1]. Over the last two decades, noise radar systems have moved towards utilizing ultrawideband (UWB) transmit waveforms [2]. The UWB nature of the transmitted waveform enables us to achieve the desired range resolutions. The use of randomly generated transmit waveforms makes noise radar signals less susceptible to detection, interception, and jamming, which adds to their benefits. Noise radar systems originally utilized analog processing for imaging targets [1, 2]. With the increased availability of accurate high rate ADCs, new UWB noise radar systems use digital processing [3, 4] for imaging in real time or with little latency. One of the early papers on compressive radar imaging [5] proposed the use of pseudorandom pulses based on random demodulators [6]. In general, the recovery performance of compressive sensing estimators depends on the system matrix satisfying certain properties. The compressive noise radar imaging problem, as described in this paper, involves a circulant system matrix generated

from a random vector. For recovering vectors with S nonzero elements, Romberg [7] showed that with specially designed Toeplitz and circulant random system matrices, recovery with probability 1 – O(N−1 ) is possible with >S log N measurements. Haupt et al. [8] studied the ∼ performance of circulant matrices generated from Bernoulli and Gaussian random vectors in the context of compressive channel estimation. They showed that based on bounds on the restricted isometry constant of the system matrix, O(S2 log N) measurements of the Toeplitz random matrix are sufficient for stable recovery using the Dantzig selector [9] solver to recover the undersampled signal. Rauhut et al. [10] derived the restricted isometry property (RIP) for circulant matrices, suggesting that O (max ((S log N)1.5 , S log2 S log2 N)) measurements guarantee stable recovery. Herman and Strohmer [11] showed that the low mutual coherence of system matrices arising from narrowband Alltop sequences and random sequences made them suitable for compressive high-resolution radar imaging. Compressive imaging using stepped frequency radar imaging was analyzed and demonstrated to be effective for ground penetrating radar (GPR) applications by Gurbuz et al. [12]. Bhattacharya et al. demonstrated the utility of compressive sensing in compressively acquiring synthetic aperture radar (SAR) images [13]. Nguyen and Tran [14] showed that SAR images can be successfully recovered from undersampled reflected signals using data from the Army Research Lab UWB SAR impulse-radar imaging system. Nguyen and Tran [15] and Kelly and Davies [16] propose techniques for radio frequency (RF) interference mitigation in SAR imaging. Gurbuz et al. [17] and Shastry et al. [18] considered the problem of imaging extended targets and proposed dictionary-learning based approaches to solve such systems. Zhang et al. [19] and Ahmad and Amin [20] proposed systems based on using impulse radar for through-the-wall imaging. Yu et al. [21] studied the problem of compressively sampled narrowband multiple input/multiple output (MIMO) radar imaging. Ender [22] presented an extensive survey of the different possible applications and use cases of compressive sensing, including analysis of the performance of sparsity-based imaging algorithms for data acquired from a 300-GHz real radar system implementing pulsed transmit waveforms. The primary goal of this paper is to verify, numerically and experimentally, the feasibility of using compressive sensing for UWB noise radar imaging. In Section II, we present the results of numerical simulations designed to predict the performance of compressively sampled radar imaging. In Section III we study experimentally the imaging performance of practical noise radar systems. For experimental data, we contrast the performance of compressive sensing with a conventional matched-filtering approach to target estimation and detection. Since the issue of resolution is important to conventional radar systems, we present experimental results to illustrate that the resolution achieved by compressive sensing is comparable to cross-correlation based imaging. Real radar

SHASTRY ET AL.: SPARSITY-BASED SIGNAL PROCESSING FOR NOISE RADAR IMAGING

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systems involve electromagnetic (EM) interactions occurring in the continuum [23]. Our results using experimental data show that for noise radar, the discrete linear systems model accurately represents the actual phenomena of noise radar imaging. For this paper, we have conducted extensive experiments resulting in around 25 data sets based on imaging performed on targets in five different scenarios. We conclude in Section IV with a summary of our contributions and a discussion of open problems in compressive noise radar imaging. II. ANALYSIS OF COMPRESSIVELY SAMPLED NOISE RADAR

In this section, we present results regarding compressively sampled noise radar as modeled by (3), with the interaction between the transmit signal and the target scene given by a linear system, y = Xs + n. Conventional noise radar systems use cross-correlation based processing so that the recovered signal is given by s∗CR = X T y. This relies on the assumption that the matrix X is orthogonal. The cross-correlation method is computationally efficient and can be easily implemented using analog tapped-delay lines [2]. The target scene can also be recovered by casting the problem as a least squares recovery problem. Least squares recovery imposes a design constraint on the system, which is that the signals be processed digitally for target scene recovery. Least squares and cross-correlation based imaging are equivalent when the matrix X is orthogonal. The least squares solution s∗LS is given by s∗LS = X † y, where X† = (XT X)−1 XT is the Moore-Penrose pseudoinverse of X. Least squares based recovery offers the advantage of enabling the use of complex signal models. Complex models of prior information can be incorporated into the least squares formalism to enhance the capabilities of the radar imaging system. One such signal model involves exploiting the sparsity of the target scene to improve imaging performance and reduce the number of samples required for recovery. The main theme of this paper is sparsity-based signal processing for noise radar systems, discussed using the framework of compressive sensing. There are significant theoretical challenges in demonstrating the viability of compressive sensing algorithms. In this paper, we validate our approach based on simulations designed to capture a wide range of possible scenarios. The signal model used in the paper is based on an undersampled version of the full model presented in (3). The undersampling operation is represented by the matrix R . The matrix R ∈ RM×N simply consists of a subset of rows of the N × N identity matrix indexed by the M-element set ⊂ {1, 2, . . ., N}. The elements of the set indicate the locations at which the signal y is sampled. Thus, the undersampled system will be yi = xi−k sk + ni , (4) k

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i ∈ ,

(5)

that is, z = R y = R Xs + R n.

(6)

The matrix X is circulant in nature. With the compressive undersampling operator, we have the undersampled signal, z ∈ RM given by z = R y = R Xs + R n.

(7)

The performance of the compressive radar ranging problem depends on the properties of the matrix R X = A ∈ RM×N . We consider arbitrary subsets of samples acquired for recovery. We assume that the continuous target scene is discretized into N grid points, with only a small percentage of the cells occupied by scattering targets. This assumption is based on prior experimental results for noise radar systems [2]. We define K as the sparsity of the vector s, that is K = = {i : si = 0}. The assumption of the target scene being highly sparse is represented by the fact that K N. We define the quantities ρ = K/M and δ = M/N. If the matrix A satisfies certain properties for vectors of given sparsity, then the problem above can be inverted by solving the following convex optimization problem, called basis pursuit de-noising (BPDN): BPDN(ρ, δ; σ ) : min s l1 subject to z − As l2 ≤ σ s∈RN

(8)

For a given (δ, ρ, σ ), we define the recovered vector as s∗ = arg min s l1 subject to z − As l2 ≤ σ s∈RN

(9)

The main requirement in order to ensure the full potential for reconstruction using BPDN is that the l2 norm of (z – As) accurately represents the standard deviation of the additive white noise. This specific formulation of the problem is chosen over other formulations [24] of compressive sensing because it is useful for practical implementations. In this paper, we use the spectral projected gradient algorithm for l1 constraints [25]. We adapted the Matlab implementation provided by van den Berg and Friedlander [26]. A. Characterizing the Performance of Compressive Noise Radar

The RIP and mutual coherence are the most common criteria used for analyzing the signal recovery performance of compressive sensing. These results offer probabilistic bounds on the recovery performance of compressive estimation algorithms. Typical results (for example, see [8]) derive the probabilistic upper bound on the mutual coherence or restricted isometry constants [9]. Practical applications of compressive sensing require more detailed analyses of the residual error. Analyzing the sensitivity of the estimation error to various nonidealities typical of practical compressive radar imaging is one of the goals of this paper.


The state evolution approach [27–29] offers an empirical and theoretical framework for analyzing the behavior of compressive sensing algorithms in an exact manner. These approaches are centered around the concept of phase transition, which is a concept borrowed from statistical physics. In this paper, following the precedent of Donoho and Tanner [30], we use phase transition diagram to refer to the boundary of the sparsity-undersampling trade-off of the compressive signal recovery of (8) over a discretized phase space generated by the ordered pair (δ, ρ). Compressive sensing is premised on regularizing least squares recovery to promote sparse solutions. This implies that the relaxed form of the recovery problem given by (8) is equivalent to brute force combinatorial search for sparse solutions to underdetermined linear equations that arise in undersampled systems. This equivalence holds for certain values of (δ, ρ) as defined clearly by a boundary on the (δ, ρ) phase space that occurs widely in many types of estimation problems [30]. The universality of phase transitions [30] provides a stable approach to characterizing the performance of compressive sensing systems. In practical scenarios, phase transition diagrams fully characterize the performance of noiseless systems and are fully determined by properties of the system matrix. This implies that each application of compressive sensing can be analyzed by looking at the phase transitions associated with the system matrix in the noiseless case. While the phase transitions formalism has been used to derive closed form error distribution for random-matrix linear systems [28, 29] in the large-system limit, the theory has not been extended to circulant-matrix linear systems. The phase transitions approach has been applied in a recent paper [31] to the problem of determining detection thresholds for constant false alarm rate radar, however the authors assume a random-matrix linear system. Our use of phase transitions is motivated by their potential application to guide radar engineers in operating real systems. We define the energy of the recovery error as p(ρ, δ; σ ) = s – s∗ 2 . We discretize the phase space of (δ, ρ) into a 32 × 32 grid with 0 < δ < 1 and 0 < ρ < 1. On each point in the grid, the vector s is generated from a Bernoulli distribution, such that the probability that si = + 1 is ρδ and the probability of si = 0 is 1 − ρδ. We solve 40 realizations of the BPDN signal recovery algorithm for each combination of ρ and δ over the grid. Over the 40 realizations, we count the fraction of successful recoveries in the 40 realizations. The criterion for successful recovery [32] is s – s∗ 2 / s 2 < 0.01. Fig. 1 shows the phase transition diagram for a simulated pseudorandom noise waveform, generated by acquiring different fractions of samples of the reflected signal at uniform intervals. The bright region indicates the values of the pair (δ, ρ) for which recovery is possible. The dark region indicates the values of (δ, ρ) for which recovery is unreliable due to the large magnitude of error. The term phase transition refers to the abrupt transition from the bright to the dark regions in these diagrams.

Fig. 1. Phase transition calibration diagram for simulated IID pseudorandom sequence. Imaging scenario was assumed to be noiseless.

Fig. 2. Block diagram for generation and acquisition of noise waveform.

Fig. 3. Phase transition diagram for transmitted waveforms generated using the W-band noise radar system described in Figure 2. The bandwidth of the signal is 500 MHz.

Phase transition diagrams were also computed for experimentally generated random noise waveforms. These waveforms were generated using a commercially available broadband RF noise source. A block diagram of the experimental system for generating and acquiring the noise waveforms is shown in Fig. 2. The noise was generated at baseband with a bandwidth of 500 MHz. The phase transition diagram for the experimentally generated noise waveform is shown in Fig. 3. We observe that the phase transition diagrams for experimentally generated waveforms closely match the simulated independent and identically distributed (IID) idealized model. From a radar-systems perspective, phase transition diagrams provide a way of calibrating compressive radar systems. Compressive radar imaging is only nonblind in the sense that the operator of the system needs to have an approximate idea of the sparsity of the target scene. The


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Fig. 4. TPSF of sample (500 MHz b/w noise) transmit waveform.

uncertainty in the knowledge of the approximate sparsity of the target scene can be quantified using the phase transition diagrams. For instance, if a radar system is intended to operate in an environment with 20% of the range cells populated by point scatterers, then the radar system can achieve reasonably good recovery by acquiring just 10% to 30% of the total samples. This information can be inferred from phase transition diagrams such as Fig. 3. We look at two other attributes of the waveform to infer the performance of the compressive imaging system. First, we look at the effect of filtering. Later on in this section, we examine the distribution underlying random variables that constitute real waveforms. Similar to conventional radar systems, the point spread function (PSF) of the system matrix determines the performance of the imaging system. For a system matrix X, the transform PSF (TPSF) is defined via the Gram matrix G = XT X. For optimal compressive sensing performance, the diagonal elements of the Gram matrix are much larger than the off-diagonal elements. Further, random off-diagonal elements imply better compressive sensing performance [33]. In practical compressive sensing systems, transmit waveforms do not fully conform to idealized models. The assumption that transmit waveforms can be discretized into IID random processes is convenient for theoretical analysis of compressive recovery. In real systems however, we need to characterize the effect of correlations that exist due to hardware-related nonidealities. We first look at the effect of correlations introduced into the transmit waveform by linear filters. We model the transmit waveform as the convolution of a realization of the white ˜ and a transfer function h(t) that noise random process x(t) represents the bandlimiting nonidealities so that ˜ ∗ h(t) x(t) = x(t)

(10)

X = X˜ H.

(11)

In this paper, we model h(t) as low-pass finite impulse response (FIR) filters. In order to quantify the effect of correlations, we look at the TPSF of the system matrix. The TPSF of a sample transmit waveform from a real radar system is plotted in Fig. 4. The PSF of an imaging 318

Fig. 5. Effect of filtering on TPSF. On y-axis is χ (G). It is seen that performance of compressive sensing deteriorates if random transmit waveforms are highly correlated (narrower filter widths).

system is a mathematical function that describes how an imaging system responds to a point-source. The TPSF is a generalization of the PSF commonly used to describe compressive sensing systems. Lustig et al. [33] defined it as a measure of “how a single transform coefficient of the underlying object ends up influencing other transform coefficients of the measured undersampled object.” In this context, the transform and the measurement matrix pair are defined by the identity matrix and the random circulant matrix. The Gram matrix G encodes this measure since each element of G represents the correlations between successive rows of the system matrix. Ideally, for effective compressive signal recovery, we desire the nondiagonal elements of the normalized Gram matrix G to be much smaller than the diagonal elements. Hence, we look at the error metric given by χ(G) = |G i,j |2 . (12) i=j In Fig. 5, the values of χ(G) for filters of different normalized bandwidths are plotted. We look at the matrix G = X Hl , where l denotes the normalized bandwidth of the filters. We characterize the matrix Hl with the parameter l = # {k : |Ph (k)| = 0}, where Ph (k) refers to the discrete power spectrum. Larger values of l indicate that the waveform has a broader spectrum. As a random process, broader power spectrum implies a lower degree of correlation. The interesting result from this simulation is that compressive sensing is fairly robust to correlations in the transmit waveform, when viewed from the perspective of the TPSF. This result conforms with the experimental observations in radar imaging experiments presented in Section III-A. We see in Fig. 5 that χ takes on values that indicate that the performance degrades gradually as the power spectrum of the transmit signal narrows. Then, we also plot the phase transition diagrams corresponding to circulant matrices generated from ˜ correlated random waveforms in Fig. 6. We start with x(t) modeled as IID random sequence, and apply FIR filters with four different cut-off frequencies. This indicates that compressive radar imaging has a certain level of robustness to correlations in the transmit waveform. The approximately exponential decay in performance


Fig. 8. QQ-plot of normalized transmit waveform for millimeter wave radar in comparison with standard normal.

Fig. 6. Effect of filtering with low-pass filters is shown in these phase transition diagrams for circulant matrix with low-pass filtered waveforms with various cut-off frequencies. (a) Cut-off frequency at 0.8Fs . (b) Cut-off frequency at 0.7Fs . (c) Cut-off frequency at 0.6Fs . (d) Cut-off frequency at 0.4Fs .

Fig. 9. Block diagram of millimeter wave radar system.

duration of around 4 ms. The power spectrum as observed across a number of trials is seen to deviate significantly from the ideal white power spectrum. 2) Compressive signal recovery is tolerant to the correlations in the system matrix. The correlations in the hardware are expected to worsen the performance of compressive signal recovery. However, as we saw in Fig. 5, the degradation may not be significant as long as the autocorrelation is reasonably close to ideal. The deviation of the generated transmit waveform from IID Gaussian distribution is shown in Fig. 8. While theoretical results assume Gaussian and

Fig. 7. Power spectrum estimates from 55 experiments (using covariance estimator) of millimeter wave transmit waveform.

indicated by 5 is confirmed by the significant difference in the shrinking area of the regions corresponding to full recovery for h(t) with cut-off frequencies of 0.8 Fs and 0.7 Fs . Compressive sensing recovery with circulant matrices is possible when the waveform that generates the circulant system matrix is IID random variables. Further, theoretically, even with the IID assumption, it is unclear how the distribution of each element of the vector generating the circulant matrix affects the performance of compressive signal recovery. Our experiments indicate the following. 1) Typical transmit waveforms generated in real systems do not have ideal autocorrelation functions as seen from the power spectrum shown in Fig. 7. The spectrum estimates were generated from 55 independent realizations of the transmit waveform, each generated using the set-up shown in Fig. 2. The noise waveforms were acquired by fully sampling the analog signal and acquiring it for a

Bernoulli distributions [10, 8] for the elements of the circulant-random matrices, this deviation does not adversely affect the recovery performance of compressive imaging as seen in Section III-A. Our claims about the suitability of noise radar for compressively sampled imaging are affirmed by the accuracy of recovery seen in Figs. 12–16 in Section III-A. III. RADAR IMAGING EXPERIMENTS A. Experimental Set-up

We used a millimeter wave radar system operating at W-band to test the suitability of using compressive sensing for noise radar imaging [34]. The bandwidth of the signal used in this system is 500 MHz. The block diagram for the radar system is shown in Fig. 9. The transmit signal is generated using a commercially available broadband noise source. This is attenuated, filtered, and split using power splitter. The first output line from the power splitter is used to record the transmit signal and goes to an ADC. The second output signal goes to a mixer where it is mixed with a millimeter wave single tone signal. The output of the mixer is high-pass filtered to eliminate the lower sideband and used to excite the antenna. The RF signal is


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Fig. 10. Photograph of experimental set-up.

Fig. 11. Close-up photograph of trihedral corner reflector target.

radiated out using a conical horn antenna. The antenna has a half power beamwidth of about 1◦ . A second identical conical horn antenna is used to receive the signal. The received signal is mixed down to baseband and low-pass filtered for processing. The analog signal is acquired using an 8-bit ADC. The ADC was operated at a rate of 1 Gs/s. The experiments were conducted in an outdoor setting. A photograph of the experimental set-up is shown in Fig. 10. A typical target scenario is shown in Fig. 11. We tested the imaging capability of the system at distances ranging from 40 ft (12.2 m) to around 108 ft (33 m). Trihedral corner reflectors and cylindrical scatterers were used as targets.

sampled received waveform y by computing the cross-correlation of the transmit and received waveforms as s∗CR = X T y. 2) Least squares based recovery from fully sampled received signal: The fully sampled received signal is inverted by solving a well-posed least squares problem, s∗LS = (X T X)−1 X T y. 3) Compressive signal recovery with fully sampled received signal: The objective of performing BPDN recovery with the fully sampled received waveform is to isolate and highlight the sparsity-promoting effect of the l1 -minimization term. These results illustrate the possibility of using l1 -norm based methods to improve the imaging performance of conventional digital radar systems even when sampling at rates above the Nyquist rate. The recovered target scene is given by s∗CS = arg mins∈R s l1 subject to the constraint that y–X s l2 ≤ σ . 4) Compressive signal recovery with undersampling: The received signal is fully sampled in the ADC, and then uniformly undersampled in software. Uniform undersampling at a rate of δ = M/N involves using only every Mth (M < N) sample for target scene recovery. The signal we use for recovery is thus, z = R X s. For a given undersampling rate ρ and sparsity δ, the recovered target ∗ scene is given by sCS (ρ, δ) = arg mins∈R s l1 subject to the constraint that z – R X s l2 ≤ σ . In the experimental results discussed in this section, we used δ = M/N = 0.25.

B. Target Recovery

In this section, we compare the performance of compressive sensing recovery with traditional correlation processing (s∗CR ) and least squares (s∗LS ). We describe five target scenes used to validate the possibility of using compressive sensing for noise radar imaging. Each experiment involved measuring the received signal for a duration of 0.4 ms, sampled at the rate of 1 GS/s. We conducted five trials for each target scenario, with the trials separated by arbitrary time durations on the order of a few minutes. The trials were all conducted in the same geographical area. The experiments were conducted in an open parking lot. Since the beam of the antenna is about 1◦ wide, clutter did not significantly alter the recovered target image. For each of the target scenes, signal recovery was performed using four recovery approaches, as follows. 1) Cross-correlation based recovery with full sampling: The target image is recovered from the fully 320

The cross-correlation based imaging was performed using all the acquired samples. The total length of the signal corresponds to a time duration of about 0.4 ms. The least squares and compressive sensing recovery was performed using 4 μs of transmitted and received signals.


Fig. 12. Corner reflector at distance of 33 m (about 108 ft) (millimeter ∗ . (b) Least squares s ∗ . wave radar). (a) Cross-correlation image sCR LS ∗ (ρ,1). (d) BPDN with 25% (c) BPDN with 100% samples sCS ∗ (ρ,0.25). undersampling sCS

For signals acquired at a rate of 1 GS/s, 0.4 ms corresponds to a linear system with 4 × 105 variables, while a duration of 4 μs corresponds to a 4 × 103 -dimensional linear system. Shorter length signals are used for compressive sensing and least squares because of the computational expense of solving large systems. The performance gains of using a larger data record for cross-correlation imaging is seen in Figs. 12–16, where the target responses are, at times, larger than those achieved by other recovery schemes. In addition to the fact that compressive sensing works in real scenarios, we saw in our experiments that the performance is comparable to conventional approaches. Further, in certain situations, the sparsity-promoting l1 -norm approach does better than conventional approaches by boosting the response from targets and suppressing the response of cells that do not contain targets. In the target recovery results shown in Figs. 12 and 13, the recovered scene shows the presence of scatterers within 10 ft (3 m) of the antenna. This response represents clutter due to the leakage of radiation from the transmit to the received antenna. In Fig. 12, the target scene consists of a single trihedral corner reflector located at around 108 ft (33 m) along the line of sight of the antennas. The corner reflector geometry is chosen to mimic point scatterers. The trihedral geometry ensures that most of the incident energy is reflected back to the antennas, thus providing an idealized reference for evaluating recovery methods. We see that for cross-correlation, one needs to sample at the Nyquist rate or higher for accurate recovery of the target scene. The least squares recovery was performed using a circulant matrix model for the system matrix. The accuracy of the recovery achieved by inverting the circulant system matrix generated from the transmit waveform validates the use of such a matrix to model the linear system.

Fig. 13. Two cylindrical targets at distance of about 14 m (40 ft) from radar, and separated from each other by 0.6 m, which is twice physical resolution corresponding to 500 MHz bandwidth EM wave in free space. ∗ . (b) Least squares s ∗ . (c) BPDN with (a) Cross-correlation image sCR LS ∗ (ρ,1). (d) BPDN with 25% undersampling s ∗ 100% samples sCS CS (ρ,0.25).

Fig. 14. Two cylindrical targets at distance of about 14 m (40 ft) from radar, and separated from each other by 0.3 m, which is physical resolution corresponding to 500 MHz bandwidth EM wave in free space. ∗ . (b) Least squares s ∗ . (c) BPDN with (a) Cross-correlation image sCR LS ∗ (ρ,1). (d) BPDN with 25% undersampling s ∗ 100% samples sCS CS (ρ,0.25).

In order to study the range resolution of the radar systems, we used cylindrical scatterers with the axis of the cylinder aligned perpendicular to the physical ground. With cylindrical scatterers, the reflected wave propagates in the same axis as the transmitted wave and the physical location of the cylinder corresponds to that obtained from the radar image. Thus, the physical distance between two cylindrical scatterers along the axis of propagation also represents the distance in the radar image. The imaging


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Fig. 15. Two corner reflectors separated by distance of 0.9m (3 ft), located at distance of 33 m (108 ft) from radar. (a) Cross-correlation ∗ . (b) Least squares s ∗ . (c) BPDN with 100% samples s ∗ image sCR LS CS ∗ (ρ,0.25). (ρ,1). (d) BPDN with 25% undersampling sCS

resolution corresponding to 500-MHz bandwidth EM radiation in free space is about 1 ft (0.3 m). The cylindrical scatterers were first placed about 2 ft (0.6 m) apart to observe the results at twice the physical resolving limit of the imaging system. The results of the target recovery performed on such a target scene is shown in Fig. 13. Two cylindrical reflectors were placed around 1ft (0.3 m), which is equal to the physical resolution corresponding to the 500-MHz bandwidth. The target recovery results are shown in Fig. 14. This set-up is used to illustrate that compressively sampled radar imaging is limited by the physical resolution of the EM radiation. The results provide experimental evidence that compressive imaging achieves range resolutions comparable to conventional methods. We note the presence of scatterers at around 0.3 m from the antenna. These artifacts represent radiation picked up by the received antenna directly from the transmit antenna. These artifacts occur consistently across different recovery schemes, and are ignored because the objective of this paper to compare different recovery schemes. Then, we consider two target scenarios with corner reflectors. In the recovery results shown in Fig. 15 and Fig. 16, we observe that compressive sensing performs comparably with conventional imaging methods when there are multiple targets in the scene. In order to characterize the performance of the different recovery techniques, we define a figure of merit with the goal of requiring minimal a priori information about the target scene. We define ζ (s∗ ) = maxi (si∗ )/ maxi∈T c (si∗ ), where Tc represents the set with coefficients corresponding to a region in the target scene that is known a priori to not contain a target as the ratio of the maximal absolute value of the entire scene and the maximal value of a region of the target scene that is known to be target free. The larger this number, the lesser 322

Fig. 16. Two corner reflectors separated by distance of 1.5 m (5 ft), ∗ . located at distance of 33 m from radar. (a) Cross-correlation image sCR ∗ . (c) BPDN with 100% samples s ∗ (ρ,1). (b) Least squares sLS CS ∗ (ρ,0.25). (d) BPDN with 25% undersampling sCS TABLE I. Table with Values of ζ for the Different Target Scenarios Discussed in this Paper. Target scenario Fig. 12 Fig. 13 Fig. 14 Fig. 15 Fig. 16

Crosscorrelation

Least squares

16.39 13.49 6.74 12.42 15.56

9.04 6.91 7.76 7.99 14.07

Fully CS with 25% sampled CS of the samples 14.77 16.89 18.90 14.01 27.24

17.79 12.70 13.17 19.25 24.70

the ambiguity between target cells and nontarget cells. If threshold detection is used to distinguish target cells from nontarget cells, then low values of ζ indicate a low probability of error or a high probability of false detection. In Table I we present ζ for the different target scenarios shown in Figs. 12–16. We observe that least squares consistently performs worse than either cross-correlation based imaging or imaging based on l1 regularized optimization. We further observe that compressive recovery with 25% of the samples compares favorably with fully sampled recovery. We are currently exploring the behavior of ζ and similar computable figures of merit for use in practical compressive radar imaging systems. IV. CONCLUSIONS A. Summary of Results

We demonstrated the feasibility of compressive noise radar imaging through theoretical arguments, numerical simulations, and experiments. This paper is one of the first attempts to demonstrate experimentally that compressive noise radar imaging is a viable technology. The performance of compressive noise radar systems was analyzed using phase transitions, which are useful tools


for calibrating compressive radar systems. We provided evidence for the applicability of compressive sensing to UWB noise radar imaging. We outlined three types of system-related nonidealities, namely: 1) nonideal autocorrelation functions for transmit waveforms, 2) non-Gaussianity of transmit waveform random process, and 3) approximation of the continuum as a discrete imaging problem. In spite of these nonidealities, the performance of compressively sampled radar compares favorably with conventional radar imaging systems. Further, the ability of compressive sensing to resolve two closely spaced targets is also comparable to that of conventional matched-filter based radar imaging. Our work extends the state of the art in formalizing compressive noise radar imaging and applying it to experimental data acquired from a real system transmitting millimeter wave radar. The system uses transmit waveforms with a bandwidth of around 500 MHz. Our analyses of practical issues relating to compressive radar imaging are intended to push the field towards real world applications. The major contributions of this paper are as follows. 1) In Section II, we studied the performance of compressive noise radar imaging by observing the properties of experimentally generated noise waveforms. Conventional approaches to analyzing radar signal processing algorithms are not directly applicable to compressive sensing. The principal reason is that compressive signal recovery is nonlinear and involves solving a convex optimization problem. We observed that experimentally generated noise waveforms perform as well as idealized theoretical models. We used two tools to justify the suitability of noise radar for compressive sensing. First, we looked at the phase transition diagrams associated with real random noise transmit waveforms. The properties of waveforms generated in real systems conformed with simulations for idealized pseudorandom IID sequences. The second property we looked at is the TPSF, which is a computationally feasible surrogate to the mutual coherence property. We show that the TPSF closely resembles idealized models, thus suggesting that system matrices that arise in practical compressive noise radar imaging are suitable for compressive signal recovery. We concluded Section II by looking at how accurately the distribution of the transmit waveforms resemble Gaussian assumptions. 2) In Section III, we presented experimental results on imaging with noise radar waveforms. We described the experimental set-up and discussed the imaging results. We analyzed recovery results using least squares formulation of the target imaging problem. Noise radar technology has conventionally used cross-correlation based imaging. We showed that least squares framework also works in the context of noise radar imaging. This justifies the appropriateness of the circulant matrix model for noise radar imaging. We see that in real systems, compressive noise radar imaging performed using only a fraction of the

samples of the received waveform compares favorably with conventional noise-radar imaging. We note that these results are significant since we use experimental data for the analysis. B. Open Problems

1) Circulant Random Matrices in Compressive Sensing: An important problem in the area of compressive sensing as applied to radar imaging is the characterization of the performance of circulant random system matrices. Consistently, in our simulations, we observed that the performance of circulant random matrices is comparable to that of random matrices. There is strong evidence to suggest that the number of samples required for accurate compressive signal recovery with random circulant system matrices scales as O(S log N). However, existing analyses of the RIP and mutual coherence properties of circulant random matrices [7, 8, 10] derive asymptotically larger bounds for the number of measurements required. Phase transition diagrams indicate the equivalence of the behavior of circulant random and fully random matrices in the context of compressive sensing. This suggests that a theoretical analysis that takes the route of phase transitions may be used to prove optimal behavior. However, phase transitions have only been shown to theoretically work for compressive sensing problems with random system matrices. 2) Compressive Noise Radar Hardware: In our work, we collected fully sampled records of transmit and received waveforms and then performed the undersampling in software. However, given the incoherent nature of random noise waveforms, we believe that it should be convenient to design sampling hardware to implement sub-Nyquist sampling. In such a scheme, the random transmit waveform would be generated digitally and then converted to the analog domain for transmission. Records of the digitally generated transmit waveform can be used for performing the signal recovery. ACKNOWLEDGMENTS

The authors would like to thank J. Sjogren of AFOSR for his valuable comments. We would also like to thank S. Smith and K. A. Gallagher from Pennsylvania State University for their help in designing and conducting the experiments. REFERENCES [1]

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Mahesh C. Shastry (S’07—M’14) received his PhD and MS degrees in electrical engineering from The Pennsylvania State University, in 2013 and 2009 respectively. His research interests are in signal processing, inverse problems, radar systems, numerical optimization, time-frequency analysis, and compressive sensing. He is currently with the 3M Company in the Corporate Research Labs. He is a member of the IEEE and the IEEE Signal Processing Society. Ram M. Narayanan (S’83—M’88—SM’93—F’01) received his B.Tech. degree from IIT, Madras, in 1976, and his Ph.D. degree from the University of Massachusetts, Amherst, in 1988, both in electrical engineering. During 1976–1983, he worked as a R&D Engineer at Bharat Electronics Ltd., Ghaziabad, where he developed microwave communications equipment. In 1988, he joined the University of Nebraska-Lincoln where he last served as Blackman and Lederer Professor. Since 2003, he is a Professor of Electrical Engineering at The Pennsylvania State University. Dr. Narayanan has coauthored over 115 journal papers and over 250 conference publications. His current areas of interest are noise radar, radar networks, compressive sensing, and radar tomography. He currently serves as Member of the IEEE Committee on Ultrawideband Radar (UWBR) Standards Development and as Associate Editor for Radar for the IEEE Transactions on Aerospace and Electronic Systems. He is also a Fellow of SPIE and IETE. Muralidhar Rangaswamy (S’89—M’93—SM’98—F’06) received the B.E. degree in electronics Engineering from Bangalore University, Bangalore, India in 1985 and the M.S. and Ph.D. degrees in Electrical Engineering from Syracuse University, Syracuse, NY, in 1992. He is presently employed as the Technical Advisor for the RF Exploitation Technology Branch within the Sensors Directorate of the Air Force Research Laboratory (AFRL). His research interests include radar signal processing, spectrum estimation, modeling non-Gaussian interference phenomena, and statistical communication theory. He has co-authored more than 150 refereed journal and conference papers in the areas of his research interests. Additionally, he is a contributor to 5 books and is a co-inventor on 3 U.S. patents. Dr. Rangaswamy is the Technical Editor (Associate Editor-in-Chief) for Radar Systems in the IEEE Transactions on Aerospace and Electronic Systems (IEEE-TAES). He has organized and chaired several sessions at various conferences. He received the 2004 (Fred Nathanson memorial) outstanding young radar engineer award from the IEEE AES Society, the 2006 Distinguished Member award from the IEEE Boston Section, the 2007 IEEE Region 1 award and the 2005 Charles Ryan basic research award from the Sensors Directorate of AFRL, in addition to more than 40 AFRL scientific achievement awards. SHASTRY ET AL.: SPARSITY-BASED SIGNAL PROCESSING FOR NOISE RADAR IMAGING

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