pulse-doppler processing has found wide applications pulse … · 2015-08-21 · pulse-doppler...

Pulse-Doppler SignalProcessing With QuadratureCompressive Sampling

CHAO LIUFENG XI, Member, IEEESHENGYAO CHEN, Member, IEEENanjing University of Science and TechnologyNanjing, Jiangsu, China

YIMIN D. ZHANG, Senior Member, IEEEVillanova UniversityVillanova, PA, USA

ZHONG LIU, Member, IEEENanjing University of Science and TechnologyNanjing, Jiangsu, China

Quadrature compressive sampling (QuadCS) is a recentlyintroduced sub-Nyquist sampling scheme for effective acquisition ofinphase and quadrature (I/Q) components of sparse radio frequencysignals. In applications to pulse-Doppler radars, the QuadCS outputscan be arranged into a two-dimensional data format, in terms of slowtime and virtual fast time, similar to that by Nyquist sampling. Thispaper develops a compressive sampling pulse-Doppler (CoSaPD)processing scheme which performs Doppler estimation/detection andrange estimation from the sub-Nyquist data without recovering theNyquist samples. The Doppler estimation is realized through aspectrum analyzer as in classical processing, whereas the detection isperformed using the Doppler bin data. The range estimation isperformed using sparse recovery algorithms only for the detectedtargets to reduce the computational load. A low detection thresholdis used to improve the detection probability and the introduced falsetargets are then removed in the range estimation stage by exploitingthe inherent target detection capability of the recovery algorithms.Simulation results verify the effectiveness of the proposed CoSaPDscheme, which requires only one-eighth of the Nyquist rate to achievesimilar performance to the classical processing with Nyquist samples,provided that the input signal-to-noise ratio (SNR) is above −25 dB.

Manuscript received July 17, 2013; revised February 11, 2014, June 19,2014, October 11, 2014; released for publication October 24, 2014.

DOI. No. 10.1109/TAES.2014.130475.

Refereeing of this contribution was handled by R. Adve.

This work was supported in part by the National Science Foundation ofChina under Grants 61171166 and 61101193.

Authors’ addresses: C. Liu, F. Xi, S.Y. Chen, and Z. Liu, Department ofElectronic Engineering, Nanjing University of Science and Technology,Nanjing, Jiangsu 210094, People’s Republic of China. E-mail:([email protected]). Y. D. Zhang, Center for AdvancedCommunications, Villanova University, Villanova, PA 19085, USA.

0018-9251/15/$26.00 C© 2015 IEEE

I. INTRODUCTION

Pulse-Doppler processing has found wide applicationsin civilian and military air surveillance radars due to itscapability to detect moving targets in strong clutterenvironments by exploiting the distinct Doppler shiftsbetween the targets and the clutter [1, 2]. A commonprocessing scheme uses a quadrature sampling structure[3, 4], as shown in Fig. 1, where the radar echoes aresampled to obtain baseband inphase and quadrature(denoted by I and Q) components. After processing thebaseband signal through a matched filter and discreteFourier transform (DFT), a detection threshold is appliedto detect targets with a constant false alarm rate (CFAR).The detected target plots are then fed into the dataprocessor to perform tracking and other functions.

Assume that the radar echoes are downconverted intoan intermediate frequency (IF) of f0 with a bandwidth ofB. Then, the Nyquist sampling rate for theanalog-to-digital conversion (ADC) is given by [5]

fs = 4fL + 2B

4l + 1

where fL = f0 − B/2, and l is a positive integersatisfying l ≤ ⌊

fL

/2B

⌋, where �·� denotes the floor

function. In wideband and ultrawideband applications,therefore, high-rate ADC and, subsequently, intensiveprocessing of high dimensional sequences are required. Asa result, the current availability of the ADC technologieslimits the development of high-resolution ultrawidebandradar systems.

The recently introduced compressed sensing (CS)[6–8], or compressive sampling, techniques bring us newconcepts to achieve sub-Nyquist data acquisition. The CStheory exploits the signal sparsity and samples signalscloser to their information rate instead of their bandwidth.With a high probability, CS techniques recover sparsesignals from far fewer samples or measurements than theNyquist samples. In radar systems, the reduced numberof samples implies a lower sampling rate requirementand, hence, a reduced processing load. While the CStheory is developed primarily for discrete-domainsignals, several schemes, such as random sampling [9],random demodulation [10, 11], random-modulationpreintegrator (RMPI) [12], segmented compressedsampling [13], and Xampling [14], have been proposedto apply CS techniques to analog signals, known asanalog-to-information conversion (AIC). These schemes,which generally handle signals that are sparse in the timedomain, frequency domain, or time-frequency domain, arestudied for bandpass signals without exploiting thecharacteristics of radar signals by extracting the I and Qcomponents from the IF waveforms.

Recently, we proposed a quadrature compressivesampling (QuadCS) scheme [15, 16] that bridges the CStheory and the digital quadrature sampling. By assumingthe sparsity of the echo signals in the waveform-matcheddictionary [17], the QuadCS can directly extract the I and

1216 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 51, NO. 2 APRIL 2015

Fig. 1. Block schematic of classical processing.

Q components of the bandpass signals and, similar to therandom demodulation scheme, demodulate the signalsthrough the chipping sequences. The chipping rate,however, is determined by the bandwidth, rather than thehighest frequency, of the bandpass signals. Therefore, theQuadCS reduces the implementation complexity ascompared with the random demodulation scheme.

It is noted that the digital signals obtained through theAIC are sparse and thus differ from those obtained fromthe uniform Nyquist sampling. As a result, theconventional signal processing techniques cannot bedirectly applied to them for information extraction. Ingeneral, there are two fundamental approaches to performthe information processing. One is to first recover theNyquist sampling signals and then process the recoveredsignals using conventional methods. It is clear, therefore,that this type of processing does not fully take advantageof sub-Nyquist sampling, i.e., it has to deal with a largevolume of recovered data represented in the Nyquist rate.The other approach is to directly process the sparsesignals, which have a much smaller size, in the CSdomain. Signal processing in the CS domain, also referredto as compressive signal processing (CSP), is attractivedue to its capability of significant reduction to the overallprocessing complexity. Some fundamental works, forexample, signal detection, parameter estimation, andfiltering, have been reported in, e.g. [18, 19]. Incomparison with conventional techniques, CSP is still inits infancy and much work should be done before it isapplied in practice.

In this paper, we discuss the applications of theQuadCS scheme to radar applications and develop thecompressive sampling pulse-Doppler (CoSaPD)processing approach. We are mainly concerned withnonfluctuating moving point targets in the presence ofadditive white Gaussian noise (AWGN). The applicationof the proposed CoSaPD in cluttered environments is alsodiscussed in less detail. It is assumed that the radartransmits repetitive pulse trains, and the target echoes aresampled at a sub-Nyquist rate in the fast-time (range)domain using the QuadCS. In the Doppler dimension, thetarget echoes are sampled at the pulse repetition frequency.Then, in a coherent processing interval (CPI), the sampleddata can be formulated in a matrix similar to that with theclassical sampling, as described in Section IV [1, 2].Because of the reduced-rate intrapulse sampling, theyielding data size becomes much smaller. The completetarget information (amplitudes, Doppler frequencies, andranges) is contained in the compressive data matrix, based

on which target detection and estimation can beperformed. The CoSaPD scheme consists of theprocedures of Doppler estimation, target detection, andrange estimation. As discussed in Section V, the order ofthe procedures is irreversible, which is different from theclassical processing. Simulation results in Section VI showthat, when the input signal-to-noise ratio (SNR) is above−25 dB, the CoSaPD scheme at one-eighth of the Nyquistrate achieves a performance similar to that obtained byclassical processing methods using Nyquist sampled data.

Applications of AIC and CSP to radar systems havebeen exploited in, e.g. [12, 20–27]. In [12], anRMPI-based radar pulse receiver is reported that extractstarget information without full signal reconstruction.Target detection from sub-Nyquist samples is consideredin [20, 21]. Target tracking and time-delay/Dopplerestimation from random measurements are respectivelystudied in [22] and [23]. In [24], the reconstructionperformance of radar echoes is examined using realexperimental data. References [25–27] present differentapproaches for the estimation of radar target parametersfrom Xampling data. All these studies demonstrate theeffectiveness of the AIC for effective radar signalacquisition with a significantly reduced complexity. In thispaper, we further consolidate the applicability of the AICto radar systems. Different from previous work, theCoSaPD is a systematic pulse-Doppler processing schemebased on QuadCS. Owing to the data structure parallel tothe Nyquist sampling, the CoSaPD takes some ideas fromthe classical processing but with different connotations ofthe QuadCS data. As such, the target detection and rangeestimation adopt different techniques, as discussed inSection IV. Other contributions of this paper includeanalyses of detection performance, joint detection andrange estimation with a low detection threshold, andextensive performance evaluations in different scenarios.

The remainder of this paper is organized as follows. InSection II we describe the radar model and theassumptions used in our discussion. Section III brieflysummarizes the fundamentals of the QuadCS scheme.Section IV describes the proposed CoSaPD processingscheme, and the target detection approach is discussed inSection V. Simulation results are presented in Section VI.We conclude this paper in Section VII.

We denote vectors by boldface lower case letters andmatrices by boldface upper case letters. (•)H denotes theconjugate transpose operation. (•)l denotes the l-th columnof matrix “•”, and (•)i,j denotes the element of “•” in thei-th row and j -th column. Re {•} and Im {•} represent thereal and the imaginary parts of “•”, respectively.

II. RADAR MODEL AND PROBLEM STATEMENT

In pulse-Doppler radar signal processing, we usuallytransmit repetitive periodic pulses and perform coherentsampling at the range bins so as to estimate desirabletarget information. Consider the case of K nonfluctuatingmoving point targets which are sparsely located in the

LIU ET AL.: PULSE-DOPPLER SIGNAL PROCESSING WITH QUADRATURE COMPRESSIVE SAMPLING 1217

radar’s field of view and satisfy the stop-and-hopassumption [1]. Assume that the radar transmits amodulated pulse train with L pulses, where the pulserepetition interval (PRI) is T and the pulsewidth is Tb.After downconverting the received signal to an IF of f0,the target echo from the k-th target corresponding to thel-th transmit pulse can be described as

rlk(t) = ρka(t − tk) cos[2πf0t + φ(t − tk)

+2πf dk (l − 1)T + ϕk], t ∈ [(l − 1)T , lT ], (1)

where a(t) and φ(t) respectively represent the amplitudeand the phase of the transmitted signal, which has abandpass spectrum with center frequency f0 andbandwidth B. In addition, ρk , tk , f d

k , and ϕk are thereflecting coefficient, time delay, Doppler frequency, andrandom phase shift of the k-th target, respectively. In thepresence of K targets, the received radar echocorresponding to the l-th transmit pulse is thesuperposition of their respective echoes and can beexpressed as

rl(t) =K∑

k=1

rlk(t) = I l(t) cos(2πf0t) − Ql(t) sin(2πf0t)

(2)where I l(t) and Ql(t) are respectively the I and Qcomponents of the signal rl(t), i.e.,

I l(t) =K∑

k=1

ρka(t − tk) cos[φ(t − tk) + ϕ′k]

(3)

Ql(t) =K∑

k=1

ρka(t − tk) sin[φ(t − tk) + ϕ′k]

and ϕ′k = 2πf d

k (l − 1)T + ϕk . Denote s0(t) =a(t)ejφ(t) asthe complex baseband signal of the transmitted radarsignal. Then the complex envelope s l(t) of rl(t) is given by

s l(t) = I l(t) + jQl(t) =K∑

k=1

ρlk s0(t−tk) (4)

where ρlk=ρk exp[j(2πf d

k (l − 1)T + ϕk)].The target information, characterized by tk , f d

k , andρk , is completely contained in the complex basebandenvelope s l(t) , l = 1, 2, · · · , L. In the radar signalprocessing, we usually sample rl(t) in (2) by thequadrature sampling scheme and then perform the analysisto obtain the target information, as shown in Fig. 1. On theother hand, this paper studies the estimation of targetinformation from the sub-Nyquist QuadCS data.

To simplify the analysis without loss of generality, weassume that the radar is operated in an unambiguoustime-frequency region, i.e., |fd | < 1/2T and tk < T , andthat the target remains in a range bin and keeps a constantvelocity in a CPI.

In practical scenarios, the received radar signalinevitably contains noise and clutter in addition to the

target echoes.1 Among various noise sources, thermalnoise is nominally dominant. Clutter is often present dueto echoes from volume or surface scatterers [28]. In ourstudy, we assume that the noise is AWGN and the surfaceclutter is Rayleigh distributed in amplitude and obeys thetwo-sided exponential law in Doppler spreading. Then, thereceived radar signal corresponding to the l-th transmittingpulse is given by

rl(t) =K∑

k=1

rlk(t) + n(t) + c(t), t ∈ [(l − 1)T , lT ], (5)

where n(t) is the bandlimited noise with power spectrumdensity N0/2 and bandwidth B, and c(t) isRayleigh-distributed clutter with an average clutter powerof ρ2

c . Define the received SNR for the k-th target as

SNRINk =

1Tb

∫ lT

(l−1)T

∣∣rlk(t)

∣∣2dt

N0B. (6)

We can obtain that SNRINk = |ρk |2

N0Bunder the

assumption of unit transmit power. Similarly, the receivedsignal-to-clutter ratio (SCR) can be defined for the k-thtarget as SCRIN

k = |ρk |2ρ2

c.

In the rest of the paper, we consider the case of targetechoes contaminated only by the thermal noise n(t). Theeffects of clutter are analyzed in Section IV and simulationresults are provided in Section VI.

III. FUNDAMENTALS OF QUADRATURECOMPRESSED SENSING

Now we introduce the QuadCS scheme that performssub-Nyquist sampling of the received radar signal asexpressed in (5). Different from the system in [15, 16],this work takes the Doppler into account in the echomodel. To simplify the notation, we consider the receivedsignal, denoted as r(t), in a single pulse interval.

We first consider the noise-free case. In radarapplications, because the transmit waveforms are knownin advance, a natural choice of the dictionary is to useentries that are matched with the transmit waveforms [17].For baseband radar CS processing with transmit waveforms0(t) of bandwidth B, the waveform-matched dictionaryconsists of time-delayed versions of s0 (t) at all integralmultiples of τ0 = 1

/B, i.e., {ψn(t)|ψn(t) = s0(t − nτ0),

n = 0, 1, · · · , N − 1}, where N = ⌈T

/τ0

⌉is the size of

the dictionary, with �· denoting the ceiling function. Assuch, the dictionary discretizes the observation time periodT with resolution τ0 = 1/B. This discretization of thetime delay is justified by the fact that the time resolutionof the bandlimited signal s0 (t) is 1

/B.

Assume that the target delays are located at the integralmultiples of τ0 = 1

/B, i.e., tk ∈ {0, τ0, · · · , (N − 1) τ0}.

Given the waveform-matched dictionary, the complex

1 Unintentional electromagnetic interference and intentional jamming areout of the scope of our discussion.


Fig. 2. Structure of QuadCS system.

envelope s (t) in (4) can be represented as

s (t) =N−1∑n=0

ρnψn (t), (7)

where ρk is the target reflection coefficient that takes anonzero value at delay tk . For K N , s (t) is said to beK-sparse in the waveform-matched dictionary. Thesparsity level K exactly equals to the number of targets.

The QuadCS system is shown in Fig. 2, which consistsof two subsystems: a sub-Nyquist sampling subsystem anda quadrature demodulation subsystem. In the firstsubsystem, the received radar signal r(t) is modulated by arandom chipping sequence p (t) of ±1s, which alternatesbetween values at or above the Nyquist rate of thebaseband signal. The mixer operation spreads thebaseband signal content to occupy the full spectrum ofp (t). Then, the mixed output is filtered by a bandpassfilter hbp (t) with a center frequency f0 and bandwidthBcs B. The filter output yields a compressive bandpasssignal, expressed as

y (t) =∫ ∞

−∞hbp (τ ) p (t − τ ) r (t − τ ) dτ

= Re{scs (t) ej2πf0t

}(8)

where

scs (t) =∫ +∞

−∞hbp (τ ) e−j2πf0τp (t − τ ) s (t − τ ) dτ (9)

is the compressive complex envelope, with Ics (t)= Re {scs (t)} and Qcs (t) = Im {scs (t)} denoting thecompressive I and Q components, respectively. The filteroutput y(t) is then sampled by a sub-Nyquist ADC togenerate a low-rate sequence y [k]. The sampling rate isset according to the bandpass sampling theorem asf cs

IF = (4fL + 2Bcs)/

(4l + 1), where fL = f0 − Bcs

/2,

and l is a positive integer satisfying l ≤ ⌊fL

/2Bcs

⌋.

The second subsystem is to extract digital compressiveI and Q sequences from the sub-Nyquist samplingsequence y [k]. Its operation is the same as in classicalquadrature sampling [3]. Because of the down-samplingoperation, the rate of the digital compressive I and Qsequences Ics [m] = Ics (mTcs) and Qcs [m] = Qcs (mTcs)is half that of y [k], i.e., Tcs = 2

/f cs

IF . In the observationinterval T , we obtain M = ⌊

T/Tcs

⌋complex samples

scs [m] = Ics [m] + jQcs [m], or 2M compressive samplesof I and Q components, from scs (t), which are much lessthan 2BT as required by the digital quadraturedemodulation.

Although the QuadCS system works on analogbandpass signals, its output scs [m] can be characterized asa linear combination of the elements of sparse coefficientvector ρ = [ρ0, ρ1, · · · , ρN−1]T . Substituting (7) into (9),we have

scs (t)

=N−1∑n=0

ρn

∫ +∞

−∞hbp (τ ) e−j2πf0τp (t − τ ) ψn (t − τ ) dτ

(10)

and the corresponding samples are expressed as

scs [m] =N−1∑n=0

ρn

∫ +∞

−∞hbp (τ ) e−j2πf0τp (mTcs − τ )

× ψn (mTcs − τ ) dτ. (11)

In the discrete CS framework, we have

scs = Mρ (12)

where scs = [scs [0] , · · · , scs [M − 1]]T andM = [

Mmn

] ∈ CM×N with

Mmn =∫ +∞

−∞hbp (τ ) e−j2πf0τp (mTcs − τ )

× ψn (mTcs − τ ) dτ. (13)

The recovery of the sparse coefficient vector ρ can beachieved through the following constrained l1-normoptimization [29] {

min ‖ρ‖1

s.t. scs = Mρ(14)

In the above expression, matrix M is referred to as thesystem measurement matrix. For radar signals with a flatspectrum, all the columns of matrix M are approximatelymutually orthogonal and have nearly the same columnenergy 2TbB

2cs/B under the assumption of unit transmit

power. Therefore, the k-th target power after the output ofthe QuadCS system becomes 2 |ρk|2 Bcs/B.

When the received signals are contaminated by noise,the QuadCS samples in (11) are corrupted by compressivenoise samples ncs [m], which are obtained by passing thereceived noise n(t) through the QuadCS system asdescribed above. For the additive white and bandlimitedGaussian noise n(t) with power spectrum density N0

/2

and bandwidth B, the compressive noise samples ncs [m]are an independent and identically distributed (IID)complex Gaussian process with zero-mean and variance2N0Bcs [30]. Then, the output SNR of the QuadCS systemfor the k-th target, denoted as SNRCS

k , stays intact.Because the QuadCS is a linear system, (12) becomes

scs = Mρ + ncs (15)


Fig. 3. Notional two-dimensional data matrix generated by (a) QuadCSsystem and (b) classical sampling.

in the noisy case, where ncs = [ncs[0], · · · , ncs[M − 1]]T .The reconstruction of the sparse coefficient vector ρ in thiscase can be obtained as the solution of the followingoptimization problem [31],

minρ

1

2

∥∥scs − Mρ∥∥2

2 + l ‖ρ‖1 , (16)

where l > 0 is the regularization parameter which is usedto tradeoff between the sparsity and the least-squareserrors.

There are a wide variety of approaches to solve (14)and (16), including the greedy iteration algorithms [32, 33]and convex optimization algorithms [31, 34] (see [35] fora review). In the simulation study, we use basis pursuitdenoising (BPDN) [31] to estimate the sparse vector ρ.

IV. PULSE-DOPPLER PROCESSING IN QUADCSDOMAIN

In this section, we discuss the extraction of the targetrange and Doppler frequency information from the dataexpressed in (15).

Consider a CPI consisting of L periodic pulses anddenote the output of the QuadCS system from the l-thecho as

slcs = Mρ

l + nlcs . (17)

Define Scs = [s1cs, s2

cs, · · · , sLcs], � = [

ρ1, ρ2, · · · , ρL],

and Ncs = [n1

cs, n2cs, · · · , nL

cs

]. Then, the sampled data of

the L consecutive echoes can be expressed in a matrixform as

Scs = M� + Ncs . (18)

Fig. 3(a) illustrates the two-dimensional data matrixgenerated by the QuadCS system, whereas Fig. 3(b)depicts that by the classical sampling [1, 2] forcomparison. It is clear that the sub-Nyquist samplesobtained by the QuadCS system correspond to the fasttime samples (range bins) in the classical sampling. Assuch, the sub-Nyquist samples are referred to as the virtualrange bins for convenience. The samples in each columnare obtained by successively sampling the echoes from asingle pulse, yielding consecutive virtual range bins. Eachcomplex element of a column represents both real andimaginary (Ics and Qcs) components for one virtual range

bin. Consequently, each row represents a series ofmeasurements from the same virtual range bin overconsecutive pulses. Because of the reduced sub-Nyquistrate in the range dimension, the data size in the rangedimension becomes much smaller than that obtained bythe classical fast-time Nyquist sampling.

As can be seen from (18), the target information iscompletely characterized by the N × L data matrix �. Infact, matrix � degenerates to the classical data matrixwhen M = IN . Then, � can be obtained by estimating thetarget information similar to traditional approaches.However, because the available data is an M × L

underdetermined data matrix Scs with M N , it isimpossible to directly obtain the target information fromScs . Ideally, each column of � is sparse because thenumber of targets is much smaller than that of the rangebins or the dictionary size. In this case, we can obtain asparse estimate of � in (18) by solving the followingl1-norm optimization problem,

minρl

1

2

∥∥∥slcs − Mρ

l∥∥∥2

2+ λ

∥∥ρl∥∥

1 l = 1, 2, · · · , L. (19)

The target information can be estimated from theestimated � by applying DFT to its rows. In practice, dueto the influence of noise and clutters, we can hardly obtainthe exact information of the targets and may yield falsetargets. In addition, the direct solution of (19) requires ahigh computational load and, thereby, may not be feasiblefor real-time processing.

It is seen that each row of the data matrix Scs

represents a series of measurements over successive pulsesfrom the same virtual range bin. Therefore, the targetDoppler frequencies can be estimated by the spectralanalysis of the slow-time data for each virtual range bin. Asimple technique is to conduct the DFT. Denote F(•) asthe DFT of “•” in row vectors. We have

F(Scs) = F(M�) + F(Ncs)

= MF(�) + F(Ncs) (20)

Each element of the matrix F(Scs) is a Dopplerspectrum sample corresponding to the virtual range binand the frequency bin. As such, the Doppler spectrumsamples can be used for target detection, i.e., determinewhether a target is present at the virtual range bin and theDoppler bin.

The DFT acts as a matched filter for slow-timesamples in the assumed scenarios. After DFT processing,the power of the k-th target becomes 2L2 |ρk|2 Bcs/B andthe noise variance is 2LN0Bcs . As such, the yielding SNRfor the k-th target SNRDFT

k is improved from the receivedSNRIN

k by a factor of L. From the point of view of targetdetection, we can further improve the detectionperformance by matched filtering the sub-Nyquist samplescorresponding to each Doppler bin. The details arediscussed in the next section.

The detection process, however, only detects theexistence of targets in a specific Doppler bin and does not


Fig. 4. Block schematic of CoSaPD processing.

Fig. 5. Mathematical procedure of CoSaPD processing.

reveal the number of targets and their correspondingranges. Note that the sparsity of F(�) can be greatlyenhanced even for practically nonsparse � afterperforming the DFT. Therefore, for the underdetermineddata Scs , we can estimate the number and the respectiveranges of the targets by obtaining the sparse solution foreach column of (20). Such estimation, however, requires ahigh computational load because the sparse estimation isperformed for each column. Since we have alreadydetected the targets from the Doppler spectrum samples,we only need to estimate the target ranges for the specificDoppler bins with detected targets. In this way, thecomputational load can be significantly reduced.

The block schematic of the processing steps involvedin the CoSaPD processing is given in Fig. 4, and themathematical procedure corresponding to the processingblocks is given in Fig. 5. It should be noted that theCoSaPD scheme must estimate the target velocities firstand the ranges afterward, and this order cannot bereversed. The details of the detection process are depictedin the next section.

We now briefly discuss the Doppler estimation in thepresence of clutter. Different from thermal noise, theclutter has a colored power spectrum as determined by theradar and the operation environment [1, 2, 28]. For astationary transceiver, the clutter spectrum is around thezero Doppler frequency. In this case, as in classicalpulse-Doppler processing, the CoSaPD scheme can isolatethe clutter from the moving target. If the target is separablefrom the clutter spectrum, its detection is affected only bythe thermal noise, but not the clutter. If the target is in theclutter-dominated area, on the other hand, the target isusually obscured by the clutter and cannot be detected. Inthis case, the Doppler spectrum samples in thecontaminated area are simply discarded. However,because of clutter sidelobes, clutter power may spreadover the entire Doppler frequencies, even though its powermay be small at high Doppler frequencies. To reduce thesidelobe effects, we can add a data window [36] to weightthe slow-time data for each virtual range bin prior tocomputing the DFT. With the windowed data, the clutterhas a negligible effect on the estimation of targets asdemonstrated by the simulated results in Section VI.

V. THRESHOLD DETECTION AND ITSPERFORMANCE

This section describes the threshold detection used inthe proposed CoSaPD processing and analyzes itsperformance.

Consider the l-th column which represents the Dopplerbin data derived from (20) as

(F(Scs)

)l = M(F(�)

)l + (F(Ncs)

)l. (21)

Our objective is to detect if there exists one or moretargets in the l-th Doppler bin, i.e., to determine if the

vector(F(�)

)lis a nonzero vector based on

(F(Scs)

)l.

First, we assume that the received noise variance isknown. For the data expressed in (21), we perform thefollowing matched filtering to further enhance thedetection performance,

MH(F(Scs)

)l = MHM(F(�)

)l + MH(F(Ncs)

)l, (22)

which corresponds to the “matched filtering” operation in

Fig. 4. For convenience, we define x = MH(F(Scs)

)l,

y = MHM(F(�)

)l, and w = MH

(F(Ncs)

)l. Then, (22)

can be simplified as

x = y + w. (23)

Note that the noise term w in (23) is Gaussian but notindependent, as a result of the matched filtering. Asillustrated in Section III, matrix M is approximatelycolumn-by-column orthogonal. Therefore, we can stillassume that w is an IID Gaussian process. For the matchedfilter output expressed in (22), the peak power of the k-thtarget is 4L2 |ρk|2 T 2

b B4cs/B

2 and the noise variance is4LN0TbB

3cs/B. As such, the SNR of the k-th target after


Fig. 6. Block schematic of detection process for Doppler bin.

the matched filtering, SNRMFk , becomes TbBcsL times the

received SNRINk .

The detection problem is to detect the targets fromdata x. The binary detection problem of each element xn,1 ≤ n ≤ N , can be formulated as

H0 : |xn| = |wn|H1 : |xn| = |yn + wn|

(24)

The detection probability and false alarm probabilityare respectively given by

P nD =

∫ ∞

TD

f|xn||H1 (|xn| |H1)d |xn| , (25)

P nF =

∫ ∞

TD

f|xn||H0 (|xn| |H0)d |xn| , (26)

where TD is the detection threshold, and f|xn||H1 (|xn| |H1)and f|xn||H0 (|xn| |H0) are the probability density functions(pdfs) of |xn| respectively for the cases when a target ispresent and absent. Then, the false alarm probability andthe detection probability of y are respectively given by

PF ≈ 1 −N∏

n=1

(1 − P nF ), (27)

PD ≈ 1 −N∏

n=1

(1 − P nD). (28)

Denote σ 2 = 2LN0TbB3cs/B and notice that |wn|

follows a Rayleigh distribution with mean√

π/

2σ and

variance (4 − π)σ 2/

2. Under hypothesis H0, i.e., a targetis absent, the conditional pdf of |xn| is given by

f|xn||H0 (|xn| |H0) = |xn|σ 2

exp

(−|xn|2

2σ 2

). (29)

Under hypothesis H1, i.e., when a target is present,xn = yn + wn is complex Gaussian distributed with meanyn and variance 2σ 2. Then, the pdf of |xn| follows theRician distribution, expressed as

f|xn||H1 (|xn| |H1)

= |xn|σ 2

exp

(−|xn|2 + |yn|2

2σ 2

)I0

( |xn| |yn|σ 2

), (30)

where I0(•) is the modified Bessel function of the firstkind [37].

With the known noise power σ 2, the Neyman-Pearsonoptimal detector can be derived from the likelihood ratiotest as

f|xn||H1 (|xn| |H1)

f|xn||H0 (|xn| |H0)= exp

(−|yn|2

2σ 2

)I0

( |xn| |yn|σ 2

)H1><

H0

l.

(31)Because I0(•) is a monotonically increasing function,

(31) is equivalent to the following simpler expression,

|xn|H1><

H0

l0σ = TD, (32)

where l0 = √−2 ln(P nF ) is the scale factor used to control

the false alarm rate. Thus, we can derive the joint detectorfor a vector data x as

N∑n=1

u (|xn| − l0σ )H1><

H0

1, (33)

where u (•) represents the unit step function. The detectionprocess for a Doppler bin is shown in Fig. 6.

From (26), (29), and (32), we can deriveP n

F = exp(−l20/2). Then, the false alarm probability is

obtained as

PF ≈ 1 − (1 − exp(−l2

0/2))N

. (34)

We are unable to derive a closed-form expression ofthe detection probability PD . However, it is noted that theprocessing gain of the CoSaPD detector is TbBcsL, whichis smaller than the gain of classical processing, T BbL.Therefore, it is expected that the performance of theproposed detector will be inferior when compared with theclassical detector at the low SNR region.

After target detection, the CoSaPD processingconducts range estimation through the sparse recoveryalgorithms, as discussed in the previous section. As is wellknown, an inherent characteristic of the sparse recoveryalgorithms [38, 39] is to detect the nonzero elements in asparse vector. To improve the system detectionperformance, we may set a low detection threshold foreach Doppler bin. However, a low threshold will increasethe false alarm probability and thus may introduce falsetargets. Nevertheless, false targets can be removed through


the detection process in the recovery algorithms. Becausethe sparse recovery algorithms strive for a minimumnumber of nonzero cells at its output, from the systempoint of view, the false alarm probability of the radarsystem does not increase despite the low detectionthreshold. This observation is verified later throughsimulations.

In practice, it is impossible to know the noiseparameter σ in advance. To maintain a CFAR, therefore,we should estimate parameter σ so as to determine anadaptive threshold for detection. Following theassumptions on the measurement matrix M, it is seen thatthe noise matrix MHF(Ncs) is IID, and the absolute valueof each element |(MHF(Ncs))i,j |, 1 ≤ i ≤ N , 1 ≤ j ≤ L,follows the Rayleigh distribution. As such, the maximumlikelihood estimate of σ is simply the average of theavailable data [40], expressed as

σ =√

2

π

∑(i,j )∈

∣∣∣(MHF(Ncs))i,j

∣∣∣|| (35)

where is the set consisting of all available i and j , and|| is its cardinality. For sparse targets, the accumulativestrength of the signals

∑(i,j )∈

|(MHMF(�))i,j | is much

smaller than that of the noise∑

(i,j )∈

|(MHF(Ncs))i,j |, and

the following approximation holds when || is large:∑

(i,j )∈

∣∣∣(MHF(Ncs))i,j

∣∣∣|| ≈

∑(i,j )∈

∣∣∣(MHF(Scs))i,j

∣∣∣|| . (36)

In the simulation study, we set || = NL and theestimated σ becomes

σ ≈√

2

π

N∑i=1

L∑j=1

∣∣∣(MHF(Scs))i,j

∣∣∣NL

, (37)

and the detection threshold is given as TD = l0σ .

VI. SIMULATIONS

In this section, we present the simulation performanceof the proposed CoSaPD processing and compare it withclassical processing [1, 2] and direct processing by (19).Subsection VI-A introduces the simulation scenarios.Subsections VI-B and VI-C respectively providesimulated results of the detection and estimationperformance in an AWGN environment. The effects ofclutter are examined in Subsection VI-D.

A. Simulation Scenarios

It is assumed that the radar transmits a linearfrequency modulation pulse train with carrier frequencyfc = 10 GHz, signal bandwidth B = 200 MHz,pulsewidth Tb = 10−5 s, and PRI T = 10−4 s. The CPIconsists of L = 100 pulses. For the assumed parameters,the unambiguous target ranges and Doppler frequenciesare 1500 m ∼ 3466.5 m and −5 KHz ∼ 5 KHz,

Fig. 7. Distribution of averaged Gram matrix MHM.

respectively. The range resolution is 0.75 m and theDoppler resolution is 0.1 KHz.

For the QuadCS system, the chipping sequence p (t) isgenerated by random ±1s with rate 1/B and the bandpassfilter is set to be an ideal one with bandwidth Bcs . Twobandpass filters with Bcs = 50 MHz and Bcs = 25 MHzare respectively considered. For the two filters, thesampling rates are one-fourth and one-eighth of theNyquist rate, respectively. The basis BPDN algorithm [31]is used for the sparse target recovery.

A flat power spectrum is assumed in the simulatedradar signal. The QuadCS measurement matrix M isapproximately column-by-column orthogonal. Fig. 7shows the distribution of the averaged Gram matrix MHMover 1000 independent trials for Bcs = 25 MHz. Themaximum off-diagonal element of the Gram matrix is0.015, thus clearly verifying the reasonableness of theassumption.

B. Detection Performance

Assume three targets with the same SNR. The targetdelays and the Doppler frequencies are randomly set in theunambiguous region. We present three simulationexamples. For the first two examples, the delays and theDoppler frequencies are on the resolution grids. For thethird example, the delays and the Doppler frequencies arearbitrarily set. To examine the detectability of the multipletargets, their Doppler frequencies are set to fall in thesame Doppler bin. All results are obtained by averagingover 1000 independent trials.

First, we show that the CoSaPD detector achieves aCFAR in a Doppler bin. Fig. 8 shows the variations of thefalse alarm probability versus the scale factor for differentvalues of SNR when Bcs = 25 MHz. It is clear that thechange of the noise power does not affect the false alarmprobability for a specified scale factor, which is consistentwith the theoretical result given in (34). The sameconclusion can be drawn for Bcs = 50 MHz.

Next, we examine the receiver operating characteristic(ROC) of the CoSaPD detector. Fig. 9 compares theaveraged ROC curves obtained by the CoSaPD and the


Fig. 8. False alarm probability versus scale factor.

Fig. 9. ROC of CoSaPD detector.

classical processing. It is seen that the performance of theCoSaPD detector is inferior to the classical detector due tothe decrease of the SNR gains. In the Nyquist-rate case,after the matched filtering and DFT processing, theprocessing gain can achieve TbBL, while the QuadCSsystem only realize a gain of TbBcsL. As the bandpasswidth Bcs increases, the processing gain TbBcsL increasesand then the detection performance is enhanced. In thesimulated example, processing gains of 53 dB, 47 dB, and44 dB are respectively achieved for classical processing,CoSaPD detector with Bcs = 50 MHz, and that withBcs = 25 MHz. The yielding SNRs for detection are23 dB, 17 dB, and 14 dB, respectively, implying an SNRreduction of 6 dB and 9 dB for the CoSaPD withBcs = 50 MHz and Bcs = 25 MHz cases as comparedwith the classical processing. For Bcs = 50 MHz, i.e.,when the compressive sampling rate equals one-fourth ofthe Nyquist rate, the detection performance of theCoSaPD detector is close to that of the classical detectorin the simulated range of PF .

As discussed in the previous section and noted inFig. 9, we can choose a high false alarm probability toincrease the detection probability. Reduction of falsetargets can be achieved in the recovery stage of the targetrange following the detector, because the sparse recoveryalgorithm has the inherent target detection ability [38, 39].

Fig. 10. System false alarm probability versus detector false alarmprobability.

Fig. 11. Detection performance versus SNR.

Fig. 10 shows the false alarm rate of the system after therecovery stage versus that of the detector. It is clear thatalthough the detector has a high false alarm probability inthe detection stage, the recovery algorithm can keep thesystem having a low false alarm probability. Processingthe system detection in this way will slightly increase thecomputational burden in the range estimation.

Fig. 11 further shows the detection performance withrespect to the SNR for the case of PF = 10−2. It is seenthat even at a low SNR of −30 dB, the CoSaPD detectorcan approach the performance of the classical detectorwith the data rate being only one-eighth of the Nyquistrate.

Finally, we consider a realistic scenario in which theranges and Doppler frequencies of the targets arecontinuous and thus may not fall on the resolution grids.Fig. 12 shows the ROC in this case. In comparison withFig. 9, the detection performance degrades. It is noted thatwhen a target is not on Doppler bin grids, the detectedtarget energy is from the Doppler leakage and thus issmaller than that of the target on the Doppler bin. Whenthe target is off from the range bin grids, the measurementmatrix contains errors, which will degenerate the matchedfilter in (22). In this case, the detection performance iscompromised as compared with the case that the targetsare on the resolution grids.


Fig. 12. ROC in realistic case.

Fig. 13. Rates of successful estimation for different methods.

C. Estimation Performance

After detecting the existence of the targets in a specificDoppler bin, we estimate the corresponding target ranges.The CoSaPD method is depicted in the last stage of Fig. 5and is realized through the sparse recovery algorithms. Therate of successful estimation is used as the performancemetric. When the targets are on the resolution grids, asuccessful estimation refers to the correct estimation ofboth ranges and Doppler frequencies. On the other hand,when the target ranges and Doppler frequencies arechosen randomly at the unambiguous region, a successfulestimation is declared if both differences between theestimated and true ranges and Doppler frequencies aresmaller than half of the respective cell resolution.

In the simulation studies, we consider five targets withthe same SNRs. To examine the discrimination capability,we assume that the first two targets belong to the samerange bin, the other two targets belong to another Dopplerbin, and the fifth target is chosen to have random rangeand Doppler. All results are obtained by averaging over1000 independent trials.

First, we depict the estimation performance when allfive targets are set on the discrete grids. Fig. 13 shows therates of successful estimation with respect to the inputSNR. For the CoSaPD method, the false alarm probabilityis set as PF = 10−2 with l0 = 5. It is seen that the

Fig. 14. Times of recovery algorithms required for range estimation.

Fig. 15. Rate of successful estimation in practical scenario.

CoSaPD method greatly outperforms the direct methodand achieves the performance of the classical method evenat a low SNR of −25 dB and the sampling rate is onlyone-eighth of the Nyquist rate. The performanceimprovement of the CoSaPD method is due to the fact thatthe range estimation is performed in the Doppler domainin which the SNR is enhanced because of the DFTprocessing. Setting PF = 10−2 will result in a high falsealarm probability in the detection stage. However, thesetting does not affect the system detection.

Another advantage of the CoSaPD method over thedirect processing method is the reduction of thecomputational burden. For the simulated parameters, thedirect method needs to execute the recovery algorithms for100 times, whereas the CoSaPD method only needs atmost 12 times, as shown in Fig. 14.

Next, we present the simulated performance when thetarget ranges and Doppler frequencies are randomly set inthe unambiguous region. Fig. 15 shows the rates ofsuccessful estimation. Compared with Fig. 13, theestimated performance degrades. However, the CoSaPDmethod is more applicable to realistic situations than thedirect processing method. As discussed in Section IV, thedirect processing method first estimates the complexamplitudes of the targets from the compressed data. Theestimation may introduce errors in both amplitude and


Fig. 16. Estimation performance of smaller target.

phase. In particular, the phase error will greatly affect theDoppler estimation in the DFT operation, thus causingperformance degradation to the direct processing method.For the CoSaPD method, the Doppler estimation isperformed in the Doppler domain data, which is obtainedfrom the DFT of the compressive data. As such, it yields arobust range estimate unaffected by the Doppler phase.

Finally, we simulate the performance of estimating aweak target that is close to a strong one. In the classicalprocessing, the output of the matched filtering will havesidelobes in range, making the weak target obscured bythe sidelobes of the strong target. We assume that the twotargets are present in the same Doppler bin and the weaktarget is randomly set in the first sidelobe of the strongtarget. Fig. 16 shows the estimation performance, whereSNRIN

1 and SNRIN2 denote the SNRs of the strong and the

weak targets, respectively. It is observed that the CoSaPDmethod outperforms the classical method when the twotargets have a large SNR difference.

D. The Effects of Clutter

We now demonstrate the performance of the CoSaPDprocessing in the presence of surface clutter. The receivedtarget signals are contaminated by both noise and clutteras described in (5). The signals and noise are set as inFig. 13 and the simulated windblown ground clutter isadded. The ground clutter is assumed to be Rayleighdistributed in amplitude and obeys the followingtwo-sided exponential law in Doppler spreading

Sc(v) = β

2exp(−β |v|), − ∞ < v < ∞,

where β corresponds to the wind conditions. It was shownin [28] that the Doppler model matches the measuredshapes of windblown ground clutter Doppler spectra muchmore closely than by the Gaussian or power-lawapproximations. In the simulation, we choose β = 4.3,which corresponds to a wind condition of about96.6 km/hr. The SCR is −40 dB. To reduce the effect ofclutter, a Taylor window, which yields 10 nearlyconstant-level sidelobes adjacent to the mainlobe with a

Fig. 17. Rates of successful estimation in clutter for discarding (a) 5Doppler bins and (b) 13 Doppler bins.

peak sidelobe level of −70 dB relative to the mainlobepeak, is used before DFT processing.

As discussed in Section IV, the Doppler spectrumsamples are discarded if the targets fall in theclutter-dominated area. We evaluate the rates of successfulestimation at different SNR levels after discardingdifferent Doppler bins around the zero Doppler frequency.Fig. 17(a) and Fig. 17(b) show the simulated results afterdiscarding 5 and 13 Doppler bins, respectively. The effectof the clutter is evident from Fig. 17(a). Because of cluttersidelobe, we cannot obtain 100% successful estimationrates for both classical and CoSaPD methods. Bydiscarding more Doppler bins, as shown in Fig. 17(b), wecan better remove the clutter effect on the signals withhigh Doppler shifts. Fig. 18 further shows the usableDoppler space fraction [2] versus the SNR. In thesimulations, we set one target with a randomly distributedrange bin and a variable Doppler bin. When the target issuccessfully estimated, we claim the Doppler bin to beusable. It is seen that there is sharp drop of the usableDoppler space when the SNR is below some threshold.This is because the recovery algorithms do not function insuch low SNRs.2 Both Fig. 17(b) and Fig. 18 indicate that

2 The l1-norm minimization algorithms in a noisy signal case yield sparsesolutions only when ‖(F(Ncs ))l‖2

2 < ‖(F(Scs ))l‖22 [41].


Fig. 18. Usable Doppler space fraction versus SNR.

the CoSaPD method achieves the performance of theclassical method even at an SNR of −25 dB and a datarate of one-eighth of the Nyquist rate. This is consistentwith the observation made in Fig. 13.

VII. CONCLUSION

In this paper, we have developed a pulse-Dopplerprocessing scheme, termed as CoSaPD, with thesub-Nyquist data delivered from the QuadCS system. Thescheme consists of two major procedures, Dopplerestimation/detection and range estimation, with the formerto be performed prior to the latter. Theoretical analysesand computer simulation results verify the performanceadvantages of the proposed CoSaPD approach. Whensampling at one-eighth of the Nyquist rate and for SNRabove −25 dB, the CoSaPD achieves the performance ofthe classical processing using Nyquist samples.

When compared with other related schemes utilizingCS data, the proposed CoSaPD scheme offers fourimportant advantages. The first one is the small sizedictionary. While other CS-based radar data processingschemes usually use a two-dimensional dictionary bydiscretizing both radar range and Doppler [23], theCoSaPD scheme adopts a one-dimensional dictionary byonly discretizing the radar range. The second advantage isthe combination of the estimation and detection processes,which has two advantages over separate estimation anddetection: improved detection performance and reducedcomputational complexity. The third advantage is itsability to detect weak targets nearby a strong target. Thelast one is the ability to cancel the clutter echoes as inclassical processing.

In comparison with the classical processing, theCoSaPD produces the system gain loss. As discussed inSection V, the CoSaPD processing gain is Bcs/B timesthat of the classical processing. That is, a narrowbandwidth Bcs and, hence, a low sampling rate f cs

IF , yieldslarge processing gain losses. There is a trade-off betweenthe unacceptable loss and the sampling rate.

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Chao Liu received the B.S. degree in electronic science and technology from NanjingUniversity of Information Science and Technology, Nanjing, China, in 2006.

He is currently pursuing his Ph.D. degree in the Department of ElectronicEngineering, Nanjing University of Science and Technology, Nanjing, China.His research interests include radar signal processing and compressive sensing.

Feng Xi (M’14) received the B.S. degree in electrical engineering and the Ph.D. degreein information and communication engineering from Nanjing University of Science andTechnology, Nanjing, China, in 2003 and 2010, respectively.

Currently, he is an Associate Professor in the Department of Electrical Engineering,Nanjing University of Science and Technology, Nanjing, China. His research interestsinclude radar signal processing, chaotic dynamical system, and wireless sensornetworks.

Shengyao Chen (M’14) received the B.S. degree in communication engineering andthe Ph.D. degree in information and communication engineering from NanjingUniversity of Science and Technology, Nanjing, China, in 2006 and 2013, respectively.

He is now a Post-Doctoral Research Fellow at Nanjing University of Science andTechnology, Nanjing, China. His research interests include radar and communicationsystems, signal processing, and chaotic dynamical systems.


Yimin D. Zhang (SM’01) received his Ph.D. degree from the University of Tsukuba,Tsukuba, Japan, in 1988.

He joined the faculty of the Department of Radio Engineering, Southeast University,Nanjing, China, in 1988. He served as a Director and Technical Manager at the OrientalScience Laboratory, Yokohama, Japan, from 1989 to 1995, and a Senior TechnicalManager at the Communication Laboratory Japan, Kawasaki, Japan, from 1995 to1997. He was a Visiting Researcher at the ATR Adaptive Communications ResearchLaboratories, Kyoto, Japan, from 1997 to 1998. Since 1998, he has been with VillanovaUniversity, Villanova, PA, USA, where he is currently a Research Professor with theCenter for Advanced Communications, and is the Director of the WirelessCommunications and Positioning Laboratory and the Director of the Radio FrequencyIdentification (RFID) Laboratory. His general research interests lie in the areas ofstatistical signal and array processing for radar, communications, and navigationapplications, including compressive sensing, convex optimization, nonstationary signaland time-frequency analysis, MIMO systems, radar imaging, target localization andtracking, wireless and cooperative networks, and jammer and clutter suppression.

Dr. Zhang has 11 book chapters and more than 250 journal articles andpeer-reviewed conference papers. He serves on the Editorial Board of the SignalProcessing journal. He was an Associate Editor for the IEEE Transactions on SignalProcessing during 2010–2014, an Associate Editor for the IEEE Signal ProcessingLetters during 2006–2010, and an Associate Editor for the Journal of the FranklinInstitute during 2007–2013. He is a member of the Sensor Array and Multichannel(SAM) Technical Committee of the IEEE Signal Processing Society. He is a TechnicalCommittee Co-chair of the IEEE Benjamin Franklin Symposium on Microwave andAntenna Sub-systems in 2014.

Zhong Liu (M’03) received the B.S. degree from Anhui University, Hefei, Anhui,China, in 1983 and the M.S. and Ph.D. degrees from the University of ElectronicScience and Technology of China, Chengdu, Sichuan, China, in 1985 and 1988, all inelectrical engineering.

He was a Postdoctoral Fellow at Kyoto University, Kyoto, Japan, from 1991 to1993. He was a Researcher at the Chinese University of Hong Kong, Hong Kong, from1997 to 1998. Currently, he is a Professor at the Department of Electronic Engineering,Nanjing University of Science and Technology, Nanjing, Jiangsu, China. His researchinterests include chaos and information dynamics, signal processing, radar, andcommunication technologies.


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