1

Multi-Carrier Agile Phased Array RadarTianyao Huang, Nir Shlezinger, Xingyu Xu, Dingyou Ma, Yimin Liu, and Yonina C. Eldar

Abstract—Modern radar systems are expected to operatereliably in congested environments. A candidate technology formeeting these demands is frequency agile radar (FAR), whichrandomly changes its carrier frequencies. FAR is known toimprove the electronic counter-countermeasures (ECCM) per-formance while facilitating operation in congested setups. Toenhance the target recovery performance of FAR in complexelectromagnetic environments, we propose two radar schemesextending FAR to multi-carrier waveforms. The first is WidebandMulti-carrier Agile Radar (WMAR), which transmits/receiveswideband waveforms simultaneously with every antenna. Tomitigate the demanding hardware requirements associated withwideband waveforms used by WMAR, we next propose multi-Carrier AgilE phaSed Array Radar (CAESAR). CAESAR usesnarrowband monotone waveforms, thus facilitating ease of im-plementation of the system, while introducing spatial agility. Wecharacterize the transmitted and received signals of the proposedschemes, and develop an algorithm for recovering the targets,based on concepts from compressed sensing to estimate the range-Doppler parameters of the targets. We then derive conditionswhich guarantee their accurate reconstruction. Our numericalstudy demonstrates that both multi-carrier schemes improveperformance compared to FAR while maintaining its practicalbenefits. We also demonstrate that the performance of CAESAR,which uses monotone waveforms, is within a small gap from thewideband radar.

Index Terms—Frequency agile radar, multi-carrier agility,compressed sensing

I. INTRODUCTION

Modern radars must be reliable, but at the same time com-pact, flexible, robust, and efficient in terms of cost and powerusage [1]–[5]. A possible approach to meet these requirementsis by exploiting frequency agility [1], namely, to utilize narrow-band waveforms, while allowing the carrier frequencies to varybetween different radar pulses. Among the main advantagesof frequency agile radar (FAR) are its excellent electroniccounter-countermeasures (ECCM) and electromagnetic com-patibility (EMC) performance [1], and the fact that it has theflexibility of supporting spectrum sharing [4]. Furthermore,FAR is compatible with phased array antennas. Finally, byutilizing narrowband signals with varying frequencies, FARsystems can synthesize a large bandwidth with narrowbandwaveforms [2], [6], which simplifies the implementation ofthe waveform generator, facilitates the receiver operation, andallows the usage of non-linear amplifiers without limiting theirpower efficiency.

Parts of this work were presented in the 2018 IEEE International Workshopon Compressed Sensing applied to Radar, Multimodal Sensing, and Imaging(CoSeRa). This work received funding from the National Natural ScienceFoundation of China under Grants 61571260 and 61801258, from the Eu-ropean Unions Horizon 2020 research and innovation program under grantNo. 646804-ERC-COG-BNYQ, and from the Air Force Office of ScientificResearch under grant No. FA9550-18-1-0208.

T. Huang, Y. Liu, X. Xu, and D. Ma are with the EE Depart-ment, Tsinghua University, Beijing, China (e-mail: huangtianyao, yimin-liu@tsinghua.edu.cn; xy-xu15, mdy16@mails.tsinghua.edu.cn).

N. Shlezinger and Y. C. Eldar are with the Faculty of Math and CS, Weiz-mann Institute of Science, Rehovot, Israel (e-mail: nirshlezinger1@gmail.com;yonina.eldar@weizmann.ac.il).

A major drawback of FAR compared to wideband radaris its reduced range-Doppler reconstruction performance oftargets. This reduced performance is a byproduct of the rela-tively small number of radar measurements processed by FAR,which stems from its usage of a single narrowband waveformfor each pulse. The performance reduction can be relieved byusing compressed sensing (CS) algorithms that exploit sparsityof the target scheme [7]. However, the degradation becomesnotable in extremely congested or contested electromagneticenvironments [8], where there may be no vacant bands in somepulses or some radar returns of the transmitted pulses may bediscarded due to strong interference [9], [10].

The performance degradation of FAR can be mitigated byusing multi-carrier transmissions. When multiple carriers aretransmitted simultaneously in a single pulse, the number ofradar measurements is increased, and the target reconstructionperformance is improved. Various multi-carrier radar schemeshave been studied in the literature, including frequency divi-sion multiple access (FDMA) multiple-input multiple-output(MIMO) (FDMA-MIMO) [11], [12], sub-Nyquist MIMOradar (SUMMeR) [13], and frequency diversity array (FDA)radar [14], [15]. In the aforementioned schemes, different arrayelements transmit waveforms at different frequencies, usuallyforming an omnidirectional beam and illuminating a largefield-of-view [16]. This degrades radar performance, especiallyin track mode, where a highly directional beam focusing onthe target is preferred [16]. In addition, frequency agility isnot exploited in FDMA-MIMO and FDA. The derivation offrequency agile multi-carrier schemes for phased array radar,which leads to a focused beam with high gain, is the focus ofthis work.

Here, we propose two multi-carrier agile phased arrayradar schemes. The first uses all the antenna elements totransmit a single waveform consisting of multiple carrierssimultaneously in each pulse. Frequency agility is inducedby randomly selecting the carriers utilized, resulting in awideband multi-carrier agile radar (WMAR) scheme. Whilethe increased number of carriers is shown to achieve improvedreconstruction performance compared to conventional FAR[8], WMAR utilizes multiband signals of large instantaneousbandwidth. Therefore, its implementation does not benefitfrom the simplifications associated with utilizing conventionalnarrowband monotone signals, and may suffer from envelopefluctuation [17].

To overcome the use of instantaneously wideband wave-forms, we next develop multi-Carrier AgilE phaSed ArrayRadar (CAESAR), which combines frequency agility andspatial agility. Specifically, CAESAR selects a small numberof carrier frequencies on each pulse and randomly allocatesdifferent carrier frequencies among its antenna elements, suchthat each array element transmits a narrowband constantmodulus waveform, facilitating system implementation. Anillustration of this transmission scheme is depicted in Fig. 1.

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Fig. 1. Transmission example of CAESAR. In every pulse of this example,two out of three carrier frequencies are emitted by different sub-arrays. Forexample, frequency 0 and 2 are selected in the 0-th pulse and are sent byantenna 0, 2, 4 and antenna 1, 3, respectively. FAR or FDMA-MIMO/FDAcan be regarded as a special case of CAESAR, with only one out of threefrequencies or all available frequencies sent in each pulse.

For each carrier frequency, dedicated phase shifts on thecorresponding sub-array elements are used to yield a direc-tional transmit beam, allowing to illuminate the tracked targetin a similar manner as phased array radar. Despite the factthat only a sub-array antenna is utilized for each frequency,the antenna-frequency hopping strategy of CAESAR resultsin array antenna gain loss and a relatively small performancegap compared to wideband radar equipped with the sameantenna array. Furthermore, the combined randomization offrequency and antenna allocation can be exploited to realize adual function radar-communications (DFRC) system [18]–[21]by embedding digital information into the selection of theseparameters. We study the application of CAESAR as a DFRCsystem in a companion paper [22], and focus here on the radarand its performance.

To present WMAR and CAESAR, we characterize thesignal model for each approach, based on which we develop arecovery algorithm for high range resolution (HRR), Doppler,and angle estimation of radar targets. Our proposed algorithmutilizes CS methods for range-Doppler reconstruction, exploit-ing its underlying sparsity, and applies matched filtering todetect the angles of the targets. We provide a detailed theo-retical analysis of the range-Doppler recovery performance ofour proposed algorithm under complex electromagnetic envi-ronments. In particular, we prove that CAESAR and WMARare guaranteed to recover with high probability a number ofscattering points which grows proportionally to the squareroot of the number of different narrowband signals used,i.e., the number of carrier frequencies that are simultaneouslytransmitted in each pulse. This theoretical result verifies thatincreasing the number of carriers improves target recovery,and reduces performance degradation due to intense interfer-ence in complex electromagnetic environments. WMAR andCAESAR are evaluated in a numerical study, where it is shownthat their range-Doppler reconstruction performance as well asrobustness to interference are substantially improved comparedto FAR. Additionally, it is demonstrated that the performanceof CAESAR is only within a small gap from that achievableusing wideband WMAR.

The remainder of the paper is structured as follows: Sec-tions II and III present WMAR and CAESAR, respectively.

Section IV introduces the recovery algorithm to estimate therange, Doppler, and angle of the targets. In Section V wediscuss the pros and cons of each scheme compared to relatedradar methods. Section VI derives theoretical performancemeasures of the recovery method. Simulation results arepresented in Section VII, and Section VIII concludes the paper.

Throughout the paper, we use C, R to denote the setsof complex, real numbers, respectively, and use | · | forthe magnitude or cardinality of a scalar number, or a set,respectively. Given x ∈ R, bxc denotes the largest integer lessthan or equal to x, and

(nk

)= n!

k!(n−k)! represents the binomialcoefficient. Uppercase boldface letters denote matrices (e.g.,A), and boldface lowercase letters denote vectors (e.g., a). The(n,m)-th element of a matrix A is denoted as [A]m,n, andsimilarly [a]n is the n-th entry of the vector a. Given a matrixA ∈ CM×N , and a number n (or a set of integers, Λ), [A]n([A]Λ ∈ CM×|Λ|) is the n-th column of A (the sub-matrixconsisting of the columns of A indexed by Λ). Similarly,[a]Λ ∈ C|Λ| is the sub-vector consisting of the elements ofa ∈ CN indexed by Λ. The complex conjugate, transpose, andthe complex conjugate-transpose are denoted (·)∗, (·)T , (·)H ,respectively. We denote ‖ · ‖p as the `p norm, ‖ · ‖0 is thenumber of non-zero entries, and ‖ · ‖F is the Frobenius norm.The probability measure is P(·), while E[·] and D[·] are theexpectation and variance of a random argument, respectively.

II. WMAR

In this section we present the proposed WMAR scheme,which originates from FAR [1], aiming to increase the numberof radar measurements and improve the range-Doppler recov-ery performance. We first briefly review FAR in SubsectionII-A. Then, we detail the proposed WMAR in Subsection II-B,and present the resulting radar signal model in Subsection II-C.

A. Preliminaries of FAR

FAR [1] is a technique for enhancing the ECCM and EMCperformance of radar systems by using randomized carrierfrequencies. In the following we consider a radar systemequipped with L antenna elements, uniformly located on anantenna array with distance d between two adjacent elements.Let N be the number of radar pulses transmitted in eachcoherent processing interval (CPI). Radar pulses are repeatedlytransmitted, starting from time instance nTr to nTr + Tp,n ∈ N := 0, 1, . . . , N−1, where Tr and Tp represent thepulse repetition interval and pulse duration, respectively, andTr > Tp. Let F be the set of available carrier frequencies,given by F := fc + m∆f |m ∈ M, where fc is the initialcarrier frequency, M := 0, 1, . . . ,M−1, M is the numberof available frequencies, and ∆f is the frequency step.

In the n-th radar pulse, FAR randomly selects a carrierfrequency fn from F . The waveform sent from each antennafor the n-th pulse at time instance t is φ(fn, t−nTr), where

φ(f, t) := rect (t/Tp) ej2πft, (1)

and rect(t) = 1 for t ∈ [0, 1) and zero otherwise, representingrectangular envelope baseband signals.

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In order to direct the antenna beam pointing towards adesired angle θ, the signal transmitted by each antenna isweighted by a phase shift wl(θ, fn) ∈ C [23], given by

wl(θ, f) := ej2πfld sin θ/c, (2)

where c denotes the speed of light. Define the vectorw (θ, f) ∈ CL whose l-th entry is [w (θ, f)]l := wl (θ, f).The transmitted signal can be written as

xF(n, t) := w(θ, fn)φ(fn, t− nTr). (3)

The vector xF(n, t) ∈ CL in (3) denotes the transmissionvector of the full array for the n-th pulse at time instance t.

The fact that FAR transmits monotone waveform facilitatesits realization. Furthermore, the frequency agility achieved byrandomizing the frequencies between pulses enhances surviv-ability in complex electromagnetic environments. However,this comes at the cost of reduced number of radar mea-surements, which degrades the target recovery performance,particularly in the presence of interference, where some of theradar returns are missed [8]. To overcome these drawbacks,in the following we propose WMAR, which extends FAR tomulti-carrier transmissions.

B. WMAR Transmit Signal ModelWMAR extends FAR to multi-carrier signalling. Broadly

speaking, WMAR transmits a single multiband waveform fromall its antennas, maintaining frequency agility by randomizinga subset of the available frequencies on each pulse.

Specifically, in the n-th radar pulse, WMAR randomlyselects a set of carrier frequencies Fn from F , Fn ⊂ F . Weassume that the cardinality of Fn is constant, i.e., |Fn| = Kfor each n ∈ N , and write the elements of this set asFn = Ωn,k|k ∈ K, K := 0, 1, . . . ,K − 1. The portionof the n-th pulse of WMAR in the k-th frequency is givenby xW,k(n, t) := 1√

Kw (θ,Ωn,k)φ (Ωn,k, t− nTr), and the

overall transmitted vector is xW(n, t)=∑Kk=1 xW,k(n, t), i.e.,

xW(n, t) =

K∑k=1

1√Kw (θ,Ωn,k)φ (Ωn,k, t− nTr) , (4)

where the factor 1√K

guarantees that (4) has the same totalpower as the FAR signal (3).

FAR is a special case of WMAR under the setting K = 1.By using multiple carriers simultaneously via wideband sig-nalling, i.e., K > 1, WMAR transmits a highly directionalbeam, while improving the robustness to missed pulses com-pared to FAR. The improved performance stems from the useof multi-carrier transmission, which increases the number ofradar measurements. To see this, we detail the received signalmodel of WMAR in the following subsection.

C. WMAR Received Signal Model

We next model the received signal processed by WMARfor target identification. To that aim, we focus on the timeinterval after the n-th pulse is transmitted, i.e., nTr + Tp <t < (n+ 1)Tr. In this period, the radar receives echoes of thepulse, which are sampled and processed in discrete-time.

To formulate the radar returns, we assume an ideal scatteringpoint, representing either target or clutter, with scatteringcoefficient β ∈ C located in the transmit beam of the radarwith direction angle ϑ, i.e., ϑ ≈ θ. Denote by r(t) therange between the target/clutter and the first radar antennaarray element at time t. The scattering point is moving at aconstant velocity v radially along with the radar line of sight,i.e., r(t) = r(0) + vt. Under the “stop and hop” assumption[24, Page 99, Ch. 2], which assumes that the target hops toa new location when the radar transmits a pulse and staysthere until another pulse is emitted, the range in the n-pulseis approximated as

r(t)≈r(nTr)=r(0)+v · nTr, nTr<t<(n+1)Tr. (5)

To model the received signal, we first consider the n-thradar pulse that reaches the target, denoted by x(n, t). Letxk(n, t) be its component at frequency Ωn,k, i.e., x(n, t) :=∑K−1k=0 xk(n, t). Note that xk(n, t) is a summation of delayed

transmissions from the corresponding antenna elements. Thedelay for the l-th array element is r(nTr)/c+ld sinϑ/c. Underthe narrowband, far-field assumption, using (2), we have that

xk(n, t)=

L−1∑l=0

[xW,k(n, t− r(nTr)/c)]l e−j2πΩn,kld sinϑ/c

= wH (ϑ,Ωn,k)xW,k(n, t− r(nTr)/c). (6)

Substituting (5) and the definition of xW,k(n, t) into (6) yields

xk(n, t)=ρW(n, k, δϑ)√

Kφ

(Ωn,k, t−nTr−

r(0)+nvTrc

), (7)

where δϑ := sinϑ − sin θ is the relative direction sinewith respect to the transmit beam, and ρW(n, k, δϑ) :=wH (ϑ,Ωn,k)w (θ,Ωn,k) is the transmit gain, expressed as

ρW(n, k, δϑ) =

L−1∑l=0

e−j2πΩn,kldδϑ/c. (8)

Note that ρW(n, k, δϑ) approaches L when δϑ ≈ 0.Having modeled the signal which reaches the target, we now

derive the radar returns observed by the antenna array. Afterbeing reflected by the scattering point, the signal at the k-thfrequency propagates back to the l-th radar array element withan extra delay of r(nTr)/c+ ld sinϑ/c, resulting in

[yW,k(n, t)]l := βxk (n, t− r(nTr)/c− ld sinϑ/c) . (9)

The echoes vector yW,k(n, t) ∈ CL can be written as

yW,k(n, t)=βw∗ (ϑ,Ωn,k) xk (n, t− r(nTr)/c)(a)=

β√Kw∗ (ϑ,Ωn,k) ρW(n, k, δϑ)

× φ (Ωn,k, t−nTr−(2r(0) + 2nvTr)/c) , (10)

where (a) follows from (7).The received echoes at all K frequencies are then separated

and sampled independently by each array element. The signalyW,k(n, t) is sampled with a rate of fs = 1/Tp at timeinstants t = nTr + i/fs, i = 0, 1, . . . , bTrfsc − 1, suchthat each pulse is sampled once. Every sample time instantcorresponds to a coarse range cell (CRC), r ∈

(i−12fs

c, i2fsc)

.

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The division to CRCs indicates coarse range information ofscattering points. We focus on an arbitrary i-th CRC, assumingthat the scattering point does not move between CRCs duringa CPI, i.e., there exists some integer i such that

i−12fs

c < r (0) < i2fsc,

i−12fs

c < r (0) + vnTr <i

2fsc, ∀n ∈ N . (11)

Collecting radar returns from N pulses and L elements atthe same CRC yields a data cube YW ∈ CL×N×K with entries

[YW]l,n,k := [yW,k(n, nTr + i/fs)]l , (12)

where i is the CRC index. The data cube YW is processedto estimate the refined range information, Doppler, and angleof the scattering point. Data cubes from different CRCs areprocessed identically and separately.

Finally, we formulate how the unknown parameters of thetargets are embedded in the processed data cube YW. Tothat aim, define δr := r(0) − ic/2fs as the high-rangeresolution distance, cn,k := (Ωn,k − fc)/∆f ∈ M as thecarrier frequency index, and ζn,k = Ωn,k/fc as the relativefrequency factor. Then, denoting by β := βe−j4πfcδr/c,r := −4π∆fδr/c and v := −4πfcvTr/c the generalizedscattering intensity, and the normalized range and velocity,respectively, and substituting (10) into (12), we have that

[YW]l,n,k=βejrcn,k√

Kejvnζn,ke−j2π

Ωn,kldsinϑ

c ρW(n, k, δϑ). (13)

The unknown parameters in (13) are β, r, v and (sinϑ, δϑ),which are used to reveal the scattering intensity |β|, HRRrange r(0), velocity v and angle ϑ of the target.

The above model can be naturally extended to noisy mul-tiple scatterers. When there are S scattering points insidethe CRC instead of a single one as assumed previously, thereceived signal is a summation of returns from all these pointscorrupted by additive noise, denoted by N ∈ CL×N×K .Following (13), the entries of the data matrix are

[YW]l,n,k=1√K

S−1∑s=0

βsejrscn,kejvsnζn,ke−j2πΩn,kldsinϑs/c

× ρW(n, k, δϑs) + [N ]l,n,k , (14)

where βs, rs, vs and ϑs represent the sets of factorsof scattering coefficients, ranges, velocities, and angles ofthe S scattering points, respectively, which are unknown andshould be estimated. A method for recovering these parametersfrom the data cube YW is detailed in Section IV.

WMAR has several notable advantages: First, as an exten-sion of FAR, it preserves its frequency agility and is suitablefor implementation with phased array antennas. Furthermore,as we discuss in Section V, its number of radar measurementsfor each CRC is increased by a factor of K comparedto FAR, thus yielding increased robustness to interference.However, WMAR transmitters simultaneously send multiplecarriers instead of a monotone as in FAR, which requireslarge instantaneous bandwidth, leading to envelope fluctuationand low amplifier efficiency. To overcome these issues, weintroduce CAESAR in the following section, which utilizes

narrowband radar transceivers while introducing spatial agility,enabling multi-carrier transmission using monotone signals ata cost of a minimal array antenna gain loss.

III. CAESAR

CAESAR, similarly to WMAR, extends FAR to multi-carrier transmission. However, unlike WMAR, CAESAR uti-lizes monotone signalling and reception, and is thus moresuitable for implementation. We detail the transmit and receivemodels of CAESAR in Subsections III-A and III-B, respec-tively.

A. CAESAR Transmit Signal Model

Broadly speaking, CAESAR extends FAR to multi-carriersignalling by transmitting monotone waveforms with varyingfrequencies from different antenna elements. The selection ofthe frequencies, as well as their allocation among the antennaelements, is randomized anew in each pulse, thus inducingboth frequency and spatial agility.

To formulate CAESAR, we consider the same pulse radarformulation detailed in Section II. Similarly to WMAR de-tailed in Subsection II-B, in the n-th radar pulse, CAESARrandomly selects a set of carrier frequencies Fn = Ωn,k|k ∈K from F . While WMAR uses the set of selected frequenciesto generate wideband waveforms, CAESAR allocates a sub-array for each frequency, such that all the antenna arrayelements are utilized for transmission, each at a single carrierfrequency. Denote by fn,l ∈ Fn the frequency used by thel-th antenna array element, l ∈ L := 0, 1, . . . , L− 1. Afterphase shifting the waveform to direct the beam, the l-th arrayelement transmission can be written as

[xC(n, t)]l := [w(θ, fn,l)]l φ(fn,l, t− nTr). (15)

The vector xC(n, t) ∈ CL in (15) denotes the full arraytransmission vector for the n-th pulse at time t. Here, unlikeFAR which transmits a single frequency from the full array (3),CAESAR assigns diverse frequencies to different sub-arrayantennas, as illustrated in Fig. 1.

The transmitted signal (15) can also be expressed by group-ing the array elements which use the same frequency Ωn,k. LetxC,k(n, t) ∈ CL with zero padding represent the portion ofxC(n, t) which utilizes Ωn,k, i.e.,

xC,k(n, t) = P (n, k)w (θ,Ωn,k)φ (Ωn,k, t− nTr) , (16)

where P (n, k) ∈ 0, 1L×L is a diagonal selection matrixwith diagonal p(n, k) ∈ 0, 1L, whose l-th entry is oneif the l-th array element transmits at frequency Ωn,k andzero otherwise, i.e., [P (n, k)]l,l = [p(n, k)]l = 1 and[xC,k(n, t)]l = [xC(n, t)]l when fn,l = Ωn,k. The transmittedsignal is thus xC(n, t) :=

∑K−1k=0 xC,k(n, t), namely

xC(n, t) =

K−1∑k=0

P (n, k)w (θ,Ωn,k)φ (Ωn,k, t− nTr) . (17)

Comparing (17) with (4), we find that each array element ofCAESAR transmits a single frequency with unit amplitudewhile in WMAR all K frequencies with amplitudes scaled bya factor 1/

√K are sent by each element.

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The diagonal selection matrices P (n, 0), . . . ,P (n,K − 1)uniquely describe the allocation of antenna elements forthe n-th pulse. CAESAR transmission scheme implies that∑K−1k=0 P (n, k) = IL, i.e., all the antenna elements are utilized

for the transmission of the n-th pulse. The trace of P (n, k)represents the number of antennas using the k-th frequency.Without loss of generality, we assume that L/K is an integerand tr (P (n, k)) = L/K, for each n ∈ N and k ∈ K.

Phased array FAR and FDA [15] are special cases ofCAESAR with K = 1 and K = M = L, respectively. Afundamental difference between these radar schemes is thetransmit beam pattern. In FAR, the same carrier frequency isutilized by all the elements, i.e., Ωn,k and fn,l are identicalfor each k ∈ K and l ∈ L, respectively, resulting inhighly directional beam. In FDA, all available frequenciesare transmitted simultaneously and one frequency correspondsto a single antenna element, leading to an omnidirectionalbeam which degrades radar performance and is not suitablefor target tracking [16]. The proposed CAESAR uses only asubset of the available frequencies in each pulse and multipleantenna elements share the same frequency, thus achievinga compromise radiation beam that only illuminates the de-sired angle. Despite the gain loss in comparison with FARdiscussed in Section V, CAESAR achieves improved range-Doppler reconstruction performance and increased robustnessto interference, as numerically demonstrated in Section VII.

B. CAESAR Received Signal ModelWe next model the received signal processed by CAESAR.

Unlike WMAR, in which each antenna receives and separatesdifferent frequency components, in CAESAR, the l-th antennaelement only receives radar returns at frequency fn,l, andabandons other frequencies. This enables the use of narrow-band receivers, simplifying the hardware requirements.

Note that the derivation of the signal component receivedat the k-th frequency in (6), xk(n, t), does not depend on thespecific radar scheme. Here, substituting (16) into (6) yields

xk(n, t)=ρC(n, k, δϑ)φ

(Ωn,k, t−nTr−

r(0)+nvTrc

), (18)

where ρC(n, k, δϑ) :=wH (ϑ,Ωn,k)P (n, k)w (θ,Ωn,k) is thetransmit gain of the selected sub-array antenna, expressed as

ρC(n, k, δϑ) =

L−1∑l=0

[p(n, k)]l e−j2πΩn,kldδϑ/c. (19)

Note that, in contrast to the transmit gain of WMAR in (8)which tends to L, ρC(n, k, δϑ) approaches L/K when δϑ ≈ 0.By repeating the arguments in the derivation of (10), the echovector yC,k(n, t) ∈ CL can be written as

yC,k(n, t)= βw∗ (ϑ,Ωn,k) ρC(n, k, δϑ)

×φ (Ωn,k, t−nTr−(2r(0) + 2nvTr)/c) .(20)

CAESAR receives and processes impinging signals by thecorresponding elements of the antenna array. In particular, onlya sub-array, whose elements are indicated by P (n, k), receivesthe impinging signal yC,k(n, t); the other array elements aretuned to other frequencies. The zero-padded received signal

at the k-th frequency, denoted by yC,k(n, t) ∈ CL, is thusyC,k(n, t) := P (n, k)yC,k(n, t). The full array received signalis given by yC(n, t) :=

∑K−1k=0 yC,k(n, t).

The observed signal yC(n, t) is sampled in a similar manneras detailed in Subsection II-C. Since CAESAR processes asingle frequency component per antenna element, the measure-ments from each CRC are collected together as a data matrixYC ∈ CL×N , as opposed to a L×N ×K cube processed byWMAR. By repeating the arguments used for obtaining (13),it holds that

[YC]l,n=βejrcn,kejvnζn,ke−j2πΩn,kldsinϑ/cρC(n, k, δϑ), (21)

which can be extended to account for multiple targets andnoisy measurements as in (14), i.e.,

[YC]l,n=

S−1∑s=0

βsejrscn,kejvsnζn,ke−j2πΩn,k

ld sinϑsc

× ρC(n, k, δϑs) + [N ]l,n , (22)

where N ∈ CL×N is the additive noise. In order to recoverthe unknown parameters from the acquired data matrix (22), inthe following section we present a dedicated recovery scheme.

IV. TARGET RECOVERY METHOD

Here, we present an algorithm for reconstructing the un-known HRR range, velocity, angle, and scattering intensityparameters of the scattering points from the radar measure-ments of both WMAR and CAESAR. Detection is performedbased on the estimated scattering intensities. The detectedscattering points may belong to either target or clutter, andthey are identified by their Doppler estimates. The motivationfor this approach is that in many ground-based radar systems,fast moving targets are of interest, while static or slow movingscatterers with zero or nearly zero Doppler are regarded asclutter. We henceforth model the Doppler of both targetsand clutter as unknown parameters, which are simultaneouslyestimated. In specific applications where the clutter Doppleris a-priori known, one can apply clutter mitigation [25] inadvance to the target recovery method. A similar procedureis also applied in pulse Doppler radars [24, Ch. 5.5.1], wherethe moving target indication filtering for gross clutter removalis placed prior to the pulse Doppler filter bank.

In order to maintain feasible computational complexity,we do not estimate all the parameters simultaneously: ourproposed algorithm first jointly recovers the range-Dopplerparameters followed by estimation of the unknown angles.When performing joint range-Doppler estimation, we assumethat all the scattering points are located within the mainlobe ofthe transmit beam, and that the difference of the angle sine isnegligible, i.e., δϑ ≈ 0. We then estimate the direction anglesof scattering points based on their range-Doppler estimates.

We divide the target recovery method into three stages:1) apply receive beamforming such that the magnitude ofthe received signal is enhanced, facilitating range-Dopplerrecovery; 2) apply CS methods for joint reconstruction ofrange and Doppler, followed by a target detection procedure;and 3) angle and scattering intensity estimation. These stepsare discussed in Subsections IV-A-IV-C, respectively. A theo-retical analysis of the range-Doppler estimation performance

6

of our algorithm is provided in Section VI, where we quan-tify how using multiple carriers improves the range-Dopplerreconstruction performance.

A. Receive BeamformingThe first step in processing the radar measurements is to

beamform the received signal in order to facilitate recovery ofthe range-Doppler parameters. This receive beamforming isapplied to radar returns at different frequencies separately. Toformulate the beamforming technique, we henceforth focus onthe k-th frequency of the n-th pulse, Ωn,k. For both CAESARand WMAR, a total of L measurements correspond to Ωn,k,and are denoted by zn,k ∈ CL. For CAESAR, zn,k is given byzn,k = P (n, k) [YC]n, of which only elements correspondingto the selected sub-array are nonzero. For WMAR, zn,kconsists of the entries [YW]l,n,k for each l ∈ L. Thesemeasurements are integrated with the weightsw (θ,Ωn,k) suchthat the receive beam is pointed towards θ, resulting in

Zk,n := wT zn,k ∈ C. (23)

Define αK := L2/K2 for CAESAR, and αK := L2/√K for

WMAR. When δϑs ≈ 0, i.e., the beam direction θ is close tothe true angle of the target, the resulting beam pattern can besimplified as stated in the following lemma:

Lemma 1. If the difference of the angle sine satisfies δϑs ≈ 0,then Zk,n in (23) can be approximated as

Zk,n ≈ αKS−1∑s=0

βsejrscn,kejvsnζn,k . (24)

Proof. See Appendix A.

The receive beamforming produces the matrix Z ∈ CK×Nwhose entries are [Z]k,n := Zk,n, for each k ∈ K, n ∈ N . Un-der the approximation (24), the obtained Z is used for range-Doppler reconstruction, as discussed in the next subsection.

B. Range-Doppler Reconstruction MethodTo reconstruct the range-Doppler parameters and detect

targets in the presence of noise and/or clutter, we first recastthe beamformed signal model of Lemma 1 in matrix form, andthen apply CS methods to recover the unknown parameters,exploiting the underlying sparsity of the resulting model. Thetargets of interest are then identified based on the estimatedparameters.

To obtain a sparse recovery problem, we start by discretiz-ing the range and Doppler domains. Recall that rs and vsdenote the normalized range and Doppler parameters, withresolutions 2π

M and 2πN , corresponding to the numbers of

available frequencies and pulses, respectively. Both parametersbelong to continuous domains in the unambiguous region(rs, vs) ∈ [0, 2π)2. We discretize rs and vs into HRR andDoppler grids, denoted by grid sets R :=

2πmM

∣∣m ∈Mand V :=

2πnN

∣∣n ∈ N, with grid intervals ∆r = 2πM and

∆v = 2πN , respectively, and assume that the targets are located

precisely on the grids. The target scene can now be representedby the matrix B ∈ CM×N with entries

[B]m,n :=

βsαK , if (rs, vs) =

(2πmM , 2πn

N

),

0, otherwise.(25)

We can now use the sparse structure of (25) to formulate therange-Doppler reconstruction as a sparse recovery problem.To that aim, let z ∈ CKN and β ∈ CMN be the vectorizedrepresentations of Z andB, respectively, i.e., [z]k+nK = Zk,nand [β]n+mN := [B]m,n. From (24), it holds that

z = Φβ, (26)

where the entries of Φ ∈ CKN×MN are given by

[Φ]k+nK,l+mN :=ej2πmM cn,k+j

2πlN nζn,k , (27)

m ∈M, l, n ∈ N , and k ∈ K. The matrix Φ is determinedby the frequencies utilized in each pulse. Consequently, Φ isa random matrix, as these parameters are randomized by theradar transmitters, whose realization is known to the receiver.

In the presence of noisy radar returns, (26) becomes

z = Φβ + n, (28)

where the entries of the noise vector n ∈ CKN are thebeamformed noise, e.g., for CAESAR these are given by[n]k+nK = wT (θ,Ωn,k)P (n, k) [N ]n.

Since in each pulse only a subset of the available frequenciesare transmitted, i.e., K ≤M , the sensing matrix Φ in (28) hasmore columns than rows, MN ≥ KN , indicating that solving(28) is naturally an under-determined problem. When β is S-sparse, which means that there are at most S non-zeroes inβ and S MN , CS algorithms can be used to solve (28),yielding the estimate β.

Particularly, CS methods aim to solve under-determinedproblems such as (26) by seeking the sparsest solution, i.e.,

β = arg minβ

‖β‖0 , s.t. z = Φβ. (29)

The `0 optimization in (29) is generally NP-hard. To re-duce computational complexity, many alternatives including`1 optimization and greedy approaches have been suggestedto approximate (29), see [26], [27].

We take `1 optimization as an example, under which weprovide a theoretical analysis and numerically evaluate theperformance in Sections VI and VII, respectively. In particular,in the absence of noise, we use the basis pursuit algorithm,which solves

β = arg minβ

‖β‖1 , s.t. z = Φβ, (30)

instead of (29). In noisy cases, recovering β can be formulatedas minimizing the `1 norm under a `2 constraint on the fidelity:

β = arg minβ

‖β‖1 , s.t. ‖z −Φβ‖2 ≤ η. (31)

Problem (31) can be solved using the Lasso method [26],which finds the solution to the `1 regularized least squaresas

β = arg minβ

λ ‖β‖1 +1

2‖z −Φβ‖22, (32)

where η and λ are predefined parameters.Having obtained the estimate β using CS methods, we can

use it to identify which of these estimated parameters corre-spond to a true target of interest. Elements with significantamplitudes in β are detected as dominant scattering points.

7

Denote by S the support set indexing these dominant scatteringpoints, whose range-Doppler parameters are recovered fromthe corresponding indices in S. For example, we may use somethreshold Th to determine whether the amplitude is significant[24, Page 295, Ch. 6]. 5 In this case, the support set is given byS = s||[β]s| > Th. According to their Doppler estimates,these dominant scattering points are categorized into target ofinterests or clutter individually.

With the recovered range-Doppler values, we can estimatethe angle and refine the scattering intensity, as detailed inthe following subsection. The procedure for detecting whichparameters correspond to targets of interest detailed above isbased on β. While this vector here represents the coarselyestimated scattering intensity, the procedure can be repeatedfor further separating targets from clutter based on the refinedestimate obtained in the sequel.

C. Angle and Scattering Intensity EstimationIn this part, we refine the angle estimation of the scattering

points, which are coarsely assumed within the transmit beamin the receive beamforming step, i.e., δϑ ≈ 0. While thefollowing formulation focuses on CAESAR, the resultingalgorithm is also applicable for WMAR as well as FAR.

We estimate the directions of the scatterers individually, asdifferent points may have different direction angles. Since afterreceive beamforming some directional information is lost inZ, we recover the angles from the original data matrix YC

(22). Using the obtained range and Doppler estimates, we firstisolate echoes for each scattering point with an orthogonalprojection, and apply a matched filter to estimate the directionangle of each scattering point. Finally, we use least squares toinfer the scattering intensities.

1) Echo isolation using orthogonal projection: In orderto accurately estimate the angle of each scattering point, itis necessary to mitigate the interference between scatteringpoints. To that aim, we use an orthogonal projection to isolateechoes from each scatterer.

Let S be the support set of β, and infer the normalizedrange and Doppler parameters rs, vs from S. According to(22), given these parameters, the original data vector from thel-th array element can be written as[

Y TC

]l

= [Ψl]S [γl]S +[NT

]l, (33)

where Ψl ∈ CN×MN has entries [Ψl]n,s := ejrscn,kejvsnζn,k ,and γl ∈ CMN denotes the effective scattering intensitiescorresponding to all discrete range-Doppler grids, with s-thentry [γl]s := βsρ(n, k, δϑs)e

−j2πΩn,kld sinϑs/c. The intensi-ties, containing unknown phase shifts and antenna gains dueto angles ϑs, are estimated as

[γl]S = arg min[γl]S

∥∥[Y TC

]l− [Ψ]S [γl]S

∥∥2

2= [Ψ]

†S

[Y T

C

]l,

where A† =(AHA

)−1AH and we assume that

∣∣S∣∣ < Nand AHA is invertible. The received radar echo from the s-thscattering point, Ys ∈ CL×N , s ∈ S, is then reconstructed bysetting the l-th row as[

Y Ts

]l

= [Ψ]s [γl]s . (34)

2) Angle estimation using matched filter: With the isolatedechoes Ys of the s-th scattering point, we use a matchedfilter to refine the unknown angle ϑs, which is coarselyassumed within the beam in the previous receive beamformingprocedure, i.e., ϑs ∈ Θ := θ+

[− π

2L ,π

2L

]. Using (22), we write

the isolated echo as Ys = βsYs(ϑs) +Ns, where Ns denotesthe noise matrix corresponding to the s-th scattering point.The entries of the steering matrix Ys(ϑs) ∈ CL×N are

[Ys(ϑs)]l,n :=ρC(n, k, δϑs)ejrscn,kejvsnζn,ke−j2πΩn,kldsinϑs/c,

which can be computed using (19) with given ϑs and theestimates of the range-Doppler parameters. Note that βs, ϑsand Ns are unknown, and ϑs is of interest. The value of theintensity βs recovered next is refined in the sequel to improveaccuracy. Here, we apply least squares estimation, i.e.,

ϑs,ˆβs = arg min

ϑs,βs

∥∥vec(Ys)− βsvec (Ys(ϑs))

∥∥2

2. (35)

Substituting ˆβs = (vec (Ys(ϑs)))

†vec(Ys)

=tr(Y Hs (ϑs)Ys)‖Ys(ϑs)‖2F

into (35) yields a matched filter

ϑs = arg maxϑs∈Θ

∣∣tr(Y Hs (ϑs)Ys

)∣∣2‖Ys(ϑs)‖2F

. (36)

The angle ϑs is estimated for each s ∈ S via (36) separately.3) Scattering intensity estimation using least squares:

When δϑs 6= 0, there exist approximation errors in (24) andthe resultant intensity estimate β. We thus propose to refinethe estimation of β from the original data matrix YC oncethe range-Doppler and angle parameters are acquired. Givenestimated angles ϑs, we concatenate the steering vectors into

C :=[vec(Ys0(ϑs0)), vec

(Ys1(ϑs1)), . . .

],

s0, s1, · · · ∈ S. The model (22) is rewritten as vec (YC) =C [β]S +N , and β can be re-estimated via least squares as[β]S=arg min

[β]S

∥∥vec (YC)−C [β]S∥∥2

2=C†vec (YC) . (37)

The overall parameter estimation method is summarized asAlgorithm 1.

Algorithm 1 CAESAR target recovery1: Input: Data matrix YC.2: Beamform YC into Z via (23).3: Use CS methods to recover the indices of the dominant

elements of β, representing the parameters of the targetsof interest, denoted by S, from Z based on the sensingmatrix Φ (27).

4: Reconstruct the normalized range-Doppler parametersrs, vs from S based on (25).

5: Isolate YC into multiple echoes Ys via (34).6: Recover the angles ϑs from Ys via (36).7: Refine the scattering intensities βs using (37).8: Output: parameters rs, vs, ϑs, βs.

V. COMPARISON TO RELATED RADAR SCHEMES

We next compare our proposed WMAR and CAESARschemes, and discuss their relationship with relevant previ-ously proposed radar methods.

8

A. Comparison of WMAR, CAESAR, and FARWe compare our proposed techniques to each other, as

well as to FAR, which is a special case of both WMARand CAESAR obtained by setting K = 1. We focus on thefollowing aspects: 1) instantaneous bandwidth; 2) the numberof measurements in a CPI; and 3) signal-to-noise ratio (SNR).A numerical comparison of the target recovery performanceof the considered radar schemes is provided in Section VII.

In terms of instantaneous bandwidth, recall that CAESARand FAR use narrowband transceivers, and only a monotonesignal is transmitted or received by each element. In WMAR,K multi-tone signals are sent and received simultaneously ineach pulse, thus it requires instantaneous wideband compo-nents.

To compare the number of obtained measurements, we notethat for each CRC, WMAR acquires a data cube with NLKsamples, while FAR and CAESAR collect NL samples inthe data matrix. After receive beamforming, the number ofobservations become N , NK and NK, for FAR, CAESAR,and WMAR, respectively, via (23). This indicates that themulti-carrier waveforms of CAESAR and WMAR increase thenumber of measurements after receive beamforming.

The aforementioned radar schemes also differ in their SNR,as the transmitted power and antenna gains differ. Here, as in[24, Page 304, Ch. 6], SNR refers to the ratio of the powerof the signal component to the power of the noise componentafter coherently accumulating the radar returns. To see thisdifference, we consider the case when there exists a targetwith range-Doppler-angle (0, 0, 0), scattering coefficient β andthe noise elements in N are i.i.d. zero-mean proper-complexGaussian with variance σ2. In this case, coherent accumulationof radar returns reduces to

∑l,n,k [YW]l,n,k in WMAR and∑

l,n [YC]l,n in CAESAR (FAR can be regarded as a specialcase of WMAR/CAESAR with K = 1), and the antenna gainsare ρW = L and ρC = L/K, respectively. It follows from(14) that the signal amplitude in radar returns of WMARis L|β|/

√K. After coherent accumulation, the amplitude

becomes NLK · L|β|/√K, leading to a signal power of

N2L4K|β|2, while the power of the noise component becomesNLK · σ2. Hence, the SNR of WMAR is NL3|β|2/σ2. InCAESAR, the amplitude of the signal component in YC isL/K|β|, which becomes NL · L/K|β| after accumulation.Since there are only NL noise elements in CAESAR, the noisepower after accumulation is NL · σ2 and the resultant SNRis NL3|β|2/(K2σ2). Letting K = 1 implies that the SNRof FAR is also NL3|β|2/σ2. The SNR calculation indicatesthat CAESAR has an SNR loss by a factor of K2 compared toWMAR and FAR. This loss stems from the fact that CAESARuses a subset of the antenna array for each carrier, and thus haslower antenna gain than FAR and WMAR. This SNR reductioncan affect the performance of CAESAR in the presence ofnoise, as numerically demonstrated in Section VII.

The above comparison reveals the tradeoff between in-stantaneous bandwidth requirement, number of observations,and post-accumulation SNR. Among these three factors, thenumber of observations is crucial to the target recovery perfor-mance especially in complex electromagnetic environments,where some observations may be discarded due to strong

interference [9], [10]. The proposed multi-carrier schemes,WMAR and CAESAR, are numerically shown to outperformFAR in Section VII, despite the gain loss of CAESAR.CAESAR also achieves performance within a relatively smallgap compared to WMAR, while avoiding the usage of instanta-neous wideband components. The resulting tradeoff betweennumber of beamformed observations and SNR, induced bythe selection of K, is not the only aspect which must beaccounted for when setting the value of K, as it also affects thefrequency agility profile. In particular, smaller K values resultin increased spectral flexibility, as different pulses are morelikely to use non-overlapping frequency sets. Consequently, inour numerical analysis in Section VII we use small values ofK, for which the gain loss between CAESAR and WMAR isless significant, and increased frequency agility is maintained.In addition, CAESAR can also exploit its spatial agilitycharacter, which is not present in FAR or WMAR, to realizea DFRC system, as discussed in our companion paper [22].

B. Comparison to Previously Proposed SchemesSimilarly to CAESAR, previously proposed FDMA-MIMO

radar [11], SUMMeR [13], and FDA radar [14], [15] schemesalso transmit a monotone waveform from each antennaelement while different elements simultaneously transmitmultiple carrier frequencies. The main differences betweenour approaches and these previous methods are beam pat-tern and frequency agility. Due to the transmission of di-verse carrier frequencies from different array elements ofFDMA-MIMO/SUMMeR/FDA, the array antenna does notform a focused transmit beam and usually illuminates a largefield-of-view [16]. This results in a transmit gain loss whichdegrades the performance, especially for track mode, where ahigh-gain directional beam is preferred [16]. By transmittingeach selected frequency with an antenna array (the full arrayin WMAR and a sub-array in CAESAR), our methods achievea focused beam pattern that facilitates accurate target recovery.

Furthermore, FDMA-MIMO and FDA transmit all availablefrequencies simultaneously, and thus do not share the advan-tages of frequency agility, e.g., improved ECCM and EMCperformance, as the multi-carrier version of SUMMeR andthe proposed WMAR/CAESAR. In addition, FDMA-MIMO,SUMMeR and WMAR receive instantaneous wideband signalswith every single antenna, as opposed to FDA [15] andCAESAR, which use narrowband receivers.

To summarize, we compare in Table I the main characteris-tics of these radar schemes. Unlike previously proposed radarmethods, our proposed techniques are based on phased arrayantenna and frequency agile waveforms to achieve directionaltransmit beam and high resistance against interference. Interms of instantaneous bandwidth, CAESAR is preferred forits usage of monotone waveforms and simple instantaneousnarrowband receiver.

VI. PERFORMANCE ANALYSIS OF RANGE-DOPPLERRECONSTRUCTION

Range-Doppler reconstruction plays a crucial role in targetrecovery. This section presents a theoretical analysis of range-Doppler recovery using CS methods. Since both WMAR and

9

TABLE ICOMPARISON BETWEEN RADAR SCHEMES

Characters Frequency agility Beam pattern # of observation Transmit bandwidth Receive bandwidthWMAR Yes Focused Moderate Large Large

CAESAR Yes Moderately focused Moderate Small SmallFAR Yes Focused Small Small Small

FDMA-MIMO No Omnidirectional Large Small LargeSUMMeR Yes Omnidirectional Moderate Small Large

FDA No Omnidirectional Large Small Small

CAESAR are generalizations of FAR, the following analysisis inspired by the study of CS-based FAR recovery in [7].In particular, we extend the results of [7] to multi-carrierwaveforms, as well as to extremely complex electromagneticenvironments, where some transmitted pulses are interferedby intentional or unintentional interference. In the presence ofsuch interference, only partial observations in the beamformedmatrix Z remain effective for range-Doppler reconstruction.To present the analysis, we first briefly review some basicanalysis techniques of CS in Subsection VI-A, followed bythe range-Doppler recovery performance analysis in Subsec-tion VI-B.

A. PreliminariesThere have been extensive studies on theoretical conditions

that guarantee unique recovery for noiseless models or robustrecovery for noisy models [26], [27]. The majority of thesestudies characterize conditions and properties of the measure-ment matrix Φ, including spark, mutual incoherence property(MIP) and restricted isometry property (RIP).

Following [7], we focus on the MIP. A sensing matrix Φ issaid to satisfy the MIP when its coherence, defined as

µ(Φ) := maxi6=j

∣∣∣[Φ]Hi [Φ]j

∣∣∣∥∥ [Φ]i∥∥

2

∥∥ [Φ]j∥∥

2

, (38)

is not larger than some predefined threshold. Bounded co-herence ensures unique or robust recovery using a variety ofcomputationally efficient CS methods. We take `1 optimizationas an example to explain the bounds on matrix coherence. Inthe absence of noise, the uniqueness of the solution to (30) isguaranteed by the following theorem:

Theorem 2 ([28]). Suppose the sensing matrix Φ has coher-ence µ(Φ) < 1

2S−1 . If β solves (30) and has support size atmost S, then β is the unique solution to (30).

In noisy cases, the following result shows that µ(Φ) <1

2S−1 also guarantees stable recovery.

Theorem 3 ([29]). Consider the model (28) with ‖z‖2 ≤ ε ≤η and ‖β‖0 ≤ S, and let sensing matrix Φ have coherenceµ(Φ) < 1

2S−1 . If β solves (31), then

‖β − β‖2≤√

3(1 + µ)

1− (2S − 1)µ(η + ε). (39)

Based on Theorems 2 and 3, we next analyze the coherencemeasure of the sensing matrix Φ in (27) for CAESAR andWMAR (whose sensing matrices are identical), and establishthe corresponding performance guarantees.

B. Performance Analysis

Here, we analyze the range-Doppler reconstruction ofWMAR and CAESAR. Since the sensing matrix Φ is random,we start by analyzing its statistics, and then derive conditionsthat ensure unique recovery by invoking Theorem 2.

We assume that the frequency set Fn is uniformly i.i.d. overX |X ⊂ F , |X | = K . For mathematical convenience, in ouranalysis we adopt the narrow relative bandwidth assumptionfrom [7], i.e., ζn,k ≈ 1, such that (27) becomes

[Φ]k+nK,l+mN = ej2πmM cn,k+j 2πl

N n. (40)

Numerical results in [7] indicate that large relative bandwidthhas negligible effect on the MIP of Φ. In addition, recallthat all the targets precisely lie on the predefined grid points,as assumed in Subsection IV-B. Here, we adopt the on-the-grid assumption for mathematical convenience. Consequently,the accuracy and actual resolutions of range and Dopplerreconstruction results are restricted by the grid intervals, i.e.,2π/M and 2π/N , respectively. In practical scenarios, we mayuse denser grid points, as will be discussed by simulationsin Subsection VII-B. Denser grid enhances the obtainableaccuracy/resolution and alleviates the performance loss whenthe real parameters are off the grid points. However, thedensity of grid points cannot go to infinity, because densergrid affects the incoherence property of the observation matrixwhile increasing the memory requirements and the computa-tional burden. An alternative approach to overcome the needto specify a range-Doppler grid is to utilize off-the-grid CSmethods, see [30], [31].

In complex electromagnetic environments, some of the radarechoes may be corrupted due to jamming or interference.Heavily corrupted echoes are unwanted and should be removedbefore processing in order to avoid their influence on theestimation of target parameters [9], [10]. In this case, thecorrupted radar returns are identified, as such echoes typicallyhave distinct characters, e.g., extremely large amplitudes.These interfered observations are regarded as missing, wherewe consider two kinds of missing patterns: 1) pulse selective,i.e., all observations in certain pulses are missing, whichhappens when the interference in these pulses is intense overall sub-bands; 2) observation selective, namely, only parts ofthe observations are missed when the corresponding pulseis interfered. We consider the first case, assuming that theradar receiver knows which pulses are corrupted, and leavethe analysis under the second case for future investigation.In particular, we adopt the missing-or-not approach [32], inwhich each pulse in z has a probability of 1− u, 0 < u < 1,

10

to be corrupted, and the missing-or-not status of the pulses arestatistically independent of each other.

After removing the corrupted returns, only part of theobservations in the beamformed vector z (28) are used forrange-Doppler recovery. Equivalently, corresponding rows inΦ can be regarded as missing, affecting the coherence of thematrix and thus the reconstruction performance. Denote byΛ ⊂ N the random set of available pulse indexes and byΛ∗ := nK + k |n ∈ Λ, k ∈ K the corresponding index setof available observations. The signal model (28) is now

z∗ = Φ∗β + n∗, (41)

where z∗ := [z]Λ∗ , Φ∗ :=[ΦT]TΛ∗

, and n∗ := [n]Λ∗ .Consider the inner product of two columns in Φ∗, denoted

[Φ∗]l1 and [Φ∗]l2 , corresponding to grid points(

2πm1

M , 2πn1

N

)and

(2πm2

M , 2πn2

N

), respectively, l1, l2 ∈ 0, 1, . . . ,MN − 1,

m1,m2 ∈ M, n1, n2 ∈ N . While there are M2N2 dif-ferent pairs of (l1, l2), the magnitude of the inner product∣∣ [Φ∗]Hl1 [Φ∗]l2

∣∣, which determines the coherence of Φ∗, takesat most MN −1 distinct random values. To see this, note that

[Φ∗]Hl1

[Φ∗]l2 =∑n∈Λ

K−1∑k=0

e−j2πm1M cn,k−j

2πn1N nej

2πm2M cn,k+j

2πn2N n

=∑n∈Λ

K−1∑k=0

e−j2πm1−m2M cn,k−j2π

n1−n2N n, (42)

indicating that the inner product depends only on the differ-ence of the grid points, i.e., m1 −m2 and n1 − n2, and noton the individual values of the column indices l1 and l2. Itfollows from (42) that the MIP of Φ∗ can be written as

µ(Φ∗) = max(∆m,∆n)6=(0,0)

1

|Λ|K∑n∈Λ

K−1∑k=0

e−j∆mcn,ke−j∆nn. (43)

where ∆m := 2πm1−m2

M , ∆n := 2π n1−n2

N take values in thesets ∆m∈± 2πm

M m∈M and ∆n ∈ ± 2πnN n∈N , respectively.

Next, we define

χn (∆m,∆n) := IΛ(n)1

K

K−1∑k=0

e−j∆mcn,k−j∆nn, (44)

where the random variable IΛ(n) satisfies IΛ(n) = 1 whenn ∈ Λ and 0 otherwise. In addition, let

χ (∆m,∆n) :=

N−1∑n=0

χn (∆m,∆n) . (45)

Some of the magnitudes |χ (∆m,∆n)| are duplicated sinceχ (∆m,∆n) = χ (∆m ± 2π,∆n ± 2π) and χ (∆m,∆n) =χ∗ (−∆m,−∆n). To eliminate the duplication and removethe trivial nonrandom value χ(0, 0), we restrict the values of∆m and ∆n to ∆m ∈ 2πm

M m∈M and ∆n ∈ 2πnN n∈N ,

respectively, and define the set

Ξ := (∆m,∆n) |(m,n) ∈M×N\(0, 0) , (46)

with cardinality |Ξ| = MN − 1, such that each value of∣∣ [Φ∗]Hl1 [Φ∗]l2

∣∣ (except the trivial case l1 = l2) correspondsto a single element of the set Ξ. We can now write (43) as

µ(Φ∗) = max(∆m,∆n)∈Ξ

1

|Λ||χ (∆m,∆n)| . (47)

The coherence in (47) is a function of the dependent randomvariables χ and |Λ|. To bound µ, we derive bounds on χ and|Λ|, respectively. To this aim, we first characterize the statisti-cal moments of χn (∆m,∆n) for some fixed (∆m,∆n) ∈ Ξ,which we denote henceforth as χn, in the following lemma:

Lemma 4. The sequence of random variables χn satisfies

E [χn] =

uej∆nn, if ∆m = 0,

0, otherwise,(48)

N−1∑n=0

D [χn] =

u(1− u)N, if ∆m = 0,M−K

(M−1)KuN, otherwise.(49)

Furthermore, for each n ∈ N ,

|χn − E[χn]| ≤ 1, w.p. 1. (50)

Proof. See Appendix B.

Using Lemma 4, the probability that the magnitude |χ| isbounded can be derived as in the following Corollary:

Corollary 5. Let V := maxu(1− u)N, M−K

(M−1)KuN

. Forany (∆m,∆n) ∈ Ξ and ε ≤ V it holds that

P(|χ| ≥

√V + ε

)≤ e− ε2

4V . (51)

Proof. Based on the definition (44) and the independence as-sumption on the frequency selection and missing-or-not statusof each pulse, it holds that χn − E [χn]n∈N are independentzero-mean complex-valued random variables. Now, since

N−1∑n=0

E [χn] =

u∑N−1n=0 e

j∆nn, if ∆m = 0,

0, otherwise,(52)

according to (48) and∑N−1n=0 e

j∆nn = 1−ej∆nN1−ej∆n equals

0 for ∆n ∈ 2πnN n∈N\0, recalling that (∆m,∆n) 6=

(0, 0), we have∑N−1n=0 E [χn] = 0. Then, it holds that∑N−1

n=0 (χn − E [χn]) =∑N−1n=0 χn = χ. Combining Bern-

stein’s inequality [33, Thm. 12] with the fact that by (50),|χn − E[χn]| ≤ 1, results in (51).

We next derive a bound on the number of effective pulses|Λ| in the following lemma:

Lemma 6. For any t > 0, it holds that

P (|Λ| ≤ uN − t) < e−2t2

N . (53)

Proof. Since, by its definition, |Λ| obeys a binomial distribu-tion, (53) is a direct consequence of [34, Thm. 1].

Based on the requirement µ(Φ∗) >1

2K−1 in Theorem 2,we now use Corollary 5 and Lemma 6 to derive a sufficientcondition on the radar parameters M , N , K, as well asthe intensity of interference 1 − u, guaranteeing that themeasurement matrix Φ∗ meets the requirement with highprobability. This condition is stated in the following theorem:

Theorem 7. For any constant δ > 0, the coherence of Φ∗satisfies P

(µ(Φ∗) ≤ 1

2S−1

)≥ 1− δ when

S ≤ uN/√V

1+√

2 (log 2|Ξ|−log δ)

1+ 12√

2Nu

2−√

N

32V+

1

2. (54)

11

Proof. See Appendix C.

Recall that the value of V depends on the quantities u, Kand M . When u is reasonably large such that 1−u ≥ M−K

(M−1)K ,we have V = M−K

(M−1)KuN . When there is no noise in radarreturns, a number of scattering points (on the grid) in the scaleof S = O

(√KuN

logMN

)guarantees a unique reconstruction of

range-Doppler parameters with high probability according toTheorems 2 and 7. Note that this rather simple asymptoticcondition assumes that M−1

M−K ≈ 1, i.e., that the overallnumber of available frequencies M is substantially larger thanthe number of frequencies utilized in each pulse K, thusensuring the agile character in frequency domain. Comparedto the asymptotic condition O

(√N

logMN

)of FAR with

full observations [7], we find that the presence of corruptedobservations, i.e. when u < 1, leads to degraded range-Doppler reconstruction performance. However, by increasingthe number of transmitted frequencies in each pulse K whilemaintaining K M , the performance deterioration due tomissing observations can be mitigated, enhancing the inter-ference immunity of the radar in extreme electromagneticenvironments. In the special case that K = 1 and u = 1, i.e.,FAR in an interference free environment, the two conditionscoincide as O

(√KuN

logMN

)= O

(√N

logMN

).

The above condition is proposed for the noiseless case, in-dicating that the inherent target/clutter reconstruction capacityincreases with

√K. In practical noisy cases, the reconstruction

performance does not monotonically increase with K, becausethe transmit power of each frequency decreases with K, thusdegrading the SNR in both CAESAR and WMAR. Particu-larly, in CAESAR, larger K means that less antennas (L/K)are allocated to each frequency, which affects the radiationbeam and enlarges the gain loss. In addition, a small Kmaintains the practical advantages of frequency agility in termsof, e.g., ECCM and EMC performance.

VII. SIMULATION RESULTS

In this section, we numerically compare the performance ofWMAR, CAESAR, and FAR in noiseless/noisy, clutter, and/orjamming environments. The performance is evaluated in termsof target detection probability, accuracy and resolution, prob-ability of correct reconstruction, and mutual interference, aspresented in Subsections VII-A - VII-D, respectively.

We consider a frequency band starting from fc = 9 GHz,with M = 4 available carriers and carrier spacing of ∆f = 1MHz. The radar system is equipped with an antenna arrayof L = 10 elements with spacing of d = c

2fc, and utilizes

N = 32 pulses focusing on θ = 0. CAESAR and WMAR useK = 2 frequencies at each pulse. In noisy scenarios, we usethe term SNR for the post-accumulation SNR of WMAR/FAR,i.e., NL3|β|2/σ2 as derived in Subsection V-A. To guaranteefair comparison, we use the same definition for all the radarschemes. In the presence of jamming, we test the pulseselective missing pattern. In order to implement target recoveryvia Algorithm 1 for the three radar schemes, we use the convexoptimization toolbox [35] to implement basis pursuit (30) innoiseless cases or the Lasso algorithm (32) with λ = 0.5 innoisy setups for range-Doppler reconstruction.

A. Target Detection in Clutter Environment

We compare the detection performance of the proposedWMAR and CAESAR schemes with FAR and conventionalfixed frequency radars, which can be regarded as a special caseof WMAR, CAESAR and FAR, with K = 1 and Ωn,k = fn =fc. To that aim, we evaluate the detection probabilities, Pd, ofa moving target under ground clutter environment. The groundclutter is modeled as radar returns from many static scatteringpoints with velocity being zero. Thus, moving targets andground clutter are distinguishable by observing their Dopplervalues. We denote by C the index set corresponding to zeroDoppler, i.e., C = n + mN |n = 0,m ∈ M, and by Cc itscomplementary set, i.e., Cc = n + mN |n ∈ N\0,m ∈M. The number of target scattering points is St = 1.The normalized range and Doppler parameter of the targetis determined by randomly selecting the index i from Cc.The target intensity |βi|, i ∈ Cc is selected to match thedesired SNR. The number of clutter scattering points is setas Sc = 1000, and the intensity of each scattering point isunity, with a random phase uniformly distributed over [0, 2π).The normalized range parameters rc of these scatterers areuniformly distributed over [0, 2π), and are not assumed tolie on the predefined grid points. Since these scatterers haveidentical velocity, the superposition of their echoes, i.e., theclutter signal, can be represented by radar returns from thegrid points indexed by C. Echoes from both target and clutterare treated as unknown, and are reconstructed simultaneouslyby CS. Particularly, we apply Lasso (32) for range-Dopplerreconstruction, yielding β, where elements indexed in C areregarded as equivalent clutter intensities and the remainingelements are regarded as intensities of moving targets. The Pdcurves are plotted versus SNR, and we choose σ2 = 1.

The simulations are carried out in two parts. In the firstpart, there are only clutter and additive noises in the radarreturns, without returns from moving targets. The result-ing radar measurements are used to determine the detectionthresholds for all radar schemes, respectively, under a givenprobability of false alarm, denoted Pfa. In the second part,these thresholds are used to detect the existence of a targetfrom the received echoes, which now include noise as wellas returns from both clutter and moving target. The targetdetection procedure is based on the estimated parameters|β|, as detailed in Subsection IV-B. In the first part, we setPfa = 10−3 and perform 105 Monte Carlo trials. Whileradar systems typically operate at lower values of Pfa, weuse this value for computational reasons. It can be faithfullysimulated under the given number of trials, and the selectedvalue of Pfa provides a characterization and understanding ofthe behavior of the considered radar schemes. A false alarmis proclaimed if any nonzero-Doppler element in |β| exceedscertain threshold Th, i.e., maxi∈Cc |βi| > Th. In the secondpart, we execute 200 Monte Carlo trials. A successful detectionis proclaimed if the estimated intensity of the moving targetis larger than the threshold, |βi| > Th, i ∈ Cc. To evaluate theinfluence of missing observation caused by jamming, we setthe survival rate u = 0.7 and compare the resulting probabilityof detection, Pd curves with those of the full observation cases

12

Pd

u= u= u=

Fig. 2. Detection probabilities Pd versus SNR. The label “Fixed” representsthe fixed frequency radar, and the Pd of FAR with u = 0.7 are zeros in thetested scenarios.

in Fig. 2. Note that the detection thresholds for these Pd curvesare calculated individually.

As shown in Fig. 2, the fixed frequency radar has higherdetection probabilities than the counterparts of frequency agileschemes. The advantage of fixed frequency radar stems fromthe property of its observation matrix Φ ∈ CKN×MN , whereK = M = 1 and Φ becomes an orthogonal matrix, benefitingthe Doppler reconstruction performance of CS methods. Whilein the frequency agile schemes, generally it holds that M > K,resulting in an incomplete observation matrix and degradationof clutter/target recovery performance. However, we note thatthe fixed frequency scheme is vulnerable in a jamming en-vironment. In the full observation cases, WMAR outperformsFAR because of the increased number of observations. ThoughCAESAR has identical number of observations with WMARafter receive beamforming, the Pd values of CAESAR areless than those achieved by WMAR with a SNR gap ofapproximately 6 dB. This follows since CAESAR suffers froman antenna gain loss of K2 as discussed in Subsection V-A(i.e., 6 dB since K = 2). When some of the observationsare missing due to jamming, the detection probabilities areaffected. Both WMAR and CAESAR suffer from an SNRloss of approximately 3 dB, while FAR fails to detect anymoving target in the scenarios under test. When FAR islacking in radar observations, the mutual coherence propertyof its observation matrix Φ∗ becomes degraded, leading tomany spurious peaks of high intensities in the recovery results|β|. These spurious peaks significantly increase the detectionthreshold, thus operating under a fixed Pfa of 10−3 results innotably reduced detection probability Pd.

To summarize, we find from the simulation results that 1)the proposed multi-tones schemes (WMAR and CAESAR)enhance the immunity against missing data over the single-tone FAR, and 2) WMAR outperforms CAESAR due to itshigher antenna gain, which comes at the cost of increasedinstantaneous bandwidth.

B. Accuracy and Resolution

Here, we compare the range, Doppler and angle estimateresults under different range-Doppler grid points. To this aim,

5 10 15 20 25 30

SNR (dB)

-6

-4

-2

0

2

RMSE

of R

ange (d

B)

CAESARFARWMARCAESAR-2FAR-2WMAR-2

Fig. 3. Range accuracy of range-Doppler reconstruction results.

two sets of range-Doppler grid points are tested: one usesthe standard grid as mentioned in Subsection IV-B, wherethe intervals of consecutive range and Doppler grid pointsare ∆r = 2π

M and ∆v = 2πN , respectively; The latter uses

a denser grid, setting ∆r = 2π2M and ∆v = 2π

2N , and theconsequent simulation results are denoted with label “-2”, e.g.,CAESAR-2. The number of scattering points is S = 1 withoutclutter. The normalized range-Doppler parameter, (r, v), of thescattering point is uniformly, randomly set over [0, 2π)2, andthe angle is randomly set within the beam ϑ ∈ Θ. Under thissetting, the ground truth of the range-Doppler parameter maybe off the grid, which leads to inevitable estimation error. TheCS method applied for range-Doppler reconstruction is basedon (32), and we estimate range-Doppler, denoted by (ˆr, ˆv),from the index of the element with maximum magnitude in β.We then use root mean squared error (RMSE) as the metric of

accuracy, defined by√

E[(r − ˆr)2], taking normalized rangeas an example. The remaining settings are the same as thoseused in Subsection VII-A. We run 500 Monte Carlo trials andthe range, Doppler and angle accuracy results are shown inFigs. 3, 4 and 5, respectively.

As expected, the RMSEs become lower when we increaseSNR, while we observe in Figs. 3 and 4 that the RMSEs ofrange and Doppler estimates reach error floors as the SNRincreases. The error floors depend on the grid intervals ∆r

or ∆v , and denser grid points lead to lower error floors. Theresults also reveal that WMAR and FAR have similar accuracyperformance, while CAESAR has an SNR loss of 6 dB inmoderate SNR levels because of its lower antenna gain. Inhigh SNR scenarios, the RMSEs of CAESAR also reach theerror floor.

We next examine the ability of CAESAR, FAR and WMARin separating closely spaced scattering points, i.e., obtainableresolution. In the simulations, we use dense grid points withintervals ∆r = 2π

5M and ∆v = 2π5N . We consider two closely

spaced scattering points, of which angles are set ϑ = 0and range-Doppler parameters are on the grid. Particularly,we fix the range-Doppler parameter of one scattering point(r1, v1) = (0, 0) and change the counterpart of the other scat-tering from [∆r,

2πM ] × [∆v,

2πN ], such that the range/Doppler

separation (∆R/∆V ) between two points are changed. In this

13

5 10 15 20 25 30

SNR (dB)

-15

-10

-5

0RMSE

of D

oppler (d

B)

CAESARFARWMARCAESAR-2FAR-2WMAR-2

Fig. 4. Doppler accuracy of range-Doppler reconstruction results.

5 10 15 20 25 30

SNR (dB)

-25.0

-22.5

-20.0

-17.5

-15.0

-12.5

RMSE

of A

ngle (d

B)

CAESARFARWMARCAESAR-2FAR-2WMAR-2

Fig. 5. Angle accuracy versus SNR.

experiment, we disregard noise and the scattering intensitiesare both |β1| = |β2| = 1 with random phase. We use(30) for range-Doppler recovery, and the two most dominantelements in the estimate β are regarded as scattering points.The indices of these two elements are compared with thecorresponding ground truth, and a successful recovery (alsoreferred to a hit) is proclaimed if both indices are correct. Thenumber of Monte Carlo trails are 500. The achievable hit rateresults versus separation between scattering points are shownin Fig. 6. The results demonstrate that all the frequency agileschemes, CAESAR, FAR and WMAR have close performancein resolution, while the hit rates of CAESAR and WMARare slightly higher than those of FAR, because they havemore observations and the measurement matrix Φ has bettercoherence property.

C. Reconstruction of Multiple Scattering PointsIn this subsection, we evaluate the proposed radar schemes

in recovering a set of S scattering points in noiseless andnoisy setups. In both scenarios, we use the standard grid pointswith grid intervals (∆r,∆v) = (2π/M, 2π/N). The range-Doppler parameters of scattering points are randomly selectedfrom the grid points, angle parameters are randomly set fromthe continuous set Θ, and scattering intensities are all set tounity. We apply CS methods for range-Doppler recovery, andthe indices of S most significant entries in β are regarded as

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Doppler se

paratio

n ∆V×N 2π

CAESAR

0.2 0.4 0.6 0.8 1.0Range separation ∆R ×M

2π

FAR

0.2 0.4 0.6 0.8 1.0

WMAR

0.75 0.80 0.85 0.90 0.95 1.00

Fig. 6. Hit rates of separating closely spaced scattering points.

S

u= u= u=

Fig. 7. Range-Doppler recovery versus S, noiseless setting.

elements of the estimated support set. Hit rates are applied asperformance metric, and a hit is proclaimed if the obtainedsupport set is identical to the ground truth, which means allthe range-Doppler parameters are reconstructed correctly.

In the noiseless experiment, we simulate different numbersof recoverable scattering points, S. We set the survival rateu = 0.4 for jamming environments. The resulting hit ratesversus S ∈ 1, . . . , N−1 are depicted in Fig. 7. As expected,the hit rates decrease as S increases. The performance ofCAESAR is within a very small gap of that achievable usingWMAR, because CAESAR and WMAR use the same amountof transmitted frequencies K, and the number of beamformedmeasurements is also the same. Hit rates of CAESAR andWMAR exceed that of FAR significantly. This gain stemsfrom the fact that transmitting multi-carriers in each pulse ofCAESAR and WMAR increases the number of observations,and thus raises the number of recoverable scattering points.

We then consider the noisy case, and compare the range-Doppler recovery performance versus SNR, which is changedby varying σ2. We set S = 10, and we let u = 0.4 forthe jamming environment. The hit rates of the range-Dopplerparameters are depicted in Fig. 8.

Observing Fig. 8, we note that, as expected, WMARachieves the best performance in range-Doppler reconstruc-tion. While WMAR and CAESAR have the same number ofobservations, CAESAR has a lower antenna gain as noted inSubsection V-A, which results in its degraded performancecompared to WMAR. In the full observation case with highSNRs, i.e., SNR ≥ 25 dB, CAESAR has higher hit ratesthan FAR due to the advantage of increased number oftransmitted frequencies, while in low SNRs of less than 20

14

u= u= u=

Fig. 8. Range-Doppler recovery versus SNR.

dB, FAR exceeds CAESAR owing to its higher antenna gain.In the jamming scenario, CAESAR outperforms FAR, andthat FAR almost fails to reconstruct scattering points (with hitrates around 0.25). The superiority of WMAR/CAESAR overFAR demonstrates the advantage of the proposed multi-carrierwaveforms.

From the experimental results in Subsections VII-A - VII-C,we find that the multi-carrier signals used by CAESAR andWMAR significantly enhance range-Doppler reconstructionperformance over the monotone waveform in FAR. The advan-tage becomes more distinct in jamming environments, wheresome radar measurements are invalid. In reasonably highSNR scenarios, the reconstruction performance of CAESAR,which uses narrowband constant modulus waveforms for eachantenna element, approach those of WMAR, which usesinstantaneously wideband waveforms.

D. Mutual Interference

One of the main advantages of frequency agile transmissionis its relatively low level of mutual interference, which impliesthat multiple transmitters can coexist in dense environments.To demonstrate this property of the proposed radar schemes,which all utilize some level of frequency agility, we nextevaluate the unintended mutual interference of closely placedradars transmitting the same waveform pattern. We comparethe frequency agile schemes of FAR, WMAR and CAESAR,with an instantaneous wideband radar which transmits allsubbands simultaneously. In the simulation, we consider ascenario with 6 radars operating independently. Mutual inter-ference occurs if a reference radar is receiving echoes whileanother radar is transmitting at the same subcarriers with theirantenna beams directed towards each other. In this case theechoes of the reference radar at the conflicted subcarriers arecorrupted. The level of mutual interference is measured by theaverage number of uncorrupted subcarriers, denoted Ku.

We use PInt to represent the probability that one radar mayinterfere the reference radar, i.e., that it is radiating duringthe reception period of the reference radar and their beamsare directed towards each other. The number of subcarrierstransmitted in each pulse varies from K = 1 to K = 4, whereK = 1 represents the FAR while K = 4 represents the radarusing full bandwidth. As WMAR and CAESAR transmit the

0.0 0.2 0.4 0.6 0.8 1.0

Interference probability ℙInt

0

1

2

3

4

Average uncorrupted su

bcarrie

rs K

u FAR(K=1)WMAR/CAESAR(K=2)WMAR/CAESAR(K=3)Wideband radar(K=4)

Fig. 9. The average number of uncorrupted subcarriers Ku versus PInt forradar schemes with different number of transmit subcarriers, which variesfrom K = 1 to K = 4.

same number of subcarriers for a specific K, their performanceon mutual interference is the same. To demonstrate the mutualinterference intensity versus interference probabilities, we sim-ulate 106 Monte Carlo trials for each interference probabilityand calculate the average number of uncorrupted subcarriers asshown in Fig. 9. From the results, we observe that, as expected,when the interference probability is small, e.g., less than0.2, radar systems transmitting more subcarriers are capableof effectively utilizing their bandwidth reliably. However, asthe probability of interference grows, wideband radar inducesevere mutual interference, resulting in a negligible averagenumber of uncorrupted subcarriers for PInt > 0.6. Thefrequency agile schemes, such as FAR and WMAR/CAESARoperating with K = 2, are still capable of reliably utilizing anotable portion of their bandwidth in the presence of such highinterference. These results indicate that the less frequency agilethe scheme is, the severer the mutual interference becomes inthe high interference probability regime.

VIII. CONCLUSION

In this work we developed two multi-carrier frequency agileschemes for phase array radars: WMAR, which uses widebandwaveforms; and CAESAR, which transmits monontone signalsand introduces spatial agility. We modeled the received radarsignal, and proposed an algorithm for target recovery. We thencharacterized theoretical recovery guarantees. Our numericalresults demonstrate that our proposed schemes achieve en-hanced survivability in extreme electromagnetic environments.Furthermore, it is shown that CAESAR is capable of achievingperformance which approaches that of wideband radar, whileutilizing narrowband transceivers. An additional benefit whichfollows from the introduction of frequency and spatial agilityis the natural implementation of CAESAR as a DFRC system,studied in a companion paper.

APPENDIX

A. Proof of Lemma 1In the following we prove (24) for CAESAR. The proof for

WMAR follows similar arguments and is omitted for brevity.

15

Substituting the definitions of w,P and Y into (23) yields

Zk,n =

L−1∑l=0

wl (θ,Ωn,k) [p(n, k)]l

S−1∑s=0

βsejrscn,k

ejvsnζn,ke−j2πΩn,kldsinϑs/cρC(n, k, δϑs)

=

S−1∑s=0

L−1∑l=0

[p(n, k)]l βsejrscn,kejvsnζn,k

e−j2πΩn,kld(sinϑs−sin θ)/cρC(n, k, δϑs)

=

S−1∑s=0

βsejrscn,kejvsnζn,kρ2

C(n, k, δϑs). (A.1)

Recall that when δϑs ≈ 0 it holds that ρC(n, k, δϑs) ≈ L/K =√αK . Then, (A.1) reduces to (24), proving the lemma.

B. Proof of Lemma 4We first prove (48) and (49), after which we address (50).1) Proof of (48) and (49): For brevity, let p = −∆m,

q = −∆n, In = IΛ(n), and B =(MK

). We set

(MK

)= 0 when

M ≤ 0 or K < 0, and(M0

)= 1 when M > 0.

We first compute E [χn] = 1KE[Ine

jqn∑K−1k=0 ejpcn,k

]. The

expectation is taken over the indicator In and frequency codescn,k. Since they are independent and E [In] = u, it holds that

E [χn] =uejqn

KE

[K−1∑k=0

ejpcn,k

]. (B.1)

Since K frequencies are selected uniformly (but not indepen-dently), it follows that

E

[K−1∑k=0

ejpcn,k

]=

1

B

B−1∑i=0

K−1∑k=0

ejpmi,k , (B.2)

where mi,k denotes the k-th frequency in the i-th combination.Out of these B combinations, there are

(M−1K−1

)that contain a

given selection m ∈M. Thus, we have thatB−1∑i=0

K−1∑k=0

ejpmi,k =

(M−1

K−1

)M−1∑m=0

ejpm=BK

M

M−1∑m=0

ejpm. (B.3)

Substituting (B.3) into (B.2) yields

E

[K−1∑k=0

ejpcn,k

]=K

M

M−1∑m=0

ejpm =K

M

1− ejpM

1− ejp. (B.4)

As p∈

2πmM

m∈M, it holds that E

[∑K−1k=0 ejpcn,k

]= K if

p = 0 and zero otherwise. Substituting this into (B.1), we have

E [χn] =

uejqn, if p = 0,

0, otherwise,(B.5)

which proves (48).To obtain D [χn] := E

[|χn−E[χn]|2

], we consider two

cases, p = 0 and p 6= 0. When p = 0, we have χn = Inejqn

and

E[∣∣Inejqn − uejqn∣∣2] = E

[(In − u)

2]

(a)= E

[In + u2 − 2Inu

] (b)= u− u2, (B.6)

where (a) holds since I2n = In and in (b) we apply E [In] = u.

When p 6= 0, the random variable χn has zero mean, andits variance is given by

E[|χn|2

]= E

InK

∣∣∣∣∣K−1∑k=0

ejpcn,k+jqn

∣∣∣∣∣2

=1

K2E

[In ·

K−1∑k=0

K−1∑k′=0

ejp(cn,k−cn,k′)

]

=u

K2E

[K−1∑k=0

K−1∑k′=0

ejp(cn,k−cn,k′)

], (B.7)

where we use I2n = In. To compute (B.7), we note that

E

[K−1∑k=0

K−1∑k′=0

ejpcn,k−jpcn,k′

]

=

(M−1K−1

)B

M−1∑m=0

ejp·0 +

(M−2K−2

)B

M−1∑m=0

M−1∑m′=0,m′ 6=m

ejp(m−m′)

(a)=

(M−1K−1

)−(M−2K−2

)B

M +

(M−2K−2

)B

M−1∑m=0

M−1∑m′=0

ejp(m−m′), (B.8)

where (a) follows sinceM−1∑m=0

M−1∑m′=0,m′ 6=m

ejp(m−m′) in the sec-

ond term can be replaced byM−1∑m=0

M−1∑m′=0

ejp(m−m′) −

M−1∑m=0

e0.

From the derivation of (B.5), it holds that for p 6= 0 the secondsummand in (B.8) vanishes, resulting in

E

[K−1∑k=0

K−1∑k′=0

ejpcn,k−jpcn,k′

]=

(M −K)K

M − 1. (B.9)

Plugging (B.9) into (B.7), we obtain

E[|χn|2

]=

M −K(M − 1)K

u, if p 6= 0. (B.10)

Finally, to prove (49), we calculate∑N−1n=0 D [χn] for p = 0

and p 6= 0. When p = 0, from (B.6), we have thatN−1∑n=0

D [χn] =

N−1∑n=0

(u− u2

)=(u− u2

)N. (B.11)

When p 6= 0, it follows from (B.10) thatN−1∑n=0

D [χn] =M −K

(M − 1)KuN. (B.12)

Combining (B.11) and (B.12) proves (49).2) Proof of (50): We again consider the two cases p = 0

and p 6= 0 separately: When p = 0, it follows from (B.5) that

|χn − E [χn]|2 =∣∣(In − u) ejqn

∣∣2 = (In − u)2

=

(1− u)2 ≤ 1, if In = 1,

u2 ≤ 1, otherwise.(B.13)

When p 6= 0, |χn − E [χn]|2 = |χn|2, which is not larger than1 by definition of χn (44), thus proving (50).

16

C. Proof of Theorem 7By fixing some positive ε ≤ V , setting t =

√N8V ε and

ε′ :=

√V + ε

uN −√

N8V ε

=

√V + ε

uN − t, (C.1)

we have that for any (∆m,∆n) ∈ Ξ

P(|χ||Λ|≥ ε′

)(a)

≤ P(|χ| ≥

√V + ε ∪ |Λ| ≤ uN − t

)≤ P

(|χ| ≥

√V +ε

)+P (|Λ| ≤ uN−t)

(b)

≤ 2e−ε2

4V . (C.2)

Here (a) holds since the event |χ||Λ| ≥√V+ε

uN−t implies that atleast one of the conditions |χ| ≥

√V + ε and |Λ| ≤ uN − t

is satisfied; and (b) follows from Corollary 5 and Lemma 6.Using the bound (C.2) on the magnitude of the normalized

correlation, we next bound the probability of µ(Φ∗) to exceedsome constant. By applying the union bound to (47), we have

P (µ(Φ∗) ≥ ε′) ≤∑

(∆m,∆n)∈Ξ

P(|χ (∆m,∆n) |

|Λ|> ε′

)≤ 2|Ξ|e− ε2

4V . (C.3)

According to (C.3), for any ε > 0, it holds that

P (µ(Φ∗) ≤ ε′) ≥ 1− 2|Ξ|e− ε2

4V , (C.4)

where ε′ is obtained from ε via (C.1). The right hand side of(C.4) is not smaller than 1− δ when δ ≥ 2|Ξ|e− ε2

4V , implyingthat P (µ(Φ∗) ≤ ε′) ≥ 1− δ when ε satisfies

ε ≥√

2V (log 2|Ξ| − log δ). (C.5)

Finally, by (C.1), fixing ε′ = 1/(2S − 1) implies that

S =uN −

√N8V ε

2√V + 2ε

+1

2=uN +

√N/8

2√V + 2ε

−√

N

32V+

1

2. (C.6)

Substituting (C.5) into (C.6) proves (54).

REFERENCES

[1] S. R. J. Axelsson, “Analysis of random step frequency radar andcomparison with experiments,” IEEE Trans. Geosci. Remote Sens.,vol. 45, no. 4, pp. 890–904, April 2007.

[2] J. Yang, J. Thompson, X. Huang, T. Jin, and Z. Zhou, “Random-frequency SAR imaging based on compressed sensing,” IEEE Trans.Geosci. Remote Sens., vol. 51, no. 2, pp. 983–994, 2013.

[3] Y. Liu, G. Liao, J. Xu, Z. Yang, and Y. Zhang, “Adaptive OFDM inte-grated radar and communications waveform design based on informationtheory,” IEEE Commun. Let., vol. 21, no. 10, pp. 2174–2177, Oct 2017.

[4] D. Cohen, K. V. Mishra, and Y. C. Eldar, “Spectrum sharing radar:Coexistence via Xampling,” IEEE Trans. Aerosp. Electron. Syst., vol. 54,no. 3, pp. 1279–1296, 2018.

[5] D. Cohen and Y. C. Eldar, “Sub-nyquist radar systems: Temporal,spectral, and spatial compression,” IEEE Signal Processing Magazine,vol. 35, no. 6, pp. 35–58, Nov 2018.

[6] T. Huang, Y. Liu, G. Li, and X. Wang, “Randomized stepped frequencyISAR imaging,” in Proc. IEEE RadarConf, May 2012, pp. 0553–0557.

[7] T. Huang, Y. Liu, X. Xu, Y. C. Eldar, and X. Wang, “Analysis ofFrequency Agile Radar via Compressed Sensing,” IEEE Trans. SignalProcess., vol. 66, no. 23, pp. 6228–6240, Dec 2018.

[8] T. Huang, Y. Liu, Y. C. Eldar, and X. Wang, “Multiple carrier agile radarvia compressed sensing,” in International Workshop on CompressedSensing applied to Radar, Multimodal Sensing, and Imaging (CoSeRa)2018, 2018.

[9] Y. Wang, J. Li, and P. Stoica, Spectral Analysis of Signals: The MissingData Case. Morgan & Claypool, 2006.

[10] W. Rao, G. Li, X. Wang, and X. Xia, “ISAR imaging of maneuveringtargets with missing data via matching pursuit,” in Proc. IEEE Radar-Con, May 2011, pp. 124–128.

[11] H. Sun, F. Brigui, and M. Lesturgie, “Analysis and comparison of MIMOradar waveforms,” in 2014 International Radar Conference, Oct 2014.

[12] D. Cohen, D. Cohen, and Y. C. Eldar, “High resolution FDMA MIMOradar,” IEEE Transactions on Aerospace and Electronic Systems, pp.1–1, 2019.

[13] D. Cohen, D. Cohen, Y. C. Eldar, and A. M. Haimovich, “SUMMeR:Sub-Nyquist MIMO Radar,” IEEE Trans. Signal Process., vol. 66,no. 16, pp. 4315–4330, Aug 2018.

[14] P. Antonik, M. C. Wicks, H. D. Griffiths, and C. J. Baker, “Frequencydiverse array radars,” in 2006 IEEE Conference on Radar, April 2006.

[15] Y. Liu, H. Ruan, L. Wang, and A. Nehorai, “The random frequencydiverse array: A new antenna structure for uncoupled direction-rangeindication in active sensing,” IEEE J. Sel. Topics Signal Process., vol. 11,no. 2, pp. 295–308, Mar 2017.

[16] E. Brookner, “MIMO radar demystified and where it makes sense touse,” in 2013 IEEE International Symposium on Phased Array Systemsand Technology, Oct 2013, pp. 399–407.

[17] S. Chen and F. Xi, “Quadrature compressive sampling for multibandradar echo signals,” IEEE Access, vol. 5, pp. 19 742–19 760, 2017.

[18] A. Hassanien, M. G. Amin, Y. D. Zhang, and F. Ahmad, “Phase-modulation based dual-function radar-communications,” IET Radar,Sonar Navigation, vol. 10, no. 8, pp. 1411–1421, 2016.

[19] C. Sturm and W. Wiesbeck, “Waveform design and signal processingaspects for fusion of wireless communications and radar sensing,” Proc.IEEE, vol. 99, no. 7, pp. 1236–1259, Jul 2011.

[20] D. Ma, T. Huang, Y. Liu, and X. Wang, “A novel joint radar andcommunication system based on randomized partition of antenna array,”in Proc. IEEE ICASSP, April 2018, pp. 3335–3339.

[21] D. Ma, N. Shlezinger, T. Huang, Y. Liu, and Y. C. Eldar, “Joint radar-communications strategies for autonomous vehicles,” arXiv preprintarXiv:1909.01729, 2019.

[22] T. Huang, N. Shlezinger, X. Xu, Y. Liu, and Y. C. Eldar, “A dual-function radar communication system using index modulation,” in theIEEE International Workshop on Signal Processing Advances in WirelessCommunications (SPAWC) 2019, 2019.

[23] S. U. Pillai, Array Signal Processing, C. S. Burrus, Ed. Springer, 1989.[24] M. A. Richards, Fundamentals of radar signal processing. McGraw-

Hill Education, 2005.[25] S. R. J. Axelsson, “Suppression of noise floor and dominant reflectors

in random noise radar,” in 2006 International Radar Symposium, May2006, pp. 1–4.

[26] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Appli-cations. Cambridge University Press, 2012.

[27] Y. C. Eldar, Sampling theory: Beyond bandlimited systems. CambridgeUniversity Press, 2015.

[28] J. Fuchs, “On sparse representations in arbitrary redundant bases,” IEEETrans. Inform. Theory, vol. 50, no. 6, pp. 1341–1344, June 2004.

[29] T. T. Cai, L. Wang, and G. Xu, “Stable recovery of sparse signals andan oracle inequality,” IEEE Trans. Inform. Theory, vol. 56, no. 7, pp.3516–3522, 2010.

[30] T. Huang, Y. Liu, H. Meng, and X. Wang, “Adaptive matching pursuitwith constrained total least squares,” EURASIP Journal on Advances inSignal Processing, vol. 2012, no. 1, p. 76, 2012.

[31] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensingoff the grid,” IEEE Transactions on Information Theory, vol. 59, no. 11,pp. 7465–7490, Nov 2013.

[32] D. S. Pham and A. M. Zoubir, “Estimation of multicomponent poly-nomial phase signals with missing observations,” IEEE Transactions onSignal Processing, vol. 56, no. 4, pp. 1710–1715, April 2008.

[33] D. Gross, “Recovering low-rank matrices from few coefficients in anybasis,” IEEE Transactions on Information Theory, vol. 57, no. 3, pp.1548–1566, March 2011.

[34] M. Okamoto, “Some inequalities relating to the partial sum of binomialprobabilities,” Annals of the Institute of Statistical Mathematics, vol. 10,no. 1, pp. 29–35, Mar 1959.

[35] M. Udell, K. Mohan, D. Zeng, J. Hong, S. Diamond, and S. Boyd,“Convex optimization in Julia,” SC14 Workshop on High PerformanceTechnical Computing in Dynamic Languages, 2014.