High Resolution FDMA MIMO Radar

High Resolution FDMAMIMO Radar

DAVID COHENRafael Advanced Defense Systems Ltd., Haifa, Israel

DEBORAH COHEN , Student Member, IEEEGoogle Research, Tel Aviv, Israel

YONINA C. ELDAR , Fellow, IEEEWeizmann Institute of Science, Rehovot, Israel

Traditional multiple-input multiple-output (MIMO) radars, whichtransmit orthogonal coded waveforms, suffer from range–azimuthresolution tradeoff. In this article, we adopt a frequency divisionmultiple access (FDMA) approach that breaks this conflict. We com-bine narrow individual bandwidth for high azimuth resolution andlarge overall total bandwidth for high range resolution. We process allchannels jointly to overcome the FDMA range resolution limitationto a single bandwidth, and address range–azimuth coupling using arandom array configuration.

Manuscript received July 9, 2019; revised November 2, 2019; released forpublication November 10, 2019. Date of publication December 9, 2019;date of current version August 7, 2020.

DOI. No. 10.1109/TAES.2019.2958193

Refereeing of this contribution was handled by Z. Davis.

The work of D. Cohen was supported by the Azrieli Foundation throughAzrieli Fellowship. This work was supported in part by the EuropeanUnion’s Horizon 2020 research and innovation program under Grant646804-ERC-COG-BNYQ and in part by the Israel Science Foundationunder Grant 0100101.

Authors’ addresses: David Cohen is with the Signal Processing De-partment, Rafael Advanced Defense Systems Ltd., Haifa 31021, Israel,E-mail: ([email protected]); Deborah Cohen is with the GoogleResearch, Tel Aviv 6789141, Israel, E-mail: ([email protected]);Y. C. Eldar is with the Faculty of Mathematics and Computer Sci-ence, Weizmann Institute of Science, Rehovot 7610001, Israel, E-mail:([email protected]). (Corresponding author: David Cohen.)

0018-9251 © 2019 IEEE

I. INTRODUCTION

Multiple-input multiple-output (MIMO) [1] radar con-sists of several transmit and receive antenna elements.Unlike phased-array systems, the emitted waveform fromeach transmitter is different, which offers more degrees offreedom [2]. Today, MIMO radars appear in many militaryand civilian applications including ground surveillance [3],[4], automotive radar [5], [6], interferometry [7], maritimesurveillance [3], [8], through-the-wall radar imaging forurban sensing [9], and medical imaging [2], [10]. MIMOradars are generally classified into two main categories,depending on the location of the transmitting and receivingelements. The first configuration is the collocated MIMO[11] in which the elements are closely spaced relatively tothe working wavelength while in the second configuration,the widely separated MIMO [12], they are distributed overa large deployment area.

MIMO radar presents significant potential for advancingstate-of-the-art modern radar in terms of flexibility andperformance. Collocated MIMO radar systems utilize themutual orthogonality of the transmitted waveforms. Eachof these waveforms can be extracted and associated at thereceiver to a single transmitting antenna. This waveformdiversity that is the heart of a MIMO radar system is the keyfor spatial channel separation, thus offering more degrees offreedom by processing the channels individually rather thanprocessing the receiver input as in a classic phased-array[2]. Consequently, the performance of MIMO systems canbe characterized by a virtual array corresponding to theconvolution of the transmit and receive antenna locations.In principle, with the same number of antenna elements,this virtual array may be much larger and, thus, achievehigher resolution than an equivalent traditional phased arraysystem [13]–[15].

The orthogonality requirement, however, poses newtheoretical and practical challenges. Choosing proper wave-forms is a critical task for the implementation of practicalMIMO radar systems. In addition to the general require-ments on radar waveforms such as high range resolutionand low sidelobes, MIMO radar waveforms must satisfygood orthogonality properties. In practice, it is difficultto find waveform families that perfectly satisfy all thesedemands [16]. Comprehensive evaluation and comparisonof different types of MIMO radar waveforms is presentedin [17]–[19]. The main waveform families considered aretime, frequency, and code division multiple access, ab-breviated as TDMA, frequency division multiple access(FDMA), and CDMA, respectively. These may either beimplemented in a single pulse, namely in the fast timedomain, referred to as intrapulse coding, or in a pulse train,that is in the slow time domain, corresponding to interpulsecoding. We focus on the former technique, which is themost popular. More details on interpulse coding may befound in [17] and [19].

An intuitive and simple way to achieve orthogonalityis using TDMA, where the transmit antennas are switchedfrom pulse to pulse, so that there is no overlap between

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two transmissions [20]. Since the transmission capabili-ties of the antennas are not fully utilized, this approachinduces significant loss of transmit power [17], resultingin signal-to-noise ratio (SNR) decrease and much shortertarget detection range. More efficient schemes have beenproposed, such as circulating MIMO waveforms [18]. How-ever, this technique suffers from loss in range resolution[18], [19].

Another way to achieve orthogonality of MIMO radarwaveforms is FDMA, where the signals transmitted bydifferent antennas are modulated onto different carrier fre-quencies. This approach has several limitations. First, dueto the linear relationship between the carrier frequencyand the index of antenna element, a strong range–azimuthcoupling occurs when using the classic virtual uniformlinear array (ULA) configuration [16], [18], [19]. To resolvethis coupling, the authors in [21]–[23] use random carrierfrequencies, which creates high sidelobe levels. These maybe mitigated by increasing the number of transmit antennas,which in turn increases system complexity. The seconddrawback of FDMA is that the range resolution is limitedto a single waveform’s bandwidth, rather than the overalltransmit bandwidth [24], [25]. To increase range resolution,the authors of [26] and [27] use an interpulse stepped fre-quency waveform (SFW), utilizing the total bandwidth overthe slow time [28]. However, SFW leads to range-Dopplercoupling [29] and the pulse repetition frequency growsproportionally to the number of steps, increasing rangeambiguities [20], [30].

In the popular CDMA approach, signals transmitted bydifferent antennas are modulated using distinct series oforthogonal codes, so that they can be separated in the radarreceiver. Although perfect orthogonality cannot be achievedin general, code families, such as Barker [31], Hadamardor Walsh [32], and Gold [33] sequences, present featuresclose to orthogonality. CDMA requires good code design [2]and may suffer from high range sidelobes depending oncross-correlation properties of the code sequence [19].

Although the CDMA range resolution is determined bythe total bandwidth as opposed to FDMA limitation to asingle waveform, high resolution systems are still not easyto achieve. One of the major challenges is preservation of thenarrowband assumption, which can be a limiting factor forhigh-resolution applications. The narrowband assumptioncreates a conflict between large desired bandwidth for highrange resolution and large virtual aperture for high azimuthresolution. The tradeoff comes from the classic beamform-ing performance degradation when using wideband signals,since this operation is frequency dependent [34].

In particular, beamforming is based on combining thearray elements to achieve constructive interference in a de-sired direction. Ideally, the signal in each element should betime delayed before summation while the exact delay of thesignal is variable according to the array configuration andthe desired direction. Traditional beamforming combinesthe elements based on a set of phase shifter values, whichhave been designed at the carrier frequency but applied

to all frequencies within the transmission bandwidth. Thephase shifter is widely adopted because it serves as a goodapproximation to a time delay for narrowband transmission,but most importantly, is easy to implement. However, thisapproximation can give rise to beam squint for a widebandsignal since the phase shifter is frequency dependent asopposed to a time delay, which produces a frequency in-dependent phase shift [35], [36]. The use of phase shiftersfor wideband beamforming leads to loss of gain as wellas distorted range–azimuth resolution. In MIMO configu-rations where the virtual array is large, this may result insevere performance degradation; the beam squint can beenough to steer off the target [37].

Although considerable research has been performed onwideband processing techniques, no consensus has beenreached in the literature regarding an appropriate widebandbeamforming scheme. Most works consider the use and im-plementation of true-time delay (TTD), which is frequencyindependent, to address the beamforming deterioration ofwideband phased arrays. TTD units are designed to providea variable time delay over the array aperture in a constructivemanner to the desired direction and may be realized inanalog or in digital.

In analog TTD implementation, delay lines are locatedin each channel path to create a variable delay over the array.While the first analog delay lines were large and bulky, todaywith technology evolution modern TTD chips are available.However, analog-based methods suffer from insertion lossas well as linearity issues [36]. Furthermore, the time delayshould be on the same length as the array aperture, whichcould be a heavy demand for large array apertures. Toaddress the analog drawbacks, digital TTD beamformingwas proposed in [38]. In digital TTD realizations, an arbi-trary delay is split into integer and fractional with respectto the sampling interval. The integer delay is obtained bydiscarding samples while the fractional sample delay (FSD)is implemented using an FSD interpolation filter. DigitalTTD beamforming requires filtering the signal received ineach element to create a variable delay, which could beresource intensive in comparison with a simple phase shift.To alleviate the system overhead using digital filtering, in[39], the digital TTDs are incorporated at the subarray levelwhile simple narrowband phase compensation is appliedat the element level. Then, stretch processing is used inwhich the received signal is mixed with a reference signalto allow low sampling rate filtering. Moreover, the filtersare implemented using the Farrow structure with reducedarithmetic complexity.

With the evolution of technology, there is increasinginterest in simultaneous generation of multiple beams re-quiring parallel computing for their generation, which isan unacceptable demand for real-time systems with TTDbeamforming. Furthermore, for MIMO radar in which allthe space is uniformly lit, the hardware resources requiredfor TTD beamforming grows rapidly since the TTDs areapplied on the channels as opposed to the phased array. In[37], a smearing filter is adopted in wideband MIMO to

COHEN ET AL.: HIGH RESOLUTION FDMA MIMO RADAR 2807


address the system complexity, which in turn leads to poorrange resolution. Another approach for TTD realization isphotonic beamforming in which the RF signal is modulatedonto an optical carrier and a long fiber is integrated to delaythe signal [36], [40], [41]. Although photonic beamforminghas recently shown rapid development [42]–[44], it stillrequires radical changes in technology in order to be in-corporated in systems [45].

In this article, a frequency diverse array (FDA) MIMOstructure [46] is adopted in which orthogonality is achievedby using an FDMA transmission scheme. As will be ex-plained later, FDA allows phase-shifting for widebandbeamforming and, thus, provides an alternative to TTD. Theterm FDA radar was first proposed in 2006 [47], however,the concepts of FDA and MIMO have earlier roots. The firstFDA MIMO radar, was the French radar RIAS, also knownas SIAR: synthetic impulse and antenna radar and wasproposed in 1984 [48]. In SIAR each transmit element sendsorthogonal signals using an FDMA transmission scheme.The received signals are synthetically combined to offerrange resolution according to the total bandwidth of thereturned echo [49].

We present an array design and processing method thatovercome FDMA drawbacks. First, to avoid range–azimuthcoupling, we randomize the carrier frequencies as well asthe transmit and receive locations within the virtual arrayaperture. The idea of randomized frequencies has beenused in single antenna radars [29] that employ SFW, toresolve range-Doppler coupling. There, hopped frequencysequences, namely with randomized steps for increasingthe carriers, are considered. Random arrays have been anobject of research since the 1960s [50]. Recently, in [51], theauthors adopt random MIMO arrays to reduce the number ofelements required for targets’ detection using sub-Nyquistspatial sampling principles [52]. Here, we use a randomarray to deal with the coupling issue while keeping thenumber of elements as in traditional MIMO. We empiricallyfound that randomizing both the antenna locations andfrequencies rather than the frequencies alone exhibits thebetter performance.

Second, we process the samples from all channelsjointly exploiting frequency diversity [53], to overcomethe range resolution limitation of a single bandwidth. Asimilar approach was used in [54] in the context of MIMOsynthetic aperture radar with orthogonal frequency-divisionmultiplexing linear frequency modulated (OFDM LFM)waveforms, where coherent processing over the channelsallows to achieve range resolution corresponding to the totalbandwidth. There, however, the MIMO array is composedof two ULAs, both with spacing equal to half the wave-length. While avoiding range-elevation coupling, this ap-proach yields poor elevation resolution [15]. Furthermore,in [54], the total bandwidth is limited by the narrowbandassumption [37] and, hence, perpetuates the range–azimuthresolution conflict.

One of the main contributions of this article is toshow that using FDMA, the narrowband assumption maybe relaxed to the individual bandwidth with appropriate

signal processing. FDMA allows to achieve a large overallreceived bandwidth over the channels while maintaining thenarrowband assumption for each channel. In particular, thebeamforming in FDMA MIMO is applied over the nar-rowband channels rather than broadband channels like inCDMA. Therefore, FDMA MIMO enables phase shiftersdesign with respect to the carrier of each narrowbandchannel while cooperative processing of the channels al-lows range resolution of broadband signals. This approachprovides an alternative to the complex TTD beamform-ing by simple phase shifting of the narrowband channelswith enhanced range–azimuth resolution capabilities. Therange–azimuth resolution conflict may, thus, be solved byenabling large aperture for high azimuth resolution alongwith large total bandwidth for high range resolution. Thenarrowband assumption holds by requiring small indi-vidual bandwidth, breaking the traditional range–azimuthtradeoff.

In order to achieve range resolution corresponding tothe overall bandwidth, we develop a recovery methodthat coherently processes all channels. This overcomes theFDMA range resolution limitation to a single bandwidth.In addition, to exploit the targets sparsity, we developed aniterative dictionary based approach. Our frequency domainprocessing is easily extended to sub-Nyquist recovery withlow-rate sampling and reduced number of elements [55],[56]. However, our principles are valid also for noniterativeapproaches and could be shown in the time domain usingclassic matched filter components and phase shifters overthe channels.

The main contributions of our approach are as follows.1) Enhanced range–azimuth resolution capabilities:

The FDA-MIMO framework provides an alternativefor TTD beamforming by simple phase shifters withrespect to each channel. Our approach simultane-ously allows large aperture for high azimuth resolu-tion along with large total bandwidth for high rangeresolution.

2) Range angle decoupling: We propose a randomarray configuration with random carrier allocation.Placing the antenna elements at random locationsadds more degrees of freedom for better control ofthe sidelobes level.

3) Iterative joint processing: We coherently process allchannels, thus enabling range resolution according tothe total bandwidth with azimuth resolution governedby the virtual array aperture. The iterative approachenables sidelobe suppression and provides robustnessto high dynamic range environments.

This article is organized as follows. In Section II,we review classic MIMO radar and processing. OurFDMA model is introduced in Section III, where range–azimuth coupling and range–azimuth beamforming are dis-cussed. Section IV presents the proposed range–azimuth-Doppler recovery. Numerical experiments are presented inSection V, demonstrating the improved performance of ourFDMA approach over classical CDMA. Finally, Section VIconcludes this article.



II. CLASSIC MIMO RADAR

This section provides an overview of the classic MIMOradar architecture, in terms of array construction and wave-forms, as well as traditional MIMO signal processing.

A. MIMO Architecture

In contrast to classic phased array, MIMO radar systemstransmit orthogonal waveforms in each of the transmit-ting antennas. Therefore, the echo signals collected by thereceiver can be reassigned to the source by performingcorrelation using a set of matched filters adapted to eachof the transmitted waveforms. For example, if a MIMOradar has five transmit antennas and three receive antennas,then 15 signals can be extracted from the receiver due tothe orthogonality property of the transmitted waveforms.Each of the extracted signals contains the information ofan individual spatial channel from one of the transmittingantennas to one of the receiving antennas. This waveformdiversity in MIMO radar results in more degrees of freedomby enabling processing the orthogonal channels in eachreceiver separately as opposed to traditional phased array.

The channel’s response to a specific direction can becharacterized by a differential phase induced by the trans-mitter and receiver array geometry, leading to the virtualarray concept. A virtual array is an equivalent array whoseresponse is the same as the channels response of a MIMOarray. Specifically, the antennas locations of the virtualarray correspond to the convolution of the transmit andreceive antenna locations. The performance of MIMO radaris characterized by a virtual array with a larger number ofelements and enlarged aperture than the physical apertureof the MIMO array.

The classic approach to collocated MIMO is based onthe virtual ULA structure [57], where T transmitters, spacedby R λ

2 and R receivers, spaced by λ2 (or vice versa), create

two ULAs. Here, λ is the wavelength. The resulting T Rchannels are coherently processed to form a virtual array inwhich each element corresponds to a single channel in theMIMO structure. The resulting virtual array is equivalentto a phased array with T R λ

2 -spaced receivers and apertureZ = T R

2 normalized to the wavelength.Denote by {ξm}T −1

m=0 and {ζq}R−1q=0 , the transmitters and

receivers’ normalized locations, respectively. For the tra-ditional virtual ULA structure, ζq = q

2 and ξm = R m2 . This

classic array structure and the resulting virtual array are il-lustrated in Fig. 1 for R = 4 and T = 4 where each color rep-resents the channels induced by a single transmit element.In our work, we consider a random array configuration [51],where the antennas’ locations are chosen uniformly atrandom within the aperture of the virtual array describedearlier, that is {ξm}M−1

m=0 ∼ U [0, Z] and {ζq}Q−1q=0 ∼ U [0, Z],

respectively. The corresponding virtual array has the sameor a greater aperture than a traditional virtual array with thesame number of elements, depending on the locations of theantennas at the far edges. The resulting azimuth resolutionis thus at least as good as that of the traditional virtual ULAstructure.

Fig. 1. Illustration of MIMO arrays and the corresponding receivervirtual array.

Each transmitting antenna emits P pulses, such that mthtransmitted signal can be expressed as

sm(t ) =P−1∑

p=0

spm(t ), 0 ≤ t ≤ Pτ, (1)

where

spm(t ) = hm

(t − pτ

)e j2π fct . (2)

Here, hm(t ), 0 ≤ m ≤ T − 1, are orthogonal waveformswith bandwidth Bh and carrier frequency fc. The pulses aretransmitted within the coherent processing interval (CPI)that is equal to Pτ , where τ denotes the pulse repetitioninterval (PRI). The pulse time width is denoted by Tw, with0 < Tw < τ .

The orthogonality property in MIMO radar places ad-ditional requirements on the transmitted waveforms. Thisproperty is added to the classic demands from the radarambiguity function such as low sidelobes and resolution[28]. Moreover, to avoid cross talk between the T signalsand to ensure T R uncorrelated channels, the orthogonalitycondition should be also invariant to time shifts, that is∫ ∞−∞ si(t )s∗

j (t − τ0)dt = δ(i − j), for i, j ∈ [0, T − 1] andfor all τ0. This implies that the orthogonal transmittedwaveforms cannot overlap in frequency (or time) [37],leading to the FDMA (or TDMA) approach. Alternatively,time invariant orthogonality may be approximately attainedusing CDMA.

Both FDMA and CDMA can be expressed by the gen-eral model [18]:

hm(t ) =Nc∑

u=1

wmue j2π fmutv(t − uδt ), (3)

where each pulse consists of Nc time slots with durationδt . Here, v(t ) denotes the fundamental waveform, wmu

represents the code, and fmu represents the frequency forthe mth transmission and uth time slot. The different wave-form families follow the general expression (3) and canbe analyzed accordingly. In particular, in CDMA, orthogo-nality is attained by the code {wmu}Nc

u=1 and fmu = 0 for all



1 ≤ u ≤ Nc. FDMA presents no coding, i.e., Nc = 1, wmu =1, and δt = 0. The orthogonality is then achieved by centerfrequencies fmu = fm that are selected in [− T Bh

2 , T Bh2 ] so that

the intervals [ fm − Bh2 , fm + Bh

2 ] do not overlap. For FDMA,transforming (3) from the time domain into the frequencydomain, {hm(t )}T −1

m=0 can be considered as frequency-shiftedversions of a low-pass pulse v(t ) = h0(t ), such that

Hm (ω) = H0

(ω − 2π fm

). (4)

Here, H0(ω) denotes the Fourier transform of h0(t ) withbandwidth Bh. For simplicity, a unified notation is adoptedfor the total bandwidth, Btot = T Bh for FDMA and Btot =Bh for CDMA.

B. Received Signal

Consider L point-targets within the area covered by theradar. Each target illuminated by the radar is with initialrange Rl to the array origin and moving with radial velocityvl along the line of sight with azimuth angle θl relative tothe array. Our objective is to recover the targets’ delay τl =2Rlc , azimuth sine ϑl = sin(θl ), and Doppler shift f D

l = 2vlλ

from the received signals. Throughout this article, we willuse interchangeably the terms range and delay, as well asazimuth angle and sine, and velocity and Doppler frequency.

The transmitted pulses are reflected from the targetsback and acquired by the receive antennas. To simplify thereceived signal expression, the following assumptions areadopted on the targets’ location and motion with respect tothe array structure as well as transmitted signal properties.

A1) Collocated array - the target scattering complexcoefficient αl and angle θl are constant over thearray (see [58] for more details).

A2) Swerling-0 model - the target radar cross section(RCS) has no fluctuation [59], allowing for constantαl .

A3) Stop-and-hop assumption - targets stop duringtransmission and reception of a single pulse [60].Specifically, the phase induced by the relative targetmotion is negligible

2v2l Pτ

λc� 1, (5)

and the Doppler phase during the pulse width Tw

can be ignored

f Dl Tw � 1. (6)

A4) Free-range migration - the target motion within theCPI is less than the range resolution [61], allowingfor constant τl

2vlPτ

c� 1

Btot. (7)

A5) Free-Doppler spread - the Doppler change withinthe CPI is less than the Doppler resolution [61],allowing for constant Doppler shift f D

l

˙f Dl Pτ � 1

Pτ. (8)

A6) Narrowband waveform - small aperture allows τl

to be constant over the channels

2Zλ

c� 1

Btot. (9)

Under Assumptions A1–A3 and A5, the received signalxq(t ) at the qth antenna is a sum of time-delayed, scaledreplica of the transmitted signals

xq(t ) =P−1∑

p=0

T −1∑

m=0

L∑

l=1

αl spm

(t − Rl,mq(pτ )

c

), (10)

where Rl,mq(t ) = 2Rl + 2vl t − (Rlm + Rlq), with Rlm =λξmϑl and Rlq = λζqϑl accounting for the array geometry.The received signal can be simplified using AssumptionsA4 and A6, as we now show.

We start with the envelope hm(t ), and consider the pthPRI and the lth target. From A4, we neglect the term 2vl pτ

cand obtain

hm

(t − Rl,mq(pτ )

c

)= hm(t − pτ − τl,mq). (11)

Here, τl,mq = τl − ηmqϑl where τl = 2Rlc is the target delay,

and ηmq = (ξm + ζq) λc follows from the respective locations

between transmitter and receiver. We then add the modula-tion term of sm(t ) and after down-conversion to basebandand ignoring constant phases, the remaining term is givenby

hm(t − pτ − τl,mq)e j2π fcηmqϑl e− j2π f Dl pτ . (12)

Finally, from A6, the delay term ηmqϑl , which stems fromthe array geometry, is neglected in the envelope whichbecomes

hm(t − pτ − τl ). (13)

Substituting (13) into (12), the received signal at the qthantenna after demodulation to baseband is given by

xq(t )=P−1∑

p=0

T −1∑

m=0

L∑

l=1

αlhm(t − pτ − τl

)e j2π fcηmqϑl e− j2π f D

l pτ .

(14)

The narrowband Assumption A6 leads to a tradeoffbetween azimuth and range resolution, by requiring eithersmall aperture Z or small total bandwidth Btot, respectively.In CDMA, Btot = Bh so that A6 limits the total bandwidthof the waveforms hm(t ) [37]. This is illustrated in Fig. 2that shows the performance of CDMA waveforms usingthe classical processing detailed as follows. We use ban-dlimited Gaussian pulses that are equivalent to CDMA,where each transmitter radiates P = 1 pulse, and considerrange–azimuth recovery in the absence of noise. We assumeL = 5 targets whose locations are generated uniformly atrandom, and adopt a hit-or-miss criterion according to theRayleigh resolution as our performance metric. A “hit” isdefined as a range–azimuth estimate that is identical to thetrue target position up to one Nyquist bin (grid point) definedas 1/Btot and 2/T R for the range and azimuth, respectively.



Fig. 2. Hit rate of MIMO radar with classical processing of CDMAwaveforms with respect to total bandwidth Btot = Bh and

aperture Z = T R/2.

Each experiment is repeated over 200 realizations. It can beseen that the recovery performance decreases with eitherincreased bandwidth or aperture since in both cases A6does not hold. In the subsequent section, We show that inthe FDMA processing framework, the assumption can berelaxed so that the aperture is required to be smaller thanthe reciprocal of Bh rather than Btot = T Bh as in (9).

C. Range–Azimuth-Doppler Recovery

The traditional collocated MIMO radar processingscheme includes the following steps.

1) Sampling: The received signal xq(t ) is sampled at itsNyquist rate Btot at each receiver element.

2) Matched filter: The sampled signal is correlatedwith a sampled version of hm(t ), for 0 ≤ m ≤ T − 1.The time resolution attained in this step is 1/Bh. InFDMA, this step leads to a limitation on the rangeresolution to a single channel bandwidth rather thanthe total bandwidth.

3) Beamforming: The matched filter output is corre-lated over the channels with steering vectors corre-sponding to each azimuth grid point. The spatial res-olution attained in this step is 2/T R. In FDMA, thisstage leads to range–azimuth coupling, as illustratedin Section III-D.

4) Doppler processing: Spectral analysis is performedover the pulses of the resulting vectors for detection ofDoppler frequencies. The Doppler resolution attainedin this step is 1/Pτ .

5) Detection: A detection process is applied on theresulting range–azimuth-Doppler map. The detectioncan follow a threshold approach [62] to preserve falsealarm requirements or select the L strongest point ofthe map, if the number of targets L is known.

CDMA is a popular MIMO approach even though itsuffers from two main drawbacks. First, the narrowbandassumption yields a tradeoff between azimuth and rangeresolution. Second, achieving orthogonality through code

Fig. 3. Maximal cross correlation of CDMA waveforms usingTw = 0.44 μs with respect to signal bandwidth Bh with T = 20transmitters (top) and number of transmitters T with bandwidth

Bh = 100 MHz (bottom).

design has proven to be a challenging task [2]. To illustratethis, we consider a set of orthogonal bandlimited Gaussianwaveforms, generated using a random search for minimiz-ing the cross correlation between pairs of waveforms. Thatis, the set of waveforms is constructed so as to minimizethe maximal cross correlation between waveforms, througha nonexhaustive search. Fig. 3 shows the maximal crosscorrelations between any pair of signals within the set. Itcan be seen that, when either the bandwidth Bh is reducedor the number of transmit antennas increases, the maximalcross correlation of the CDMA waveforms increases.

Meanwhile, classic FDMA has been almost neglectedowing to its two main drawbacks. First, a strong range–azimuth coupling results from frequency diversity along thechannels [16], [18], [19], as we illustrate in Section III-D.The second drawback of FDMA is that the range resolu-tion is dictated by a single waveform’s bandwidth, namelyBh, rather than the overall transmit bandwidth Btot = T Bh

[24], [25]. In the following section, we adopt the FDMAapproach, in order to exploit the narrowband property ofeach individual channel to achieve both high range and



azimuth resolution. To address the coupling issue, the an-tenna elements are uniformly distributed over the aperture,while keeping the random carrier frequencies on a gridwith spacing Bh. Next, by joint processing of the channels,we attain a range resolution of 1/Btot = 1/T Bh rather than1/Bh. This way, we utilize the overall received bandwidthfor range resolution enhancement, while preserving thenarrowband assumption for each channel, paving the wayto high azimuth resolution. In addition, no code design isrequired, which may be a challenging task [2], as shown inFig. 3.

III. FDMA SYSTEM

We now describe our MIMO system based on jointchannel processing of FDMA waveforms. Our FDMA pro-cessing differs from the classic CDMA approach introducedin Section II in several aspects. First, the single channel pro-cessing, which is equivalent to matched filtering in step (2),is limited to 1/Bh = T/Btot whereas in CDMA, we achieveresolution of 1/Btot. In addition, in FDMA, the range de-pends on the channels while in CDMA, it is decoupled fromthe channels domain. Therefore, our processing involvesrange–azimuth beamforming while the classic approach forCDMA uses beamforming on the azimuth domain onlyas in step (3). The range dependency on the channels inFDMA is exploited to enhance the poor range resolutionof the single channel 1/Bh to 1/T Bh = 1/Btot, as explainedin the remainder of this section. Finally, combining the useof FDMA waveforms with our proposed processing recon-ciles the narrowband assumption with large total bandwidthfor range resolution, enhancing range–azimuth resolutioncapabilities.

A. Received Signal Model

Our FDMA framework followed by appropriate pro-cessing, described in Section IV, allows to alleviate the rigidrestrictions induced by A6. In FDMA, the strict neglect ofthe delay term can be substantially reduced in the transitionfrom (12) to (13). Specifically, we confine the ηmqϑl termremoval to the envelope h0(t ) rather than hm(t ). Then, (13)becomes

hm(t − pτ − τl )ej2π fmηmqϑl . (15)

Here, the delays governed by the array geometry are main-tained over the channels, relaxing A6 to 2Zλ

c � 1Bh

. Werecall that under the CDMA framework, A6 leads to acompromise between azimuth and range resolution, bylimiting the array aperture or the total bandwidth. Here,using the FDMA framework and the approximation (15),the total bandwidth Btot = T Bh is not limited and we needonly the single bandwidth Bh to be narrow. As we showin the following, the resolution is governed by the totalbandwidth, thus eliminating the tradeoff between range andazimuth resolution.

The received signal at the qth antenna after demodula-tion to baseband is in turn given by

xq(t ) =P−1∑

p=0

T −1∑

m=0

L∑

l=1

αlhm(t − pτ − τl

)e j2πβmqϑl e− j2π f D

l pτ ,

(16)where βmq = (ζq + ξm)( fm

λc + 1). In comparison with (14),

neglecting the delay term only in the narrowband envelopeh0(t ), (16) results in the extra term e j2π (ζq+ξm )ϑl fmλ/c. Thiscorrects the time of arrival differences between channels,so that the narrowband assumption is required only onh0(t ) with bandwidth Bh and not on the entire bandwidthBtot. Intuitively, the waveforms are aligned to eliminatethe arrival differences resulting from the array geometrywith respect to fm, thus enabling to detect the azimuth withrespect to the central carrier fc. This assumption does notrequire additional steps on the processing and allows keep-ing simple phase shifting beamforming over the channels.The phases are simply updated to account for the carrierfrequency of each narrowband channel, thus providing analternative to the complicated TTD beamforming.

It will be convenient to express xq(t ) as a sum of singlePRIs

xq(t ) =P−1∑

p=0

xpq (t ), (17)

where

xpq (t ) =

T −1∑

m=0

L∑

l=1

αlhm(t − pτ − τl )ej2πβmqϑl e− j2π f D

l pτ .

(18)Our goal is to estimate the targets range, azimuth, andvelocity, i.e., to estimate τl , ϑl , and f D

l from xq(t ).

B. Frequency Domain Analysis

We begin by deriving an expression for the Fourier coef-ficients of the received signal, and show how the unknownparameters, namely τl , ϑl , and f D

l are embodied in thesecoefficients. We next turn to range–azimuth beamformingand its underlying resolution capabilities and discuss therange–azimuth coupling. Finally, we present our proposedrecovery algorithm, which is based on FDMA waveforms.To introduce our processing, we start with the special caseof P = 1, namely a single pulse is transmitted by eachtransmit antenna. We show how the range–azimuth mapcan be recovered from the Fourier coefficients in time andspace. Subsequently, we treat the general case where a trainof P > 1 pulses is transmitted by each antenna, and presenta joint range–azimuth-Doppler recovery algorithm from theFourier coefficients.

The pth PRI of the received signal at the qth antenna,namely xp

q (t ), is limited to t ∈ [pτ, (p + 1)τ ] and, thus, canbe represented by its Fourier series

xpq (t ) =

∑

k∈Zcp

q [k] e j2πkt/τ , t ∈ [pτ, (p + 1)τ

], (19)



where for −NT2 ≤ k ≤ NT

2 − 1, with N = τBh

cpq [k] = 1

τ

T −1∑

m=0

L∑

l=1

αl ej2πβmqϑl e− j 2π

τkτl e− j2π f D

l pτ Hm

(2π

τk

).

(20)

Once the Fourier coefficients cpq[k] are computed, we

separate them into channels for each transmitter, by exploit-ing the fact that they do not overlap in frequency. Applyinga matched filter, we have

c̃pq,m [k] = cp

q [k] H∗m

(2π

τk

)

= 1

τ

∣∣∣∣Hm

(2π

τk

)∣∣∣∣2 L∑

l=1


τkτl e− j2π f D

l pτ .

(21)

Let ypm,q[k] = τ

|H0( 2πτ

k)|2 c̃pq,m[k + fmτ ] be the normalized and

aligned Fourier coefficients of the channel between the mthtransmitter and qth receiver. Then,

ypm,q[k] =

L∑

l=1


τkτl e− j2π fmτl e− j2π f D

l pτ , (22)

for −N2 ≤ k ≤ N

2 − 1. Our goal then is to recover the tar-gets’ parameters τl , θl ϑl , and f D

l from ypm,q[k].

C. Range–Azimuth Beamforming

Let us now pause to discuss the range and azimuth reso-lution capabilities of the described model and processingas well as the coupling issue between the two parame-ters. Since the Doppler frequency is decoupled from therange–azimuth domain, we assume that P = 1 for the sakeof clarity. Then, (22) can be simplified to

ym,q[k] =L∑

l=1


τkτl e− j2π fmτl , (23)


2 − 1. The azimuth is embodied in thefirst term and its resolution is related to the virtual arraygeometry governed by βmq as discussed in [15]. The delayis embodied in the second and third terms, which allowsboth high range resolution and large unambiguous range.The second term leads to a poor range resolution of 1/Bh

corresponding to a single channel, while the resolutioninduced by the third term, which measures the effect ofthe delay on the transmit carrier, is dictated by the totalbandwidth, namely 1/T Bh. On the other hand, since fm is amultiple of Bh in our configuration, the last term is periodicin τl with a limited period of 1/Bh, whereas the second termis periodic in τl with period τ so that the correspondingunambiguous range is cτ/2. Therefore, by jointly process-ing both terms, we overcome the resolution and ambiguousrange limitations, and thus, achieve a range resolution of1/T Bh with unambiguous range of τ , as summarized inTable I.

Both the first and third terms, which contain the azimuthand delay, respectively, depend on the channels indexed by

TABLE IRange Resolution and Ambiguity

Fig. 4. Illustration of the resolution obtained by processing a singlechannel and by joint processing of all channels, using

range–azimuth beamforming.

m, q and, thus, need to be resolved simultaneously. Thisprocessing step is referred to as range–azimuth beamform-ing and will be discussed in the following section. The jointprocessing, which combines single-channel processing andrange–azimuth beamforming, is illustrated in Fig. 4. Thepoor range resolution that would be obtained by processingeach channel separately can be seen in Fig. 4(a). Range–azimuth beamforming, illustrated in Fig. 4(b), achievesa higher resolution of 1/T Bh, corresponding to the totalbandwidth. However, the resulting range ambiguity can beclearly observed. Combining the single channel process-ing with range–azimuth beamforming yields joint range–azimuth recovery [see Fig. 4(c)]. Note that the processingis not divided into these two steps, which are provided forillustration purposes only.



Fig. 5. Range–azimuth map in noiseless settings for antennas locatedon the conventional ULA and carrier frequencies selected on a grid, withL = 1 target. The highest peak with the red circle corresponds to the true

target. The other peaks results from the range–azimuth coupling [55].

D. Range–Azimuth Coupling

As explained earlier, range–azimuth beamforming overthe channels involves the estimation of both parametersfrom one dimension, the channel dimension. Therefore, itrequires a one-to-one correspondence between the phasesover the channels to range–azimuth pairs in order to pre-vent coupling. This ensures that each phase over the chan-nels, expressed by the first and third terms of (23), corre-sponds to a unique azimuth-range pair. We next illustratethe range–azimuth coupling which occurs, in particular,in the case where the antennas are located according tothe conventional virtual ULA structure shown in Fig. 1,and the carrier frequencies are selected on a grid such thatfm = (m − T −1

2 )Bh.In the single pulse case, where P = 1, we can rewrite

(23) with respect to the channel γ = mR + q as

yγ [k] =L∑

l=1

alej2πβγ ϑl e− j2π fγ τl e− j 2π

τkτl , (24)

where βγ = γ /2, fγ = (γ mod T )Bh, and al is equal to αl

up to constant phases. Assume that τl and ϑl lie on theNyquist grid such that

τl = τ

T Nsl ,

ϑl = −1 + 2

T Rrl , (25)

where sl and rl are integers satisfying 0 ≤ sl ≤ T N − 1 and0 ≤ rl ≤ T R − 1, respectively. Then, (24) becomes

yγ [k] =L∑

l=1

alej2π

γ

T R (rl−sl R)e− j 2πT N ksl , (26)


2 − 1. We observe that the first term isambiguous with respect to the range and azimuth, andtherefore, the range–azimuth beamforming is ambiguous.In noiseless settings, we can recover the range and azimuthwith no ambiguity from the second term (single channel).This is illustrated in Fig. 5, where the highest peak in therange–azimuth map corresponds to the true target. However,

this configuration may lead to ambiguity between bothparameters in the presence of noise, due to other high peaks.We, thus, choose to adopt a random array, as discussed inSection V.

IV. RANGE–AZIMUTH-DOPPLER RECOVERY

We now describe our recovery approach from theFourier coefficients of the FDMA received waveforms (16).We first consider the case where P = 1 and derive range–azimuth recovery from the coefficients (23). We next turnto range–azimuth-Doppler recovery from (22).

A. Range–Azimuth Recovery

In practice, as in traditional MIMO, suppose we nowlimit ourselves to the Nyquist grid with respect to the totalbandwidth T Bh so that τl and ϑl lie on the grid defined in(25). Let Ym be the N × R matrix with qth column given byym,q[k − N/2], defined in (23), for 0 ≤ k ≤ N − 1. We canwrite Ym as

Ym = AmX(Bm

)T. (27)

Here, Am denotes the N × T N matrix whose (k, n)th ele-

ment is e− j 2πT N (k− N

2 )ne− j2πfmBh

nT and Bm is the R × T R matrix

with (q, p)th element e j2πβmq (−1+ 2T R p). The matrix X is a

T N × T R sparse matrix that contains the values αl at the Lindices (sl , rl ).

Our goal is to recover X from the measurement matri-ces Ym, 0 ≤ m ≤ M − 1. The time and spatial resolutioninduced by X are τ

T N = 1T Bh

, and 2T R , respectively, as in

classic CDMA processing.Define

A = [A0TA1T · · · A(T −1)T

]T , (28)

and

B = [B0TB1T · · · B(T −1)T

]T . (29)

The matrices A and B are generally described as dictionar-ies, whose columns correspond to the range and azimuthgrid points, respectively. To better understand the structureof A and B, consider the case when the carriers fm lieon the grid fm = (m − T −1

2 )Bh and the array constructionsis according to virtual array configuration as illustratedin Fig. 1. In this case, under the Assumption A6 the re-sulting A and B matrices are Fourier matrices up to rowor column permutation. However, this configuration leadsto range–azimuth coupling as discussed in Section III-D.In Section V, we use a random array and random carrierallocation to avoid range–azimuth coupling.

One approach to solving (27) is based on CS [52], [63]techniques that exploits the sparsity of the target scene. Oneof CS recovery advantages is that it allows to reduce thenumber of required samples, pulses, and channels whilepreserving the underlying resolution [55]. In particular, weadopt an iterative reconstruction approach that is beneficialwhen dealing with high dynamic range with both weak andstrong targets, especially since the sidelobes are slightlyraised due to the random array configuration. Our recovery



Algorithm 1: Simultaneous 2D Recovery BasedOMP.Input: Observation matrices Ym, measurement

matrices Am, Bm, for all 0 ≤ m ≤ T − 1Output: Index set � containing the locations of the

non zero indices of X, estimate for sparse matrix X̂1: Initialization: residual Rm

0 = Ym, index set�0 = ∅, t = 1

2: Project residual onto measurement matrices:

� = AH RB̄

where A and B are defined in (28) and (29),respectively, and R = diag([R0

t−1 · · · RT −1t−1 ]) is

block diagonal3: Find the two indices λt = [λt (1) λt (2)] such

that

[λt (1) λt (2)] = arg maxi, j

∣∣�i, j

∣∣

4: Augment index set �t = �t⋃{λt }

5: Find the new signal estimate

α̂ = [α̂1 . . . α̂t ]T = (DT

t Dt )−1DTt vec(Y)

6: Compute new residual

Rmt = Rm

0 −t∑

l=1

αlam�t (l,1)

(bm

�t (l,2)

)T

7: If t < L, increment t and return to step 2,otherwise stop

8: Estimated support set �̂ = �L

9: Estimated matrix X̂: (�L(l, 1), �L(l, 2))-thcomponent of X̂ is given by α̂l for l = 1, . . . , Lwhile the rest of the elements are zero

algorithm is based on orthogonal matching pursuit (OMP)[52], [63]. Similar subtraction techniques are used in manyiterative algorithms such as the CLEAN process [64].

To recover the sparse matrix X from the set of equations(27), for all 0 ≤ m ≤ M − 1, where the targets’ range andazimuth lie on the Nyquist grid, we consider the followingoptimization problem

min ||X||0 s.t. AmX(Bm

)T = Ym, 0 ≤ m ≤ T − 1.

(30)It has been shown in [55] that the minimal numberof channels required for perfect recovery of X in (30)with L targets in noiseless settings is T R ≥ 2L with aminimal number of T N ≥ 2L samples per receiver. Tosolve (30), we extend the matrix OMP from [65] tosimultaneously solve a system of CS matrix equations,as shown in Algorithm 1. In the algorithm description,vec(Y) � [vec(Y0)T · · · vec(YT −1)T ]T , dt (l ) = [(d0

t (l ))T

· · · (dT −1t (l ))T ]T where dm

t (l ) = vec(am�t (l,1)(b

m�t (l,2))

T )with �t (l, i) the (l, i)th element in the index set �t at thet th iteration, and Dt = [dt (1) . . . dt (t )]. Here, am

j denotesthe jth column of the matrix Am and bm

j denotes the jth

column of the matrix Bm. B̄ denotes the conjugate ofmatrix B. Once X is recovered, the delays and azimuths areestimated as

τ̂l = τ

T N�L(l, 1), (31)

ϑ̂l = −1 + 2

T R�L(l, 2). (32)

The projection performed in step 2 of the algorithm com-bines single channel processing with range–azimuth beam-forming. The former coherently processes the second termof (23), which appears in the matrix A, while range–azimuthbeamforming over the channels coherently processes thefirst and third terms of (23), which are contained in Aand B, respectively. The FDMA narrowband assumptionreconciliation is due to the additional third term of (15),contained in B, thus enhancing range–azimuth resolutioncapabilities by simply updating the phase term for eachchannel.

The improved performance of the iterative approachover noniterative target recovery with high dynamic rangeis illustrated in simulations in Section V. There, we alsocompare our FDMA approach with classic CDMA, whenusing a noniterative recovery method in the former. Inparticular, we only use one iteration of Algorithm 1, which isequivalent to the classic approach. This demonstrates thatour FDMA method outperforms CDMA due to the highrange–azimuth resolution capabilities stemming from thereconciliation between the individual narrowband assump-tion and the large overall bandwidth.

B. Range–Azimuth-Doppler Recovery

Besides τl and ϑl lying on the grid defined in (25), weassume that the Doppler frequency f D

l is limited to theNyquist grid as well, defined by the CPI as

f Dl = − 1

2τ+ 1

Pτul , (33)

where ul is an integer satisfying 0 ≤ ul ≤ P − 1. Let Zm

be the NR × P matrix with qth column given by the ver-tical concatenation of yp

m,q[k], such that the (k + qN, p)thelement of Zm is given by (Zm)(k+qN,p) = yp

m,q[k − N/2],defined in (22), for 0 ≤ k ≤ N − 1 and 0 ≤ q ≤ R − 1. Wecan then write Zm as

Zm = (Bm ⊗ Am

)XDFT , (34)

where the N × T N matrix Am and the R × T R matrix Bm

are defined as in Section IV-A and F denotes the P × PFourier matrix up to column permutation. The matrix XD isa T 2NR × P sparse matrix that contains the values αl at theL indices (rlT N + sl , ul ).

Our goal is now to recover XD from the measure-ment matrices Zm, 0 ≤ m ≤ T − 1. The time, spatial, andfrequency resolution stipulated by XD are τ

T N = 1Btot

with

Btot = T Bh, 2T R , and 1

Pτ, respectively, as in classic CDMA

processing.To jointly recover the range, azimuth, and Doppler

frequency of the targets, we apply the concept of Doppler



focusing from [66] to our setting. Once the Fourier coeffi-cients (22) are processed, we perform Doppler focusing fora specific frequency ν, that is

�νm,q[k] =

P−1∑

p=0

ypm,q[k]e j2πνpτ

=L∑

l=1


τ(k+ fmτ )τl

P−1∑

p=0

e j2π (ν− f Dl )pτ ,

(35)

for −N2 ≤ k ≤ −N

2 − 1. Following the same argument asin [66], it holds that

P−1∑

p=0

e j2π (ν− f Dl )pτ ∼=

{P |ν − f D

l | < 12Pτ

0 Otherwise.(36)

Therefore, for each focused frequency ν, (35) reduces toa two-dimensional (2-D) problem. We note that Dopplerfocusing increases the SNR by a factor a P, as can be seenin (36).

Algorithm 2 solves (34) for 0 ≤ m ≤ T − 1 us-ing Doppler focusing. Note that step 1 can be per-formed using the fast Fourier transform. In the algo-rithm description, vec(Z) is defined as in the previoussection, et (l ) = [(e0

t (l ))T · · · (eT −1t (l ))T ]T where em

t (l ) =vec((Bm ⊗ Am)�t (l,2)T N+�t (l,1)(Fm

�t (l,3))T ) with �t (l, i) the

(l, i)th element in the index set �t at the t th iteration, andEt = [et (1) . . . et (t )]. Once XD is recovered, the delays andazimuths are given by (31) and (32), respectively, and theDoppler frequencies are estimated as

f̂ Dl = − 1

2τ+ �L(l, 3)

Pτ. (37)

Similarly to the one-pulse case, the minimal number ofchannels required for perfect recovery of XD with L targetsin noiseless settings is T R ≥ 2L with a minimal numberof T N ≥ 2L samples per receiver and P ≥ 2L pulses pertransmitter [55].

V. SIMULATIONS

In this section, we present numerical experiments illus-trating our FDMA approach and compare our method withclassic MIMO processing using CDMA.

A. Preliminaries

Throughout the experiments, the standard MIMO sys-tem is based on a virtual array, as depicted in Fig. 1 gen-erated by T = 20 transmit antennas and R = 20 receiveantennas, yielding an aperture λZ = 6 m. We consider arandom array configuration where the transmitters and re-ceivers’ locations are selected uniformly at random over theaperture Z for each realization. We use FDMA waveformshm(t ) such that fm = (im − T −1

2 )Bh, where im are integerschosen uniformly at random in [0, T ), for 0 ≤ m ≤ T − 1,and all frequency bands within [− T

2 Bh,T2 Bh] are used for

transmission. We consider the following parameters: PRIτ = 100 μs, bandwidth Bh = 5 MHz and carrier frequency

Algorithm 2: Simultaneous 3D Recovery BasedOMP With Focusing.Input: Observation matrices Zm, measurement

matrices Am, Bm, for all 0 ≤ m ≤ T − 1Output: Index set � containing the locations of the

non zero indices of XD, estimate for sparse matrixX̂D

1: Perform Doppler focusing for 0 ≤ i ≤ N − 1,0 ≤ j ≤ R − 1 and 0 ≤ ν ≤ P − 1:

�(m,ν )i, j = (ZmF̄)i+ jN,ν .

2: Initialization: residual R(m,ν )0 = �(m,ν ), index set

�0 = ∅, t = 13: Project residual onto measurement matrices for

0 ≤ ν ≤ P − 1:

�ν = AH RνB̄,

where A and B are defined in (28) and (29),respectively, andRν = diag([R(0,ν )

t−1 · · · R(T −1,ν )t−1 ]) is block

diagonal4: Find the three indices λt = [λt (1) λt (2) λt (3)]

such that

[λt (1) λt (2) λt (3)] = arg maxi, j,ν

∣∣�νi, j

∣∣

5: Augment index set �t = �t⋃{λt }

6: Find the new signal estimate

α̂ = [α̂1 . . . α̂t ]T = (ET

t Et )−1ETt vec(Z)

7: Compute new residual

R(m,ν )t =R(m,ν )

0 −t∑

l=1

αlam�t (l,1)

(bm

�t (l,2)

)T (f�t (l,3)

)Tfν

8: If t < L, increment t and return to step 3,otherwise stop

9: Estimated support set �̂ = �L

10: Estimated matrix X̂D:(�L(l, 2)T N + �L(l, 1), �L(l, 3))-thcomponent of X̂D is given by α̂l for l = 1, . . . , Lwhile rest of the elements are zero

fc = 10 GHz. We simulate targets from the Swerling-0model with identical amplitudes and random phases. Thereceived signals are corrupted by uncorrelated additiveGaussian noise with power spectral density N0. The SNR isdefined as

SNR =1

Tw

∫ Tw

0 |h0(t )|2dt

N0Bh, (38)

where Tw is the pulse time.We consider a hit-or-miss criterion according to the

Rayleigh resolution as performance metric. A “hit” is de-fined as a range–azimuth estimate that is identical to thetrue target position up to one Nyquist bin (grid point)defined as 1/T Bh and 2/T R for the range and azimuth,respectively. In pulse-Doppler settings, a “hit” is proclaimed



Fig. 6. Range–azimuth-Doppler recovery for L = 6 targets andSNR = −10 dB.

if the recovered Doppler is identical to the true frequencyup to one Nyquist bin of size 1/Pτ , in addition to the twoprevious conditions.

B. Numerical Results

We first consider a sparse target scene with L = 6 targetsincluding a couple of targets with close ranges, a couple withclose azimuths and another couple with close velocities.We use P = 10 pulses and the SNR is set to −10 dB. Ascan be seen in Fig. 6, all targets are perfectly recovered,demonstrating high resolution in all dimensions. Here, therange and azimuth are converted to 2-D x and y locations.

We next turn to the range–azimuth coupling issue anddiscuss the impact of the choice of antennas’ locationsand transmissions’ carrier frequencies. As discussed inSection III-D, the conventional ULA array structure shownin Fig. 1 with carrier frequency selected on a grid, leadsto ambiguity in the range–azimuth domain. In order toovercome the ambiguity issue, we adopt a random arrayconfiguration [51]. We found heuristically that a configu-ration with random antennas’ locations along with randomcarriers on a grid provides better results than random car-riers with a ULA structure. This is reasonable as we addmore degrees of freedom for better sidelobe control. Fig. 7shows a typical result of sidelobes for both configurations.The peak sidelobe level for the configuration with randomantennas’ locations is consistently lower.

We then compare our FDMA processing with classicMIMO processing using CDMA waveforms. Fig. 8 showsthe hit rate of both techniques with respect to bandwidthBh so that the total bandwidth Btot is identical for both.The experiment was performed without noise so that thedecrease in the performance in CDMA is due to the violationof the narrowband assumption (9). In our configuration,the delay ηmqϑl resulting from the array geometry cannotbe ignored since A6 does not hold, as the array apertureλZ = 6m is 4 times the range resolution 1/T Bh = 1.25m.When the received signal is modeled such as to artificiallyremove the delay differences between antennas, that issynthetically generated so that τl replaces τl,mq in (12), thenthe performance of both methods is identical, as expected. InFig. 9, we observe that the targets with small azimuth angle

Fig. 7. Range–azimuth map in noiseless settings for random carrierfrequencies along (a) range axis (b) azimuth axis, and for random

antennas’ locations along (c) range axis and (d) azimuth axis, for L = 1target. The red dotted line indicates the peak sidelobe level for this target.

Fig. 8. Hit rate of FDMA and classic CDMA versus bandwidth.

Fig. 9. Range–azimuth recovery for L = 4 targets using classic (a)CDMA and (b) FDMA.

θl are detected by both techniques, whereas targets on theend-fire direction (θl = ±90◦, corresponds to the broadsidedirection) are missed by the CDMA approach. This happensbecause the delay differences between channels are toolarge, which violates the narrowband Assumption A6.



Fig. 10. Hit rate of noniterative FDMA and classic CDMA versusbandwidth.

Fig. 11. Hit rate of iterative FDMA and noniterative FDMA versusRCS ratio.

In order to demonstrate that the performance gain ofour FDMA approach over the classic CDMA is due tothe relaxed narrowband assumption rather than our specificiterative processing, we consider a noniterative recovery ap-proach. Fig. 10 shows our FDMA method with noniterativerecovery, corresponding to one iteration of Algorithm 1.This constitutes evidence that our approach outperformsconventional CDMA processing from the relaxed narrow-band assumption.

The iterative approach does boost performance of multi-ple targets recovery with high RCS dynamic range, allowingdetection of weak targets masked by the strong ones. Indoing so, we further decrease the effect of the sidelobelevel and, thus, improve the detection performance. To com-pare both iterative and noniterative recovery, we considerL = 2 targets whose locations are generated uniformly atrandom with varying RCS ratios defined as 10 log10( αlmax

αlmin).

In Fig. 11, we can see that the noniterative approach attains50% hit rate for an RCS ratio of 8 dB, which means thatthe weak target is totally masked by the strong one. The

Fig. 12. FDMA MIMO prototype and user interface [68].

iterative approach detects the weak target up to an RCSratio of 20 dB.

Each iteration of our proposed FDMA approach takes3.9 s for 40 million range–azimuth-Doppler grid points us-ing an Intel Core i7 PC without GPU components. We haveimplemented a hardware prototype realizing the FDMAMIMO processing presented here. The prototype, shownin Fig. 12, proves the hardware feasibility of our FDMAMIMO radar. Further details can be found in [67]–[69].

VI. CONCLUSION

In this article, we considered a MIMO radar configura-tion based on FDMA waveforms. Using FDMA allows usto relax the traditional narrowband assumption that createsa tradeoff between range and azimuth resolution. We areable to combine a large overall bandwidth for high rangeresolution and a large virtual array aperture for high az-imuth resolution. This is enabled by the relaxed narrow-band assumption that applies to the individual bandwidthof each transmitter rather than the total bandwidth as forCDMA. We thus allow widebeam beamforming while us-ing simple phase-shifiting over the channels, providing analternative to the classic TTD beamforming. In order toovercome one of the main FDMAs drawbacks, which limitsthe range resolution to the individual bandwidth, we pro-posed a joint processing algorithm of the channels achievingrange resolution with respect to the overall bandwidth.Our system and subsequent processing copes with range–azimuth coupling, which occurs when using FDMA, byusing a random array configuration along with random car-rier allocation. The digital processing is a feasible iterativeCS-based approach for simultaneous sparse recovery withrobustness to a multiple target scenario with high dynamicrange. Simulations illustrated the increased resolution ob-tained by our approach in comparison with classic CDMA,leading to the better detection performance.

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David Cohen received the B.Sc. degree in electrical engineering from Bar Ilan University,Ramat Gan, Israel, in 2013 and the M.Sc degree in electrical engineering in 2018 from theTechnion—Israel Institute of Technology, Haifa, Israel.

In 2016, he joined Rafael Advanced Defense Systems Ltd., Haifa, Israel, where heis currently a Research Associate with the Signal Processing Department. His researchinterests include theoretical aspects of signal processing, sampling theory, compressedsensing, and estimation theory with applications to advanced radar systems.

Deborah Cohen received the B.Sc. degree (summa cum laude) in electrical engineeringin 2010 and the Ph.D. degree in electrical engineering (signal processing) in 2016 fromthe Technion—Israel Institute of Technology, Haifa, Israel, in 2010.

She is currently a Senior Research Scientist with the Machine Learning and NaturalLanguage Processing Teams, Google Research Israel, Haifa, Israel. Her research inter-ests include theoretical aspects of signal processing, compressed sensing, reinforcementlearning, and conversational bots.

Dr. Cohen was awarded the Meyer Foundation Excellence Prize, in 2011. She was therecipient of the Sandor Szego Award and the Vivian Konigsberg Award for Excellence inTeaching from 2012 to 2016, the David and Tova Freud and Ruth Freud-Brendel MemorialScholarship in 2014 and the Muriel and David Jacknow Award for Excellence in Teachingin 2015. Since 2014, she is an Azrieli Fellow.



Yonina C. Eldar (S’98–M’02–SM’07–F’12) received the B.Sc. degree in physics in 1995and the B.Sc. degree in electrical engineering in 1996 both from Tel-Aviv University (TAU),Tel-Aviv, Israel, and the Ph.D. degree in electrical engineering and computer science in2002 from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA.

She is currently a Professor with the Department of Mathematics and ComputerScience, Weizmann Institute of Science, Rehovot, Israel. She was previously a Professorwith the Department of Electrical Engineering, Technion—Israel Institute of Technology(Technion), where she held the Edwards Chair in Engineering. She is also a VisitingProfessor with the MIT, a Visiting Scientist with the Broad Institute, and an AdjunctProfessor with Duke University, and was a Visiting Professor with the Stanford University.She is the author of the book Sampling Theory: Beyond Bandlimited Systems (CambridgeUniversity Press) and co-author of four other books published by Cambridge UniversityPress. Her research interests are in the broad areas of statistical signal processing, samplingtheory and compressed sensing, learning and optimization methods, and their applicationsto biology, medical imaging, and optics.

Dr. Eldar is a member of the Israel Academy of Sciences and Humanities (elected 2017)and a EURASIP Fellow. She was the recipient of many awards for excellence in researchand teaching, including the IEEE Signal Processing Society Technical Achievement Award(2013), the IEEE/AESS Fred Nathanson Memorial Radar Award (2014), and the IEEE KiyoTomiyasu Award (2016). She was a Horev Fellow of the Leaders in Science and Technologyprogram with the Technion and an Alon Fellow, the Michael Bruno Memorial Award fromthe Rothschild Foundation, the Weizmann Prize for Exact Sciences, the Wolf FoundationKrill Prize for Excellence in Scientific Research, the Henry Taub Prize for Excellencein Research (twice), the Hershel Rich Innovation Award (three times), the Award forWomen with Distinguished Contributions, the Andre and Bella Meyer Lectureship, theCareer Development Chair with the Technion, the Muriel and David Jacknow Award forExcellence in Teaching, and the Technion’s Award for Excellence in Teaching (two times)and several best paper awards and best demo awards together with her research studentsand colleagues including the SIAM Outstanding Paper Prize, the UFFC Outstanding PaperAward, the Signal Processing Society Best Paper Award and the IET Circuits, Devicesand Systems Premium Award, was selected as one of the 50 most influential women inIsrael and in Asia, and is a highly cited researcher. She was a member of the Young IsraelAcademy of Science and Humanities and the Israel Committee for Higher Education. Sheis the Editor-in-Chief of Foundations and Trends in Signal Processing, a member of theIEEE Sensor Array and Multichannel Technical Committee and serves on several otherIEEE committees. In the past, she was a Signal Processing Society Distinguished Lecturer,member of the IEEE SIGNAL PROCESSING THEORY AND METHODS and Bio Imaging SignalProcessing technical committees, and was an Associate Editor for the IEEE TRANSACTIONS

ON SIGNAL PROCESSING, the EURASIP Journal of Signal Processing, the SIAM Journalon Matrix Analysis and Applications, and the SIAM Journal on Imaging Sciences. She wasa Co-Chair and Technical Co-Chair of several international conferences and workshops.



Documents

High Resolution FDMA MIMO Radar