1

Opportunistic sensing using mmWavecommunication signals: a subspace approach

Emanuele Grossi, Senior Member, IEEE, Marco Lops, Fellow, IEEE, Antonia Maria Tulino, Fellow, IEEE,Luca Venturino, Senior Member, IEEE

Abstract—In this work, we study the joint detection and local-ization of multiple delay- and Doppler-spread targets through anopportunistic radar exploiting mmWave communication signals.The problem is formulated as the identification of an unknownnumber of active subspaces in a large family of subspaces,accounting for the possible positions of potential targets in thedelay-Doppler domain: the resulting testing problem is compositeand multi-hypothesis. At first, we derive a solution based onthe generalized information criterion (GIC), whose complexityis however prohibitive; then, we propose an approximatedform of the GIC-based receiver and derive two iterative data-adaptive strategies, both extracting and eliminating the superim-posed back-scattered subspace signals one-by-one. Leveraging theIEEE 802.11ad standard, a short-range low-mobility applicationis discussed to validate the merits of the proposed proceduresin terms of detection and localization capabilities, robustness tomulti-target interference, and achievable resolution.

Index Terms—Opportunistic sensing, dual-function radar com-munication, radar-aided communication, mmWaves, spread tar-gets, subspace signals, generalized information criterion, gener-alized likelihood ratio test.

I. INTRODUCTION

Millimeter wave (mmWave) has been recognized as a keyenabling technology for fifth-generation (5G) and beyond-5Gwireless communication systems, as it grants a bandwidthof about 250 GHz in between 30 and 300 GHz [1]–[3];the exploitation of mmWaves is made possible by the recentadvances in cost-effective CMOS integrated circuits, high-gain steerable antennas, and new spatial processing techniques,such as massive MIMO and adaptive beamforming [4]–[6].

The availability of an unlicensed spectrum band around60 GHz have spurred several international organizations to

©20XX IEEE. Personal use of this material is permitted. Permission fromIEEE must be obtained for all other uses, in any current or future media,including reprinting/republishing this material for advertising or promotionalpurposes, creating new collective works, for resale or redistribution to serversor lists, or reuse of any copyrighted component of this work in other works.

The work of E. Grossi and L. Venturino was supported by the researchprogram “Dipartimenti di Eccellenza 2018–2022” sponsored by the ItalianMinistry of Education, University, and Research (MIUR).

Emanuele Grossi and Luca Venturino are with the Department of Electricaland Information Engineering (DIEI), University of Cassino and SouthernLazio, 03043 Cassino, Italy, and with Consorzio Nazionale Interuniversitarioper le Telecomunicazioni, 43124 Parma, Italy (e-mail: e.grossi@unicas.it;l.venturino@unicas.it).

Marco Lops and Antonia Maria Tulino are with the Department of Electricaland Information Technology (DIETI), University of Naples Federico II,80138 Naples, Italy, and with Consorzio Nazionale Interuniversitario perle Telecomunicazioni, 43124 Parma, Italy (e-mail: lops@unina.it; antonia-maria.tulino@unina.it).

A preliminary version of this article was presented at the 2020 IEEE RadarConference, Florence, Italy.

develop commercial solutions for gigabit-per-second (Gb/s)short-range communication, supporting high speed internetaccess, real time data streaming (video on demand, HDTV,home theater, etc.), and wireless data bus for cable replace-ment. There are now several WLAN and WPAN standardsoperating in the 57−66 GHz band, including the WirelessHD(released in Jan. 2008 and supporting rates of up to 4 Gb/s) [7],the ECMA-387 (released in Dec. 2008 and supporting ratesof up to 6.35 Gb/s) [8], the IEEE 802.15.3c (released inSep. 2009 and supporting rates of up to 5.3 Gb/s) [9], [10],the IEEE 802.11ad (released in Dec. 2012 and supportingrates of up to 4 Gb/s) [11], [12], and the IEEE 802.11ay (aforthcoming evolution of the 802.11ad which supports rates ofup to 100 Gb/s) [13], [14]. In particular, the IEEE 802.11ad/ystandards are part of the WiGig technology, which allows Wi-Fi devices a seamless transfer between the 2.4, 5, or 60 GHzbands while maintaining compatibility with existing equip-ments [15]. The use of mm-Waves has also been proposedfor vehicular communications to support the massive dataexchange required for autonomous driving [16].

To cope with the severe path loss, mmWave links are usuallydirectional and rely on training/tracking algorithms to alignthe transmit and receive beams [17]. These wide-band narrow-beam transmissions offer the opportunity of turning an existingmmWave communication device into a high-resolution radarby adding a dedicated receive chain aimed at detecting thereflections (if any) generated by nearby scatterers, thus real-izing a low-cost dual-function radar communication (DFRC)system. Monitoring applications may benefit from this plug-in feature, as they could be implemented without the useof a dedicated device [18], [19]. Opportunistic sensing mayeven be exploited by a communication source node to dis-cover and track the destination nodes [20]–[23] and, in full-duplex data transceivers [24]–[26], to identify and localizenearby scatterers producing indirect self-interference (SI) atthe receiver;1 hence, future radar-aided communication pro-tocols might rethink user access control, channel estimation,indirect SI cancellation in full-duplex data transfer, and beamtraining/tracking procedures to reduce the signaling overhead.More generally, communication and radar functions are be-coming more and more intermingled in forthcoming networkevolutions [27]–[30], and a low-cost integration may be helpful

1In this work, we distinguish between direct and indirect SI [24]–[26]. Theformer is due to the direct path from the transmitter to the receiver and it isalso referred to as line-of-sight or near-field SI in the literature; instead, thelatter is due to the indirect paths from the transmitter to nearby scatterers tothe receiver and it is also referred to as non-line-of-sight or far-field SI.

2

in the deployment of smart cities, roads, and factories, whereinexchanging on-demand data, sensing the environment, andsharing the acquired radar images or a selected number ofextracted features are key basic operations to enable higherlevel services and applications [31], [32].

The idea of implementing an opportunistic mmWave radarby using the 802.11ad standard has been discussed in [33] andfurther elaborated in [34], [35]; initial experimental resultshave also been reported in [36]. As probing signal, we canexploit either the control physical (CPHY) packet transmittedin the sector level sweep (SLS) or the single-carrier physical(SCPHY) packet transmitted in the data transmission interval(DTI). Each choice has its own merit: the SLS mimics ascanning radar and allows patrolling a wide angular sector,while the DTI may contain multiple packets to be jointlyprocessed. These solutions are not mutually exclusive; forexample, we argue that the SLS can be used to get aninitial map of the environment, while the subsequent DTI toconfirm and refine previous decisions, thus realizing an alert-confirm logic [37], [38]; this procedure could be facilitatedby reserving some service periods of the DTI to send dummypackets towards the alerts found in the SLS.

The studies in [33]–[36] rely on the use of a correlation-based receiver. However, since the probing signal provided bythe underlying communication system has not been optimizedfor radar operations, the ambiguity function of the processeddata segment may present heavy side-lobes [39], thus leadingto a near-far problem in the presence of multiple targets.2

In order to improve to some extent the velocity estimationaccuracy, the use of Doppler-resilient Golay complementarysequences is suggested in [40], while an adaptive virtual wave-form design is considered in [41]: however, these solutionsrequire a modification of the operational mode of the standard,which is not an option in current off-the-shelf commercialdevices, and still may not guarantee a satisfactory detectioncapability with a constant false alarm rate. To combat theside-lobe masking problem, a data-adaptive procedure for thedetection and localization of point-like targets has instead beenpresented in [42]–[44], which is based upon a successive-interference-cancellation mechanism and nor does it requireany modification of the underlying communication standard.

The present study is motivated by the evidence that, giventhe bandwidth and the duration of the processed signals, phys-ical objects are typically delay- and Doppler-spread. Indeed,a bandwidth in the order of 1 GHz at 60 GHz would yielda range resolution in the order of decimeters, whereby evenobjects of limited size would be resolved into several scatteringpoints with possibly different radial velocities. A natural fixcould be to first design the radar detector for point-like targetsand then undertake a post-detection automatic data associationprocedure: this would in any case represent a sub-optimum so-lution, with the inconvenient of a remarkable complexity if thenumber of scattering centers grows large. Instead, in this paperwe take a different approach: starting from the preliminaryresults reported in [45], the reverberation of a spread target is

2This is akin to the near-far problem encountered in code-division or space-division multiple-access communication systems.

deemed as a vector of a linear subspace whose dimensionalityis dictated by the delay and Doppler resolutions granted by theavailable probing signal and by the target extension [46]–[48].As a consequence, the problem of detecting and localizingan unknown number of targets is reduced to that of deter-mining the set of active linear subspaces in a large familyof candidate linear subspaces matched to all feasible delayand Doppler locations.3 This is a composite multi-hypothesistesting problem, wherein the number of targets and theirextension determine the model order. Starting from a solutionbased on the generalized information criterion (GIC) rule [49],[50], we propose two approximate forms thereof leading to asequence of generalized likelihood ratio tests (GLRTs) and,eventually, to an iterative subspace-based detector/estimatorwhich extracts the prospective spread targets one-by-one fromthe noisy data, after eliminating the previously-detected signalcomponents. The proposed formulation subsumes the GLRTderived in [48] to detect and identify a single subspace signaland complements the results established in [44] with referenceto point-like targets. A thorough numerical analysis showsthat the proposed approach represents a valid solution for theconsidered joint detection-localization problem, particularlysuited for short-range low-mobility sensing applications.

In line with previous related studies, we leverage here thenomenclature of the IEEE 802.11ad standard and employ itsSCPHY packet; however, the proposed design methodologycan also be adapted to other packets of the 802.11ad or topackets of other mmWave standards. As in [44], the integrationof the radar function does not require to modify the operationalmode of the underlying communication standard and onlyrelies on the signal processing implemented at the dedicatedreceive chain. As a consequence of this design choice, theperformance of the active communication links is not affectedin any way by the opportunistically-added radar function,which remains fully transparent to the network operator andusers and even to any external eavesdropper.

The remainder of this work is organized as follows. Sec. IIdescribes the considered opportunistic radar system and spec-ifies the design assumptions. Sec. III contains the formulationand the theoretical study of the joint detection and localizationproblem, while Sec. IV presents the proposed approximatesolutions. Sec. V contains the numerical analysis. Finally,Sec. VI contains the conclusions and hints for future work.

Notation: Column vectors and matrices are denoted bylowercase and uppercase boldface letters, respectively. (·)Tand (·)H denote the transpose and the conjugate-transposeoperation, respectively. 1M and 0M denote a vector withM all-one and all-zero entries, respectively. IM and OM,N

denote a M × M identity and a M × N all-zero matrix,respectively. tr{A} and ‖a‖ denote the trace of the squarematrix A and the Frobenius norm of the vector a, respectively.vec{a1, . . . ,aN} is the MN -dimensional vector obtained bypiling up the M -dimensional vectors {an}Nn=1. The symbols ?,�, and ⊗ denote convolution, Schur product, and Kroneckerproduct, respectively. Two subspaces S1 and S2 of CM aresaid essentially disjoint if S1 ∩ S2 = {0M}. For P ≥ 2,

3This family can also be regarded as a dictionary.

3

0 dB

-17 dB-22 dB

-30 dB

+0.94 +1.2 +2.7 +3.06

frequency [GHz]

-0.94-1.2-2.7-3.06

Figure 1. Spectrum mask of the IEEE 802.11ad standard.

the subspaces {Sp}Pp=1 of CM are said independent if Spand

∑Pj=1, j 6=p Sj are essentially disjoint for p = 1, . . . , P ;

otherwise, they are said dependent. Finally, E[·] denotes thestatistical expectation, while i is the imaginary unit.

II. OPPORTUNISTIC MMWAVE RADAR

Consider an 802.11ad device sending L SCPHY packets tothe same destination and, therefore, towards the same angularsector under inspection; the packets are sent on the samefrequency channel during as many service periods of theDTI, which may be non-consecutive. Each packet contains apreamble, a header, and a payload (see [11] for details). Up toa scaling factor accounting for the transmit energy (includednext in the target response), the baseband transmit signal iswritten as s(t) =

∑L`=1 s`(t), where

s`(t) =

J∑j=1

b`(j)ψtx(t− (j − 1)T − T`

). (1)

In the above equation, {b`(j)}J`j=1 are the unit-energy symbolscontained in the `-th packet starting at T`, T is the symbolinterval, and ψtx(t) is a unit-energy causal pulse.

The following remarks are now in order. The first J = 3328symbols of each packet correspond to the common preamble:hence, b1(j) = · · · = bL(j) = b(j) for j = 1, . . . , J . Also, thestandard specify the symbol rate, namely, 1/T = 1760 MHz,and requires that the power spectral density of the transmitsignal s`(t) complies with the mask reported in Fig. 1. Finally,the specific choice of ψtx(t) is left to the implementer.

A. Design assumptions

The source node is equipped with a dedicated radar receivechain, sharing the timing with the co-located communicationtransmitter and aimed at discovering the echoes generated byany physical obstacle (hereafter simply referred to as a target)in the probed angular sector.4

We assume that no delay/Doppler migration occurs duringthe illumination time: this limits the duration of s(t), say T ,determined by the number of packets, the spacing among theservice periods, the number of bits conveyed in each payload,

4In opportunistic sensing, the indirect SI is exploited to obtain informationabout the environment; instead, in full-duplex data transfer, the indirect SI maymask the data message, whereby it is deemed as a harmful signal [24]–[26].If both these technologies are integrated in the same device, the radar receivechain could actually provide important information about nearby scatterers tothe full-duplex data transceiver.

and the adopted modulation and coding scheme.5 Specifically,let vmax ≥ 0 and amax ≥ 0 be the maximum radial velocityand acceleration of a prospective scatterer, we require that [39]

2vmaxT

co� ∆τ ,

2amaxT

λo� ∆ν (2)

where co is the speed of light, λo is the carrier wavelength(5 mm), ∆τ = 1/W and ∆ν = 1/T are approximately thedelay and Doppler resolutions, respectively, and W is the (two-sided) bandwidth of the transmitted signal. The constraintsin (2) result into a fundamental trade-off between the max-imum velocity to be measured and the achievable velocityresolution. For example, if W = 1/T , amax = 0.5 m/s2, andvmax = 6.25 or 30 m/s, then we need T � 13.6 or 2.84 ms,which implies λo∆ν/2� 0.18 or 0.88 m/s, respectively.

We denote by Se =[τmin, τmax] × [νmin, νmax

]the in-

spected delay-Doppler region, where τmin, τmax, νmin, andνmax are the minimum and maximum delay and the minimumand maximum Doppler shift of a echo, respectively. Also, weassume that a prospective target has a delay and Dopplerspread Dτ∆τ and Dν∆ν , respectively, where Dτ and Dν

are positive integers tied to the dimension and the motion ofits constituent parts and, possibly, to the presence of multi-path propagation; hence, this target can be decomposed intoD = DτDν scattering points in the inspected delay-Dopplerregion, which produce as many resolvable echoes [46], [53].In a heterogeneous environment, Dτ and Dν must be chosenas a compromise between the following contrasting effects.If the actual spread of the observed object is larger than thedesign value, then this object may be resolved into multipledistinct targets; if it is smaller, only a coarse localization willbe possible, tied to the prior design uncertainty. As common inradars [54], a subsequent logic (not included in this study) mayassociate adjacent detected spread targets to a unique physicalobject and/or further refine their estimated parameters.

Finally, we assume that the direct SI has been perfectlyremoved in previous stages of the processing chain. Currentstudies envision a number of solutions to mitigate the directSI at mm-Waves [25], [26], [55], [56], which include theuse of highly directive antennas, polarization separation, self-reflectors, absorptive shields, analog and digital spatial beam-forming, and analog and digital cancellation techniques alongeach radio-frequency chain. Any residual direct SI from theprevious filtering stages would of course increase the distur-bance (i.e., noise plus direct SI) power and, if simply ignoredat the design stage, may reduce the capability of detectingand localizing weaker (usually more distant) targets. Moregenerally, the design framework considered in the followingsections can provide additional degrees of freedom to combatthe residual direct SI. Indeed, the latter could be regarded as adummy point-like target with known delay and zero Dopplerand, in principle, canceled out either by resorting to a zero-forcing-like filter or by embedding its statistical propertiesinto the structure of the disturbance covariance matrix (seeFootnote 7). However, we feel that properly studying the

5To extend the processing time, coherent motion compensation proceduresand/or incoherent track-before-detect algorithms can be used [51], [52].

4

s2(t) s3(t)s1(t)

α1e2πiν1ts(t− τ1)

α2e2πiν2ts(t− τ2)

received

echoes

s(t)

T1 T2 T3

service periods 1, 2, 3, and 4 (from left to right)

︸ ︷︷ ︸ ︸ ︷︷ ︸︸ ︷︷ ︸

i(t)

T

Figure 2. Examples of two echoes generated by the signal transmitted in theservice periods 1, 2, and 4 (when L = 3, P = 2, Dτ = 1, Dν = 1) and ofone echo generated by the another packet transmitted in the service period 2.

effectiveness of these solutions would require to also accountfor other non-ideal effects in the overall processing chain (suchas, for example, I/Q imbalance, quantization errors, phase andfrequency offsets, non-linear distortions, leakage from adjacentfrequency channels, etc.): this is outside the scope of thiscontribution and left as a future work.

B. Continuous-time received signal

Under the above assumptions, the received baseband signalcan be modeled as (see also Fig. 2)

r(t) =

P∑p=1

∫ +∞

0

s(t− τ)hp(t, τ)dτ + i(t) + w(t) (3)

where• P ∈ {0, 1, . . . ,K} is the unknown number of targets andK is an upper bound to P ;

• hp(t, τ) is the time-variant impulse response of the p-thtarget [28], namely,

hp(t, τ) =

Dτ−1∑dτ=0

Dν−1∑dν=0

αp,dτ ,dν

e2πi(νp+dν∆ν)tδ(τ − τp − dτ∆τ ) (4)

where the pair (τp, νp) specifies its location in the region

S =[τmin, τmax − (Dτ − 1)∆τ ]×

[νmin, νmax − (Dν − 1)∆ν

]⊆ Se (5)

and αp,dτ ,dν is the response of the scatterer with delayand Doppler shift (τp + dτ∆τ , fp + dν∆ν) ∈ Se;

• i(t) contains the echoes (if any) generated by packetstransmitted in other service periods than those employedfor the probing signal s(t);

• w(t) is a circularly-symmetric Gaussian process account-ing for the additive noise.

The signal r(t) is sent to a causal filter with impulseresponse ψrx(t) to remove the out-of-bandwidth noise and theinterference from adjacent frequency channels; as such, thetwo-sided bandwidth of ψrx(t) should be less than the channel

T` T` + JT

I`

τmax

I`,off ∆I`

T` + J`TI`︷ ︸︸ ︷preamble

preamble

Figure 3. Description of the processing interval I`.

spacing (namely, 2.16 GHz for the IEEE 802.11ad standard).The filtered signal r(t) = r(t) ? ψrx(t) can be written as

r(t) =

P∑p=1

∫ +∞

0

s(t− τ)hp(t, τ)dτ + i(t) + w(t) (6)

where s(t) = s(t) ? ψrx(t), i(t) = i(t) ? ψrx(t), and w(t) =w(t)?ψrx(t). In (6), we have exploited the fact that e2πiνts`(t−τ) ? ψrx(t) ' e2πiνt [s`(t− τ) ? ψrx(t)], since ψrx(t) has aneffective duration in the order of 1/W and, for any Dopplershift ν of interest, we have ν �W .

C. Processing intervals

We elaborate the signal received in the time intervals I` =[I`, I`+∆I`) ⊆ [T`, T`+K`T ], for ` = 1, . . . , L, as shown inFig. 3. To simplify the design of the radar receiver, we makethe following assumptions:• I`,1 > T` + τmax, where τmax ≥ τmax is the maximum

traveling time of any echo generated during the DTI andhitting the radar receiver;6

• I` + ∆I` < T` + JT .These conditions are not stringent; indeed, even for a maxi-mum traveling distance of 300 m (way beyond the radius cov-ered by most 802.11ad devices [12]), then τmax ' 1760T '1 µs < JT ' 1.89 µs and I` can span up to 0.89 µs.

The first assumption implies that only the echoes generatedby s`(t) fall in I`; as a consequence, we can write

r(t) =

P∑p=1

∫ +∞

0

s`(t− τ)hp(t, τ)dτ + w(t), t ∈ I`. (7)

The second assumption implies that the segment of s`(t− τ)falling in I` only contains the symbols from the preamble; asa consequence, we can write

s`(t− τ) = g(t− τ − T`), t ∈ I` (8)

where

g(t) =

J∑j=1

b(j)ψtx(t− (j − 1)T

)? ψrx(t) (9)

is a packet-independent waveform; in particular, including thesymbols of the channel estimation field (CEF) of the preamblehas been proved effective in [34], [44]. Finally, under the aboveassumptions, we have ν∆I` � 1 for any Doppler shift ν of

6Recall that the radar receiver may also be hit by echoes generated bythe SCPHY packets transmitted (possibly by other devices) in other serviceperiods than those included in s(t); such echoes are contained in i(t). Forthis reason, τmax can possibly be larger than τmax.

5

practical interest: indeed, even assuming a radial velocity of200 Km/h, we have ν∆I` < νJT ' 0.04; as a consequence,we can write

e2πiνt ' e2πiνI` , t ∈ I`. (10)

Summing up (4) and (7)-(10), we obtain

r(t) =

P∑p=1

Dτ−1∑dτ=0

Dν−1∑dν=0

αp,dτ ,dν e2πi(νp+dν∆ν)I`

g(t− τp − T` − dτ∆τ ) + w(t), t ∈ I`. (11)

D. Discrete-time model

The baseband signal r(t) is sampled at the time instantsI`+ (m`− 1)Tc for m` = 1, . . . ,M` and ` = 1, . . . , L, whereM` = d∆I`/Tce is the number of samples taken in I` andTc ≤ ∆τ is the sampling interval. We now organize thesemeasurements into a vector structure; after collecting {r(I` +(m − 1)Tc)}M`

m=1 into r` ∈ CM` and piling up {r`}L`=1 intor ∈ CM , with M =

∑L`=1M`, we can write

r =

P∑p=1

Dτ−1∑dτ=0

Dν−1∑dν=0

αp,dτ ,dνx(τp + dτ∆τ , νp + dν∆ν

)︸ ︷︷ ︸

yp

+w.

(12)In the above equation, x(τ, ν) is the signature of an echo

hitting the radar receiver with delay and Doppler shift (τ, ν) ∈Se, defined as follows

x(τ, ν) = d(ν)� g(τ) (13)

where d(ν) = vec{e2πiνI11M1, . . . , e2πiνIL1ML

}, g(τ) =vec{g1(τ), . . . , gL(τ)}, and g`(τ) contains the samples{g(I`,off + (m` − 1)Tc − τ

)}M`

m=1with I`,off = I` − T`.

Also, yp is the signal reflected by the p-th target, whichbelongs to the D-dimensional subspace spanned by the vectors{x(τp + dτ∆τ , νp + dν∆ν

), dτ = 0, . . . , Dτ − 1, dν =

1, . . . , Dν − 1}

. Finally, w contains the samples of the noise;in particular, it is a circularly-symmetric Gaussian vector withcovariance matrix Cw, assumed to be full rank and known.7

A choice of relevant practical interest in Fig. 3 is ∆I` = ∆Iand I`,off = Ioff, for ` = 1, . . . , L; in this case, we have

x(τ, ν) = d(ν)⊗ g(τ) (14)

where d(ν) = vec{

e2πiνI1 , . . . , e2πiνIL}

, g(τ) =(g(Ioff −

τ)· · · g

(Ioff − τ + (M − 1)Tc)

)T, and M = d∆I/Tce.

E. Radar receiver

The radar receiver has to detect the superimposed subspacesignals of the prospective targets and estimate their localizationparameters based on the observation of r and on the knowl-edge of the probing signal and of the target spread. In the nextsection, we formulate this problem as a multiple compositehypothesis testing and study a solution based on a GIC.

7In principle, the structure of the covariance matrix Cw might even accountfor the presence of some residual direct SI.

III. SIGNAL DETECTION AND PARAMETER ESTIMATION

Consider the observed vector r = y + w ∈ CM , wherey =

∑Pp=1 yp is a linear combination of P ∈ {0, 1, . . .K}

subspace signals and w is a circularly-symmetric Gaussiannoise vector with a full-rank covariance matrix Cw. At thedesign stage, we assume that the unknown component signals{y1, . . . ,yP } belong to as many independent subspaces ofCM selected from a family F = {X1, . . . ,XQ} which containsQ distinct D-dimensional subspaces, with Q ≥ K andDK≤M . We refer to the elements of F as the elementarysubspaces, since they are used as building blocks to reconstructy.

Let Xq ∈ CM×D be a steering matrix whose columns forma basis of Xq , for q = 1, . . . , Q. Then, any non-zero signalcomponent in Xq can be expressed asXqαq , where αq ∈ CD\{0D}, is the gain vector containing the unique coordinateswith respect to the given basis [46]. Accordingly, we can writey =

∑Qq=1Xqαq , wherein αq 6= 0D if a signal component is

present in Xq; in this latter case, we refer to Xq as an activeelementary subspace. Also, we denote by

Nk ={

(a1 . . . ak)T ∈ {1, . . . , Q}k : a1 < a2 < · · · < ak

and Xai ∩k∑

j=1, j 6=i

Xaj = {0M}, i = 1, . . . , k}

(15)

the set of ordered vectors indexing k independent elemen-tary subspaces, for k = 1, . . . ,K, and we indicate withXnk =

∑ki=1Xnk,i the sum of the elementary subspaces

indexed by nk = (nk,1 · · · nk,k)T ∈ Nk. Accordingly,

any element of Xnk can be expanded as Xnkαnk , whereXnk =

(Xnk,1 · · · Xnk,k

)is a rank-kD matrix and αnk =

vec{αnk,1 , . . . ,αnk,k} is the corresponding gain vector.Given r and {Xq}Qq=1, we aim at detecting the number

of observed signals and at estimating the active elementarysubspaces and the corresponding gain vectors. Otherwisestated, we have to distinguish among the following compositehypotheses

Hk : αq 6= 0D, for q ∈ {nk,1, . . . , nk,k} andnk = (nk,1 · · · nk,k)

T ∈ Nkαq = 0D, otherwise

(16)

for k = 0, . . . ,K, with the understanding that n0 is an emptyvector andN0 = ∅. In the following, we first discuss the choiceof F and then the solution to the above testing problem.

A. Choice of the family FIf {Xq}Qq=1 are dependent, then the set of active elementary

subspaces containing the signal of interest y may be notunique. Hence, even in the absence of the additive noise,the joint detection/estimation problem may have multiplesolutions. To provide more insights into this point, consider thefollowing examples, where M = 9, ei denotes the canonicalbasis vector of CM with one in the i-th position and zero inthe other positions, and y = β4e4 + β5e5 + β6e6.Example 1. Assume D = 3, Q = 3, and Xq =(e(q−1)3+1 e(q−1)3+2 e(q−1)3+3), for q = 1, . . . , 3. In this

6

case, the elementary subspaces are independent and, as long asat least one of the coefficients {βi}6i=4 is non zero, the activeelementary subspace is necessarily X2.

Example 2. Assume D = 1, Q = 11, Xq = (eq), forq = 1, . . . , 9, and X10 = (e5 + e4), X11 = (e5 − e4). Theelementary subspaces are now pairwise essentially disjoint butnot independent. If β4 6= 0, β5 6= 0, and β6 6= 0, the setof active elementary subspaces can be either {X4,X5,X6}or {X6,X10,X11}; if β4 = 0, β5 6= 0, and β6 6= 0, theset of active elementary subspaces can be either {X5,X6} or{X6,X10,X11}; finally, if β4 = β5 = 0, and β6 6= 0, the activeelementary subspace is necessarily X6.

Example 3. Assume D = 3, Q = 7, and Xq =(eq eq+1 eq+2), for q = 1, . . . , 7. The elementary subspacesare now distinct but not pairwise essentially disjoint. If β4 6= 0,β5 6= 0, and β6 6= 0, the set of active elementary subspaces canbe either {X4} or {X2,X5} or {X3,X6}; if β4 = 0, β5 6= 0,and β6 6= 0, the set of active elementary subspaces can beeither {X4} or {X5} or {X3,X6}; finally, if β4 = β5 = 0, andβ6 6= 0, the active elementary subspace can be either X4 orX5 or X6.

As common in the framework of model order selection (see[49], [50] and references therein), we aim at identifying thesmallest possible number of subspace signals which best fitthe observed data according to some design criterion. For agiven model order, the estimation uncertainty remains tied tothe elementary subspaces in F [46]–[48], which are designchoices tailored to the specific application. In the consideredradar problem, {Xq}Qq=1 are matched to the possible delaysand Doppler shifts of a prospective target [46]–[48], [57], [58],as explained in more detail in Section IV-B.

B. Generalized information criterion

We resort here to a GIC to solve the testing problem [49],[50]. To proceed, we first write the negative log-likelihoodlikelihood function under each hypothesis. Under H0, we have

− ln f0(r) =∥∥∥C−1/2

w r∥∥∥2

+ ln(πMdet (Cw)

). (17)

Under Hk, for k = 1, . . . ,K, and assuming that nk and αnkare deterministic unknown parameters, we have

− ln fk(r;nk,αnk) =∥∥∥C−1/2

w (r −Xnkαnk)∥∥∥2

+

ln(πMdet (Cw)

). (18)

In order to identify the active elementary subspaces underHk, we compute the maximum likelihood (ML) estimate ofthe unknown parameters, say nk and αnk , for k = 1, . . . ,K.For any feasible nk, we have [59], [60]

αnk = arg maxαnk∈CkD

ln fk(r;nk,αnk)

=(XHnkC−1w Xnk

)−1XHnkC−1w r. (19)

Moreover, after plugging (19) into (18), we have

nk = arg maxnk∈Nk

ln fk(r;nk, αnk)

= arg maxnk∈Nk

∥∥∥ΠnkC−1/2w r

∥∥∥2

(20)

where

Πnk = C−1/2w Xnk

(XHnkC−1w Xnk

)−1XHnkC−1/2w . (21)

Notice that Πnk is the orthogonal projector onto the aggregatewhitened subspace Xnk =

∑ki=1Xnk,i , with Xq being the

subspace spanned by the columns of C−1/2w Xq; also∥∥∥ΠnkC

−1/2w r

∥∥∥2

= αHnkXHnkC−1w Xnkαnk (22)

is the energy contained into Xnk .Finally, we compute an estimate of the number of subspace

signals, say P , according to the following rule [49], [50]

P = arg mink∈{0,...,K}

GICk (23)

where

GICk =

{−2 ln f0(r), if k = 0

−2 ln fk(r; nk, αnk) + 2kDγ, if k > 0.(24)

Here 2kD is the model order (i.e., the number of unknownreal-valued coordinates) under Hk, while γ is a penalty factor(more on this infra). After plugging (19) and (20) into (24)and neglecting the irrelevant terms, we obtain

P = arg mink∈{0,...,K}

{− Ek + kDγ

}(25)

where

Ek =

0, if k = 0

maxnk∈Nk

∥∥∥ΠnkC−1/2w r

∥∥∥2

, if k > 0.(26)

is the energy contained into the best sum of k whitenedelementary subspaces indexed by nk under Hk.

The computation of Πnk in (21) requires inverting a kD×kD matrix, which costs O

(k3D3

)flops, and evaluating two

matrix multiplications, one between matrices of size kD×kDand kD × M , which costs O

(k2D2M

)flops, and another

between matrices of size M × kD and kD×M , which costsO(kDM2

)flops. Even assuming that all orthogonal projectors

{Πnk , nk ∈ Nk} can be precomputed and stored into adedicated memory, the in-line calculation of Ek still entailsa complexity O

(M2|Nk|

); indeed, the multiplication between

a matrix of size M×M (namely, Πnk ) and a vector of size M(namely, C−1/2

w r) and the squared Euclidean norm of a vectorof size M have to be computed for any set of k independentelementary subspaces. Hence, the implementation of (23) isnot affordable when Q and/or K is large.

In the remaining part of this section, we recast the rulein (23) in a different way and discuss the choice of γ. In thenext section, instead, we discuss low-complexity approxima-tions of the rule in (23).

7

E1 − E0

Hc0

≷Hc

1

γD

E2 − E0

Hc0

≷Hc

2

γ2D

E3 − E1

Hc1

≷Hc

3

γ2D E3 − E2

Hc2

≷Hc

3

γDE3 − E0

Hc0

≷Hc

3

γ3D

{H0,H2,H3}

E2 − E1

Hc1

≷Hc

2

γD

{H1,H2,H3}

{H0,H3} {H2,H3} {H2,H3}{H1,H3}

H0 H1 H2 H3

Figure 4. Decision logic implementing the rule in (23) for K = 3. When the test statistic exceeds the threshold, the right branch is taken; otherwise, the leftone is taken. The branch label shows the set of surviving hypotheses. As consequence of (29), the dashed paths are never taken.

C. Implementation of the GIC rule as a sequence of GLRTs

Notice that, for K ≥ d > m ≥ 0, we can write

(GICm − GICd) /2 = (Ed − Em)− (d−m)Dγ (27)

where Ed − Em is the statistic of a GLRT discriminatingbetween Hd and Hm [60]. Hence, the hypothesis selectedin (23) is the one surviving the following tests

Ed − EmHcm

≷Hcd

(d−m)Dγ, K ≥ d > m ≥ 0 (28)

where Hcm is the complement of Hm. For each pair of

hypotheses, the test in (28) compare the energy contained inthe estimated sum of whitened elementary subspaces. If theenergy increment granted by considering a larger aggregatesubspace exceeds a given threshold (i.e., it is significant), thenthe lower-order model is discarded; otherwise, the higher-ordermodel is discarded. As each test excludes one hypothesis, Kbinary decisions are required, as shown in Fig. 4 [61].

Interestingly, the event E1 < Dγ excludes here multiplehypothesis. Indeed, let nd = (nd,1, . . . , nd,d)

T be the ML es-timate of nd underHd and denote by E(nd,i; nd,1, . . . , nd,i−1)the energy contained into the projection of Xnd,i onto to theorthogonal complement of

∑i−1j=1 Xnd,j , for i = 1, . . . , d; then,

for any d > m ≥ 1, we have

Ed =

m∑i=1

E(nd,i; nd,1, . . . , nd,i−1)︸ ︷︷ ︸≤Em

+

d∑i=m+1

E(nd,i; nd,1, . . . , nd,i−1)︸ ︷︷ ︸≤Ed−m≤(d−m)E1

. (29)

Exploiting (29), a decision for H0 can readily be made ifE1 < γ: hence, the dashed paths in the decision logic shownin Fig. 4 are never taken.

Several choices of the penalty term have been proposedin the literature (see [49] and the references therein). In-terestingly, Fig. 4 also suggests that we can set γ to get

a desired probability of false alarm, defined as Pfa =Pr(reject H0 under H0) = Pr(E1 > γ under H0). This latterchoice appears relevant in radar applications wherein weusually want to control the number of false alarms in eachscan under the null hypothesis [62].

IV. PROPOSED ALGORITHMS

In order to simplify the implementation of (25), we proposehere to compute a sub-optimal estimate of nk under Hk,say nk, by augmenting the estimate nk−1 = (n1 · · · nk−1)

T

obtained under Hk−1, for k = 1, . . . ,K and with the under-standing that n0 is an empty vector. To this end, denote by

Ak =

{a ∈ {1, . . . , Q} : Xa ∩

k−1∑i=1

Xni = {0M}

}(30)

the set of integers indexing the elements of F which areessentially disjoint by the sum of the elementary subspacesindexed by nk−1. Then, we have

nk = arg maxnk∈Nk

∥∥∥ΠnkC−1/2w r

∥∥∥2

(31)

where8 Nk = {n1}× · · · × {nk−1}×Ak. Also, the energy ofa prospective signal contained in Xnk is

Lk =∥∥∥ΠnkC

−1/2w r

∥∥∥2

. (32)

Upon defining L0 = 0, we now approximate the rule in (28)by replacing Ek with Lk, for k = 0, . . . ,K.

To get more insights into this approximation, we nowillustrate some properties of the test statistics {Lk}Kk=0. Toproceed we first introduce two projection matrices. We denoteby Π⊥nk−1

= IM − Πnk−1the projector on the orthogonal

complement of Xnk−1, with Xn0

= {0M} and Π⊥n0= IM ;

also, for nk ∈ Ak, we denote by

T (nk; nk−1) = Π⊥nk−1C−1/2w Xnk

8Notice that, if ak ∈ Nk , then there exists a unique k × k permutationmatrix, say Pk , such that Pkak ∈ Nk; in particular, ak and Pkak indexthe same set of k independent elementary subspaces taken from F .

8

H2 H3

H1

H0

L3 − L2 ≷ γD

L2 − L1 ≷ γD

L1 − L0 ≷ γD

Figure 5. Decision logic approximating the rule (23) for K = 3.

(XHnkC−1/2w Π⊥nk−1

C−1/2w Xnk

)−1

XHnkC−1/2w Π⊥nk−1

(33)

the projector on the part of Xnk which is orthogonal to Xnk−1.

At this point, the energy contained in the projection of Xnkonto the orthogonal complement of Xnk−1

can be computedas follows [63]

L(nk; nk−1) =∥∥∥T (nk; nk−1)C−1/2

w r∥∥∥2

= αHnkXHnkC−1/2w Π⊥nk−1

C−1/2w Xnkαnk (34)

for k = 1, . . . ,K, where

αnk =(XHnkC−1/2w Π⊥nk−1

C−1/2w Xnk

)−1

XHnkC−1/2w Π⊥nk−1

C−1/2w r. (35)

In particular, it is verified that Lk − Lk−1 = L(nk; nk−1),which is non negative and decreasing with k by construction.As consequence, if Ld − Lm < (d −m)Dγ for d > m, thenLd′ − Lm < (d′ − m)Dγ for any d′ > d. This allows tosimplify the implementation of the approximated logic. Fork = 1, . . . ,K, it is sufficient to solve the sequence of tests

L(nk; nk−1) ≷ γD (36)

withnk = arg max

nk∈AkL(nk; nk−1) (37)

and declare P = k−1 targets if no threshold crossing is firstlyobserved at the k-th step; otherwise, P = K. Fig. 5 illustratesthis logic, while Algorithm 1 summarizes its implementation.Remark 1. At the k-th iteration of Algorithm 1, Xnk isobtained by augmenting Xnk−1

, thus avoiding a joint search ofmultiple active elementary subspaces: this comes at the priceof a possible energy loss, since Lk ≤ Ek. Notice that the teststatistic in (36) can be decomposed into a noise-whiteningfilter C−1/2

w , a filter Π⊥nk−1for rejecting the previously-

estimated components in Xnk−1(treated here as interference),

a filter T (nk; nk−1) matched to the prospective subspacesignal, and an energy detector. Notice that αnk is here the MLestimate of the gain vector of the newly added subspace signalafter the aforementioned noise-whitening and interference-rejection stages. The decision logic progressively augmentsthe interference subspace until no additional signal is found;in particular, we can exploit here the Gram-Schmidt algorithmto update Πnk at each iteration [64]. We refer to this procedureas the matched subspace detector with iterative estimation ofthe interference subspace (MSD-IS).

Algorithm 1 Matched subspace detector with iterative estima-tion of the interference subspace

1: Set A1 = {1, . . . , Q}, Π⊥n0= IM , and P = 0

2: for k = 1, . . . ,K do3: Compute L(nk; nk−1) from (34)4: Compute nk = arg max

nk∈AkL(nk; nk−1)

5: if L(nk; nk−1) > γD then6: P = P + 17: Compute αnk from (35)8: Compute Πnk from (21) and Π⊥nk = IM −Πnk

9: Compute Ak+1 ={a ∈ Ak : Xa ∩

∑ki=1Xni =

{0M}}

10: else11: break12: end if13: end for14: If P ≥ 1, the active elementary subspaces are {Xnp}Pp=1

and the estimated gain vectors are {αnp}Pp=1

Remark 2. Algorithm 1 is optimal—as far as the GIC rulein (23) is concerned—for K = 1 or, for K > 1, if thewhitened elementary subspaces {Xq}Qq=1 are independent. ForK = Q = 1, Algorithm 1 reduces to the matched subspacedetector in colored Gaussian noise discussed in [46], [60],[63], [65]. Instead, the case K = 1 and Q > 1 have beenrecently studied in [48] when the elementary subspaces arepairwise essentially disjoint; a major result in [48] is that theprobability of correct identification of the active elementarysubspace increases with increasing principal angles betweenthe elements of F .

A. Further insights into the decision logic

At the k-th iteration of Algorithm 1 we aim at detecting aprospective subspace signal contained in Xnk , with nk ∈ Ak,in the presence of a subspace interference contained in Xnk−1

and colored Gaussian noise. Accordingly, the test in (36) canbe regarded as the GLRT discriminating between the followingtwo hypotheses [46], [60], [63], [65]

H(k)1 : αq 6= 0D, for q ∈ {n1, . . . , nk−1, nk}, nk ∈ Ak

αq = 0D, otherwise

H(k)0 : αq 6= 0D, for q ∈ {n1, . . . , nk−1}

αq = 0D, otherwise(38)

when nk, αnk , and {αni}k−1i=1 are deterministic unknown

parameters; in particular, H(k)1 is declared if the test statistic

exceeds the threshold; otherwise, H(k)0 is declared.

While the iterative procedure outlined in Algorithm 1 nat-urally arises upon approximating the implementation of theGIC rule, the above interpretation as a sequence of GLRTs ishelpful to devise a convenient modification, which avoids zero-forcing the interfering echoes, as this operation may cause anunnecessary noise enhancement. To proceed, we assume at thedesign stage that only the parameters nk and αnk (related tothe signal to be detected) are deterministic quantities, while the

9

parameters {αni}k−1i=1 (related to the underlying interference)

are modeled as independent circularly-symmetric Gaussianvectors with E

[αniα

Hni

]= σ2

α,niID. In this case, let

Cnk−1=

k−1∑i=1

σ2α,niXniX

Hni +Cw (39)

be the data covariance matrix under H(k)0 ; then, the ML

estimate of αnk under H(k)1 is

αnk =(XHnkC−1nk−1

Xnk

)−1

XHnkC−1nk−1

r (40)

for any nk ∈ Ak. Also, let

T (nk; nk−1) =

C−1/2nk−1

Xnk

(XHnkC−1nk−1

Xnk

)−1

XHnkC−1/2nk−1

(41)

be the orthogonal projector onto the column space ofC−1/2nk−1

Xnk ; then, the ML estimates of nk under H(k)1 is

nk = arg maxnk∈Ak

M(nk; nk−1) (42)

where

M(nk; nk−1) =∥∥∥T (nk; nk−1)C

−1/2nk−1

r∥∥∥2

= αHnkXHnkC−1nk−1

Xnkαnk (43)

is the energy contained in the column space of C−1/2nk−1

Xnk .Hence, the GLRT discriminating between H(k)

1 and H(k)0 now

becomes [46], [60], [63], [65]

M(nk; nk−1)H(k)

1

≷H(k)

0

γD. (44)

At this point, it appears natural to modify the decisionlogic in Fig. 5 by replacing the test statistic L(nk; nk−1)with M(nk; nk−1). Since σ2

α,ni is unknown, we replace itin (39) by ‖αni‖2/D, where αni is the estimated gain vectorobtained when testing H(i)

1 against H(i)0 , for i = 1, . . . , k− 1.

Algorithm 2 summarizes the resulting detector/estimator.Remark 3. Both Algorithm 1 and 2 identify one subspacesignal at the time. However, they handle the previouslyestimated components in a different way. In particular, thetest statistic in (44) can be decomposed into a filter C−1/2

nk−1

whitening the noise plus the previously-estimated componentscontained in Xnk−1

, a filter T (nk; nk−1) matched to theprospective subspace signal, and an energy detector. Thedecision logic iteratively updates the interference covariancematrix until no additional subspace signal is found; in partic-ular, we can exploit the matrix inversion lemma to computeC−1nk , namely, C−1

nk = C−1nk−1

+C−1nk−1

Xnk

(D/‖αnp‖2ID +

XHnkC−1nk−1

Xnk

)−1XHnkC−1nk−1

. We refer to this procedure asthe matched subspace detector with iterative estimation of theinterference covariance matrix (MSD-ICM).Remark 4. The estimated gain vector in (40) can alsobe regarded as the output of the multi-rank mini-mum variance distortionless response (MVDR) beamformerW =

(XHnkC−1nk−1

Xnk

)−1XHnkC−1nk−1

, which minimizes the

Algorithm 2 Matched subspace detector with iterative estima-tion of the interference covariance matrix

1: Set A1 = {1, . . . , Q}, Cn0= Cw, and P = 0

2: for k = 1, . . . ,K do3: Compute M(nk; nk−1) from (43)4: Compute nk = arg max

nk∈AkM(nk; nk−1)

5: if M(nk; nk−1) > γD then6: P = P + 17: Compute αnk from (40)8: Compute Cnk from (39) with σ2

α,ni = ‖αni‖2/Dfor i = 1, . . . , k

9: Compute Ak+1 ={a ∈ Ak : Xa ∩

∑ki=1Xni =

{0M}}

10: else11: break12: end if13: end for14: If P ≥ 1, the active elementary subspaces are {Xnp}Pp=1

and the estimated gain vectors are {αnp}Pp=1

interference-plus-noise power, defined as tr{WCnk−1

WH}

,under the constraint WXnk = ID [66].

B. Implementation of the radar receiver

As common in radars, we divide the inspected region Seinto rectangular bins with size equal to the delay and Dopplerresolution along the corresponding axes [39]; thus, we obtainthe following grid

Ge ={(τmin + (dτ − 0.5) ∆τ , νmin + (dν − 0.5) ∆ν

),

dτ = 1, . . . , Qe,τ , dν = 1, . . . , Qe,ν

}(45)

where Qe,τ = b(τmax−τmin)/∆τ+0.5c and Qe,ν = b(νmax−νmin)/∆ν + 0.5c. Similarly, we divide S into resolution bins,thus obtaining a grid G ⊆ Ge with Q = QτQν points, whereQτ = Qe,τ −Dτ +1 and Qν = Qe,ν−Dν +1 are the numberof bins along the delay and Doppler dimensions, respectively.

For each (τ , ν) ∈ G, we construct the D-dimensional ele-mentary subspace Xτ ,ν spanned by the signatures associatedwith the points {(τ + dτ∆τ , ν + dν∆ν), dτ = 0, . . . , Dτ −1, dτ = 0, . . . , Dτ − 1} ⊆ Ge, which in turn are organized asthe columns of the M × D matrix Xτ ,ν ; also, we constructthe family F containing all these elementary subspaces.9 Atthis point, Algorithms 1 and 2 could directly be employed ifthe observed echoes were located on Ge.

In practice, the inherent small-scale localization errors (alsodue to the off-grid target placement) may be detrimental.Indeed, at each iteration of Algorithms 1 and 2, the estimatedinterference subspace may not fully span the previously-detected signal components; this may lead to a non-negligibleinterference spillover, which in turn may cause spurious

9We retain here a double index to list the elementary subspaces and thecorresponding matrices containing the basis vectors, as it is easily related to aposition on the delay/Doppler plane; this double index can straightforwardlybe mapped into a single index.

10

GeBτi,νi

delay

Dop

ple

r

∆τ

∆ν

τi

νi

Figure 6. The cross markers are the points of the fine-grid Bτi,νi around(τi, νi), when Dτ = 4, Dν = 2, and U = 4; the square markers indicatethe points in Ge; the boldface square markers are the estimated positions ofthe scatterers of the i-th detected target.

threshold crossings and masking of weaker targets [44], [67],[68]. A reasonable fix involves a slight modification of theconstruction of the interference subspace accounting for suchsmall-scale localization errors. Denote by (τi, νi) ∈ G the esti-mated location of the i-th detected target, for i = 1, . . . , k−1,and let ατi,νi be the corresponding estimated gain vector withrespect to the columns of Xτi,νi . As shown in Fig. 6, weconsider a locally-oversampled grid around (τi, νi), namely,

Bτi,νi ={(τi + (dτ/U − 1) ∆τ , νi + (dν/U − 1) ∆ν

),

dτ = 1, . . . , Bτ , dν = 1, . . . , Bν

}(46)

where Bτ = (Dτ + 1)U − 1, Bν = (Dν + 1)U − 1, andU ≥ 1 is the oversampling factor, containing the positions ofB = BτBν ≥ D virtual scattering points for the i-th detectedtarget; also, we construct the M ×B matrix Ξτi,νi containingthe signatures corresponding to the locations in Bτi,νi . Then,at the k-th iteration of Algorithm 2, the observed vector r iswhitened by using the following augmented interference-plus-noise covariance matrix

k−1∑i=1

(‖ατi,νi‖2/B)Ξτi,νiΞHτi,νi +Cw. (47)

The resulting procedure extends the iterative detector/estimatorin [44] to the case of delay/Doppler spread targets.

Likewise, at the k-th iteration of Algorithm 1, we project thewhitened vector C−1/2

w r onto the orthogonal complement ofthe augmented interference subspace spanned by the columnsof {C−1/2

w Ξτi,νi}k−1i=1 . Since the matrix C−1/2

w Ξτi,νi may beill-conditioned in practice, we compute its singular values, sayλi,1 ≥ . . . ≥ λi,B , and consider as the interference subspaceof the i-th target only the span of the left singular vectorscorresponding to the j largest singular values such that

‖ατi,νi‖2

B

B∑m=j+1

λ2i,m < η (48)

where η is a design parameter.

Figure 7. Normalized ambiguity function of the signature vector x(τ, ν)in (13) versus cτ1/2 and λ0(ν1 − ν2)/2 when cτ2/2 = 7 m

V. NUMERICAL RESULTS

We consider a 802.11ad-based radar inspecting the rangeinterval [4, 10] m. We choose ψtx(t) = ψrx(t) to be a truncatedsquare-root raised-cosine with roll-off factor 0.3 and support in[0, 16T ]; the resulting probing signal has 98% of its power inthe frequency interval [−0.5/T, 0.5/T ], whereby we set W =1/T . The receiver elaborates L = 6 packets, equally-spaced ofTp = 0.2 ms, and inspects the non-ambiguous radial velocityinterval [−λ0/(4Tp), λ0/(4Tp)), with λ0/(4Tp) ' 6.25 m/s.Under this setup, the range and radial velocity resolutionsare about 8.5 cm and 2 m/s, respectively, and the conditionsin (2) remain satisfied. This scenario may be reasonable in alow-mobility indoor DFRC system wherein a ceiling-mountedaccess point (AP) serves a corridor or a large room.

As to the processing intervals, we set I`,off = 1536T , ∆I` =80T , and Tc = T , whereby M` = 80 sample/packet are taken.To illustrate the correlation properties of (13), we report inFig. 7 its ambiguity function, defined as Φ(τ1, τ2, ν1 − ν2) =|x(τ1, ν1)Hx(τ2, ν2)|, normalized by its peak value; along thedelay and Doppler axes, the main lobe has a full-width athalf-maximum approximately equal to the delay and Dopplerresolutions, respectively; also, it is evident the presence ofseveral side-lobes, which make a correlator-based receiver notsuitable for the detection of multiple targets, and of periodicpeaks along the Doppler axis at multiples of 1/Tp, which limitthe non-ambiguous radial velocity interval.

The noise process w(t) is assumed white, with a powerspectral density normalized to one. The prospective targets arerandomly displaced in the (continuous) delay-Doppler domain,whereby they are not required to lie on the search grid; also,they span an interval of 34 cm in range and of 4 m/s in radialvelocity (whereby we have Dτ = 4 and Dν = 2); finally, theresponse of each scatterer is modeled as [5], [46]

αp,dτ ,dν = e−jφp,dτ ,dν√Aslowp Afast

p,dτ ,dνζp,dτ ,dν (49)

for p = 1, . . . , P , dτ = 0, . . . , Dτ−1, and dν = 0, . . . , Dν−1.In the above expression, φp,dτ ,dν is a random phase uniformly

11

-6

-4

-2

0

2

4

4 5 6 7 8 9-6

-4

-2

0

2

4

4 5 6 7 8 9 4 5 6 7 8 9

16

18

20

22

24

26

28

30

Figure 8. Evolution of Algorithm 2 when P = 6. At each iteration, we report the value of scoring metric versus the inspected range cτ/2 and radial velocityλ0ν/2, for (τ , ν) ∈ G. The white cross marker denotes the detection in the current iteration, while the red square markers are the previous detections.

4 5 6 7 8 9 10

-5

0

5

18

20

22

24

Figure 9. Positions of the true scatterers (diamond markers) and of theestimated spread targets (red rectangles) in the range-Doppler plane. The blackdotted rectangles denote the resolution bins associated with the grid Ge.

distributed in [0, 2π); 10 log10(Aslowp ) is a Gaussian random

variable with standard deviation of 3 dB, accounting for log-Normal shadowing; (Afast

p,dτ ,dν)1/2 is a Rice random variable

with a unit-power and a shape parameter of 15 dB accountingfor fast fading; finally, ζp,dτ ,dν is a scaling factor, accountingfor the transmit power, the two-way antenna gain, the path loss,the radar cross-section of the scatterer, and the noise power,which rules the signal-to-noise ratio (SNR) of the observedecho, defined as

E[|αp,dτ ,dν |2

]∥∥C−1/2w x

(τp + dτ∆τ , νp + dν∆ν

)∥∥2. (50)

In the following, we assume that all scatterers of p-th targethave the same SNR, say SNRp. Also, in the implementationof the proposed algorithms we set K = 10, U = 10, η = 0.05,and Pfa = 10−6.

A. Performance measures

To measure the performance of the proposed algorithms,we compare the location of their output detections and ofthe true targets. An output detection is counted as a targetdetection if there are true targets within a delay distance Dτ∆τ

and a Doppler distance Dν∆ν . Otherwise, it is counted as afalse alarm. Finally, let yp ∈ CML be the estimated signalassociated with the p-th target; then, we consider the following

normalized root mean square error (NRMSE) to measure theaccuracy in the estimation of the true signal yp:(

E[‖yp − yp‖2 | a detection has occurred

])1/2E [‖yp‖ | a detection has occurred]

. (51)

As benchmark, we consider two genie-aided procedures todetect and localize the target of interest. In the first one, agenie removes the echoes generated by all other interferingtargets; then, the GLRT in [48] is employed on this cleaneddata: we refer to this strategy as the “GLRT with cleaneddata” (GLRT-CD). In the second one, a genie provides theexact covariance matrix of the echoes generated by all otherinterfering targets; then, the GLRT in [48] is implemented byusing such a covariance matrix: we refer to this strategy asthe “GLRT with known-covariance” (GLRT-KC).

B. Output of the MSD-ICM in a single snapshot

Fig. 8 shows six iterations of Algorithm 2 in a singlesnapshot when P = 6. At each iteration, we report thescoring metric M versus the inspected range cτ/2 and theradial velocity λ0ν/2, for (τ , ν) ∈ G: the white cross markerdenotes the detection in the current iteration, while the redsquare markers are the previous detections. It is seen thatthe scoring metric may present several peaks, some of themcaused by true targets, while others by secondary lobes (relatedto the secondary lobes of the ambiguity function in Fig. 7):at each step, the MSD-ICM just detects the current highestpeak, which is likely to correspond to the strongest undetectedtarget. As we proceed, the effect of eliminating the previously-detected components is evident, as their associated peaksand secondary lobes are not present anymore. The algorithmdetects six targets and stops at the iteration 7, not shown hereas no threshold crossing is observed. Fig. 9 reports the location(diamond markers) and the SNR (encoded in the marker color)of the true scattering points; the location of detected targets(red rectangles) is also shown. It is seen that all targets havebeen localized within one resolution bin and no false alarm ispresent. Similar results are observed by using Algorithm 1.

12

0 3 6 9 12 15 180

0.2

0.4

0.6

0.8

1

6 9 12 15 18 21 240.48

0.52

0.56

0.6

0.64

0.68

Figure 10. Detection probability of the first target (left) and NRMSE forits reconstructed signal (right) versus SNR1, when P = 6 and SNR2:P ∈[10, 20] dB.

0 3 6 9 12 15 184.6

4.8

5

5.2

5.4

5.6

5.8

0 3 6 9 12 15 1897

97.25

97.5

97.75

98

98.25

98.5

Figure 11. Average number of detections per snapshot among all targets (left)and percentage of detections within one bin from a target (right) versus SNR1,when P = 6 and SNR2:P ∈ [10, 20] dB.

C. Widely-spaced targets

We evaluate here the average performances over 4000independent snapshots; at each run, the prospective targetsare randomly displaced in the inspected delay-Doppler region,with a separation of at least Dτ∆τ along the delay axis andSNR2:P randomly chosen in the interval [10, 20] dB.

Fig. 10 reports the probability of detecting the first targetand the NRMSE for its reconstructed signal versus SNR1,when P = 6. Remarkably, the proposed solutions performquite close to the genie-aided benchmarks; in particular, fora detection probability of 0.9, the MSD-ICDM is less than1 dB away from the GLRT-CD. Also, the MSD-ICDM slightlyoutperforms the MSD-IS, since zero-forcing some signal com-ponents may lead to a detrimental noise enhancement [46],[59], [60], [69]. Finally, notice that the NRMSE converges hereto a non-zero floor, as the estimation accuracy of the unknownparameters is inherently limited by the finite resolution ofthe adopted search grid. If needed, a refined estimate of thetarget parameters can be obtained by using a locally over-sampled grid as in [44]; alternatively, gridless super-resolution

1 2 3 4 5 6 7 80.82

0.84

0.86

0.88

0.9

0.92

1 2 3 4 5 6 7 80.54

0.55

0.56

0.57

0.58

0.59

0.6

Figure 12. Detection probability of the first target (left) and NRMSE for itsreconstructed signal (right) versus P , when SNR1 = 12 dB and SNR2:P ∈[10, 20] dB.

8 10 12 14 16 18 200

5

10

15

20

25

30

8 10 12 14 16 18 200

2

4

6

8

10

12

Figure 13. Detection (left) and NRMSE (right) loss with respect with respectto the GLRT-CD solution versus SNR1 and P , when SNR2:P ∈ [10, 20] dB.

algorithms might be employed to zoom-in around each regionwherein a detection was found [70].

Fig. 11 reports the average number of detections per snap-shot among all targets versus SNR1 and the percentage ofsuch detections which are within one resolution bin fromboth the delay and Doppler of a true target; in the simulatedsnapshots no false alarm was observed, whereby the numberof prospective targets has never been overestimated. The gapin term of detection capability among the proposed solutionsand the GLRT-CD is here more evident, while the MSD-ICDMstill performs close to the genie-aided detector with a perfectknowledge of the interference covariance matrix. Also, it isinteresting to notice that more than 97% of the detections arewithin one resolution bin from a true target.

Finally, we study the impact of the number of targets onthe system performance. Fig. 12 reports the probability ofdetecting the first target and the NRMSE for its reconstructedsignal versus P , when SNR1 = 12 dB. Also, Fig. 12 reportsthe detection and NRMSE loss of the proposed solutions withrespect to the GLRT-CD versus SNR1, for P = 4, 6, 8. It isseen that the performances of the GLRT-KC, MSD-ICM, and

13

0 2 4 61.55

1.6

1.65

1.7

1.75

1.8

1.85

0 2 4 670

75

80

85

90

95

100

0 2 4 60.55

0.6

0.65

0.7

0.75

0.8

0.85

Figure 14. Average number of detections (left), percentage of detectionswithin one bin from a target (middle), and average NRMSE for the recon-structed signals (right) versus δτ/∆τ , when P = 2 and SNR1:2 = 12 dB.

MSD-IS procedures gracefully degrade when the number oftargets is increased. The gap between GLRT-KC and MSD-ICDM increases with P , since the MSD-ICDM employs anaugmented interference covariance matrix. The gap betweenMSD-ICM and MSD-IS is also increasing, since a zero-forcingprocessing suffers a larger noise enhancement [69].

D. Closely-spaced targets

Finally, we study the effect of varying the target separation.To this end, we simulate P = 2 targets with SNR1 = SNR2 =12 dB and a delay separation δτ . At each run, the target pairis randomly displaced in the inspected delay region with arandomly-chosen and common Doppler shift. Performance areaveraged over 104 independent snapshots.

Fig. 14 reports the average number of detections, thepercentage of detections within one bin from a target, and theaverage NRMSE for the reconstructed subspace signals versusδτ . When the separation drops below the target extension,the performance of the GLRT-KC, MSD-ICM, and MSD-IS procedures progressively deteriorate as the targets getcloser: indeed, the principal angle between the target subspacesreduces and their discrimination becomes more difficult [48].

VI. CONCLUSIONS

In this paper we have considered the problem of detectingand localizing multiple delay- and Doppler-spread targets hitby mmWave communication signals through an opportunisticradar receiver, co-located with the communication transmitter.The problem has been stated as the identification of an un-known number of active subspaces (which contain the signalsback-scattered by as many prospective targets) in a familyof candidate subspaces (matched to the possible scattererpositions over a grid of admissible points in the delay-Dopplerdomain). A GIC-based solution has at first been derived;subsequently, we have devised two iterative approximations ofthe GIC rule to reduce complexity. The proposed proceduresextract one active subspace at the time and eliminate it fromfurther processing. The cancellation step consists of either a

projection onto the orthogonal complement of the interferencesubspace or an interference-plus-noise whitening transforma-tion. A thorough performance assessment has been offered tovalidate the proposed procedures under diverse instances oftarget number, separation, and strength, when the commercialIEEE 802.11ad standard is employed.

The proposed methodology lends itself to further generaliza-tions and applications. The present study could be extendedto scenarios where the prospective echoes are produced bya multiplicity of sources (e.g., GSM, DVB-T, 4/5G, GNSS,and so on) without a full knowledge of their time references,which is the typical scenario encountered in passive sensingsystems. Also, the proposed solutions could be adapted andemployed in other contexts, such as, for example, the jointdetection and direction-of-arrival estimation of multiple angle-spread sources or user discovery in code-division and/or space-division multiple-access systems.

Finally, as also mentioned in Section II-A, future worksshould include in the model some of the non-ideal effectsinevitably present in the overall processing chain (e.g., residualdirect SI, I/Q imbalance, quantization errors, phase and fre-quency offsets, non-linear distortions, leakage from adjacentfrequency channels, etc.) and evaluate their impact on theachievable performance.

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Emanuele Grossi (Senior Member, IEEE) was bornin Sora, Italy, on May 10, 1978. He received the Dr.Eng. degree in Telecommunication Engineering in2002 and the Ph.D. degree in Electrical Engineeringin 2006, both from the University of Cassino andSouthern Lazio, Italy, where he is an assistant pro-fessor since 2006. In 2005 he was visiting Scholarwith the Department of Electrical & Computer En-gineering of the University of British Columbia,Canada, and in 2009 he had a visiting appointmentin the Digital Technology Center, University of Min-

nesota, MN. Since 2017, he serves as an associate editor for EURASIP SignalProcessing, Elsevier. His research interests concern wireless communicationsystems, radar detection and tracking, and statistical decision problems withemphasis on sequential analysis.

Marco Lops (Fellow, IEEE) is a Professor withthe Department of Electrical and Information Tech-nology (DIETI) at the University “Federico II” ofNaples, Italy. He obtained his “Laurea” and hisPh.D. degrees from “Federico II” University, wherehe was first assistant and then associate Professor. InMarch 2000 he moved to University of Cassino andSouthern Lazio as a full professor, and he returned to“Federico II” in 2018. Meanwhile, in 2009–2012, hewas also with ENSEEIHT (Toulouse, France), firstas full professor (on leave of absence from Italy) and

then as visiting professor. In fall 2008 he was a visiting professor at Universityof Minnesota, and in spring 2009 at Columbia University. Previously, he hadalso held visiting positions at University of Connecticut, at Rice University,and at Princeton University. In 2009–2015 he served two terms in the SensorArray and Multichannel Signal Processing Technical Committee (SAM).He has served as Associate Editor for the Journal of Communications andNetworks, for IEEE Transactions on Information Theory (Area: Detectionand Estimation), for IEEE Signal Processing Letters, for IEEE Transactionson Signal Processing (two terms), and is now serving as a Senior Area Editorfor IEEE Transactions on Signal Processing. He was co-recipient (with EzioBiglieri) of the 2014 best paper award from the Journal of Communicationsand Networks. He was selected to serve as a Distinguished Lecturer for theSignal Processing Society during 2018–2020. His research interests are indetection and estimation, with emphasis on communications and radar signalprocessing, and he has authored or co-authored over 90 scientific paperspublished on refereed journals.

Antonia M. Tulino (Fellow, IEEE) received thePh.D. degree in electrical engineering from SecondaUniversita’ degli Studi di Napoli, Italy, in 1999. Sheheld research positions with the Center for WirelessCommunications, Princeton University, Oulu, Fin-land, and also with the Universita’ degli Studi delSannio, Benevento, Italy. From 2002 to 2016, shehas been an Associate Professor with the Universita’degli Studi di Napoli Federico II, where she has beena Full Professor since 2017. Since 2002, she hasbeen collaborating with Nokia Bell Labs. Starting

from October 2019, she is, also, Research Professor at the at Dep. of Electricaland Computer Engineering NYU Tandon School of Engineering, NY, 11201,USA. In September 2020 she is appointed as teaching director of the 5GAcademy jointly organized by the Università degli Studi di Napoli, Federico IIin collaboration with Capgemini, Nokia and TIM. From 2011 to 2013, she hasbeen a member of the Editorial Board of the IEEE Transactions on InformationTheory and in 2013, she was elevated to IEEE Fellow. She was selected by theNational Academy of Engineering for the Frontiers of Engineering program in2013. From 2019, she is the chair of the Information Theory society FellowsCommittee. She has received several paper awards and among the others the2009 Stephen O. Rice Prize in the Field of Communications Theory for thebest paper published in the IEEE Transactions on Communications in 2008.She was a recipient of the UC3M-Santander Chair of Excellence from 2018to 2019.

Luca Venturino (Senior Member, IEEE) was bornin Cassino, Italy, on August 26, 1979. He receivedthe Ph.D. degree in Electrical Engineering in 2006from the University of Cassino and Southern Lazio,Italy. In 2004 and 2009, he was Visiting Researcherat the Columbia University, New York (NY). Be-tween 2006 and 2008, he spent nine months at NECLaboratories America, Princeton (NJ). Currently, heis an associate professor at the University of Cassinoand Southern Lazio. Since 2017, he serves as anassociate editor for IEEE Transactions on Signal

Processing and IEEE Signal Processing Letters. His research interests arein detection, estimation, and resource allocation, with emphasis on commu-nications and radar signal processing.