JOINT RADAR-COMMUNICATIONS PROCESSING FROM A DUAL-BLINDDECONVOLUTION PERSPECTIVE

Edwin Vargas†, Kumar Vijay Mishra‡, Roman Jacome†, Brian M. Sadler‡, and Henry Arguello†

†Universidad Industrial de Santander, Bucaramanga, Colombia, 680002‡United States CCDC Army Research Laboratory, Adelphi, MD 20783 USA

ABSTRACTWe consider a general spectral coexistence scenario, wherein thechannels and transmit signals of both radar and communications sys-tems are unknown at the receiver. In this dual-blind deconvolution(DBD) problem, a common receiver admits the multi-carrier wirelesscommunications signal that is overlaid with the radar signal reflected-off multiple targets. When the radar receiver is not collocated with thetransmitter, such as in passive or multistatic radars, the transmittedsignal is also unknown apart from the target parameters. Similarly,apart from the transmitted messages, the communications channelmay also be unknown in dynamic environments such as vehicularnetworks. As a result, the estimation of unknown target and com-munications parameters in a DBD scenario is highly challenging. Inthis work, we exploit the sparsity of the channel to solve DBD bycasting it as an atomic norm minimization problem. Our theoreticalanalyses and numerical experiments demonstrate perfect recoveryof continuous-valued range-time and Doppler velocities of multipletargets as well as delay-Doppler communications channel parametersusing uniformly-spaced time samples in the dual-blind receiver.

Index Terms— Atomic norm, dual-blind deconvolution, channelestimation, joint radar-communications, passive sensing.

1. INTRODUCTION

With the advent of new wireless communications systems and novelradar technologies, electromagnetic spectrum has become a contestedresource [1]. This has led to the development of various system en-gineering and signal processing approaches for an optimal spectrumsharing performance [2]. In general, spectrum-sharing technologiesfollow three major approaches: co-design [3], cooperation [4], andcoexistence [5]. While co-design requires designing new systemsand waveforms to efficiently utilize the spectrum [6], spectral coop-eration requires exchange of additional information between radarand communications to enhance their respective performances [2].The coexistence approach, on the other hand, does not mandate anyadditional hardware redesign or new waveforms. However, among allthree approaches, separating the radar and communications signalsat the coexistence receiver is the most challenging because of lackof any degrees-of-freedom in distinguishing the two signals. In thispaper, we focus on the coexistence problem.

In general, coexistence systems employ different radar and com-munications waveforms and separate receivers, wherein the manage-ment of interference from different radio systems is key to retrievinguseful information [4]. When the received radar signal reflected-offfrom the target is overlaid with communications messages occupyingthe same bandwidth, knowledge of respective waveforms is useful indesigning matched filters to extract the two signals [2]. Usually, theradar signal is known and the goal of the radar receiver is to extract

unknown target parameters from the received signal. In a typicalcommunications receiver, the channel is known and the unknowntransmit message is of interest in the receive processing. However,these assumptions do not extend to a general scenario. For example,in a passive [7] or multistatic [8] radar system, the receiver may nothave information about the radar transmit waveform. Similarly, incommunications over dynamic channels such as millimeter-wave [1]or terahertz-band [9], the channel coherence times are very short.As a result, the channel state information changes rapidly and thechannel cannot be considered perfectly known.

In this paper, we consider this general spectral coexistence sce-nario, where both radar and communications channels and the re-spective transmit signals are unknown to the common receiver. Ac-cordingly, we model the extraction of all four of these quantities as adual-blind deconvolution (DBD). The formulation is an extension ofthe blind deconvolution — a longstanding problem that occurs in avariety of engineering and scientific applications, such as astronomy,communication, image deblurring, system identification and optics— wherein two unknown signals are estimated from an observationof their convolution [10–12]. This problem is ill-posed and, usually,structural constraints on signals are imposed to obtain algorithmswith provable performance guarantees. The underlying ideas in thesetechniques are based on compressed sensing and low-rank matrixrecovery, wherein signals lie in the low-dimensional random subspaceand/or in high signal-to-noise ratio (SNR) regime [13–16].

Previously, [17] studied the dual deconvolution problem, wherethe radar transmit signal and communications channel were known.In this paper, unlike prior works, we examine the overlaid radar-communications signal as an ill-posed DBD problem. Our approachtoward this more challenging problem is inspired by some recentworks [18, 19] that have analyzed the basic blind deconvolution foroff-the-grid sparse scenarios. The radar and communications channelsare usually sparse [20], more so at higher frequency bands, and theirparameters are continuous-valued [21]. Hence, sparse reconstructionin the off-the-grid or continuous parameter domain through tech-niques based on atomic norm minimization (ANM) [22] are appro-priate for our application. While ANM has been extended to higherdimensions [23] and multiple parameters (e.g. delay-Doppler) [24],prior works have not dealt with mixed radar-communications signalstructures. We formulate an ANM-based recovery of both continuous-valued target parameters (range-time and Doppler velocities) as wellas delay-Doppler communications channel estimates from a DBDreceiver signal. We assume both channels are sparse, radar transmitsa train of pulses and communications is a multi-carrier signal. Nu-merical experiments validate our approach through perfect recovery.

The rest of the paper is organized as follows. In the next section,we introduce the system model and DBD problem. We describe ourANM-based semi-definite program (SDP) to solve DBD in Section 3.

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We validate our model and methods in Section 4 through numericalexperiments. We conclude in Section 5. Throughout the paper, wereserve boldface lowercase, boldface uppercase, and calligraphicletters for vectors, matrices, and index sets, respectively. We denotethe transpose, conjugate, Hermitian, and trace by (·)T , (·)∗, (·)H , andTr(·), respectively. The identity matrix of size N ×N is IN . || · ||p isthe `p norm. For notational convenience, the variables with subindexr refer to the signals and parameters related to the radar system, whilethose with subindex c refer to the communications system.

2. SYSTEM MODEL

Consider a continuous-time linear system model that receives anoverlaid radar-communications signal

y(t) = xr(t) ∗ hr(t) + xc(t) ∗ hc(t), (2.1)where xr(t) is a train of P transmitted pulses s(t) with a pulse repeti-tion interval (PRI) T ; xc(t) is the transmitted communications signalcomposed of P messages gp with symbol duration T ; and hr(t)and hc(t) are, respectively, the radar and communications channelsmodeled as a delay-Doppler channel with attenuation α, time de-lay τ and shifting frequencies ν. Assume L targets/sources and Qpropagation paths for the radar and communications channels, respec-tively. Then, the set of vectors {αr ∈ CL, τr ∈ RL, νr ∈ RL} and{αc ∈ CQ, τc ∈ RQ, νc ∈ RQ} contain the parameters of all Ltargets and Q paths, respectively. The communications messages aremodulated by orthogonal frequency-division multiplexing (OFDM)waveform with K sub-carriers, each separated by ∆f .

The communications messages and radar pulses are transmittedin the same PRI. Then, the continuous-time received signal in thecoherent processing interval (CPI) comprising P pulse, i.e. t ∈[0, (P − 1)T ], is

y(t) =

P−1∑p=0

L−1∑`=0

[αr]`s(t− pT − [τr]`)e−j2π[νr ]`pT

+

P−1∑p=0

Q−1∑q=0

K−1∑k=0

[αc]q[gp]kej2πk∆f(t−pT−[τc]q)e−j2π[νc]qpT .

Expressing the received signal as y(t) =∑P−1p=0 yp(t), our measure-

ments are determined in terms of shifted signals yp(t) = yp(t+ pT ),such that the signals yp(t+ pτ) are time-aligned with y0(t). There-fore, the signal y0(t) and the shifted signals yp(t) contain the same setof parameters. Take the continuous-time Fourier Transform (CTFT)of yp(t) in f ∈ [−B

2, B

2], with p = 0, . . . , P − 1 and uniformly

sample at fn = BnM

= n∆f , with n = −N, . . . , N , M = 2N + 1.Assume M = K, i.e. sample in frequency domain at the OFDM

separation frequency ∆f [25] to obtain

Yp(fn) =

L−1∑`=0

[αr]`S(fn)e−j2πn∆f [τr ]`e−j2π[νr ]`pT

+

Q−1∑q=0

[αc]qe−j2πn∆f [τc]qe−j2π[νc]qpT [gp]n+N , (2.2)

where S(f) is the Fourier transform of s(t). Concatenating Yp(fn)in the vector yp, i.e., [yp]n+N = Yp(fn) for n = −N, · · · , N , thediscrete values of the Fourier transform in (2.2) are

[yp]n+N =

L−1∑`=0

[αr]`[s]n+Ne−j2π(n[τr ]`+p[νr ]`)

+

Q−1∑q=0

[αc]q[gp]n+Ne−j2π(n[τc]`+p[νc]`), (2.3)

where [τr]` = [τr ]`T

, [νr]` = [νr ]`∆f

, [τc]` = [τc]`T

, [νc]` = [νc]`∆f∈

[0, 1) are normalized sets of target parameters and [s]n+N = S(fn).

Define the vector y =[yT0 , · · · ,yTP−1

]T. Then, all samples of the

P received blocks of information are

[y]v =

L−1∑`=0

[αr]`[s]n+Ne−j2π(n[τr ]`+p[νr ]`) (2.4)

+

Q−1∑q=0

[αc]q[g]ve−j2π(n[τc]q+p[νc]q), v = 0, · · · ,MP − 1,

where v = n+N +Np is the sequence index with n = −N, . . . , Nand p = 0, . . . , P − 1, and g = [gT0 , ...,g

TP−1]. Note that [y]v =

[yp]n+N and [g]v = [gp]n+N .Our goal is to estimate the set of parameters αr, τr,νr,αc, τc

and νc when the radar pulses s and communications symbols g arealso unknown. To this end, we exploit the sparsity of the channelsin our ANM formulation. We first employ the lifting trick [18] thatconsists on representing the unknown signals in a low-dimensionalsub-space, which follows the isotropy and incoherence properties[26]. This implies spectral flatness over the represented signals.Highly spectrum-efficient waveforms such as OFDM [27] or certainmodulating Gaussian radar waveforms [28] satisfy this property.

We represent the pulse signal s and the set of symbols gp forp = 0, · · · , P −1 as s = Bu, where B = [b−N , . . . ,bN ]H , bn ∈CJ , gp = Dpvp, Dp =

[[Dp]−N , · · · , [Dp]N

]H , [Dp]n ∈ CJ ,and [Dp]n denotes the n-th column of matrix Dp. Define thefull set of symbols g as g = Dv, D = [d0, · · · ,dMP−1]H =blockdiag

[DH

0 , · · · ,DHP−1

], where blockdiag [X1, · · · ,Xn] is a

block diagonal matrix with matrices X1, · · · ,Xn. The matricesB ∈ CM×J and D ∈ CMP×PJ are known representation basis forthe pulses and symbols and u ∈ CJ and v = [vT0 , · · · ,vTP ]T ∈ CPJare the corresponding coefficients vector with J � N . Using thisrepresentation, (2.4) becomes:

[y]v =

L−1∑`=0

[αr]`bHn ue−j2π(n[τr ]`+p[νr ]`)

+

Q−1∑q=0

[αc]qdHv ve−j2π(n[τc]q+p[νc]q). (2.5)

Denote the channel vectors hr =∑L−1`=0 [αr]`a(r`) and hc =∑L−1

`=0 [αc]qa(cq) where r` = [[τr]`, [νr]`], cq = [[τc]q, [νc]q] arethe parameter-tuples, and the vector a(r) is

a([τ, ν]) =[ej2π(τ(−N)+ν(0)), . . . , ej2π(τ(N)+ν(0)),

. . . , ej2π(τ(N)+ν(P−1))] ∈ CMP . (2.6)Then, (2.5) becomes

[y]v = hHr evbHn u + hHc evd

Hv v. (2.7)

Denote the matrices Zr = uhHr ∈ CJ×MP , Zc = vhHc ∈CPJ×MP , Gv = evb

Hn ∈ CMP×J and Av = evd

Hv ∈ CMP×PJ ,

where ev is the v-th canonical vector of RMP . Therefore, weformulate (2.7) as y = ℵr(Zr) + ℵc(Zc) where the linear operatorsℵr : CJ×MP → CMP and ℵc : CPJ×MP → CMP are

[ℵr(Zr)]v = Tr(GvZr), [ℵc(Zc)]v = Tr(AvZc). (2.8)Finally, we write the measured vector as

y = ℵr(Zr) + ℵc(Zc). (2.9)

Note that the matrices Zr and Zc contain all the unknown variablesthat we want to estimate.

3. DUAL-BLIND DECONVOLUTION ALGORITHM

The radar and communications channels are often characterized by afew continuous-valued parameters. Leveraging the sparse nature ofthese channels, we use ANM framework [22] for super-resolved esti-mations of continuous-valued channel parameters. For the overlaidradar-communications signal, we formulate the parameter recoveryas the minimization of two atomic norms, each corresponding to theradar and communications signal trails. Define the sets of atoms forthe radar and communications signals as, respectively,

Ar ={

ua(r)H : r ∈ [0, 1)2, ||u||2 = 1}

(3.1)

Ac ={

va(c)H : c ∈ [0, 1)2, ||v||2 = 1}. (3.2)

The corresponding atomic norms are

||Zr||Ar = inf[αr ]`∈C,r`∈[0,1]2

||u||2=1

{∑`

|[αr]`|∣∣∣Zr =

∑`

[αr]`ua(r`)H

}

||Zc||Ac = inf[αc]q∈C,cq∈[0,1]2

||v||2=1

{∑q

|[αc]q|∣∣∣Zc =

∑q

[αc]qva(cq)H

}.

Consequently, our proposed ANM problem isminimize

Zr,Zc

||Zr||Ar + ||Zc||Ac

subject to y = ℵr(Zr) + ℵc(Zc). (3.3)

In order to formulate the semidefinite program (SDP) of theabove-mentioned ANM, we find its dual problem [21, 23] via standardLagrangian analysis. The Lagrangian function of (3.3) isL(Zr,Zc,q) = ||Zr||Ar + ||Zc||Ac + 〈q,y − ℵr(Zr)− ℵc(Zc)〉

(3.4)

where q is the dual variable. The dual function g(q) is obtained fromthe Lagrangian function asg(q)

= infZr,Zc

L(Zr,Zc,q)

= 〈q,y〉 (3.5)+ inf

Zr,Zc

(||Zr||Ar + ||Zc||Ac)− 〈q,ℵr(Zr)〉 − 〈q,ℵc(Zc)〉)

= 〈q,y〉− sup

Zr

(〈ℵ∗r(q),Zr〉 − ||Zr||Ar )− supZc

(〈ℵ∗c(q),Zc〉 − ||Zc||Ac) .

The supremum values in (3.5) correspond to the convex conjugatefunction of the atomic norms ||Zr||Ar and ||Zc||Ac . For the atomicnorm, f = || · ||A the conjugate function is the indicator function ofthe dual norm unit ball

f∗(Z) =

{0 if ||Z||∗A ≤ 1,

∞ otherwise,

where the dual norm is defined as ||Z||∗A = sup||U||A≤1〈U,Z〉.Based on the dual function, the dual optimization problem of (3.3) is

maximizeq

〈q,y〉R

subject to ‖ℵ?r(q)‖?Ar≤ 1,

‖ℵ?c(q)‖?Ac≤ 1, (3.6)

where ℵ?r : CMP → CJ×MP and ℵ?c : CMP → CPJ×MK

are adjoint operators of ℵr and ℵc respectively, i.e. ℵ?r(q) =∑P−1p=0

∑Nn=−N [q]vG

Hv ,ℵ?c(q) =

∑P−1p=0

∑Nn=−N [q]vA

Hv .

The SDP of this dual problem is found using the trigonometricvector polynomials

fr(r) =

P−1∑p=0

N∑n=−N

[q]vGHv a(r) ∈ CJ , (3.7)

fc(c) =

P−1∑p=0

N∑n=−N

[q]vAHv a(c) ∈ CJP . (3.8)

In particular, the SDP relaxation is achieved through the BoundedReal Lemma [29] to convert the constraints on (3.6) into linear matrixinequalities and using the fact that polynomials (3.7) and (3.8) areparameterized by positive definite matrices. The relaxation of (3.6)leads to the following equivalent SDP problem

maximizeq,Q

〈q,y〉R

subject to Q � 0,[Q QH

r

Qr IJ

]� 0,[

Q QHc

Qc IJP

]� 0,

Tr(ΘnQ) = δn, (3.9)

where Qr =∑P−1p=0

∑Nn=−N [q]vG

Hv ∈ CMP×J and Qc =∑P−1

p=0

∑Nn=−N [q]vA

Hv ∈ CMP×PJ are the coefficients of two-

dimensional (2-D) trigonometric polynomials, the matrix Θn,n = [n1, n2] for the 2-D case is Θn = Θn2 ⊗Θn1 , where Θn isthe Toeplitz matrix with ones in the n-th diagonal with 0 < n1 < m1

and −m2 < n2 < m2. Here, we define m1 = P − 1 andm2 = N − 1. Finally, δn = 1 if n = [0, 0] and 0 otherwise. ThisSDP formulation is solved by employing off-the-shelf solvers.

Based on the strong duality implied by Slater’s conditions, thefollowing proposition states the conditions for exact recovery of theradar and communications channels parameters.

Proposition 3.1. Let y be as defined in (2.9) and the atomic setsArandAc as defined in (3.2). LetR = {r`}L−1

`=0 and C = {cq}Q−1q=0 and

the solution of (3.3) be Zr and Zc. Then, Zr = Zr and Zc = Zcare the primal optimal solutions of (3.3) if the following condition issatisfied: There exist two 2-D trigonometric polynomials

fr(r) =

P−1∑p=0

N∑n=−N

[q]vGHv a(r) ∈ CJ (3.10)

fc(c) =

P−1∑p=0

N∑n=−N

[q]vAHv a(c) ∈ CPJ (3.11)

with complex coefficients q such that

fr(r`) = sign([αr]`)u if ∀r` ∈ R (3.12)fc(cq) = sign([αc]q)v if ∀cq ∈ C (3.13)

‖fr(r)‖22 < 1 ∀r ∈ [0, 1]2 \ R (3.14)

‖fc(c)‖22 < 1 ∀c ∈ [0, 1]2 \ C (3.15)where sign(c) = c

|c| .

Proof. The variable q is dual feasible.

〈q,y〉R = 〈ℵ∗r(q),Zr〉R + 〈ℵ∗c(q),Zc〉R

=

L−1∑`=0

[αr]∗` 〈ℵ∗r(q),ua(r`)

H〉R +

Q−1∑q=0

[αc]∗q〈ℵ∗c(q),va(cq)

H〉R

=

L−1∑`=0

[αr]∗` 〈fr(r`),u〉R +

Q−1∑q=0

[αc]∗q〈fc(cq),v〉R

=

L−1∑`=0

[αr]∗` sign([αr]`) +

Q−1∑q=0

[αc]∗qsign([αc]q)

=

L−1∑`=0

|[αr]`|+Q−1∑q=0

|[αc]q| ≥ ||Zr||Ar + ||Zc||Ac . (3.16)

On the other hand, it follows from Holder inequality that

〈q,y〉R = 〈ℵ∗r(q),Zr〉R + 〈ℵ∗c(q),Zc〉R (3.17)≤ ||ℵ∗r(q)||∗Ar

||Zr||Ar+ ≤ ||ℵ∗c(q)||∗Ac||Zc||Ac (3.18)

≤ ||Zr||Ar + ||Zc||Ac , (3.19)

where the first inequality is due to Cauchy-Schwarz inequality andthe last inequality follows from (3.12), (3.13), (3.14), and (3.15).Therefore, based on (3.16) and (3.19), we conclude that 〈q,y〉R =||Zr||Ar + ||Zc||Ac showing that the pair (Zr,Zc) is primal optimaland, from strong duality, q is dual optimal.

The existence of the polynomial has been previously demon-strated in 2-D (non-blind) deconvolution of a radar channel [24] aswell as in the blind case [30]. In our problem, two different trigono-metric polynomials are directly related through the dual variable q.Thus, to prove the existence of these polynomials under this con-straint implies a more extended analysis that the above-mentionedworks [24], [30]. Finally, even though this paper focuses on thechannel estimation, the information embedded in the atoms Zr andZc also allow for exact estimation of the radar waveform and thecommunications symbols. We omit the proof of the existence of thesepolynomials and the guarantees for the unique solution of Zr and Zcbecause of paucity of space. Briefly, the proof follows from [18, 19,31], where the polynomials are formulated as linear combination offast decaying random kernels (e.g. randomized Fejer kernel) and theirderivatives with the constraint that the polynomials have commonkernels.

4. NUMERICAL EXPERIMENTS

To numerically validate the proposed method, we set M = 13, P =9, Q = L = 3 and J = 3. The delay-Doppler parameters weretaken from a random uniform distribution, which results in τr =[0.23, 0.68, 0.87], νr = [0.45, 0.42, 0.71], τc = [0.12, 0.21, 0.95],and νc = [0.09, 0.25, 0.87]. The columns of the transformationmatrices B and Dp were generated following the distribution de-scribed in [18], i.e. bn = [1, ej2πσn , . . . , ej2π(J−1)σn ], whereσn ∼ N (0, 1). The parameters αr and αc are drawn from a nor-mal distribution with |[αr]`| = |[αc]q| = 1 ∀q,∀`. The coefficientvectors u,v are generated from a normal random distribution andnormalized ‖u‖ = ‖v‖ = 1.

We solve the SDP optimization problem in (3.9) using CVXSDP3 solver [32]. With the dual solution, we build the dual trigono-metric polynomials by evaluating them in a discrete 2-D time-delayand doppler domain with a sampling step of 1e − 3. The resultingdual polynomials are depicted in Fig. 1, showing that the set of theradar and communications channels parameter are exactly locatedwhen ‖fr(r)‖ = 1 and ‖fc(c)‖ = 1.

Fig. 1. Dual polynomials corresponding to estimating the (a) radarand (b) communications channels in dual-blind deconvolution set-up. The locations of the parameters are given when the maximummodulus of the polynomials is unity.

Fig. 2. Exact localization probability of (a) radar and (b) communica-tions channels as a function of the number of the targets and pathsQ = L and the subspace size J .

Fig. 2 shows the phase transition of the proposed method byvarying the number of the targets and paths Q = L and the subspacesize J for 10 realizations of L and J . The other parameters are sameas before. We declare a successful estimation when ‖r− r‖2 < 1e−3

and ‖c− c‖2 < 1e−3.

5. SUMMARY

We proposed a dual-blind deconvolution method for jointly estimatingthe channel parameters of radar and communications systems from theinformation acquired with a single receiver. The proposed approachleverage the sparse structure of the channels to formulate the recoveryproblem as the minimization of two atomic norms corresponding tothe radar and communications signals. The dual problem leads tothe construction of two trigonometric polynomials corresponding tothe radar and communications signal directly coupled through thedual variable. This problem is reformulated as an equivalent SDP byharnessing the parametrization of the dual trigonometric polynomialsusing positive semidefinite matrix and efficiently solved using off-the-shelf solvers. Numerical experiments validate the proposed approachthat estimates the radar and communications channels parametersperfectly.

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