Optimum Co-Design for Spectrum Sharing Between Matrix...

1

Optimum Co-Design for Spectrum Sharing BetweenMatrix Completion Based MIMO Radars and a

MIMO Communication SystemBo Li, Student Member, IEEE, Athina P. Petropulu, Fellow, IEEE, and Wade Trappe, Fellow, IEEE

Abstract—Spectrum sharing enables radar and communicationsystems to share the spectrum efficiently by minimizing mutualinterference. Recently proposed multiple input multiple outputradars based on sparse sensing and matrix completion (MIMO-MC), in addition to reducing communication bandwidth and pow-er as compared to MIMO radars, offer a significant advantage forspectrum sharing. The advantage stems from the way the sam-pling scheme at the radar receivers modulates the interferencechannel from the communication system transmitters, renderingit symbol dependent and reducing its row space. This makes iteasier for the communication system to design its waveforms in anadaptive fashion so that it minimizes the interference to the radarsubject to meeting rate and power constraints. Two methods areproposed. First, based on the knowledge of the radar samplingscheme, the communication system transmit covariance matrixis designed to minimize the effective interference power (EIP) tothe radar receiver, while maintaining certain average capacityand transmit power for the communication system. Second, ajoint design of the communication transmit covariance matrixand the MIMO-MC radar sampling scheme is proposed, whichachieves even further EIP reduction.

Index Terms—Collocated MIMO radar, matrix completion,spectrum sharing

I. INTRODUCTION

THE operating frequency bands of communication andradar systems often overlap, causing one system to exert

interference to the other. For example, the high UHF radarsystems overlap with GSM communication systems, and theS-band radar systems partially overlap with Long Term Evo-lution (LTE) and WiMax systems [2]–[5]. Spectrum sharingis an emerging technology that can be applied to enable radarand communication systems to share the spectrum efficientlyby minimizing mutual interference [4]–[14].

In this paper we study spectrum sharing between a specialclass of collocated MIMO radars and a MIMO communicationsystem. The rationale behind considering a MIMO-type radarsystem is the high resolution which such systems can achievewith a relatively small number of transmit (TX) and receive(RX) antennas [15]–[18]. A MIMO radar system lends itselfto a networked implementation, which is very desirable inboth military and civilian applications. A networked radar is a

This work was supported by NSF under Grant ECCS-1408437. Parts of thiswork have been presented at the IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP’15) [1].

The authors are with Department of Electrical and Computer Engineering,Rutgers, The State University of New Jersey, Piscataway NJ 08854, USA.(E-mails: paul.bo.li,[email protected]; [email protected])

configuration of TX and RX antennas. The TX antennas trans-mit probing waveforms, and target information is extracted byjointly processing the measurements of all RX antennas. Thisprocessing can be done at a fusion center, i.e., a network nodeendowed with more computational power than the rest of thenodes. Reliable surveillance requires collection, communica-tion and fusion of vast amounts of data from various antennas.This is a power and bandwidth consuming task, which can beespecially taxing in scenarios in which the antennas are onbattery operated devices and are connected to the fusion centervia a wireless link. Recently, MIMO radars using compressivesensing (MIMO-CS) [19]–[22], and MIMO radars via matrixcompletion (MIMO-MC) [23]–[26] have been proposed tosave power and bandwidth on the link between the receiversand the fusion center, thus facilitating the network implemen-tation of MIMO radars. MIMO-MC radars transmit orthogonalwaveforms from their multiple TX antennas. Each RX antennasamples the target returns in a pseudo-random sub-Nyquistfashion and forwards the samples to the fusion center, alongwith the seed of the random sampling sequence. By collectingthe samples of all RX antennas, and based on knowledge ofeach antenna’s sampling scheme, the fusion center constructsa matrix, refereed to as the data matrix (see [24] SchemeI), in which only the entries corresponding to sampled timescontain non-zero values. Subsequently, the missing entries,corresponding to non-sampled times, are provably recoveredvia MC techniques. In MIMO-MC radars the interference isconfined to the sampled entries of the data matrix, while aftermatrix completion the target echo power is preserved. UnlikeMIMO-CS, MIMO-MC does not require discretization of thetarget space, thus does not suffer from grid mismatch issues[27].

Spectrum sharing between a MIMO radar and a com-munication system has been considered in [5]–[9], wherethe radar interference is eliminated by projecting the radarwaveforms onto the null space of the interference channelbetween the MIMO radar transmitters and the communicationsystem. In [10], a radar receive filter was proposed to mitigatethe interference from the communication systems. However,null space projection-type or spatial filtering-type techniquesmight miss targets aligned with the the interference channel. Ingeneral, the existing literature on MIMO radar-communicationsystems spectrum sharing addresses interference mitigationfor either solely the communication system [5]–[9] or solelythe radar [10]. While joint design of traditional radar andcommunication systems for spectrum sharing has been consid-

2

ered in [4], [13], [14], co-design of MIMO radar and MIMOcommunication systems for spectrum sharing has not beenaddressed before, with the exception of our preliminary resultsin [1], [28]. In practice, however, the two systems are oftenaware of the existence of each other, and they could shareinformation, which could be exploited for co-design. Recentdevelopments in cognitive radios [29] and cognitive radars [30]could provide the tools for information sharing and channelfeedback, thus facilitating the cooperation between radar andcommunication systems.

Motivated by the cooperative methods in cognitive radionetworks [31]–[33], we propose ways via which a MIMO-MC radar and a MIMO communication system, in a coop-erative fashion, negotiate spectrum use in order to mitigatemutual interference. In addition to reducing communicationbandwidth and power, MIMO-MC radars offer a significantadvantage for spectrum sharing. The advantage stems fromthe way the sampling scheme at the radar receivers modu-lates the interference channel from the communication systemtransmitters, rendering it symbol dependent and reducing itsrow space. This makes it easier for the communication systemto design its waveforms in an adaptive fashion so that itminimizes the interference to the radar subject to meeting rateand power constraints. Two methods are proposed. The firstmethod is a cooperative design; for a fixed radar samplingscheme, which is known to the communication system, thecommunication system optimally selects its precoding matrixto minimize the interference to the radar. The second methodis a joint design, whereby the radar sampling scheme as wellas the communication system precoding matrix are optimallyselected to minimize the interference to the radar. For the firstmethod, an efficient algorithm for solving the correspondingoptimization problem is proposed based on the Lagrangiandual decomposition (see Algorithm 1). For the second method,alternating optimization is employed to solve the correspond-ing optimization problem. The candidate sampling schemeneeds to be such that the resulting data matrix can be com-pleted. Recent work [34] showed that for matrix completion,the sampling locations should correspond to a binary matrixwith large spectral gap. Since the spectral gap of a matrixis not affected by column and row permutations, we proposeto search for the optimum sampling matrix among matriceswhich are row and column permutations of an initial samplingmatrix with large spectral gap.

The paper is organized as follows. Section III introduces thesignal model when the MIMO-MC radar and communicationsystems coexist. The problem of a MIMO communicationsystem sharing the spectrum with a MIMO-MC radar systemis studied in Section IV. Numerical results, discussions andconclusions are provided in Section V-VII.Notation: CN (µ,Σ) denotes the circularly symmetric complexGaussian distribution with mean µ and covariance matrix Σ.| · |, Tr(·), ‖·‖∗ and ‖·‖F denote the matrix determinant, trace,nuclear norm and Frobenius norm, respectively. The set N+

L isdefined as 1, . . . , L. N (A) and R(A) denote the null androw spaces of matrix A, respectively. Ai· and A·j respectively,denote the i-th row and j-th column of matrix A. [A]i,jdenotes the element on the i-th row and j-th column of matrix

A. x+ is defined as max(0, x).

II. BACKGROUND ON MIMO-MC RADARS

Consider a collocated MIMO radar system with Mt,R TXantennas and Mr,R RX antennas. The targets are in the far-field of the antennas and are assumed to fall in the same rangebin. The radar operates in two phases; in the first phase theTX antennas transmit waveforms and the RX antennas receivetarget returns, while in the second phase, the RX antennasforward their measurements to a fusion center. In each pulse,the m-th, m ∈ N+

Mt,R, antenna transmits a coded waveform

containing L symbols sm(1), · · · , sm(L) of duration TReach. Each RX antenna samples the target returns every TRseconds, i.e., samples each symbol exactly once. Followingthe model of [23]–[25], the data matrix at the fusion centercan be formulated as

YR = γρDS + WR, (1)

where the m-th row of YR ∈ CMr,R×L contains the L samplesforwarded by the m-th antenna; γ and ρ respectively denotethe path loss corresponding to the range bin of interest, andthe radar transmit power; D ∈ CMr,R×Mt,R denotes the targetresponse matrix, which depends on the target reflectivity, angleof arrival and target speed (details can be found in [24]);S = [s(1), · · · , s(L)], with s(l) = [s1(l), · · · , sMt,R

(l)]T

being the l-th snapshot across the transmit antennas. Thetransmit waveforms are assumed to be orthogonal, i.e., itholds that SSH = I [24]; WR denotes additive noise. Aftermatched filtering at the fusion center, target estimation can beperformed based on YR via standard array processing schemes[35].

If the number of targets is smaller than Mr,R and L, matrixDS is low-rank and can be provably recovered based on asubset of its entries [24], [26]. This observation gave rise toMIMO-MC radars [23]–[26], where each RX antenna sub-samples the target returns and forwards the samples to thefusion center. The sampling scheme could be a pseudo-randomsequence of integers in [1, L], with the fusion center knowingthe random seed of each RX antenna. In MIMO-MC radars,the partially filled data matrix at the fusion center can bemathematically expressed as follows (see [24] Scheme I)

Ω YR = Ω (γρDS + WR), (2)

where denotes Hadamard product and Ω is a matrix con-taining 0’s and 1’s; the 1’s in the m-th row correspond to thesampled symbols of the m-th TR antenna. The sub-samplingrate, p, equals ‖Ω‖0/(LMr,R). When p = 1, the Ω matrixis filled with 1’s, and the MIMO-MC radar is identical to thetraditional MIMO radar. At the fusion center, the completionof γρDS is formulated as the following problem [36]

minM‖M‖∗ s.t. ‖Ω M−Ω YR‖F ≤ δ, (3)

where δ > 0 is a parameter related to the noise over thesampled noise matrix entries, i.e., Ω WR. On denoting byM the solution of (3), the recovery error ‖M − γρDS‖Fis determined by the noise power in Ω WR, i.e., the noiseenters only through the sampled entries of the data matrix. It is

3

… …

… …

Collocated MIMO radar

Communication TX Communication RX

Fig. 1. A MIMO communication system sharing spectrum with a colocatedMIMO radar system

important to note that, assuming that the reconstruction erroris small, the reconstructed M has the same received targetecho power as γρDS of (1).

Early studies on matrix completion theory suggested thatthe low-rank matrix reconstruction requires that the entriesare sampled uniformly at random. However, recent works [34]showed that non-uniform sampling would still work, as longas the sampling matrix has large spectral gap (i.e., large gapbetween the largest and second largest singular values).

III. SYSTEM MODEL

Consider a MIMO communication system which coexistswith a MIMO-MC radar system as shown in Fig. 1, sharing thesame carrier frequency. The MIMO-MC radar operates in twophases, i.e., in Phase 1 the RX antennas obtain measurementsof the target returns, and in Phase 2, the RX antennas forwardthe obtained samples to a fusion center. The communicationsystem interferes with the radar system during both phases.In the following, we will address spectrum sharing during thefirst phase only. The interference during the second phase canbe viewed as the interference between two communicationsystems; addressing this problem has been covered in theliterature [31], [32].

Suppose that the two systems have the same symbol rateand are synchronized in terms of sampling times (see SectionV for the mismatched case). We do not assume perfect carrierphase synchronization between the two systems. The datamatrix at the radar fusion center, and the received matrix atthe communication RX antennas during L symbol durationscan be respectively expressed as

Radar fusion center:Ω YR = Ω (γρDS)︸︷︷︸

signal

+ Ω (G2XΛ2)︸︷︷︸interference

+ Ω WR︸︷︷︸noise

, (4a)

Communication receiver:YC = HX︸︷︷︸

signal

+ ρG1SΛ1︸︷︷︸interference

+ WC︸︷︷︸noise

, (4b)

where

• YR, ρ,D, S, WR, and Ω are defined in Section II.• X , [x(1), . . . ,x(L)]; x(l) ∈ CMt,C×1 denotes the

transmit vector by the communication TX antennas dur-ing the l-th symbol duration. The rows of X are code-words from the code-book of the communication system.

• WC and WR denote the additive noise; their elementsare assumed to be independent identically distributed asCN (0, σ2

C) and CN (0, σ2R), respectively.

• H ∈ CMr,C×Mt,C denotes the communication channel,where Mr,C and Mt,C denote respectively the numberof RX and TX antennas of the communication system;G1 ∈ CMr,C×Mt,R denotes the interference channel fromthe radar TX antennas to the communication system RXantennas; G2 ∈ CMr,R×Mt,C denotes the interferencechannel from the communication TX antennas to theradar RX antennas. All channels are assumed to be flatfading and remain the same over L symbol intervals [5],[6], [8], [31].

• Λ1 and Λ2 are diagonal matrices. The l-th diagonalentry of Λ1, i.e., ejα1l , denotes the random phase offsetbetween the MIMO-MC radar carrier and the commu-nication receiver reference carrier at the l-th samplingtime. The l-th diagonal entry of Λ2, i.e., ejα2l , denotesthe random phase offset between the communicationtransmitter carrier and the MIMO-MC radar referencecarrier at the l-th sampling time. The phase offsetsarise due to random phase jitter of the radar oscillatorand the oscillator at the communication receiver Phase-Locked Loops. In the literature [37]–[39], the phase jitterof oscillator α(t) is modeled as a zero-mean Gaussianprocess. In this paper, we model α1lLl=1 as a sequenceof zero-mean Gaussian random variables with varianceσ2α. Modern CMOS oscillators exhibit very low phase

noise, e.g., −94 dB below the carrier power per Hz (i.e.,−94dBc/Hz) at an offset of 2π × 1 MHz, which yieldsphase jitter variance σ2

α ≈ 2.5× 10−3 [40].

The following assumptions are made:

• About the synchronization of sampling times- In theabove model, we assume that the radar receivers andthe communication system sample in a time synchronousmanner. Although this assumption is later relaxed inSection V, we next provide an example of radar andcommunication parameter settings suggesting that theaforementioned assumption is applicable in real worldsystems. The typical range resolution for an S-bandsearch and acquisition radar is between 100m and 600m[41], [42]. Thus, for range resolution of cTb/2 = 300m,where c = 3 × 108m/s denotes the speed of light, theradar sub-pulse duration is Tb = 2µs. In order to haveidentical symbol rate for two systems, the communicationsymbol duration should be 2µs, which corresponds tosignal bandwidth of 0.5 MHz. This symbol interval valuefalls in the typical range of symbol interval values in LTEsystems [43].

• About channel fading- We assume that H, G1 and G2

are flat fading, which is valid when the channel coherencebandwidth is larger than the signal bandwidth [44]–[46],i.e., when the transmitted signals are narrowband. Con-sider the symbol interval value 2µs and signal bandwidth0.5 MHz given above. In a LTE macro-cell, the coherencebandwidth is in the order of 1 MHz [43], [47]. The typicalvalues of LTE channel coherence bandwidth are much

4

larger than the signal bandwidth of 0.5 MHz, thus makingthe flat fading channel assumption valid. Since the radarand communication systems use the same carrier andsignal bandwidth, the flat fading assumption is valid forall H, G1 and G2. In the radar-communication systemcoexistence literature [5]–[9], the flat fading assumptionis quite typical. If the narrowband assumption is notvalid then perhaps one could consider an OFDM scenario,where the flat fading model would apply on each carrier[13], [14].

• About channel information feedback- The channels Hand G2 are also assumed to be perfectly known at thecommunication TX antennas. In practice, such channelinformation can be obtained at the radar RX antennasthrough the transmission of pilot signals [5], [48]. View-ing the radar system as the primary user of a cognitiveradio system and the MIMO communication system asthe secondary user, techniques similar to those of [31]–[33], [49] can be used to estimate and feed back thechannel information between the primary and secondarysystems.

Let Sobs , ρG1S be the radar interference as viewed bythe communication system. This can be obtained during timesthat the communication system does not transmit. Since theradar transmission power ρ is very high, ρG1S can be esti-mated with high accuracy. Based on Sobs, the communicationreceivers can eliminate some of the interference via directsubtraction. However, due to the high power of the radar [3]and the unknown phase offset, there will always be residualinterference, i.e.,

ρG1S(Λ1 − I) ≈ ρG1SΛα≡ SobsΛα, (5)

where Λα = diag(jα11, · · · , jα1L), and the approximation isbased on the fact that α1lLl=1 are small. In the above, we as-sume that the radar waveforms have not changed between thetime the interference is estimated and used in (5). The signalat the communication receiver after interference cancellationequals

YC = HX + SobsΛα + WC . (6)

This residual interference is not circularly symmetric, andthus the communication channel capacity is achieved by non-circularly symmetric Gaussian codewords, whose covarianceand complementary covariance matrix would need to bedesigned simultaneously [50]. Here, we consider circularlysymmetric complex Gaussian codewords x(l) ∼ CN (0,Rxl),which achieve a lower bound of the channel capacity [50],[51]. This reduces the complexity of the design since we onlyneed to design the transmit covariance matrix Rxl.

Unless special measures are taken, the interference fromthe radar transmissions, i.e., SobsΛα, will reduce the com-munication system capacity, and the interference from thecommunication system transmission, i.e., Ω (G2XΛ2) willdegrade the completion of the data matrix and as a resultthe target detection/estimation. The application of traditionalspatial filtering on ΩYR for eliminating the communicationsystem interference is not straightforward and to the best ofour knowledge has not been previously addressed. For the

case with complete samples, the optimal detector to maximizethe SINR is matched filtering following a whitening filter.However, in the case of partially sampled YR, i.e., Ω YR,S cannot be fully matched due to the sub-sampling operator.Also, the interference plus noise at the whitening filter outputwould not be white anymore. Further, note that the recoveryof γρDS via matrix completion in (3) is based on (Ω YR).Even if we somehow find the spatial filter W that maximizesthe SINR, the filter output W(Ω YR) cannot be used bythe matrix completion formulation in (3), which follows theformulation in the MIMO-MC radar [23]–[26] and generalmatrix completion literature [36]. The extension of the matrixcompletion working with the additional filtering matrix is outthe scope of this paper. Of course, one could apply filteringon the recovered data matrix DS as post processing. However,such post-filtering would first need the matrix completion tobe successful.

The approach that we propose here for addressing theradar and comm systems interference is a design for thecommunication TX signals, or a co-design of the commu-nication TX signals and the radar sub-sampling scheme, sothat we minimize the interference at the radar RX antennasfor successful matrix completion, while satisfying certaincommunication system rate requirements.

IV. SPECTRUM SHARING BETWEEN MIMO-MC RADARAND A MIMO COMMUNICATION SYSTEM

First, let us provide expressions for the communication TXpower and channel capacity, and the interference power at theMIMO-MC radar receiver. The total transmit power of thecommunication TX antennas equals

ETr(XXH) = E

Tr

(L∑l=1

x(l)xH(l)

)=

L∑l=1

Tr(Rxl),

where Rxl , Ex(l)xH(l).Due to the sampling performed at the MIMO-MC radar

receiver, the effective interference power (EIP) at the radarRX nodes can be expressed as:

EIP , E

Tr(Ω (G2XΛ2) (Ω (G2XΛ2))

H)

=E

Tr([G21x(1) . . .G2Lx(L)]Λ2Λ

H2 [G21x(1) . . .G2Lx(L)]H

)=E

Tr

(L∑

l=1

G2lx(l)xH(l)GH

2l ]

)

=

L∑l=1

Tr(G2lRxlG

H2l

)=

L∑l=1

Tr(∆lG2RxlG

H2

),

(7)where G2l , ∆lG2, with ∆l = diag(Ω·l). We note that the

EIP at sampling time l contains the interference correspondingonly to 1’s in Ω·l. Thus, the effective interference channelduring the l-th symbol duration is G2l. In the following, theEIP is used as the figure of merit for MIMO-MC radars as itaffects the performance of matrix completion and further targetestimation (see simulation results in Section VI-A). Beforematrix completion and any target estimation, the EIP should beminimized. From another perspective, the EIP is a reasonablesurrogate of the radar SINR, which is widely used as figure

5

of merit in the literature [52], [53], as in this paper we do notassume any prior information on target parameters.

In the coexistence model of (4a) and (6), both the ef-fective interference channel G2l, and the interference co-variance matrix at the communication receiver, i.e., Rintl ,σ2αSobs(l)S

Hobs(l), vary between sampling times. Thus, the

optimum scheme for the communication transmitter would beadaptive/dynamic transmission. A symbol dependent covari-ance matrix, i.e., Rxl, would need to be designed for eachsymbol duration in order to match the variation of G2l andRintl.

The channel G2l can be equivalently viewed as a fast fadingchannel with perfect channel state information at both thetransmitter and receiver [45], [54]. Similar to the definitionof ergodic capacity [54], the achieved capacity is the averageover L symbols, i.e.,

Cavg(Rxl) ,1

L

∑L

l=1log2

∣∣I + R−1wl HRxlHH∣∣ , (8)

where Rxl denotes the set of all Rxl’s and Rwl , Rintl +σ2CI for all l ∈ N+

L .The adaptive transmission could be implemented using the

V-BLAST transmitter architecture [45, Chapter 7], where theprecoding matrix for symbol index l is set to R

1/2xl . This idea

is also used in the transceiver architecture for achieving thecapacity of a fast fading MIMO channel with full channel stateinformation [45, Chapter 8.2.3], and is also discussed in [46,Chapter 9]. The adaptive transmission in response to highlymobile, fast fading channels requires the transmitter to vary therate, power and even the coding strategy. The main bottleneckof the system is not due to the complexity of designingand implementing the variable transmission parameters, butrather due to the feedback delay of the fast fading channel.In our paper, the latter issue is not relevant because thechannel variations are introduced by the MC technique andradar waveforms, which are available at the communicationtransmitter.

In this section, spectrum sharing between the commu-nication system and the MIMO-MC radar is achieved byminimizing the interference power at the MIMO-MC radar RXnode, while satisfying the communication rate and TX powerconstraints of the communication system. The design variablesare the communication TX covariance matrices and/or theradar sub-sampling scheme. In the following we will considertwo approaches, namely, a cooperative and a joint designapproach.

A. Cooperative Spectrum Sharing

In the cooperative approach, the MIMO-MC radar sharesits sampling scheme Ω with the communication system. Thespectrum sharing problem can be formulated as

(P1) minRxl0

EIP(Rxl) s.t.∑L

l=1Tr (Rxl) ≤ Pt (9a)

Cavg(Rxl) ≥ C, (9b)

where the constraint of (9a) restricts the total transmit power atthe communication TX antennas to be no larger than Pt. The

constraint of (9b) restricts the communication average capacityduring L symbol durations to be at least C, in order to providereliable communication and avoid service outage. Rxl 0imposes the positive semi-definiteness on the solution.

Problem (P1) is convex and involves multiple matrix vari-ables, the joint optimization with respect to which requireshigh computational complexity. Fortunately, we observe thatboth the objective and constraints are separable functions ofRxl. An efficient algorithm for solving the above problemcan be implemented based on the Lagrangian dual decompo-sition [55] as follows.

1) An Efficient Algorithm Based on Dual Decomposition:The Lagrangian of (P1) can be written as

L(Rxl, λ1, λ2) =EIP(Rxl) + λ2 (C − Cavg(Rxl))

+λ1

(∑L

l=1Tr (Rxl)− Pt

),

where λ1 ≥ 0 is the dual variable associated with the transmitpower constraint, and λ2 ≥ 0 is the dual variable associatedwith the average capacity constraint. The dual problem of (P1)is

(P1-D) maxλ1,λ2≥0

g(λ1, λ2),

where g(λ1, λ2) is the dual function defined as

g(λ1, λ2) = infRxl0

L(Rxl, λ1, λ2).

The domain of the dual function, i.e., dom g, is λ1, λ2 ≥ 0such that g(λ1, λ2) > −∞. It is also called dual feasible if(λ1, λ2) ∈ dom g. It is interesting to note that g(λ1, λ2) canbe obtained by solving L independent subproblems, each ofwhich can be written as follows

(P1-sub) minRxl0

Tr((

GH2 ∆lG2 + λ1I

)Rxl

)− λ2 log2

∣∣I + R−1wl HRxlHH∣∣ . (10)

Before giving the solution of (P1-sub), let us first state someobservations.Observation 1) The average capacity constraint should beactive at the optimal point. This means that the achievedcapacity is always C and λ2 > 0. To show this, let us assumethat the optimal point R∗xl achieves Cavg(R∗xl) > C.Then we can always shrink R∗xl until the average capacityreduces to C, which would also reduce the objective. Thus,we end up with a contradiction. By complementary slackness,the corresponding dual variable is positive, i.e., λ2 > 0.Observation 2)

(GH

2 ∆lG2 + λ1I)

is positive definite for alll ∈ N+

L . This can be shown by contradiction. Suppose thatthere exists l such that GH

2 ∆lG2 + λ1I is singular. Then itmust hold that GH

2 ∆lG2 is singular and λ1 = 0. Therefore,we can always find a nonzero vector v lying in the null spaceof GH

2 ∆lG2. At the same time, it holds that R−1/2wl Hv 6= 0

with very high probability, because H is a realization of therandom channel. If we choose Rxl = αvvH and α→∞, theLagrangian L(Rxl, 0, λ2) will be unbounded from below,which indicates that λ1 = 0 is not dual feasible. This meansthat λ1 is strictly larger than 0 if GH

2 ∆lG2 is singular forany l. Thus, the claim is proven.

6

Based on the above observations, we have the followinglemma.

Lemma 1 ( [32], [33]). For given feasible dual variablesλ1, λ2 ≥ 0, the optimal solution of (P1-sub) is given by

R∗xl(λ1, λ2) = Φ−1/2l UlΣlU

Hl Φ−1/2l , (11)

where Φl , GH2 ∆lG2 +λ1I; Ul is the right singular matrix

of Hl , R−1/2wl HΦ

−1/2l ; Σl = diag(βl1, . . . , βlr) with βli =

(λ2 − 1/σ2li)

+, r and σli, i = 1, . . . , r, respectively being therank and the positive singular vales of Hl. It also holds that

log2

∣∣I + R−1wl HR∗xlHH∣∣ =

∑r

i=1

(log(λ2σ

2li))+. (12)

Based on Lemma 1, the solution of (P1) can be obtainedby finding the optimal dual variables λ∗1, λ

∗2. The cooperative

spectrum sharing problem (P1) can be solved via the proce-dure outlined in Algorithm 1. The convergence of Algorithm1 is guaranteed by the convergence of the ellipsoid method[56].

Algorithm 1 Cooperative Spectrum Sharing (P1)

1: Input: H,G1,G2,Ω, Pt, C, σ2C

2: Initialization: λ1 ≥ 0, λ2 ≥ 03: repeat4: Calculate R∗xl(λ1, λ2) according to (11) with the

given λ1 and λ2;5: Compute the subgradient of g(λ1, λ2) as∑L

l=1 Tr (R∗xl(λ1, λ2))− Pt andC − Cavg(R∗xl(λ1, λ2)) respectively for λ1 and λ2;

6: Update λ1 and λ2 accordingly based on the ellipsoidmethod [56];

7: until λ1 and λ2 converge to a prescribed accuracy.8: Output: R∗xl = R∗xl(λ1, λ2)

Based on Lemma 1, the coexistence model can be equiv-alently viewed as a fast fading MIMO channel Hl. Thecovariance of the waveforms transmitted on Hl is Rxl ,Φ

1/2l RxlΦ

1/2l . It is well-known that the optimum Rxl equals

UlΣlUHl with power allocation obtained by the water-filling

algorithm [54]. The achieved capacity is the average overall realization of the channel, i.e., HlLl=1. This justifiesthe definition of average capacity in (8). Lemma 1 showsthat the communication transmitter will allocate more powerto directions determined by the left singular vectors of Hcorresponding to larger eigenvalues and by the eigenvectorsof Φl corresponding to smaller eigenvalues. In other words,the communication will transmit more power in directionsthat convey larger signal at the communication receivers andsmaller interferences to the MIMO-MC radars.

2) Spectrum Sharing without knowledge of the radar’ssampling scheme: If the MIMO-MC radar does not share Ωwith the communication system, the expression of EIP of (7)is also not available at the communication system. In this case,the communication system can design its covariance assuming

that Ω is full of 1’s, i.e., for the worst case of interference

(P0) minRxl0

L∑l=1

Tr(G2RxlG

H2

)s.t.

L∑l=1

Tr (Rxl) ≤ Pt,Cavg(Rxl) ≥ C.

(13)

The same design would hold for the case in which a traditionalMIMO radar is used instead of a MIMO-MC radar. Problem(P0) is also convex and has exactly the same constraints as(P1). The efficient algorithm based on the dual decompositiontechnique in Algorithm 1 could also be applied to solve (P0).

The following theorem compares the minimum EIPachieved by the cooperative approaches of (P0) and (P1)under the same communication constraints.

Theorem 1. For any Pt and C, the EIP achieved by thecooperative approaches of (P1) is less or equal than thatachieved by the approach of (P0).

Proof: Let R∗0xl and R∗1xl denote the solution of(P0) and (P1), respectively. We know that R∗0xl satisfiesthe constraints in (P1), which means that R∗0xl is a feasiblepoint of (P1). The optimal R∗1xl achieves an objective valueno larger than any feasible point, including R∗0xl . It holdsthat EIP(R∗1xl ) ≤ EIP(R∗0xl ), which proves the claim.

There are certain scenarios in which (P1) outperforms (P0)significantly in terms of EIP. Let us denote by φ1 the intersec-tion of null space N (G2l) and range space R(R

1/2wl H), and

by φ2 the intersection of null space N (G2) and range spaceR(R

1/2wl H). It holds that φ2 ⊆ φ1. Consider the case where

φ1 is nonempty while φ2 is empty. This happens with highprobability when Mr,R ≥ Mt,C but pMr,R is much smallerthan Mt,C . Problem (P1) will guide the communication sys-tem to focus its transmission power along the directions inφ1 to satisfy both communication system constraints, whileintroducing zero EIP to the radar system. On the other hand,since φ2 is empty, Problem (P0) will guide the communicationsystem transmit power along directions that introduce nonzeroEIP. In other words, the sub-sampling procedure in the MIMO-MC radar essentially modulates the interference channel fromthe communication transmitter to the radar receiver by mul-tiplying ∆l. Compared to the original interference channelG2, the dimension of the row space of modulated channel G2l

may be greatly reduced. The cooperative approach allows thecommunication system to optimally design the communicationprecoding matrices with respect to the effective interferencechannel G2l. Therefore, it is expected that the cooperativeapproach based on the knowledge of Ω, i.e., (P1), introducessmaller EIP than its counterpart approach without knowledgeof Ω, i.e., (P0), does under the same the communicationconstraints.

B. Joint Communication and Radar System Design for Spec-trum Sharing

In the above described spectrum sharing strategies, theMIMO-MC radar operates with a predetermined pseudo ran-dom sampling scheme. In this section, we consider a joint

7

design of the communication system transmit covariance ma-trices and the MIMO-MC radar random sampling scheme, i.e.,Ω. The candidate sampling scheme needs to ensure that theresulting data matrix can be completed. This means that Ωis either a uniformly random sub-sampling matrix [36], or amatrix with a large spectral gap [34].

Recall that EIP =∑Ll=1 Tr

(∆lG2RxlG

H2

). The joint

design scheme is formulated as

(P2) minRxl0,Ω

∑L

l=1Tr(∆lG2RxlG

H2

)s.t.

L∑l=1

Tr (Rxl) ≤ Pt,Cavg(Rxl) ≥ C,

∆l = diag(Ω·l),Ω is proper.

The above problem is not convex. A solution can be obtainedvia alternating optimization. Let (Rn

xl,Ωn) be the variablesat the n-th iteration. We alternatively solve the following twoproblems:

Rnxl = arg min

Rxl0

∑L

l=1Tr(∆n−1l G2RxlG

H2

), (14a)

s.t.L∑l=1

Tr (Rxl) ≤ Pt,Cavg(Rxl) ≥ C,

Ωn = arg minΩ

∑L

l=1Tr(∆lG2R

nxlG

H2

), (14b)

s.t. ∆l = diag(Ω·l),Ω is proper.

The problem of (14a) is convex and can be solved efficiently.By simple algebraic manipulation, the EIP can be reformulatedas EIP = Tr(ΩTQ), where the l-th column of Q containsthe diagonal entries of G2RxlG

H2 . Based on the above

reformulation of EIP, we can rewrite (14b) as

Ωn = arg minΩ

Tr(ΩTQn) s.t. Ω is proper, (15)

where the l-th column of Qn contains the diagonal entries ofG2R

nxlG

H2 . Therefore, the EIP can be reduced by carefully

choosing Ω. Recall that the sampling matrix Ω is propereither if it is a uniformly random sampling matrix, or it haslarge spectral gap. However, it is difficult to incorporate suchconditions in the above optimization problem.

Noticing that row and column permutation of the samplingmatrix would not affect its singular values and thus thespectral gap, we propose to optimize the sampling scheme bypermuting the rows and columns of an initial sampling matrixΩ0, i.e.,

Ωn = arg minΩ

Tr(ΩTQn) s.t. Ω ∈ ℘(Ω0), (16)

where ℘(Ω0) denotes the set of matrices obtained by arbitraryrow and/or column permutations. The Ω0 is generated withbinary entries and bpLMr,Rc ones. One good candidate forΩ0 would be a uniformly random sampling matrix, as suchmatrix exhibit large spectral gap with high probability [34].Brute-force search can be used to find the optimal Ω. However,the complexity is very high since |℘(Ω0)| = Θ(Mr,R!L!).By alternately optimizing w.r.t. row permutation and column

permutation on Ω0, we can solve (16) using a sequence oflinear assignment problems [57].

To optimize w.r.t. column permutation, we need to find thebest one-to-one match between the columns of Ω0 and thecolumns of Qn. We construct a cost matrix Cc ∈ RL×L with[Cc]ml , (Ω0

·m)TQn·l. The problem turns out to be a linear

assignment problem with cost matrix Cc, which can be solvedin polynomial time using the Hungarian algorithm [57]. LetΩc denote the column-permutated sampling matrix after theabove step. Then, we permute the rows of Ωc to optimallymatch the rows of Qn. Similarly, we construct a cost matrixCr ∈ RMr,R×Mr,R with [Cr]ml , Ωc

m·(Qnl·)T . Again, the

Hungarian algorithm can be used to solve the row assignmentproblem. The above column and row permutation steps arealternately repeated until Tr(ΩTQn) becomes smaller than acertain predefined threshold δ1.

The complete joint-design spectrum sharing algorithm issummarized in Algorithm 2. The proposed algorithm stopswhen the value of EIP changes between two iterations dropsbelow a certain threshold δ2. It is easy to show that theobjective function, i.e., EIP, is nonincreasing during the al-ternating iterations between (14a) and (14b), and is lowerbounded by zero. According to the monotone convergencetheorem [58], the alternating optimization is guaranteed toconverge. The proposed joint-design spectrum sharing strategyis expected to further reduce the EIP at the MIMO-MC radarRX node compared to the cooperative methods in SectionIV-A. However, (P2) has higher computational complexitythan (P1) and (P0) (see detailed complexity analysis inSection IV-C). (P1) and (P0) could be preferable in casesof limited computing resources.

Algorithm 2 Joint design based spectrum sharing betweenMIMO-MC radar and a MIMO comm. system

1: Input: H,G1,G2, Pt, C, σ2C , δ1, δ2

2: Initialization: Ω0 is a uniformly random sampling matrix3: repeat4: Rn

xl ← Solve problem (14a) using Algorithm 1while fixing Ωn−1

5: Ωprev ← Ωn−1

6: loop7: Ωc ← Find the best column permutation of Ωprev

by solving the linear assignment problem with costmatrix Cc

8: Ωr ← Find the best row permutation of Ωc by solv-ing the linear assignment problem with cost matrixCr

9: if |Tr((Ωr)TQn)− Tr((Ωprev)TQn)| < δ1 then10: Break11: end if12: Ωprev ← Ωr

13: end loop14: Ωn ← Ωr; n← n+ 115: until |EIPn − EIPn−1| < δ216: Output: Rxl = Rn

xl,Ω = Ωn

8

C. Complexity

The adaptive communication transmission in the proposedspectrum sharing methods involves high complexity. A naturalquestion would be how much would one lose by usinga sub-optimal transmission approach of constant rate, i.e.,Rxl = · · · = RxL ≡ Rx, which has lower implementationcomplexity. In such case, the spectrum sharing problem (P1)can be reformulated as

(P′1) minRx0

EIP(Rx) s.t. LTr (Rx) ≤ Pt,Cavg(Rx) ≥ C,(17)

where EIP(Rx) , Tr(∆G2RxG

H2

)and ∆ is diagonal

and with each entry equal to the sum of the entries in thecorresponding row of Ω. We can see that (P′1) is much easierto solve because there is only one matrix variable. However,as it will be seen in the simulations of Section VI-B, theconstant rate transmission based on solving (9) is inferior tothe adaptive transmission based on solving (17).

It is clear that (P0) and (P1) have the same computationalcomplexity, because the objectives and the constraints aresimilar. If an interior-point method [55] is used directly tothe problems, the complexity is polynomial (cubic or slightlyhigher orders) in the number of real variables in each problem.For both (P0) and (P1), the semidefinite matrix variablesRxl have LM2

t,C real scalar variables. For the sub-optimal(P′1), there is only one semidefinite matrix variable Rx,which results in M2

t,C real scalar variables. Therefore, thecomputational costs of (P0) and (P1) are at least L3 times ofthat of (P′1), which are prohibitive if L is large. Fortunately,when (P0) and (P1) are solved using Algorithm 1 basedon dual decomposition, the computation complexity is greatlyreduced and scales linearly with L. Furthermore, the overallcomputation time of (P0) and (P1) using dual decompositioneven becomes independent of L and thus equal to that of (P′1)if all L sub-problems (P1-sub) are solved simultaneously inparallel using the same computational routine [31].

To solve (P2), several iterations of solving problems in(14a) and (14b) are required. The computational complexityof (14a) is identical to that of (P1), which has been consid-ered previously. Problem (14b) is in turn solved via severaliterations of linear assignment problem, whose complexitycubically scales with L. Simulations show that the numbersof both inner and outer iterations in Algorithm 2 are relativesmall. In summary, the computational complexity of (P2) isthe sum of L times of a polynomial of M2

t,C and O(L3).

V. MISMATCHED SYSTEMS

In Section III, the waveform symbol duration of the radarsystem is assumed to match that of the communication system.In this section, we consider the mismatched case, and showthat the proposed techniques presented in the previous sectionscan still be applied. Let fRs = 1/TR and fCs denote theradar waveform symbol rate and the communication symbolrate, respectively. Also, let the length of radar waveformsbe denoted by LR. The number of communication symbolstransmitted in the duration of LR/fRs is LC , dLRfCs fRs e.The communication average capacity and transmit power can

be expressed in terms of RxlLC

l=1 as in Section IV. In thefollowing, we will only focus on the effective interference tothe MIMO-MC radar receiver.

If fRs < fCs , the interference arrived at the radar receiverwill be down-sampled. Let I1 ⊂ N+

LCbe the set of indices

of communication symbols that are sampled by the radarin ascending order. It holds that |I1| = LR. Followingthe derivation in previous sections, we have the followinginterference power expression:

EIP =∑

l∈I1Tr(∆l′G2RxlG

H2

),

where l′ ∈ N+LR

is the index of l in ordered set I1. We observethat the communication symbols indexed by N+

LC\ I1, which

are not sampled by the radar receiver, would introduce zerointerference power to the radar system.

If fRs > fCs , the interference at the radar receiver willbe over-sampled. One individual communication symbol willintroduce interference to the radar system in bfRs /fCs c con-secutive symbol durations. Let Il be the set of radar samplingtime instances during the period of the l-th communicationsymbol. Note that Il is with cardinality bfRs /fCs c, and thecollection of sets I1, . . . , ILC

is a partition of N+LR

. Theeffective interference power for both schemes of MIMO-MCradar is respectively

EIP =∑LC

l=1Tr(∆lG2RxlG

H2

),

where ∆l =∑l′∈Il ∆l′ . We observe that each individual

communication transmit covariance matrix will be weightedby the sum of interference channels for bfRs /fCs c radar symboldurations instead of one single interference channel.

We conclude that in the above mismatched cases, the EIPexpressions have the same form as those in the matched caseexcept the diagonal matrix ∆l. To calculate the correspondingdiagonal matrices, the communication system only needs toknow the sampling time of the radar system. Therefore, thespectrum sharing problems in such cases can still be solvedusing the proposed algorithms of Section IV.

VI. NUMERICAL RESULTS

For the simulations, we set the number of symbols toL = 32 and the noise variance to σ2

C = 0.01. The MIMOradar system consists of colocated TX and RX antennasforming half-wavelength uniform linear arrays, and transmit-ting Gaussian orthogonal waveforms [23]. The channel H istaken to have independent entries, distributed as CN (0, 1).The interference channels G1 and G2 are generated withindependent entries, distributed as CN (0, σ2

1) and CN (0, σ22),

respectively. The channels are Rayleigh fading and are con-sistent with a flat fading model assumption [5]–[9], [31],[32], [44], [45]. We fix σ2

1 = σ22 = 0.01 unless otherwise

stated. The maximum communication transmit power is setto Pt = L (the power is normalized w.r.t. the power ofradar waveforms). The propagation path from the radar TXantennas to the radar RX antennas via the far-field targetintroduces a much more severe loss of power, γ2, which isset to −30dB in the simulations. The transmit power of the

9

radar antennas is fixed to ρ2 = ρ0 , 1000L/Mt,R unlessotherwise stated, and noise in the received signal is addedat SNR= 25dB. The phase jitter variance is taken to beσ2α = 10−3. The same uniformly random sampling scheme

Ω0 is adopted by the radar in both the cooperative spectrumsharing (SS) methods of (P0) and (P1). The joint-designspectrum sharing method uses the same sampling matrix asits initial sampling matrix. Recall that (P0) is the cooperativespectrum sharing method when Ω is not shared with thecommunication system. In (P0), the communication systemdesigns its waveforms by assuming Ω as the all 1’s matrix.Based on the obtained communication waveforms, an EIPvalue is calculated for (P0) using the true Ω for the easeof comparison. In the following figures, we denote the coop-erative spectrum sharing method of (P0) without knowledgeof Ω by “cooperative SS w/ Ω unknown”. We denote thecooperative spectrum sharing method of (P1) by “cooperativeSS”; and denote the joint-design spectrum sharing methodof (P2) by “joint-design SS”. The TFOCUS package [59] isused for low-rank matrix completion at the radar fusion center.The communication covariance matrix is optimized accordingto the criteria of Section IV. The obtained Rxl is used togenerate x(l) = R

1/2xl randn(Mt,C , 1). We use the EIP and

MC relative recovery error as the performance metrics. Therelative recovery error is defined as ‖DS − DS‖F /‖DS‖F ,where DS is the completed result of DS. For comparison,we also implement a “selfish communication” scenario, wherethe communication system minimizes the transmit power toachieve certain average capacity without any concern aboutthe interferences it exerts to the radar system.

Effective Interference Power

0.5 1 1.5 2

Rel

ativ

e R

ecov

ery

Err

or

0.14

0.16

0.18

0.2

0.22

0.24

Effective Interference Power

0.5 1 1.5 2

DO

A E

stim

atio

n S

ucce

ss R

ate

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. MC relative recovery errors and target angle estimation success ratesunder different levels of EIP for the MIMO-MC radar. Mt,R = 16,Mr,R =32,Mt,C = 4,Mr,C = 4.

A. The Impact of EIP on Matrix Completion and Target AngleEstimation

In the following we provide simulation results in supportof the use of EIP as a design objective. We take Mt,R =16,Mr,R = 32,Mt,C = 4,Mr,C = 4. We consider two far-field targets at angle 30 and 32.5 w.r.t. the radar arrays,with target reflection coefficients equal to 0.2 + 0.1j. Thesub-sampling rate of MIMO-MC radar is fixed to 0.5. Wesimulate different levels of EIP by setting the communicationTX covariance matrices equal to identity matrix and varyinga scaling parameter. In Fig. 2, we show the MC relativerecovery errors and target angle estimation success rates underdifferent levels of EIP. The angle estimation is achieved by theMUSIC method based on 5 pulses [24]. A success occurs ifthe angle estimation error is smaller than 0.25. The results arecalculated based on 200 independent trials. One can see thatthe EIP indeed greatly affects the matrix completion accuracyand further the target angle estimation. In particular, a 0.5 unitincrease of EIP causes a sharp 30% drop of the target angleestimation success rate. Therefore, in order to guarantee thefunction of the MIMO-MC radar, the EIP has to be maintainedat a small level.

B. Spectrum Sharing Based on Adaptive Transmission andConstant Rate Transmission

0.2 0.4 0.6 0.8 1Subsampling Rate p

0

0.5

1

Effe

ctiv

e In

terf

eren

ce P

ower Selfish Comm. + Adaptive

Cooperative SS + AdaptiveSelfish Comm. + Const. RateCooperative SS + Const. Rate


10-1

100

Rel

ativ

e R

ecov

ery

Err

or

Selfish Comm. + Adaptive

Cooperative SS + Adaptive

Selfish Comm. + Const. Rate

Cooperative SS + Const. Rate

Fig. 3. Spectrum sharing based on adaptive transmission and constantrate transmission for the MIMO-MC radar. Mt,R = 4,Mr,R = Mt,C =8,Mr,C = 4.

In this simulation, we compare the performance of thecooperative scheme of (P1) based on adaptive transmissionand the constant rate transmission scheme of (P′1). We alsoimplement the selfish communication scenario using constantrate transmission. We take Mt,R = 4,Mr,R = Mt,C =8,Mr,C = 4, and one far-field stationary target at angle 30

w.r.t. the radar arrays, with target reflection coefficient equal

10

20 30 40 50 60Waveform Length L

0

50

100

150

CP

U ti

me

(s)

Cooperative SS w/ Ω UnkownCooperative SS + AdaptiveCooperative SS + Const. RateJoint-Design SSSelfish Comm. + AdaptiveSelfish Comm. + Const. Rate

Fig. 4. CPU time comparison for various spectrum sharing algorithms underdifferent values of waveform length L.

to 0.2 + 0.1j. For the communication capacity constraint, weconsider C = 12 bits/symbol. Fig. 3 shows the EIP andMC relative recovery error as functions of the sub-samplingrate at the MIMO-MC radar. We observe that the cooperativescheme of (P1) (labeled as “Cooperative SS + Adaptive”)achieves much smaller EIP and MC errors than the constantrate transmission scheme of (P′1) (labeled as “CooperativeSS + Const. Rate”) does. It can also be seen that the constantrate transmission scheme is inferior even to the adaptive trans-mission based selfish communication scheme. This impliesthat the adaptive transmission technique plays an importantrole in reducing the EIP and MC errors. In the following,the performance of adaptive transmission based schemes isevaluated in more detail. As we already mentioned in SectionII, when the sub-sampling rate p equals 1, the MIMO-MCradar becomes the traditional MIMO radar. Therefore, theabove comparison between the adaptive and the constant ratetransmission scheme for MIMO-MC radars also holds fortraditional MIMO radars.

To get an idea of the complexity involved in the aforemen-tioned simulations, we recorded the CPU times for the variousspectrum sharing algorithms executed on a laptop with IntelCore i7 CPU and 8GB memory. Fig. 4 shows the CPU timesunder different values of waveform length. One can observethat 1) the constant rate algorithms are the fastest and theirrunning times are independent of L; 2) the running times ofthe adaptive rate algorithms, both selfish and cooperative ones,scale linearly with L; 3) the joint-design spectrum sharingmethod takes about 2-3 times of the cooperative spectrumsharing methods’ running time. These observations match ourcomplexity analysis in Section IV-C.

C. Spectrum Sharing between a MIMO-MC radar and aMIMO Communication System

1) Performance under different sub-sampling rates: Thereis a far-field stationary target at angle 30 w.r.t. the radararrays, with target reflection coefficient equal to 0.2+0.1j. Forthe communication capacity constraint, we consider C = 12bits/symbol. The sub-sampling rate of MIMO-MC radar variesfrom 0.2 to 1. The following two scenarios are considered.

In the first scenario, we take Mt,R = 4,Mr,R = Mt,C =8,Mr,C = 4. In Fig. 5(a) we plot the EIP results for 4 differentrealizations of Ω0. For better visualization, Fig. 5(b) shows the


0

0.1

0.2

0.3

0.4

0.5

Effe

ctiv

e In

terf

eren

ce P

ower Selfish Communication

Cooperative SS w/ Ω UnkownCooperative SSJoint-Design SS


0.1

0.2

0.3

0.40.5

Rel

ativ

e R

ecov

ery

Err

or

Selfish CommunicationCooperative SS w/ Ω UnkownCooperative SSJoint-Design SS

Fig. 5. Spectrum sharing with the MIMO-MC radar under different sub-sampling rates. Mt,R = 4,Mr,R = Mt,C = 8,Mr,C = 4. Dashed curvescorrespond to EIP results using different realization of Ω0.


0.5

1

1.5

2

2.5

Effe

ctiv

e In

terf

eren

ce P


Cooperative SS w/ Ω UnkownCooperative SSJoint-Design SS


0.15

0.2

0.25

Rel

ativ

e R

ecov

ery

Err

or

Selfish CommunicationCooperative SS w/ Ω UnkownCooperative SSJoint-Design SS

Fig. 6. Spectrum sharing with the MIMO-MC radar under different sub-sampling rates. Mt,R = 16,Mr,R = 32,Mt,C = Mr,C = 4. Dashedcurves correspond to EIP results using different realization of Ω0.

relative recovery errors averaged over all 4 realizations of Ω0.The cooperative scheme (see P1) outperforms its counterpartwithout knowledge of Ω (see P0) in terms of both the EIP andthe MC relative recovery error. As discussed in Section IV, theEIP is significantly reduced by the cooperative method whenp < 0.6, i.e., when pMr,R is much smaller than Mt,C . Thejoint-design scheme in this scenario performs the same as thecooperative scheme, possibly because the row dimension of

11

Ω is too small to generate sufficient difference in EIP amongthe various permutations of Ω.

In the second scenario, we take Mt,R = 16,Mr,R =32,Mt,C = 4,Mr,C = 4. In Fig. 6(a), we plot the EIPcorresponding to 4 different realizations of Ω0, taken asuniformly random sampling matrices. Again, Fig. 6(b) showsthe relative recovery errors averaged over all 4 realizationsof Ω0. The cooperative scheme outperforms the cooperativescheme without knowledge of Ω only marginally. This is dueto the fact that both G2 and G2l are full rank. The joint-design scheme (see Section IV-B) optimizes Ω starting fromthe same sampling matrix used by the other three methods. Inthis case, the joint-design scheme achieves smaller EIP andrelative recovery errors than the other three methods.

In the above scenarios, we would like p ≥ 0.5 for a smallrelative recovery error during matrix completion. However,values of p > 0.7 require more samples while achieving littleor even no improvement on the recovery accuracy. Therefore,the optimal range of p is [0.5, 0.7], where the proposed joint-design scheme significantly outperforms the “selfish communi-cation method” and the “SS method w/o knowledge of Ω”. Weconclude that the proposed co-design based spectrum sharingmethods utilize the sub-sampling procedure in the MIMO-MCradar to achieve small EIP and high matrix recovery accuracy.

Interestingly, for the joint-design based spectrum sharingmethod, the relative recovery error achieved by the sub-sampling rate p ∈ [0.5, 0.9] is even smaller than that byfull sampling p = 1. This indicates that due to the achievedsmall EIP, the full signal matrix is accurately completed. Thecompletion process smoothes out the noise, and as result, thecompleted matrix enjoys higher SIR that the initial full signalmatrix. Therefore, the sub-sampling procedure in MIMO-MCradar is beneficial for radar-communication spectrum sharingin terms of improving radar SINR as well as reducing theamount of data to be sent to the fusion center.

In addition, simulations indicate that the communication av-erage capacity constraint holds with equality in both scenarios,confirming observation (1) of Section IV-A.

2) Performance under different capacity constraints: Inthis simulation, the constant C in the communication ca-pacity constraint of (9b) varies from 6 to 14 bits/symbol,while the sub-sampling rate p is fixed to 0.5. Four differentrealizations of Ω0 are considered. Fig. 7 shows the resultsfor Mt,R = 4,Mr,R = Mt,C = 8,Mr,C = 4. For the“selfish communication” scheme and the cooperative schemewithout knowledge of Ω, the EIP and relative recovery errorsincrease as the communication capacity increases. In contrast,the cooperative and joint-design schemes achieve significantlysmaller EIP and relative recovery errors under all values of C.This indicates that the latter two spectrum sharing methodssuccessfully allocate the communication transmit power indirections that result in high communication rate, but smallEIP to the MIMO-MC radar.

The results for Mt,R = 16,Mr,R = 32,Mt,C = Mr,C = 4are shown in Fig. 8. Since Mr,R is much larger than Mt,C ,the cooperative scheme with the knowledge of Ω outperformsits counterpart without knowledge of Ω only marginally.Meanwhile, the joint-design scheme can effectively further

6 8 10 12 14Capacity Constraint C

0

0.05

0.1

0.15

0.2

Effe

ctiv

e In

terf

eren

ce P


Cooperative SS w/ Ω UnkownCooperative SS with p=0.5Joint-Design SS with p=0.5


0.1

0.15

0.2

Rel

ativ

e R

ecov

ery

Err

or

Fig. 7. Spectrum sharing with the MIMO-MC radar under different capacityconstraints C. Mt,R = 4,Mr,R = Mt,C = 8,Mr,C = 4. Dashed curvescorrespond to EIP results using different realization of Ω0.


0.5

1

1.5

2

2.5E

ffect

ive

Inte

rfer

ence

Pow

er Selfish CommunicationCooperative SS w/ Ω UnkownCooperative SS with p=0.5Joint-Design SS with p=0.5


0.1

0.15

0.2

0.25

Rel

ativ

e R

ecov

ery

Err

or

Fig. 8. Spectrum sharing with the MIMO-MC radar under different capacityconstraints C. Mt,R = 16,Mr,R = 32,Mt,C =Mr,C = 4. Dashed curvescorrespond to EIP results using different realization of Ω0.

reduce the EIP and relative recovery errors.3) Performance under different number of targets: In this

simulation, we fix p = 0.5 and C = 12 and evaluate theperformance when multiple targets are present. The targetreflection coefficients are designed such that the target returnshave fixed power, independent of the number of targets. Weobserve that the EIPs of different methods remain constantfor different number of targets. This is because the design of

12

1 2 3 4 5 6 7 8The Number of Targets

0.15

0.2

0.25

0.3

0.35

0.40.45

Rel

ativ

e R

ecov

ery

Err

or

Selfish CommunicationCooperative SS w/ Ω UnkownCooperative SS with p=0.5Joint-Design SS with p=0.5

Fig. 9. Spectrum sharing with the MIMO-MC radar when multiple targetspresent. Mt,R = 16,Mr,R = 32,Mt,C =Mr,C = 4, p = 0.5 and C = 12bits/symbol.

the communication waveforms is not affected by the targetnumber. Fig. 9 shows the results of the relative recoveryerror, which increases as the number of targets increases. Allmethods have large recovery error for large number of targets,because the retained samples are not sufficient for reliablematrix completion under any level of noise. The proposedjoint-design scheme can work effectively for the MIMO-MCradar when a moderate number of targets are present.

0.01 0.2 0.4 0.8 1 1.5 2Radar Transmit Power ρ

2/ρ0

0.8

1

1.2

1.4

1.6

1.8

Effe

ctiv

e In

terf

eren

ce P

ower


0.01 0.2 0.4 0.8 1 1.5 2Radar Transmit Power ρ

2/ρ0

0.2

0.4

0.6

0.8

1

Rel

ativ

e R

ecov

ery

Err

or


Fig. 10. Spectrum sharing with the MIMO-MC radar under different levelsof radar TX power. Mt,R = 16,Mr,R = 32,Mt,C =Mr,C = 4.

4) Performance under different levels of radar TX power:In this simulation, we evaluate the effect of radar TX powerρ2, while fixing p = 0.5, C = 12 and the target number to be1. Fig. 10 shows the results of EIP and relative recovery errorsfor Mt,R = 16,Mr,R = 32,Mt,C = Mr,C = 4. Again, we seethat the joint-design scheme performs the best, followed bythe cooperative scheme with the knowledge of Ω and then thecooperative scheme without knowledge of Ω. When the radarTX power increases, the EIP increases but with a much slowerrate. Therefore, increasing the radar TX power improves the

relative recovery errors.

0.05 0.1 0.15 0.2 0.25 0.3 0.35Interference channel G1 variance σ

2

1

0.8

1

1.2

1.4

1.6

1.8

Effe

ctiv

e In

terf

eren

ce P

ower


0.1 0.2 0.3 0.4Interference channel G1 variance σ

2

1

0.12

0.14

0.16

0.18

Rel

ativ

e R

ecov

ery

Err

or


Fig. 11. Spectrum sharing with the MIMO-MC radar under differentchannel variance σ2

1 for the interference channel G1. Mt,R = 16,Mr,R =32,Mt,C =Mr,C = 4.

5) Performance under different interference channelstrength: In this simulation, we evaluate the effectthe interference channel G1 with different σ2

1 , whilefixing p = 0.5, C = 12 and the target number tobe 1. As the communication RX gets closer to theradar TX antennas, σ2

1 gets larger. Fig. 11 showsthe results of EIP and relative recovery errors forMt,R = 16,Mr,R = 32,Mt,C = Mr,C = 4. For allthe spectrum sharing methods, when the interference channelG1 gets stronger, the communication TX increases its transmitpower in order to satisfy the capacity constraint. Therefore,the EIP and the relative recovery errors increases with thevariance σ2

1 . We also observe that the joint-design schemeperforms the best, followed by the cooperative scheme withthe knowledge of Ω and then the cooperative scheme withoutknowledge of Ω.

VII. CONCLUSIONS

This paper has considered spectrum sharing between aMIMO communication system and a MIMO-MC radar system.In order to reduce the effective interference power (EIP) atradar RX antennas, we have first considered the cooperativespectrum sharing method, which designs the communicationtransmit covariance matrix based on the knowledge of theradar sampling scheme. We have also formulated the spectrumsharing method for the case where the radar sampling schemeis not shared with the communication system. Our theoreticalresults guarantee that the cooperative approach can effectivelyreduce the EIP to a larger extent as compared to the spectrumsharing method without the knowledge of the radar sampling

13

scheme. Second, we have proposed a joint design of the com-munication transmit covariance matrix and the radar samplingscheme to further reduce the EIP. The EIP reduction and thematrix completion recovery errors have been evaluated undervarious system parameters. We have shown that the MIMO-MC radars enjoy reduced interference by the communicationsystem when the proposed spectrum sharing methods areconsidered. In particular, the sparse sampling at the radar RXantennas can reduce the rank of the interference channel. Oursimulations have confirmed that significant EIP reduction isachieved by the cooperative approach; this is because in thatapproach, the communication power is allocated to directionsin the null space of the effective interference channel. Oursimulations have suggested that the joint-design scheme canachieve much smaller EIP and relative recovery errors thanother methods when the number of radar TX and RX antennasis moderately large.

The adaptive communication transmission has been shownto be the optimal scheme for the considered spectrum shar-ing scenario. Compared to the constant rate transmission,the adaptive transmission requires higher computational andimplementation complexity. To reduce the computation com-plexity, efficient algorithms have been provided based on theLagrangian dual decomposition. As more and more powerfuldigital signal processors are used in modern communicationterminals, advanced adaptive transmission approaches ought toweigh heavily due to the increasing demand on high spectralefficiency. Nevertheless, the adaptive transmission approachconsidered in the paper provides useful insights on the optimaldesign of the MIMO communication system coexisting withMIMO-MC radars, which deserves research attention despitethe computational and implementation complexity.

REFERENCES

[1] B. Li and A. P. Petropulu, “Spectrum sharing between matrix completionbased MIMO radars and a MIMO communication system,” in IEEEInternational Conference on Acoustics, Speech and Signal Processing,April 2015, pp. 2444–2448.

[2] “Radar spectrum regulatory overview,” [online] 2013, http://www.darpa.mil/WorkArea/DownloadAsset.aspx?id=2147486331, (Accessed: July2014).

[3] F. H. Sanders, R. L. Sole, J. E. Carroll, G. S. Secrest, and T. L. Allmon,“Analysis and resolution of rf interference to radars operating in theband 2700–2900 MHz from broadband communication transmitters,”US Dept. of Commerce, Tech. Rep. NTIA Technical Report TR-13-490,2012.

[4] A. Lackpour, M. Luddy, and J. Winters, “Overview of interference mit-igation techniques between WiMAX networks and ground based radar,”in 20th Annual Wireless and Optical Communications Conference, April2011, pp. 1–5.

[5] S. Sodagari, A. Khawar, T. C. Clancy, and R. McGwier, “A projectionbased approach for radar and telecommunication systems coexistence,”in IEEE Global Telecommunication Conference, Dec 2012, pp. 5010–5014.

[6] A. Babaei, W. H. Tranter, and T. Bose, “A practical precoding approachfor radar/communications spectrum sharing,” in 8th InternationalConference on Cognitive Radio Oriented Wireless Networks, July 2013,pp. 13–18.

[7] S. Amuru, R. M. Buehrer, R. Tandon, and S. Sodagari, “MIMOradar waveform design to support spectrum sharing,” in IEEE MilitaryCommunication Conference, Nov 2013, pp. 1535–1540.

[8] A. Khawar, A. Abdel-Hadi, and T. C. Clancy, “Spectrum sharingbetween S-band radar and LTE cellular system: A spatial approach,” inIEEE International Symposium on Dynamic Spectrum Access Networks,,April 2014, pp. 7–14.

[9] C. Shahriar, A. Abdelhadi, and T. C. Clancy, “Overlapped-MIMO radarwaveform design for coexistence with communication systems,” in IEEEWireless Communications and Networking Conference, 2015, pp. 223–228.

[10] H. Deng and B. Himed, “Interference mitigation processing forspectrum-sharing between radar and wireless communications systems,”IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no.3, pp. 1911–1919, July 2013.

[11] A. Aubry, A. De Maio, M. Piezzo, and A. Farina, “Radar waveformdesign in a spectrally crowded environment via nonconvex quadraticoptimization,” IEEE Transactions on Aerospace and Electronic Systems,vol. 50, no. 2, pp. 1138–1152, 2014.

[12] A. Aubry, A. De Maio, Y. Huang, M. Piezzo, and A. Farina, “A newradar waveform design algorithm with improved feasibility for spectralcoexistence,” IEEE Transactions on Aerospace and Electronic Systems,vol. 51, no. 2, pp. 1029–1038, April 2015.

[13] S. C. Surender, R. M. Narayanan, and C. R. Das, “Performance analysisof communications & radar coexistence in a covert UWB OSA system,”in IEEE Global Telecommunications Conference, 2010, pp. 1–5.

[14] A. Turlapaty and Y. Jin, “A joint design of transmit waveforms forradar and communications systems in coexistence,” in IEEE RadarConference, 2014, pp. 0315–0319.

[15] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valen-zuela, “MIMO radar: an idea whose time has come,” in Proceedings ofthe IEEE Radar Conference, April 2004, pp. 71–78.

[16] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE SignalProcessing Magazine, vol. 24, no. 5, pp. 106–114, 2007.

[17] J. Li, P. Stoica, L. Xu, and W. Roberts, “On parameter identifiabilityof MIMO radar,” IEEE Signal Processing Letters, vol. 14, no. 12, pp.968–971, Dec 2007.

[18] C. Chen and P. P. Vaidyanathan, “MIMO radar space time adaptiveprocessing using prolate spheroidal wave functions,” IEEE Transactionson Signal Processing, vol. 56, no. 2, pp. 623–635, Feb 2008.

[19] C. Chen and P. P. Vaidyanathan, “MIMO radar ambiguity properties andoptimization using frequency-hopping waveforms,” IEEE Transactionson Signal Processing, vol. 56, no. 12, pp. 5926–5936, 2008.

[20] M. A. Herman and T. Strohmer, “High-resolution radar via compressedsensing,” IEEE Transactions on Signal Processing, vol. 57, no. 6, pp.2275–2284, June 2009.

[21] Y. Yu, A. P. Petropulu, and H. V. Poor, “MIMO radar using compressivesampling,” IEEE Journal of Selected Topics in Signal Processing, vol.4, no. 1, pp. 146–163, Feb 2010.

[22] M. Rossi, A. M. Haimovich, and Y. C. Eldar, “Spatial compressivesensing for MIMO radar,” IEEE Transactions on Signal Processing,vol. 62, no. 2, pp. 419–430, Jan 2014.

[23] S. Sun, A. P. Petropulu, and W. U. Bajwa, “Target estimation incolocated MIMO radar via matrix completion,” in IEEE InternationalConference on Acoustics, Speech and Signal Processing, May 2013, pp.4144–4148.

[24] S. Sun, W. Bajwa, and A. P. Petropulu, “MIMO-MC radar: A MIMOradar approach based on matrix completion,” Aerospace and ElectronicSystems, IEEE Transactions on, vol. 51, no. 3, pp. 1839–1852, July2015.

[25] D. S. Kalogerias and A. P. Petropulu, “Matrix completion in colocatedMIMO radar: Recoverability, bounds and theoretical guarantees,” IEEETransactions on Signal Processing, vol. 62, no. 2, pp. 309–321, Jan2014.

[26] S. Sun and A. P. Petropulu, “Waveform design for MIMO radarswith matrix completion,” IEEE Journal of Selected Topics in SignalProcessing, vol. PP, no. 99, pp. 1–1, December 2015.

[27] Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity tobasis mismatch in compressed sensing,” IEEE Transactions on SignalProcessing,, vol. 59, no. 5, pp. 2182–2195, May 2011.

[28] B. Li and A. P. Petropulu, “Radar precoding for spectrum sharingbetween matrix completion based MIMO radars and a MIMO commu-nication system,” in IEEE Global Conference on Signal and InformationProcessing, December 2015.

[29] K. Letaief and W. Zhang, “Cooperative communications for cognitiveradio networks,” Proceedings of the IEEE, vol. 97, no. 5, pp. 878–893,May 2009.

[30] S. Haykin, “Cognitive radar: A way of the future,” IEEE SignalProcessing Magazine, vol. 23, no. 1, pp. 30–40, Jan 2006.

[31] R. Zhang and Y. Liang, “Exploiting multi-antennas for opportunisticspectrum sharing in cognitive radio networks,” IEEE Journal of SelectedTopics in Signal Processing, vol. 2, no. 1, pp. 88–102, Feb 2008.

[32] R. Zhang, Y. Liang, and S. Cui, “Dynamic resource allocation incognitive radio networks,” IEEE Signal Processing Magazine, vol. 27,

14

no. 3, pp. 102–114, May 2010.[33] S. J. Kim and G. B. Giannakis, “Optimal resource allocation for MIMO

Ad Hoc cognitive radio networks,” IEEE Transactions on InformationTheory, vol. 57, no. 5, pp. 3117–3131, May 2011.

[34] S. Bhojanapalli and P. Jain, “Universal matrix completion,” inProceedings of The 31st International Conference on Machine Learning,2014, pp. 1881–1889.

[35] H. Krim and M. Viberg, “Two decades of array signal processingresearch: The parametric approach,” IEEE Signal Processing Magazine,vol. 13, no. 4, pp. 67–94, 1996.

[36] E. J. Candes and Y. Plan, “Matrix completion with noise,” Proceedingsof the IEEE, vol. 98, no. 6, pp. 925–936, June 2010.

[37] F. M. Gardner, Phaselock techniques, John Wiley & Sons, 2005.[38] R. Poore, “Phase noise and jitter,” Agilent EEs of EDA, 2001.[39] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility of

distributed beamforming in wireless networks,” IEEE Transactions onWireless Communications, vol. 6, no. 5, pp. 1754–1763, 2007.

[40] B. Razavi, “A study of phase noise in CMOS oscillators,” IEEE Journalof Solid-State Circuits, vol. 31, no. 3, pp. 331–343, 1996.

[41] C. Kopp, “Search and acquisition radars (S-band, X-band),” TechnicalReport APA-TR-2009-0101, [online] 2009, http://www.ausairpower.net/APA-Acquisition-GCI.html, (Accessed: July 2015).

[42] “Radar performance,” Radtec Engineering Inc., [online], http://www.radar-sales.com/PDFs/Performance\ RDR\%26TDR.pdf, (Ac-cessed: July 2015).

[43] Telesystem Innovations, “LTE in a nutshell: The physical layer,” Whitepaper, 2010.

[44] T Rappaport, “Wireless communications principles and practice edition,”2001.

[45] D. Tse and P. Viswanath, Fundamentals of wireless communication,Cambridge university press, 2005.

[46] A. Goldsmith, Wireless communications, Cambridge university press,2005.

[47] R. P. Jover, “LTE PHY fundamentals,” [online], http://www.ee.columbia.edu/∼roger/LTE\ PHY\ fundamentals.pdf, (Accessed: July2015).

[48] M. Filo, A. Hossain, A. R. Biswas, and R. Piesiewicz, “Cognitive pilotchannel: Enabler for radio systems coexistence,” in 2nd InternationalWorkshop on Cognitive Radio and Advanced Spectrum Management,May 2009, pp. 17–23.

[49] L. Lu, X. Zhou, U. Onunkwo, and G. Y. Li, “Ten years of research inspectrum sensing and sharing in cognitive radio.,” EURASIP J. WirelessComm. and Networking, vol. 2012, pp. 28, 2012.

[50] G. Taubock, “Complex-valued random vectors and channels: Entropy,divergence, and capacity,” IEEE Transactions on Information Theory,vol. 58, no. 5, pp. 2729–2744, 2012.

[51] S. N. Diggavi and T. M. Cover, “The worst additive noise under acovariance constraint,” IEEE Transactions on Information Theory, vol.47, no. 7, pp. 3072–3081, Nov 2001.

[52] C. Chen and P. P. Vaidyanathan, “MIMO radar waveform optimizationwith prior information of the extended target and clutter,” IEEETransactions on Signal Processing, vol. 57, no. 9, pp. 3533–3544, 2009.

[53] G. Cui, H. Li, and M. Rangaswamy, “MIMO radar waveform designwith constant modulus and similarity constraints,” IEEE Transactionson Signal Processing, vol. 62, no. 2, pp. 343–353, 2014.

[54] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacitylimits of MIMO channels,” IEEE Journal on Selected Areas inCommunications, vol. 21, no. 5, pp. 684–702, 2003.

[55] S. Boyd and L. Vandenberghe, Convex optimization, Cambridgeuniversity press, 2004.

[56] R. G. Bland, D. Goldfarb, and M. J. Todd, “The ellipsoid method: Asurvey,” Operations research, vol. 29, no. 6, pp. 1039–1091, 1981.

[57] H. W. Kuhn, “The Hungarian method for the assignment problem,”Naval research logistics quarterly, vol. 2, no. 1-2, pp. 83–97, 1955.

[58] J. Yeh, “Real analysis,” in Theory of measure and integration. 2006,Singapore: World Scientific.

[59] S. R. Becker, E. J. Candes, and M. C. Grant, “Templates for convex coneproblems with applications to sparse signal recovery,” MathematicalProgramming Computation, vol. 3, no. 3, pp. 165–218, 2011.

Optimum Co-Design for Spectrum Sharing Between Matrix...

Documents

Transcript of Optimum Co-Design for Spectrum Sharing Between Matrix...