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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/JSEN.2015.2424393, IEEE Sensors Journal

Target Detection Performance of Spectrum SharingMIMO Radars

Awais Khawar, Student Member, IEEE, Ahmed Abdelhadi, Member, IEEE, and T. Charles Clancy, SeniorMember, IEEE

Abstract—Future wireless communication systems are envi-sioned to share radio frequency spectrum with radars in orderto meet the growing spectrum demands. In this paper, weaddress the problem of target detection by radars that projectwaveform onto the null space of interference channel in order tomitigate interference to cellular systems. We consider a multiple-input multiple-output (MIMO) radar and a MIMO cellularcommunication system with K base stations (BS). We considertwo spectrum sharing scenarios. In the first scenario the degreesof freedom (DoF) available at the radar are not sufficient enoughto simultaneously detect target and mitigate interference to KBSs. For this case we select one BS among K BSs for waveformprojection on the basis of guaranteeing minimum waveformdegradation. For the second case, the radar has sufficient DoFto simultaneously detect target and mitigate interference to all KBSs. We study target detection capabilities of null-space projected(NSP) waveform and compare it with the orthogonal waveform.We derive the generalized likelihood ratio test (GLRT) for targetdetection and derive detector statistic for NSP and orthogonalwaveform. The target detection performance for both waveformsis studied theoretically and via Monte Carlo simulations.

Index Terms—MIMO Radar, Null Space Projection, CellularSystem Coexistence, Target Detection, GLRT, ML Estimation.

I. INTRODUCTION

Spectrum sharing between wireless communication systemsand radars is an emerging area of research. In the past,spectrum has been shared primarily between wireless com-munication systems using opportunistic approaches by usersequipped with cognitive radios [1]. This type of spectrumsharing has been made possible with the use of spectrumsensing [2], or geolocation databases [3], or a combination ofboth in the form of radio environment maps (REM) [4]. Somerecent efforts have explored co-channel sharing approachesamong secondary network entities, please see [5] and referencetherein. However, in contrast, co-channel spectrum sharing be-tween wireless systems and radars has received little attentionthus far because of regulatory concerns.

Spectrum policy regulators, in the past, have not allowedcommercial wireless services in radar bands, except in few

Awais Khawar ([email protected]), Ahmed Abdelhadi ([email protected]),and T. Charles Clancy ([email protected]) are with Virginia Polytechnic Instituteand State University, Arlington, VA, 22203.

This work was supported by Defense Advanced Research Projects Agency(DARPA) under the SSPARC program. Contract Award Number: HR0011-14-C-0027. The views, opinions, and/or findings contained in this arti-cle/presentation are those of the author(s)/presenter(s) and should not beinterpreted as representing the official views or policies of the Departmentof Defense or the U.S. Government.

Approved for Public Release, Distribution Unlimited

cases, due to the fear of harmful interference from theseservices to radar systems [6]. Recently, in the United States(U.S.), the Federal Communications Commission (FCC), hasproposed to use the 3550-3650 MHz band for commercialbroadband use [7]. The incumbents in this band are radarand satellite systems [8]. The Commission has proposed thatincumbents share this band with commercial communicationsystems. The Commission’s spectrum sharing initiative ismotivated by many factors including the President’s NationalBroadband plan, which called to free up to 500 MHz offederal-held spectrum by 2020 [9]; surge in consumers’ de-mand for access to mobile broadband, which operators can’tmeet with current spectrum allocations; the report on efficientspectrum utilization by President’s Council of Advisers onScience and Technology (PCAST), which emphasized to share1000 MHz of government-held spectrum [10], and the lowutilization of the 3550-3650 MHz band by federal incumbents[11].

In future, when radio frequency (RF) spectrum will beshared among many different systems, e.g., radars and cellularsystems, it is important to access the interference scenario. Ofcourse, radars will cause interference to communication sys-tems and vice versa if proper interference mitigation methodsand novel sharing algorithms are not employed. In a studyconducted by the National Telecommunications and Informa-tion Administration (NTIA), it was observed that in orderto protect commercial cellular communication systems, fromhigh power radar signal, large exclusion zones are required[11]. These exclusion zones cover a large portion of the U.S.where majority of the population lives and, thus, does not makea business case for commercial deployment in radar bands. Inorder to share radar bands for commercial operation we have toaddress interference mitigation techniques at both the systems.In this work, we focus on interference caused by radar systemsto communication system and propose methods to mitigate thisinterference.

The federal-commercial spectrum sharing is not a newpractice. In fact, in the past, commercial wireless systems haveshared government bands on a low transmit power basis, inorder to protect incumbents from interference. An example ofsuch a scenario is wireless local area network (WLAN) in the5250-5350 MHz and 5470-5725 MHz radar bands [12]. So,the Commission’s latest initiative to share the 3.5 GHz radarband with small cells, i.e. wireless base stations operating ona low power, is in harmony with previous practices [7].



A. Related Work

On going research efforts have shown that there are nu-merous ways to share spectrum between radars and commu-nication systems. Cooperative sensing based spectrum sharingapproaches can be utilized where radar’s allocated bandwidthis shared with communication systems [13]–[15]. A jointcommunication-radar platform can be envisioned in whichspectrally-agile radar performs an additional task of spectrumsensing and upon finding of unused frequencies it can changeits operating frequency. In addition of spectrum sharing such asetting can enable co-located radar and communication systemplatforms for integrated communications and radar applica-tions [16]–[19]. Radar waveforms can be shaped in a way thatthey don’t cause interference to communication systems [20]–[24]. Moreover, database-aided sensing at communication sys-tems [25] and beamforming approaches at MIMO radars canalso be realized for spectrum sharing [26].

B. Our Contributions

The problem of target estimation, detection, and trackinglies at the heart of radar signal processing. This problembecomes critically important when we talk about sharingradar spectrum with other systems, say cellular systems. Theregulatory work going on in the 3.5 GHz band to share theradar spectrum with commercial systems is the motivationof this work. The focus of this work is to study targetdetection performance of radar that is subject to share itsspectrum with cellular system. We consider spectrum sharingbetween a MIMO radar and a cellular system with manybase stations. In our previous work, we have addressed theproblem of radar waveform projection onto the null space ofinterference channel, in order to mitigate radar interferenceto communication system, and the problem of selection ofinterference channel for projection when we have a cellularsystem with K base stations [22]. In this work we consider twospectrum sharing scenarios. In the first scenario the degrees offreedom (DoF) available at the radar are not sufficient enoughto simultaneously detect target and mitigate interference toK BSs. For this case we select one BS among K BSs forwaveform projection on the basis of guaranteeing minimumwaveform degradation. For the second case, the radar hassufficient DoF to simultaneously detect target and mitigateinterference to all K BSs. For both the scenarios we studythe target detection performance of radar for the null-spaceprojected (NSP) waveform and compare it with that of theorthogonal radar waveform. We use the generalized likelihoodratio test (GLRT) for target detection and derive detectorstatistic for NSP and the orthogonal waveform.

C. Notations

Matrices are denoted by bold upper case letters, e.g. A,and vectors are denoted by bold lower case letters, e.g. a.Transpose, conjugate, and Hermitian operators are denoted by(·)T , (·)∗, and (·)H , respectively. Moreover, notations usedthroughout the paper are provided in Table I along withdescriptions for a quick reference.

TABLE ITABLE OF NOTATIONS

Notation Description

x(t) Transmitted radar (orthogonal) waveform

a(θ) Steering vector to steer signal to target angle θ

y(t) Received radar waveform

Rx Correlation matrix of orthogonal waveforms

sUEj (t) Signal transmitted by the jth UE in the ith cell

LUEi Total number of user equipments (UEs) in the ith cell

K Total number of BSs

M Radar transmit/receive antennas

NBS BS transmit/receive antennas

Hi ith interference channel

ri(t) Received signal at the ith BS

Pi Projection matrix for the ith channel

D. Organization

This paper is organized as follows. Section II discussMIMO radar, target channel, orthogonal waveforms, interfer-ence channel, and our cellular system model. Moreover, italso discusses modeling and statistical assumptions. Section IIIdiscusses spectrum sharing between MIMO radar and cellularsystem and introduces sharing architecture and projectionalgorithms. Section IV presents the generalized likelihood ratiotest (GLRT) for target detection and derives detector statisticfor NSP and orthogonal waveform. Section V discusses numer-ical results and compares performance of NSP and orthogonalwaveform. Section VI concludes the paper.

II. SYSTEM MODEL

In this section, we introduce preliminaries of MIMO radar,point target in far-field, orthogonal waveforms, interferencechannel, and cellular system model. Moreover, we also discussmodeling and statistical assumptions along with RF environ-ment assumptions used throughout the paper.

A. Radar Model

The radar we consider in this paper is a colocated MIMOradar with M transmit and receive antennas and is mountedon a ship. The colocated MIMO radar has antennas that havespacing on the order of half the wavelength. Another class ofMIMO radar is widely-spaced MIMO radar where elementsare widely-spaced which results in enhanced spatial diversity[27]. The colocated radar gives better spatial resolution andtarget parameter identification as compared to the widely-spaced radar [28].

B. Target Model/Channel

In this paper, we consider a point target model which isdefined for targets having a scatterer with infinitesimal spatialextent. This model is a good assumption and is widely usedin radar theory for the case when radar elements are colocatedand there exists a large distance between the radar array and



the target as compared to inter-element distance [29]. Thesignal reflected from a point target with unit radar cross-section (RCS) is mathematically represented by the Dirac deltafunction.

C. Signal Model

Let x(t) be the signal transmitted from the M -elementMIMO radar array, defined as

x(t) =[x1(t)ejωct x2(t)ejωct · · · xM (t)ejωct

]T(1)

where xk(t)ejωct is the baseband signal from the kth transmitelement, ωc is the carrier angular frequency, t ∈ [0, To], withTo being the observation time. We define the transmit steeringvector as

aT (θ) ,[e−jωcτT1

(θ) e−jωcτT2(θ) · · · e−jωcτTM

(θ)]T.

(2)Then, the transmit-receive steering matrix can be written as

A(θ) , aR(θ)aTT (θ). (3)

Since, we are considering M transmit and receive elements,we define a(θ) , aT (θ) , aR(θ). The signal received froma single point target, in far-field with constant radial velocityvr, at an angle θ can be written as

y(t) = α e−jωDtA(θ) x(t− τ(t)) + n(t) (4)

where τ(t) = τTk(t) + τRl

(t), denoting the sum of propaga-tion delays between the target and the kth transmit elementand the lth receive element, respectively; ωD is the Dopplerfrequency shift, α represents the complex path loss includingthe propagation loss and the coefficient of reflection, and n(t)is the zero-mean complex Gaussian noise.

D. Modeling Assumptions

In order to keep the analysis tractable we have made thefollowing assumptions about our signal model:• The path loss α is assumed to be identical for all transmit

and receive elements, due to the far-field assumption [30].• The angle θ is the azimuth angle of the target.• After compensating the range-Doppler parameters, we

can simplify equation (4) as

y(t) = αA(θ) x(t) + n(t). (5)

E. Statistical Assumptions

We make the following assumptions for our received signalmodel in equation (5):• θ and α are deterministic unknown parameters repre-

senting the target’s direction of arrival and the complexamplitude of the target, respectively.

• The noise vector n(t) is independent, zero-mean complexGaussian with known covariance matrix Rn = σ2

nIM , i.e.n(t) ∼ Nc(0M , σ

2nIM ), where Nc denotes the complex

Gaussian distribution.• With the above assumptions, the received signal model

in equation (5) has independent complex Gaussian distri-bution, i.e., y(t) ∼ Nc(αA(θ) x(t), σ2

nIM ).

F. Orthogonal Waveforms

In this paper, we consider orthogonal waveforms transmittedby MIMO radars, i.e.,

Rx =

∫To

x(t)xH(t)dt = IM . (6)

The transmission of orthogonal signals gives MIMO radaradvantages in terms of digital beamforming at the transmitterin addition to receiver, improved angular resolution, extendedarray aperture in the form of virtual arrays, increased numberof resolvable targets, lower sidelobes [31], and lower proba-bility of intercept as compared to coherent waveforms [30].

G. Communication System

In this paper, we consider a MIMO cellular system, with K

base stations, each equipped with NBS transmit and receiveantennas, with ith BS supporting LUE

i user equipments (UEs).The UEs are also multi-antenna systems with NUE transmitand receive antennas. If sUE

j (t) is the signals transmitted bythe jth UE in the ith cell, then the received signal at the ith BSreceiver can be written as

ri(t) =∑j

HNBS×NUE

j sUEj (t) + w(t) 1 ≤ j ≤ LUE

i (7)

where w(t) is the additive white Gaussian noise.

H. Interference Channel

In this section, we characterize the interference channel thatexists between MIMO cellular base station and MIMO radar.In our paper, we are considering K cellular BSs that’s why ourmodel has Hi, i = 1, 2, . . . ,K, interference channels, wherethe entries of Hi are denoted by

Hi =

h

(1,1)i · · · h

(1,M)i

.... . .

...

h(NBS,1)i · · · h

(NBS,M)i

(NBS ×M) (8)

where h(l,k)i denotes the channel coefficient from the kth

antenna element at the MIMO radar to the lth antenna el-ement at the ith BS. We assume that elements of Hi areindependent, identically distributed (i.i.d.) and circularly sym-metric complex Gaussian random variables with zero-meanand unit-variance, thus, having a i.i.d. Rayleigh distribution.A more thorough treatment of interference channel modelingbetween radar and cellular system, including two- and three-dimensional channel models, can be found in [32]–[35].

I. Cooperative RF Environment

In the wireless communications literature, it is usuallyassumed that the transmitter (mostly BS) has channel stateinformation (CSI) either by feedback from the receiver (mostlyUE), in FDD systems [36], or transmitters can reciprocate thechannel, in TDD systems [36]. The feedback and reciprocityare valid and practical as long as the feedback has a reasonableoverhead and coherence time of the RF channel is larger thanthe two-way communication time, respectively.



Fig. 1. Spectrum Sharing Scenario: A seaborne MIMO radar detecting apoint target while simultaneously sharing spectrum with a MIMO cellularsystem without causing interference to the cellular system.

In the case of radars sharing their spectrum with communi-cations systems one way to get CSI is that radar estimates Hi

based on the training symbols sent by communication receivers(or BSs in this case) [37]. Another approach is that radaraids communication systems in channel estimation, with thehelp of a low-power reference signal, and they feed back theestimated channel to radar [38]. Since, radar signal is treatedas interference at communication system, we can characterizethe channel as interference channel and refer to informationabout it as interference-channel state information (ICSI).

Spectrum sharing between radars and communications sys-tems can be envisioned in two domains: military radars sharingspectrum with military communication systems, we call itMil2Mil sharing; another possibility is military radars sharingspectrum with commercial communication systems, we call itMil2Com sharing. In Mil2Mil sharing, ICSI can be acquiredby radars fairly easily as both systems belong to military. InMil2Com sharing, ICSI can be acquired by giving incentivesto commercial communication system. The biggest incentivein this scenario is null-steering and protection from radarinterference. Thus, regardless of the sharing scenario, Mil2Milor Mil2Com, we have ICSI for the sake of mitigating radarinterference at communication systems.

III. RADAR-CELLULAR SYSTEM SPECTRUM SHARING

After introducing our radar and cellular system models wecan now discuss the spectrum sharing scenario between radarand cellular system. In our sharing architecture, MIMO radarand cellular systems are the co-primary users of the 3550-3650 MHz band under consideration. In the following sections,we will discuss the architecture of spectrum sharing problemwhich is followed by our spectrum sharing algorithm.

A. Architecture

We illustrate our coexistence scenario in Figure 1 where themaritime MIMO radar is sharing K interference channels withthe cellular system. Considering this scenario, the receivedsignal at the ith BS receiver can be written as

ri(t) = HNBS×Mi x(t) +

∑j

HNBS×NUE

j sUEj (t) + w(t). (9)

The goal of the MIMO radar is to map x(t) onto the null-space of Hi in order to avoid interference to the ith BS, i.e.,Hix(t) = 0, so that ri(t) has equation (7) instead of equation(9).

We consider two spectrum sharing scenarios which arediscussed as follows.Case 1 (M KNBS but M > NBS): Consider a scenarioin which a MIMO radar has a very small antenna arrayas compared to the combined antenna array of K BSs, i.e.M KNBS, but is larger than individual BS antenna array,i.e. M > NBS. In such a scenario, it is not possible for theMIMO radar to simultaneously mitigate interference to all theK BSs present in the network because of insufficient degreesof freedom (DoF) available. However, the available DoF allowsimultaneous target detection and interference mitigation toone of the BS among K BSs. The choice of BS selectiondepends upon the performance metric which radar wants tooptimize. In this paper, our performance metric is minimumdegradation of radar waveform in a minimum norm sense.

A drawback of this approach is that interference is notmitigated to K− 1 BSs present in the network and the radarhas to utilize higher transmit power to achieve the sameperformance level which can increase the level of interferenceat BSs not part of the mitigation scheme. This drawback isaddressed in the literature by moving K − 1 BSs to non-radar frequency bands by using resource allocation and carrieraggregation techniques [39], [40].

It is worth mentioning that when M KNBS tradi-tional colocated MIMO radar architecture is not suitable formitigation of interference by using NSP approaches becausesufficient DoF are not available and doing so will resultin performance degradation of radar systems. However, theMIMO radar architecture can be modified into an overlapped-MIMO radar architecture where the transmit array of colocatedMIMO radar is partitioned into a number of subarrays that areallowed to overlap. The overlapped-MIMO radar architectureincreases the DoF and enjoys the advantages of the MIMOradar while mitigating interference to communication systemswithout sacrificing the main desirable characteristics for itsown transmission [41].Case 2 (M KNBS): Consider a scenario in which aMIMO radar has a very large antenna array as compared tothe combined antenna array of K BSs, i.e. M KNBS.In such a scenario, it is feasible for the MIMO radar tosimultaneously mitigate interference to all the K BSs presentin the network while reliably detecting targets. This is becausesufficient degrees of freedom are available for both the tasks.In such a scenario the combined interference channel that theMIMO radar shares with K BSs in the networks is given as

H =[H1 H2 · · · HK

]· (10)

B. Projection Matrix

In this section, we introduce formation of projection matri-ces for Case 1 and Case 2.Projection for Case 1 (M KNBS but M > NBS): Inthis section, we define the projection algorithm for ‘Case 1’which projects radar signal onto the null space of interference



channel Hi. Assuming, the MIMO radar has channel stateinformation of all Hi interference channels, through feedback,in Mil2Mil or Mil2Com scenario, we can perform singularvalue decomposition (SVD) to find the null space and thenconstruct a projector matrix. We proceed by first finding SVDof Hi, i.e.,

Hi = UiΣiVHi . (11)

Now, let us define

Σi , diag(σi,1, σi,2, . . . , σi,p) (12)

where p , min(NBS,M) and σi,1 > σi,2 > · · · > σi,q >σi,q+1 = σi,q+2 = · · · = σi,p = 0 are the singular values ofHi. Next, we define

Σ′i , diag(σ′i,1, σ

′i,2, . . . , σ

′i,M ) (13)

where

σ′i,u ,

0, for u ≤ q,1, for u > q.

(14)

Using above definitions we can now define our projectionmatrix, i.e.,

Pi , ViΣ′iV

Hi . (15)

In order to show that Pi is a valid projection matrix we provetwo results on projection matrices below.

Property 1. Pi ∈ CM×M is a projection matrix if and onlyif Pi = PH

i = P2i .

Proof: Let’s start by showing the ‘only if’ part. First, weshow Pi = PH

i . Taking Hermitian of equation (15) we have

PHi = (ViΣ

′iV

Hi )H = Pi. (16)

Now, squaring equation (15) we have

P2i = ViΣiV

Hi ×ViΣiV

Hi = Pi (17)

where above equation follows from VHi Vi = I (since they

are orthonormal matrices) and (Σ′i)

2 = Σ′i (by construction).

From equations (16) and (17) it follows that Pi = PHi = P2

i .Next, we show Pi is a projector by showing that if v ∈ range(Pi), then Piv = v, i.e., for some w,v = Piw, then

Piv = Pi(Piw) = P2iw = Piw = v. (18)

Moreover, Piv − v ∈ null(Pi), i.e.,

Pi(Piv − v) = P2iv −Piv = Piv −Piv = 0. (19)

This concludes our proof.

Property 2. Pi ∈ CM×M is an orthogonal projection matrixonto the null space of Hi ∈ CNBS×M .

Proof: Since Pi = PHi , we can write

HiPHi = UiΣiV

Hi ×ViΣ

′iV

Hi = 0. (20)

The above results follows from noting that ΣiΣ′i = 0 by

construction.

For ‘Case 1’ we are dealing with K interference channels.Therefore, we need to select the interference channel whichresults in least degradation of radar waveform in a minimumnorm sense, i.e.,

imin , arg min1≤i≤K

∣∣∣∣∣∣Pix(t)− x(t)∣∣∣∣∣∣

2(21)

P , Pimin . (22)

Once we have selected our projection matrix it is straight for-ward to project radar signal onto the null space of interferencechannel via

x(t) = P x(t). (23)

The correlation matrix of our NSP waveform is given as

Rx =

∫To

x(t)xH(t)dt (24)

which is no longer identity, because the projection does notpreserve the orthogonality, and its rank depends upon the rankof the projection matrix.Projection for Case 2 (M KNBS): In this section, wedefine the projection algorithm for ‘Case 2’ which projectsradar signal onto the null space of combined interferencechannel H. The SVD of H is given as

H = UΣVH . (25)

Now, let us define

Σ , diag(σ1, σ2, . . . , σp) (26)

where p , min(NBS,M) and σ1 > σ2 > · · · > σq > σq+1 =σq+2 = · · · = σp = 0 are the singular values of H. Next, wedefine

Σ′i , diag(σ′1, σ

′2, . . . , σ

′M ) (27)

where

σ′u ,

0, for u ≤ q,1, for u > q.

(28)

Using above definitions we can now define our projectionmatrix, i.e.,

P , VΣ′VH . (29)

It is straightforward to see that P is a valid projection matrixby using Properties 1 and 2.

C. Spectrum Sharing and Projection Algorithms

In this section, we explain spectrum sharing and projectionalgorithms for ‘Case 1’ and ‘Case 2’.Algorithms for Case 1 (M KNBS but M > NBS):For this case, the process of spectrum sharing by formingprojection matrices and selecting interference channels isexecuted with the help of Algorithms (1) and (2). First, ateach pulse repetition interval (PRI), the radar obtains ICSIof all K interference channels. This information is sent toAlgorithm (2) for the calculation of null spaces and formationof projection matrices. Algorithm (1) process K projectionmatrices, received from Algorithm (2), to find the projectionmatrix which results in least degradation of radar waveform in



a minimum norm sense. This step is followed by the projectionof radar waveform onto the null space of the selected BS, i.e,the BS to the corresponding selected projection matrix, andwaveform transmission.

Algorithm 1 Spectrum Sharing Algorithm for Case 1loop

for i = 1 : K doGet CSI of Hi through feedback from the ith BS.Send Hi to Algorithm (2) for the formation of projec-tion matrix Pi.Receive the ith projection matrix Pi from Algorithm(2).

end forFind imin = arg min

1≤i≤K

∣∣∣∣∣∣Pix(t)− x(t)∣∣∣∣∣∣

2.

Set P = Pimin as the desired projector.Perform null space projection, i.e., x(t) = Px(t).

end loop

Algorithm 2 Projection Algorithm for Case 1if Hi received from Algorithm (1) then

Perform SVD on Hi (i.e. Hi = UiΣiVHi )

Construct Σi = diag(σi,1, σi,2, . . . , σi,p)

Construct Σ′i = diag(σ′i,1, σ

′i,2, . . . , σ

′i,M )

Setup projection matrix Pi = ViΣ′iV

Hi .

Send Pi to Algorithm (1).end if

Projection for Case 2 (M KNBS): For this case, theprocess of spectrum sharing is executed with the help ofAlgorithms (3) and (4). First, at each pulse repetition interval(PRI), the radar obtains ICSI of all K interference channels.This information is sent to Algorithm (4) for the calculationof null space of H and the formation of projection matrix P.The projection of radar waveform onto the null space of H isperformed by Algorithm 3.

Algorithm 3 Spectrum Sharing Algorithm for Case 2loop

Get CSI of H through feedback from K BSs.Send H to Algorithm (4) for the formation of projectionmatrix P.Receive the projection matrix P from Algorithm (4).Perform null space projection, i.e., x(t) = Px(t).

end loop

IV. STATISTICAL DECISION TEST FOR TARGETDETECTION

In this section, we develop a statistical decision test fortarget illuminated with the orthogonal radar waveforms andthe NSP projected radar waveforms. The goal is to compareperformance of the two waveforms by looking at the testdecision on whether the target is present or not in the range-Doppler cell of interest. We present a system-level architecture

Algorithm 4 Projection Algorithm for Case 2if H received from Algorithm (3) then

Perform SVD on H (i.e. H = UΣVH )Construct Σ = diag(σ1, σ2, . . . , σp)

Construct Σ′i = diag(σ′1, σ

′2, . . . , σ

′M )

Setup projection matrix P = VΣ′VH .

Send P to Algorithm (3).end if

of the spectrum sharing radar in Figure 2. In our architecture,the transmitter performs the functions of waveform generation,channel selection, and projection; and the receiver performsthe functions of signal detection and estimation.

For target detection and estimation, we proceed by con-structing a hypothesis test where we seek to choose betweentwo hypothesis: the null hypothesis H0 which represents thecase when the target is absent or the alternate hypothesis H1

which represents the case when the target is present. Thehypothesis for a single target model in equation (5) can bewritten as

y(t) =

H1 : αA(θ) x(t) + n(t), 0 ≤ t ≤ To,H0 : n(t), 0 ≤ t ≤ To.

(30)

Since, θ and α are unknown, but deterministic, we use thegeneralize likelihood ratio test (GLRT). The advantage ofusing GLRT is that we can replace the unknown parameterswith their maximum likelihood (ML) estimates. The ML esti-mates of α and θ are found for various signal models, targets,and interference sources in [30], [42] when using orthogonalsignals. In this paper, we consider a simpler model with onetarget and no interference sources in order to study the impactof NSP on target detection in a tractable manner. Therefore, wepresent a simpler derivation of the ML estimation and GLRT.

The received signal model in equation (5) can be written as

y(t) = Q(t, θ)α+ n(t) (31)

where

Q(t, θ) = A(θ)x(t). (32)

We use Karhunen-Loeve expansion for derivation of thelog-likelihood function for estimating θ and α. Let Ω denotethe space of the elements of y(t), Q(t, θ), and n(t).Moreover, let ψz , z = 1, 2, . . . , be an orthonormal basisfunction of Ω satisfying

< ψz(t), ψz′(t) >=

∫T0

ψz(t), ψ∗z′(t) = δzz′ (33)

where δzz′ is the Kronecker delta function. Then, the followingseries can be used to expand the processes, y(t), Q(t, θ),



MIMO Radar Array

Waveform Generation Detection and Estimation

∑ ∑ ∑ ∑

x x x x

Channel State Information

𝐇𝑖

Projector

𝐏 = 𝐏𝑖min

Projection Algorithm

𝐏𝑖 = 𝐕𝑖 𝚺 ′𝐕𝑖

𝐻

. . . . . . . . . . . . . .

. . . . . . . . . . . . .

𝜆 = 3𝜋4

y1(t) yM(t)

x1(t) xM(t) xM(t) x1(t)

Receiver Transmitter

𝐱(t)

𝐱 (t) y(t)

𝑖min SVD

Fig. 2. Block diagram of spectrum sharing radar. The transmitter is modifiedto perform the functions of ICSI collection, projection matrix formation,interference channel selection, and radar waveform projection on to theselected interference channel for spectrum sharing. On the other hand, thereceiver is a traditional radar receiver performing functions of parameterdetection and estimation on radar returns.

and n(t), as

y(t) =

∞∑z=1

yzψz(t) (34)

Q(t, θ) =

∞∑z=1

Qz(θ)ψz(t) (35)

n(t) =

∞∑z=1

nzψz(t) (36)

where yz,Qz , and nz are coefficients in the Karhunen-Loeveexpansion of the considered processes obtained by taking thecorresponding inner product with basis function ψz(t). Thus,an equivalent discrete model of equation (31) can be obtainedas

yz = Qz(θ)α+ nz, z = 1, 2, . . . (37)

For white circular complex Gaussian processes, i.e,E[n(t)n(t − τ(t))] = σ2

nIMδ(τ(t)), the sequence nzis i.i.d. and nz ∼ Nc(0M , σ

2nIM ). Thus, we can express the

log-likelihood function as

Ly(θ, α) =

∞∑z=1

(−M log(πσ2

n)− 1

σ2n

∣∣∣∣∣∣yz −Qz(θ)α∣∣∣∣∣∣2).

(38)Maximizing equation (38) with respect to α yields

Ly(θ, α) = Γ− 1

σ2n

(Eyy − eHQyE−1

QQeQy

)(39)

where

Γ , −M log(πσ2n) (40)

Eyy ,∞∑z=1

∣∣∣∣∣∣yz∣∣∣∣∣∣2 (41)

eQy ,∞∑z=1

QHz yz (42)

E−1QQ ,

∞∑z=1

QHz Qz. (43)

Note that, in equation (39), apart from the constant Γ, theremaining summation goes to infinity. However, due to thenon-contribution of higher order terms in the estimation of θand α the summation can be finite. Using the identity∫

To

v1(t)vH2 (t)dt =

∞∑z=1

v1zvH2z (44)

for vi(t) =∑∞z=1 v1zψz(t), i = 1, 2, equations (41)-(43) can

be written as

Eyy ,∫To

∣∣∣∣∣∣y(t)∣∣∣∣∣∣2dt (45)

eQy ,∫To

QH(t, θ)y(t)dt (46)

EQQ ,∫To

QH(t, θ)Q(t, θ)dt. (47)

Using the definition of Q(t, θ) in equation (32), we can writethe f th element of eQy as

[eQy]f = aH(θf )ETa(θf ) (48)

whereE =

∫To

y(t)xH(t)dt. (49)

Similarly, we can write the fgth element of EQQ as

[EQQ]fg = aH(θf )a(θg)aH(θf )RT

x a(θg). (50)

Since, eQy and EQQ are independent of the received signal,the sufficient statistic to calculate θ and α is given by E. Usingequation (48)-(50) we can write the ML estimate in matrix-vector form as

Ly(θML) = arg maxθ

∣∣∣aH(θML)Ea∗(θML)∣∣∣2

MaH(θML)RTxa(θML)

· (51)

Then, the GLRT for our hypothesis testing model in equa-tion (30) is given as

Ly = maxθ,αlog fy(y, θ, α;H1) − log f(y;H0)

H1

≷H0

δ (52)

where fy(y, θ, α;H1) and f(y;H0) are the probability den-sity functions of the received signal under hypothesis H1 andH0, respectively. Hence, the GLRT can be expressed as

Ly(θML) = arg maxθ


MaH(θML)RTxa(θML)

H1

≷H0

δ. (53)



The asymptotic statistic of L(θML) for both the hypothesisis given by [43]

L(θML) ∼

H1 : χ2

2(ρ),

H0 : χ22,

(54)

where• χ2

2(ρ) is the noncentral chi-squared distributions with twodegrees of freedom,

• χ22 is the central chi-squared distributions with two de-

grees of freedom,• and ρ is the noncentrality parameter, which is given by

ρ =|α|2

σ2n

|aH(θ)RTxa(θ)|2. (55)

For the general signal model, we set δ according to a desiredprobability of false alarm PFA, i.e.,

PFA = P (L(y) > δ|H0) (56)

δ = F−1χ22

(1− PFA) (57)

where F−1χ22

is the inverse central chi-squared distributionfunction with two degrees of freedom. The probability ofdetection is given by

PD = P (L(y) > δ|H1) (58)

PD = 1− Fχ22(ρ)

(F−1χ22

(1− PFA))

(59)

where Fχ22(ρ) is the noncentral chi-squared distribution func-

tion with two degrees of freedom and noncentrality parameterρ.

A. PD for Orthogonal Waveforms

For orthogonal waveforms RTx = IM , therefore, the GLRT

can be expressed as

LOrthog(θML) =


MaH(θML)a(θML)

H1

≷H0

δOrthog (60)

and the statistic of L(θML) for this case is

LOrthog(θML) ∼

H1 : χ2

2(ρOrthog),

H0 : χ22,

(61)

where

ρOrthog =M2|α|2

σ2n

· (62)

We set δOrthog according to a desired probability of false alarmPPF-Orthog, i.e.,

δOrthog = F−1χ22

(1− PPF-Orthog) (63)

and then the probability of detection for orthogonal waveformsis given by

PD-Orthog = 1− Fχ22(ρOrthog)

(F−1χ22

(1− PPF-Orthog)). (64)

B. PD for NSP Waveforms

For spectrum sharing waveforms RTx = RT

x , therefore, theGLRT can be expressed as

LNSP(θML) =


MaH(θML)RTxa(θML)

H1

≷H0

δNSP (65)

and the statistic of L(θML) for this case is

LNSP(θML) ∼

H1 : χ2

2(ρNSP),

H0 : χ22,

(66)

where

ρNSP =|α|2

σ2n

|aH(θ)RTxa(θ)|2. (67)

We set δNSP according to a desired probability of false alarmPPF-NSP, i.e.,

δNSP = F−1χ22

(1− PPF-NSP) (68)

and then the probability of detection for orthogonal waveformsis given by

PD-NSP = 1− Fχ22(ρNSP)

(F−1χ22

(1− PPF-NSP)). (69)

V. NUMERICAL RESULTS

In order to study the detection performance of spectrumsharing MIMO radars, we carry out Monte Carlo simulationusing the radar parameters mentioned in Table II.

TABLE IIMIMO RADAR SYSTEM PARAMETERS

Parameters Notations Values

Radar/Communication System RF band - 3550− 3650 MHz

Carrier frequency fc 3.55 GHz

Wavelength λ 8.5 cm

Inter-element antenna spacing 3λ/4 6.42 cm

Radial velocity vr 2000 m/s

Speed of light c 3 × 108 m/s

Target distance from the radar r0 500 Km

Target angle θ θ

Doppler angular frequency ωD 2ωcvr/c

Two way propagation delay τr 2r0/c

Path loss α α

A. Analysis of Case 1

For this case, at each run of Monte Carlo simulation we gen-erate K Rayleigh interference channels each with dimensionsNBS×M , calculate their null spaces and construct correspond-ing projection matrices using Algorithm (2), determine the bestchannel to perform projection of radar signal using Algorithm(1), transmit NSP signal, estimate parameters θ and α fromthe received signal, and calculate the probability of detectionfor orthogonal and NSP waveforms.



−10 −5 0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

= 10−1

Pd for NSP Waveforms to BS 1





Pd for Orthogonal Waveforms

6dB 13dB

(a) Probability of detection when dimN(Hi) = 2. Note that 6 dB to 13 dBof additional gain in SNR is required to detect target with 90% probability,depending upon the NSP waveform transmitted.

−8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

= 10−5

PD for NSP Waveforms to BS 1





PD for Orthogonal Waveforms

3dB 5dB

(b) Probability of detection when dimN(Hi) = 6. Note that 3 dB to 5 dBof additional gain in SNR is required to detect target with 90% probability,depending upon the NSP waveform transmitted.

Fig. 3. Case 1 – Performance of Algorithms (1) and (2): Using our spectrumsharing and projection algorithms, we can select interference channel for radarsignal projection to maximize detection probability and minimize gain in SNRrequired as a result of NSP of radar waveforms. For example, Algorithms (1)and (2) select BS#5 and BS#2 for dimN(Hi) = 2 and dimN(Hi) = 6cases, respectively, as they require minimum additional gain in SNR.

Performance of Algorithms (1) and (2): In Figure 3, wedemonstrate the use of Algorithms (1) and (2) in improvingtarget detection performance when multiple BSs are presentin detection space of radar and the radar has to reliably detecttarget while not interfering with communication system ofinterest. As an example, we consider a scenario with fiveBSs and the radar has to select a projection channel whichminimizes degradation in its waveform, thus, maximizing itsprobability of detection of the target.

In Figure 3(a), we consider the case when dimN(Hi) = 2.We show detection results for five different NSP signals, i.e,radar waveform projected onto five different BSs. Note that,in order to achieve a detection probability of 90%, we need 6dB to 13 dB more gain in SNR as compared to the orthogonalwaveform, depending upon which channel we select. Using

Algorithms (1) and (2) we can select interference channel thatresults in minimum degradation of radar waveform and resultsin enhanced target detection performance with the minimumadditional gain in SNR required. For example, Algorithms (1)and (2) would select BS#5 because in this case NSP waveformrequires least gain in SNR to achieve a detection probabilityof 90% as compared to other BSs.

In Figure 3(b), we consider the case when dimN(Hi) =6. Similar to Figure 3(a) we show detection results for fivedifferent NSP signals but now MIMO radar has a larger arrayof antennas as compared to the previous case. In this case, inorder to achieve a detection probability of 90%, we need 3dB to 5 dB more gain in SNR as compared to the orthogonalwaveform. As in the previous case, using Algorithms (1) and(2) we can select interference channel that results in minimumdegradation of radar waveform and results in enhanced targetdetection performance with the minimum additional gain inSNR required. For example, Algorithms (1) and (2) wouldselect BS#2 because in this case NSP waveform requires leastgain in SNR to achieve a detection probability of 90% ascompared to the other BSs.

The above two examples demonstrate the importance ofAlgorithms (1) and (2) in selecting interference channel forradar signal projection to maximize detection probability andminimize gain in SNR required as a result of NSP of radarwaveforms for spectrum sharing.Case 1(a): dimN(Hi) = 2: In Figure 4, we plot thevariations of probability of detection PD as a function ofsignal-to-noise ratio (SNR) for various values of probabilityof false alarm PFA. Each sub-plot represents the PD for afixed PFA. We choose to evaluate PD against PFA values of10−1, 10−3, 10−5 and 10−7 when the interference channel Hi

has dimensions 2× 4, i.e., the radar has M = 4 antennas andthe communication system has NBS = 2 antennas, thus, wehave a null space dimension of ‘dimN(Hi) = 2’. When wecompare the detection performance of two waveforms we notethat in order to get a desired PD for a fixed PFA we need moreSNR for NSP than orthogonal waveforms. For example, saywe desire PD = 0.9, then according to Figure 4 we need 6 dBmore gain in SNR for NSP waveform to get the same resultproduced by the orthogonal waveform.Case 1(b): dimN(Hi) = 6: In Figure 5, similar to Figure 4,we do an analysis of PD against the same values of PFA butfor interference channel Hi having dimensions 2×8, i.e., nowthe radar has M = 8 antennas and the communication systemhas NBS = 2 antennas, thus, we have a null space dimensionof ‘dimN(Hi) = 6’. Similar to Case 1, when we comparethe detection performance of two waveforms we note that inorder to get a desired PD for a fixed PFA we need more SNRfor NSP than the orthogonal waveforms. For example, say wedesire PD = 0.9, then according to Figure 5 we need 3.5 to4.5 dB more gain in SNR for the NSP waveform to get thesame result produced by the orthogonal waveform.Comparison of Case 1(a) and Case 1(b): As expected,when SNR increases detection performance increases for bothwaveforms. However, when we compare the two waveformsat a fixed value of SNR, the orthogonal waveforms performmuch better than the NSP waveform in detecting target. This



−10 −5 0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

=10−1

PD

for Orthogonal Waveforms

PD

for NSP Waveforms

−10 −5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

=10−3

PD


PD

for NSP Waveforms

−10 −5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

=10−5

PD


PD

for NSP Waveforms

−10 −5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

=10−7

PD


PD

for NSP Waveforms

6dB 6dB

6dB 6dB

Fig. 4. ’Case 1(a): dimN(Hi) = 2’: PD as a function of SNR for various values of probability of false alarm PFA, i.e., PFA = 10−1, 10−3, 10−5 and10−7. The interference channel Hi has dimensions 2× 4, i.e., the radar has M = 4 antennas and the communication system has NBS = 2 antennas, thus,we have a null space dimension of ‘dimN(Hi) = 2’. Note that we need 6 dB more gain in SNR for the NSP waveform to get the same result produced bythe orthogonal waveform.

−10 −5 0 5 10 150.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

P

D for P

FA = 10−1

PD


PD

for NSP Waveforms

−10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

P

D for P

FA = 10−3

PD


PD

for NSP Waveforms

−10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

P

D for P

FA = 10−5

PD


PD

for NSP Waveforms

−10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

P

D for P

FA = 10−7

PD


PD

for NSP Waveforms

3.5dB4dB

4.5dB 4.5dB

Fig. 5. ’Case 1(b): dimN(Hi) = 6’: PD as a function of SNR for various values of probability of false alarm PFA, i.e., PFA = 10−1, 10−3, 10−5 and10−7. The interference channel Hi has dimensions 2× 8, i.e., the radar has M = 8 antennas and the communication system has NBS = 2 antennas, thus,we have a null space dimension of ‘dimN(Hi) = 6’. Note that we need 3.5 to 4.5 dB more gain in SNR for the NSP waveform to get the same resultproduced by the orthogonal waveform.



−40 −35 −30 −25 −20 −15 −10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR

PD

PD

for PFA

= 10−5

PD

for NSP Waveforms when NBS = 2

PD


PD


PD


PD


Fig. 6. Case 2: PD as a function of SNR for PFA = 10−5. The MIMOradar mitigates interference to all the BSs in the network. As an example, weconsider M = 100, K = 5, and NBS = 2, 4, 6, 8.

is because our transmitted waveforms are no longer orthogonaland we lose the advantages promised by orthogonal waveformswhen used in MIMO radars as discussed in Section II-F, but,we ensure zero interference to the BS of interest, thus, sharingradar spectrum at an increased cost of target detection in termsof SNR.

In Case 1(a), in order to achieve a desired PD for a fixed PFAwe need more SNR for NSP as compared to Case 1(b). This isbecause we are using more radar antennas, while the antennasat the BS remain fixed in Case 1(b) which increases thedimension of the null space of the interference channel. Thisyields better detection performance even for NSP waveform.So, in order to mitigate the effect of NSP on radar performanceone way is to employ a larger array at the radar transmitter.

B. Analysis of Case 2

For this case, at each run of Monte Carlo simulation wegenerate K Rayleigh interference channels, combine theminto one interference channel with dimensions KNBS ×M ,calculate its null space and construct corresponding projectionmatrix using Algorithm (4), perform projection of radar signalusing Algorithm (3), transmit NSP signal, estimate parametersθ and α from the received signal, and calculate the probabilityof detection for orthogonal and NSP waveforms.

In Figure 6, we consider the case when the radar hasa very large antenna array as compared to the combinedantenna array of K BSs. In such a scenario, we have enoughdegrees of freedom at the radar for reliable target detectionand simultaneously nulling out interference to all the BSspresent in the network. As an example, in Figure 6, weconsider M = 100, K = 5, and NBS = 2, 4, 6, 8. Wedo an analysis of PD against PFA = 10−5 for the combinedinterference channel H having dimensions KNBS×M . Whenwe compare the detection performance of original waveformand NSP waveform onto the combined channel we note thatin order to get a desired PD for a fixed PFA we need moreSNR for NSP than the orthogonal waveforms. For example,say we desire PD = 0.95, then according to Figure 6 we need1, 2, 3.5, and 4.5 dB more gain in SNR for the NSP waveform

when NBS is 2, 4, 6, and 8, respectively, to get the same resultproduced by the orthogonal waveforms.

VI. CONCLUSION

In future, radar RF spectrum will be shared with wirelesscommunication systems to meet the growing bandwidth de-mands and mitigate the effects of spectrum congestion forcommercial wireless services. In this paper, we analyzed asimilar spectrum sharing scenario between radars and cellularsystems. We evaluated the detection performance of spectrumsharing MIMO radars. We formulated the statistical detectionproblem for target detection and used generalized likelihoodratio test to decide about the presence of target when usingorthogonal waveforms and null-space projected (NSP) wave-forms. We proposed two spectrum sharing cases in whichMIMO radar is sharing spectrum with cellular system.

For ‘Case 1’, when M KNBS but M > NBS, theradar had not sufficient degrees of freedom (DoF) availableto simultaneously detect target and mitigate interference to K

BSs in the network. For this case, we mitigated interferenceto one of the BS in the network and proposed algorithmsfor interference channel selection and projection of radarwaveform onto the selected interference channel in order tomitigate interference to the selected BS. We showed thatby using our spectrum sharing and projection algorithms theradar can maximize target detection probability and minimizeadditional gain in SNR required to detect the target. Our resultsshowed that, about 6 dB of gain in SNR is required whendimN(Hi) = 2 and 3.5 to 4.5 dB of gain in SNR is requiredwhen dimN(Hi) = 6 when NSP waveforms are used insteadof orthogonal waveforms for spectrum sharing. Our analysisshowed that this degradation in performance can be mitigatedby using a larger array at the MIMO radar transmitter.

For ‘Case 2’, when M KNBS, the radar had sufficientDoF available to simultaneously detect target and mitigateinterference to all K BSs in the network. We proposedspectrum sharing and projection algorithms for this case. Ourresults showed that when the interference channel has a largenull space the amount of SNR required to achieve a desireddetection probability is low and vice versa.

For both the cases we showed that when using the NSPwaveforms, the detection performance degraded as comparedto the orthogonal waveforms and we needed more SNR todetect reliably. However, this resulted in co-channel spectrumsharing with communication systems which solved the spec-trum congestion problem.

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