1

Information Theoretic Bounds and WaveformOptimization for Joint Radar and Communication

with Semi-Passive TargetsGanesan Thiagarajan and Sanjeev Gurugopinath

Abstract—In this paper, we derive the information theoreticperformance bounds on communication data rates and errorsin parameter estimation, for a joint radar and communication(JRC) system. We assume that targets are semi-passive, i.e. theyuse active components for signal reception, and passive com-ponents to communicate their own information. Specifically, welet the targets to have control over their passive reflectors inorder to transmit their own information back to the radar viareflection-based beamforming or backscattering. We derive theCramer-Rao lower bounds (CRBs) for the mean squared errorin the estimation of target parameters. The concept of a targetambiguity function (TAF) arises naturally in the derivation CRBs.Using these TAFs as cost function, we propose a waveformoptimization technique based on calculus of variations. Further,we derive lower bounds on the data rates for communicationon forward and reverse channels, in radar-only and joint radarand communications scenarios. Through numerical examples,we demonstrate the utility of this framework for transmitwaveform design, codebook construction, and establishing thecorresponding data rate bounds.

Index Terms—Cramer-Rao lower bounds, joint radar and datacommunications, passive targets, radar information theory, targetambiguity functions, transmit waveform design.

I. INTRODUCTION

A. Background and Motivation

The tremendous increase in demand for wireless applica-tions and a sharp rise in the number of connected devices hascaused severe spectrum shortage and congestion of existingbands. Towards this end, research attention on coexistence ofother communication systems in radar bands has significantlygrown in the past decade or so. For instance, DARPA funded aresearch project called shared spectrum access for radar andcommunications (SSPARC), for eliciting technologies whereboth radar and communication can co-exist [1]. Joint radarand communications (JRC) finds its utility in several civilianapplications such as millimeter wave communications, WiFi-based localization, unmanned aerial vehicles communications,RFID, etc., and also in military applications. Excellent re-cent surveys on applications and future research directionson JRC are given in [2]–[4]. As discussed in [4], thereare two main research directions in JRC, namely (i) radar-communication coexistence (RCC), and (ii) dual-functionalradar-communication (DFRC) systems [2], [5]. The goal in

Ganesan Thiagarajan is with MMRFIC Technology Private Limited, Ben-galuru 560043, India.

Sanjeev Gurugopinath is with the Department of Electronics and Commu-nication Engineering, PES University, Bengaluru 560085, India.

Emails: gana@mmrfic.com, sanjeevg@pes.edu

RCC is to design efficient interference management techniquesfor coexistence of a communication system in the radar band[6]–[12]. On the other hand, the goal in a DFRC system isthe joint design of sensing and signaling operations througha single hardware for both radar and communications inapplications such as indoor radars [13]–[15] and radars forvehicular networks [16], [17].

Recent radar applications such as automotive radars [18],[19] and drone tracking [20], [21] have introduced new set ofchallenges and design considerations for spectrum sharing inJRC. Tracking of unmanned aerial vehicles (UAV) and dronesfinds applications in security and surveillance [22]. Latestdevelopments in joint data transmission and target parameterestimation, by resource sharing, have added new dimensionsto this problem [23]. In many such applications, it is necessaryto identify a UAV as a friend or a foe, to ensure securecommunication [24]. Since UAVs such as drones operate onbattery power, passive communication over active is preferredfrom a UAV to a JRC transceiver [25], [26], for the followingtwo reasons. First, this enables a UAV to hide its presenceand identity from adversaries. Second, employing technologiessuch as opportunistic ambient backscattering communication(ABC) [27] and intelligent reflecting surfaces (IRS) [28],[29] that require only a small fraction of power of activetransmission, enables the battery power of a UAV to be utilizedefficiently.

To the best of our knowledge, a study on the fundamentallimits on the performance of JRC with passive targets interms of joint parameter estimation and communication datarates has not been addressed in literature so far. In thiscase, crucial design trade-offs between the performance ofparameter estimation and data transmission, must be derivedand studied based on fundamental performance metrics suchas mutual information achievable over the JRC communicationlinks and limits on the target parameter estimates, in terms ofthe Cramer-Rao lower bounds (CRB).

B. Related Work

Studies on information theory for radar started with theseminal works by Woodward [30]–[33], and Davies [34].Woodward and Davies showed that a receiver which max-imizes the a posteriori probability is optimal, and it re-duces to a correlation receiver in the case of additive whiteGaussian noise. In particular, as opposed to the signal-to-noise (SNR) maximization, the goal is to maximize the ‘a

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posteriori probability’, which improves the detection proba-bility. The Wigner-Ville transformation was proposed as theunnormalized ambiguity function, which was used to analysethe simultaneous range and velocity estimation resolution [35].Klauder used this ambiguity function to design better radarwaveforms which results in improved simultaneous range andvelocity resolution, although their practical use case is limited[36], [37]. The goal in these early works on informationtheory for radars was information theoretic formulation inwaveform design, to improve the performances in terms oftarget detection and parameter estimation [36], [38], [39]. Therelationship between the mutual information between the targetparameters of interest and the transmitted signal, and accuracyof parameter estimation was exploited in [38], to designoptimal radar waveforms. Transmit waveform optimizationfor frequency diverse array (FDA) radar was considered in[40], where target localization was done by analyzing theCRBs, and the knowledge of location of the targets wasused to design the transmit waveform for FDA and to createrange-angle dependent beams. Transmit array sub-aperturingwas employed in FDA radar [9], where each sub-array wasassigned with different carrier frequencies and their weightswere chosen adaptively via cognitive beamforming.

Even in the context of radars, the mean squared error(MSE) for parameter estimation is well-studied in literature.In [41], CRB for the target range, velocity, and angle-of-arrival (AoA) with a narrow band assumption on the transmitwaveform was derived. Further, it was shown that the CRBfor AoA is independent of the delay and velocity parameters.Moreover, CRB for AoA was shown to be a function of sensorlocations only, through the moment-of-inertia parameters ofthe sensory array. It was noted in [42] that the CRB andambiguity functions were related to each other and impactthe performance of parameter estimation, as follows – “Theambiguity function establishes global conditions under whichthe local bounds are accurate predictions of the expected errorperformance and identifies the regions of the parameter spacewhere large errors may occur.” [42]. Exploiting this key result,we extend the notion of ambiguity functions to widebandsignals and general array geometries for our problem at hand.

Bounds on data rate between a JRC receiver and a com-munication transceiver was studied in [43], where the radaraids to relay the data from the communication transmitterto the receiver which is not co-located with the radar.1 Thissystem follows a DFRC model, in which the radar performsjoint data decoding and parameter estimation of targets us-ing the combined received signal and then modulates theinformation onto the radar signal. Thus, the communicationand radar receivers are able to decode the message andestimate the target parameters, respectively. Further, the datarate and parameter estimation rate were balanced similar tothe case of a two user multiple access channel at the radarreceiver, and the corresponding rate region was derived. In

1Note that this setup is different from our scenario. In our setup, theradar transceiver is capable of sending and receiving data from the targets.The model in [43] assumes a different data transmission node, which isindependent of radar and targets assist in relaying the information to thereceiver which is collocated with radar receiver.

[4], a massive MIMO DFRC architecture was proposed, thatemploys hybrid beamforming to enable joint radar trackingand data communication. A radar transmission signal wasdesigned using orthogonal frequency division multiplexing(OFDM) in [44]. Here, uncertainties in the received data dueto data symbols at the radar were mitigated, by dividingthe target echo signal by apriori known or decoded datasymbols in the frequency domain. In this setup, the transmitterand receiver are not co-located as in the case of bistaticradars. In [11], a joint waveform design using preambles ofthe communication signal as radar detection waveforms wereemployed to achieve a trade-off in the performance of datatransmission and parameter estimation. Even though the usageof communication preambles as radar waveforms allows oneto reuse the spectral and temporal resources, it results in non-optimal radar performance due to poor ambiguity function inthe transmit waveform. For the preamble duration, multiplesmaller periodic waveforms can be designed to achieve a betterperformance, especially when the targets move rapidly. Anexperimental validation of data communication using BPSKmodulation on radar with FMCW waveforms via backscatter-ing by semi-passive targets, is reported in [45], where singleantenna was used at both ends. This results in a limiteddata rate for a given distance between the targets, due to thepotential overlap of the spectrum between the transmissionfrom two adjacent targets.

C. Contributions

In this work, we derive lower bounds on the mutual in-formation for the forward and reverse channels between aradar transceiver and targets, and derive the CRB on theminimum MSE of target parameters, namely range, velocityand AoA. We consider generic linear models for forward andreverse channels and target response function, for analyticaltractability. The proposed model accommodates both narrow-band and wideband transmit signals, any receiver antennaarray geometry, and applicable for both monostatic and bistaticradars. For brevity, we consider monostatic radars in this paper.Motivated by the relationship between CRB and the ambiguityfunction studied in [42], we propose novel target ambiguityfunctions (TAFs) for each target parameter, which can beoptimized to achieve the corresponding CRB. We illustrate theutility of the proposed framework, which includes (a) optimalwaveform design for JRC that minimizes TAFs, (b) JRC usingoverlaid data pulses on radar waveforms. In summary,• We derive the CRB on the minimum MSE of the esti-

mates of relevant target parameters, for radar-only andJRC scenarios. Three novel TAFs for range, velocity andazimuth – which embed parameters such as elevationangle, target impulse responses and array geometry – areintroduced, which are important for the MSE bounds.

• Using calculus of variations, we propose an optimalwaveform design procedure that minimizes a chosen TAF.

• We derive lower bounds on data rates over the forwardand reverse channels between a JRC transceiver and apassive target. Further, we show that communication onthe forward channel does not affect the CRB. We also

3

Radar / CommunicationTransmitter

Radar / CommunicationReceiver

ForwardChannel

ReverseChannel

Target

+ +j(t)

w(t)

x(t)

n(t)

Transceiver Effective Channel

Figure 1: Block diagram of a joint radar and communication system with atransceiver and a single passive target.

elaborate on the inter-target interference suppression andits effect on parameter estimation on reverse channel.

• Through numerical examples, we demonstrate the prac-tical utility of the proposed methods in a JRC systemin terms of transmit waveform optimization, beamformerdesign, Gaussian codebook design and data rate bounds.

D. Organization

The remainder of the paper is organized as follows. InSection II, we discuss the data model and formulate the JRCproblem with semi-passive targets. In Section III, we computethe CRB for estimates of radar parameters for both radar-only,and JRC scenarios. In Section IV, we present the optimaltransmit waveform design using calculus of variations. InSection V, we derive lower bounds on communication data ratefor forward and reverse channels. Section VI demonstrates thepractical application of the proposed methods via numericalexamples in JRC, receiver beamformer design, codebook de-sign and transmit waveform optimization. Concluding remarksare provided in Section VII.

II. DATA MODEL AND PROBLEM FORMULATION

A. Joint Radar and Communication Transmit Signal

Consider a radar system as depicted in Fig. 1, consistingof a transceiver equipped with a single antenna joint radarand communication (JRC) transmitter, and a JRC receiverwith L antennas. The radar transmits a signal xRF

i (t) atcenter frequency fc, for i = 1, . . . , N with N pulses of Tsduration each. The average transmit power is Pavg, and thetotal transmit power PT = NTsPavg. The baseband componentof xRF

i (t), denoted by xi(t), is assumed to be band-limited toB Hz. In other words, magnitude spectrum of xi(t), denotedby |Xi(f)|2, is positive for |ω| ≤ B, and zero otherwise.Throughout this paper, we consider our analysis on xi(t),even though all the channel effects such as Doppler shift,target impulse response, etc. are applied on xRF

i (t). Further, weassume that the transmitted signal comprises of N Gaussianpulses with BTs > 1 [35].2 That is,

xi(t) =

√π

βexp

(−π

2(t− iTs)2

β2

),−Ts

2≤ t ≤ Ts

2, (1)

2We consider the Gaussian pulse for the ease of analysis, as opposed toother popular choices such as a rectangular pulse with a given duty cycle.Moreover, Gaussian modulated pulses are often used in practice [46], [47].

where the choice of β determines the 3 dB pulse bandwidth.The magnitude of the Gaussian pulse in frequency-domainrepresentation can be written as

|Xi(f)| = exp(−β2f2

). (2)

Following [38], we assume that the characteristics of kth targetis captured in its random target channel response, gk(t), k =1, . . . ,K, with the following assumptions.

(i) Finite energy: gk(t) satisfies∫∞−∞ E|gk(t)|2dt < 1, where

EX denotes the expectation of a random variable X .Further, the response gk(t) does not include parameterssuch as the reflection coefficient and the effective radarcross section (RCS) of the kth target.

(ii) Causality: gk(t) is causal, i.e. gk(t) = 0, t < 0.(iii) Fourier transform of gk(t), denoted by Gk(f), exists.(iv) Independence: gk(t) is independent across k = 1, . . . ,K,

and is independent of xi(t), for i = 1, . . . , N .In the considered setup, data communication along with

radar target detection is achieved through a random code cof length N , whose ith component is multiplied with xi(t)before transmission. Vector c is determined by a sequenceof Gaussian random variables {ai, i = 1, . . . N}.3 Therefore,a total of NBTs resource dimensions are available to beshared between radar and communication. We assume thatc is an N -dimensional Gaussian random vector with zeromean vector and a covariance matrix σ2

xIN , where IN denotesthe identity matrix of size N , and σ2

x = Pavg. We assumethat the radar transmitter and targets follow time divisionmultiplexing for communication, where the transmitter sendsdata on odd frames and targets respond on even frames. Next,we present the models for the reflected signal from K semi-passive targets.

B. Reflected Signal from Semi-Passive Targets

Figure 2 depicts the linear models of the forward and reversechannels between the radar transmitter and the kth target, andthe corresponding target response channel. In this work, welet the targets to be semi-passive, i.e. they can receive andmodulate the data sent by radar transmitter – using techniquessuch as ABC, or IRS-aided beamforming – by using the activeelements, but cannot transmit data using active components.Further, we assume that a strong direct line-of-sight path existsbetween the radar and targets. Therefore, the signal receivedvia the multi-path reflections are relatively weak and can beignored. However, our analyis can be extended to a generalsetup in a straightforward manner. Note that the random codec can be applied either at the transmitter or by the passivetargets, or both. First, let us consider a single antenna at theradar receiver, i.e. L = 1. The reflected signal from kth targetfor the ith pulse can be written as

wk,i(t) = aiα(f)k γk

∫gk(t′)xi

(t− t′ − τ (f)

k

)dt′, (3)

3In radar-only mode, the pulses can be directly transmitted with maximumamplitude, without this modulation. In such cases, these unknown pulseamplitudes of are considered as nuisance parameters. Additionally, it is alsopossible that some fraction of the N pulses can be retained for radar-only,while the remaining pulses can be modulated for JRC.

4

where τ (f)k is the propagation delay in the forward channel

– from the transmitter to the target, α(f)k , ε r−ek denotes

the pathloss in the forward channel with e as the pathlossexponent, and γk denotes the reflection coefficient for kth

target including the RCS of the target. For convenience, thiscontinuous time system can be approximated by a discretemodel as∫

gk(t′)xi

(t− t′ − τ (f)

k

)dt′ ≈

D∑p=1

ζk,pδ (t− tk,p) , (4)

with D denoting the number of delay taps, and ζk,p being theamplitude in the pth tap due to kth target. The delays betweenthe multiple taps are negligible compared to the round tripdelay between the transmitter and targets, and hence can beapproximated as a single point reflection integrating all theenergy from the taps. However, the extended targets which arein close proximity of the radar can be split into multiple targetsmoving together, without loss of generality. Optimal radarwaveforms can be designed either in a multi-path environmentor for detection of specific targets with known target responsemodels, using this tap-delay model [38]. The signal receivedat the radar due to non-moving targets can be written as

zi(t) = aiα(r)k

(K∑k=1

α(f)k γkwk,i(t− τ

(r)k )

+ji(t− τ (r)k ))

+ ni(t),−Ts

2≤ t ≤ Ts

2, (5)

where τ (r)k and α(r)

k denote the propagation delay and thepathloss factor in the reverse channel, ni(t) denotes the addi-tive white Gaussian noise (AWGN) with zero mean and unitvariance, and ji(t) denotes the jamming signal. For the easeof analysis, we assume that the transmitter and receiver arecollocated, in which case τfk = τ rk = τk and α(r)

k = α(f)k = αk.

Therefore, (5) simplifies to

zi(t) = ai

(K∑k=1

α2kγkwk,i(t−τk)+ji(t−τk)

)+ni(t). (6)

On the other hand, if the kth target is moving at a constantspeed vk radially from the radar transmitter, we can write

zi(t) = ai

(K∑k=1

α2kγkwk,i([t− τk]Tk)

+ji(t− τk)) + ni(t),−Ts

2≤ t ≤ Ts

2, (7)

where

Tk =

√c+ 2vkc− 2vk

≈ c+ vkc− vk

, (8)

denotes the time-axis compression factor due the Dopplershift in frequency from a moving target.4 Now, separating thereflected signal from kth target, we get (9), which is givenin the top of next page. Note that the reflection from each

4Note that this is a generic model which includes both narrow-bandand wide-band transmitted signals. Moreover, for L > 1, this model isindependent of the array geometry. It is easy to show that for vk � c,the difference between the observed frequency fo and fc simplifies tofc − fo = (1− 1

Tk)fc ≈ 2 vkfc

c.

target may undergo different time compression, depending onthe velocity of the target. Moreover, as discussed earlier, itis possible that some of the transmitted data pulses fromthese passive targets can be used for sending data information,while the remaining can be used to detect the presence of atarget and estimate its parameters. The trade-off between theperformance of radar target parameter estimation and the datatransmission is viable by proportioning the available resourcessuch as energy, bandwidth and codes. In this sense, it is notnecessary that all the N transmitted pulses need to be identical.However, set of all possible realizations of xi(t) comes fromthe finite Cartesian product of waveforms selected from twoensembles; one corresponding to radar parameter estimationand another for data transmission.

Next, we consider the general case where the receiver hasL > 1 antennas, and the received signal vector is written as

zi(t) = aiXi(t)Γi(t) + ni(t), (10)

where Xi(t) is a L×K matrix, whose (l, k)th entry is

X(l,k)i (t) = x([t− iTs − 2τk − ϕl]Tk), (11)

for k = 1, . . . ,K, l = 1, . . . , L, where ϕl ,pTk plc , with pl

denoting the relative Cartesian coordinates — with respect toa reference element in the receive antenna array as origin —of the lth antenna element and pk are the direction cosines ofthe kth target with respect to the antenna. Using (4), the kth

entry of the K-length vector Γi(t) can be written as

Γ(k)i (t) = γkα

2k

D∑p=1

ζk,pδ (t− tk,p) . (12)

The L-length vector ni(t) is a spatio-temporally white Gaus-sian random vector with mean being the zero vector andcovariance matrix σ2

nIL. The mean vector of zi(t) for alli = 1, . . . , N is also assumed to be the zero vector. In the nextsection, we derive lower bounds on parameter estimation forrange, velocity and angle in the forward and reverse channels.

III. BOUNDS ON PARAMETER ESTIMATION

A. Radar-Only TransmissionThe Cramer-Rao lower bounds (CRB) on the variance of the

radar target parameters in our setup can be obtained by findingthe inverse of the Fisher information matrix (FIM). Following(10), recall that the L-length vector zi follows a Gaussiandistribution with mean vector aiXiΓi, and covariance matrixRn , E

(nin

Ti

), independent across i = 1, . . . , N . Let

Θ , [r v φ]T denote the vector of desired parameters,and rk, vk and φk denote the kth entries of vectors r, v andφ, respectively, which correspond to the range, Doppler andazimuth angle of the kth target.5

Towards this end, the second order partial derivatives of thelogarithm of joint PDFs of zi, i = 1, . . . , N computed withrespect to the parameters rk, vk and φk, k = 1, . . . ,K arederived in Appendix A, and the final expressions are given inequations (13)-(15) at the top of this page. Here, ◦ denotes

5In a radar-only scenario, nuisance parameters such as θ, c, i.e., {ai}, tapcoefficients and amplitude of gk(t) exist. Tighter CRBs can be computedwhen these are conditioned out in the computation of FIM, which will not bediscussed in this paper.

5

zi(t) = aiα2kγkwk,i([t− τk]Tk) + ai

∑j 6=k

α2j γj wj,i([t− τj]Ti) + ji(t− τj)

︸ ︷︷ ︸

Interference for kth target

+ni(t). (9)

× ×xi(t)

ai α(f)k

×

γk

+

jk(t)

g(t′)

wk,i(t)

×Delayτ(r)k

Delayτ(f)k

α(r)k

zi(t)+

ni(t)

Forward Channel Model Target Response Model Reverse Channel Model

Figure 2: Linear target response model.

E[∂2 log fz(zi)

∂r2k

]= −

16ρTNπ4T4

k E[γ2k] E[‖ζk‖2]ε4r−4ek

β4 c2E(‖xi,k ◦ tk‖2

). (13)

E[∂2 log fz(zi)

∂v2k

]= −

16ρTNπ4c2T2

k E[γ2k] E[‖ζk‖2]ε4r−4ek

β4 (c− vk)2E(‖xi,k ◦ tk ◦ tk‖2

). (14)

E[∂2 log fz(zi)

∂φ2k

]= −

4ρTNπ2T4

k E[γ2k] E[‖ζk‖2]ε4r−4ek

β4E(‖xi,k ◦ tk ◦Φk‖2

). (15)

the Hadamard (or element-wise) product of two matrices ofsame size, ρT =

(E[a2i ]σ2n

)is the transmit SNR, c is the speed

of light, xi,k is the kth column of Xi, and Φk is defined as

Φk ,{sin θk (yl cosφk − xl sinφk) + zl cos θk}

c, (16)

for l = 1, . . . , L is a vector of dimension L×1. The Cartesiancoordinates of the lth receive antenna is denoted by (xl, yl, zl)and tk is the vector of relative time instants at which thewavefront from kth target hits the receiver antenna elements.

It can be observed that the lower bound on the MSE acrossall the parameters depend on the key factors such as N , ρT,RCS value E[γ2k], extension nature of the target E[‖ζk‖2], andthe target ambiguity functions (TAF). Therefore, by designinga radar transit waveform which minimizes the TAFs, lowerbounds on the minimum MSE can be written as

E[(rk − rk)2] ≥ −1

E[∂2 log fz(zi)

∂r2k

] , (17)

E[(vk − vk)2] ≥ −1

E[∂2 log fz(zi)

∂v2k

] , (18)

E[(φk − φk)2] ≥ −1

E[∂2 log fz(zi)

∂φ2k

] . (19)

Note that the cross terms in the FIM are ignored to derivethe lower bounds in (13)-(15). In general, the achievableCRBs (local bounds) are influenced by the ambiguity functions(global conditions), as observed in [42]. That is, if one designswaveforms that achieve the least possible ambiguity functions,

the corresponding CRBs listed in (13)-(15) becomes achiev-able. Therefore, the approach considered here – optimizing theambiguity functions – is flexible, in the sense that the perfor-mance of the radar can be designed by individual optimizationof the TAF for range, velocity and angle, for selected set oftargets within the given range, velocity or angle-of-arrivals.This is advantageous in contrast to the conventional ambiguityfunctions defined to optimize all parameters using one transmitwaveform function. As as example, the process the optimizingthe range TAF is detailed later in Sec. IV-A. Other TAFs can beoptimized along similar lines. See Appendix A for a detaileddiscussion on TAFs.

1) MSE on Range Estimation: Note that the CRB for rangeestimation given in (13) decreases with ρT, N , mean RCSvalue E[γ2k], mean energy in the target impulse response ζkas well as the effective BW of the signal

∫ Ts2

t=−Ts2

(dx(t)dt

)2dt,

as observed in [48]. Another interesting observation is thatthe bound increases significantly with large velocity values,i.e. when Tk < 1, but only increases as 1/T4

k for lowvelocities. The parameter β is inversely proportional to B, andhence the bound decreases with fourth power on B, which isimportant to note.

2) MSE on Velocity Estimation: From (14), it can beobserved that the velocity estimation also improves withthe previously mentioned parameters. The key difference is

observed in the new term∫ Ts

2

t=−Ts2

(d2x(t)d t2

)2dt in place of the

effective bandwidth term in (13). With large bandwidth, thisquantity will increase exponentially and improves the velocityestimation, provided the wide-band array configuration is used

6

as in the case of space-time adaptive processing (STAP) filters[49]. That is, using narrow-band pulses, or shortening the arrayaperture to meet the narrow-band conditions greatly impact thevelocity and angle estimation.

3) MSE on Angle Estimation: Apart from the effect fromthe expected parameters, the key difference in this case isthe influence of Φk due to array geometry, as seen in (15).Further, it is important to note that it may appear that the arraygeometry may not influence the bounds in (13) and (14). Thisis not true, as the TAFs corresponding to range and velocity dodepend on the array geometry, as discussed in Appendix A.This is an interesting result emerging from our analysis, incontrast to the existing results, e.g. [41]. In most of the analysisin the existing literature, the array geometry does not influencethe CRBs for range and velocity estimation, and the CRB onthe angle estimation error does not depend on the range andvelocity estimation errors.

B. Joint Radar and Communication

It can be observed from the equations (13)-(15) that thebounds on MSE are linear in N . The corresponding CRBs onMSE on the forward and reverse channels of an JRC systemcan be computed as follows.

1) JRC on the Forward Channel: Consider the scenariowhere radar pulses are modulated by the N -dimensional Gaus-sian code c designed in Appendix D. At the radar receiver, theamplitudes {ai, i = 1, . . . , N} are known and hence can beused in the parameter estimation. Therefore, the previouslycomputed parameter estimation bounds given in (13)-(15) areapplicable even in this case. This result is in contrast with thework in [43], where the JRC transmitter and receiver are notco-located, i.e. bi-static radar. However, our results can alsobe extended to the scenario in [43] as follows. The parameterestimation at the JRC receiver, and information transmitted tothe communication receiver using data pulses overlaid withthe radar pulses can be treated as part of a two-user multipleaccess channel, and the CRBs can be computed [50].

2) JRC on the Reverse Channel: In this case, radar trans-mits constant modulus pulses during the frame in whichthe targets are expected to send data, and the targets mod-ulate them using their backscattering control elements. Let{bki, i = 1, . . . , N } denote the reflection amplitudes dueto data modulation from the kth target to the radar receiver.To obtain the parameter estimation bounds for this case, weneed to compute the partial derivatives with respect to symbolsbki additionally, and use them along with the derivatives forthe radar parameters to invert the larger FIM. The secondderivative with respect to the symbols bki can be shown as

E[∂2 log fz(zi)

∂b2ki

]= −ρTE[b2ki]E[γ2k] E[‖ζk‖2]

ε4r−4ek E[‖xi,k‖2], (20)

and the cross-term second order derivative can be written as

E[∂2 log fz(zi)

∂bki∂rk

]= −ρT 4π2E[b2ki]T2

kE[γ2k] E[‖ζk‖2]

ε4r−4ek E[(xi,k ◦ tik)Txi,k], (21)

for one parameter. The derivatives in a general case can bederived similarly. It can be noticed that the cross-terms arenot zero and hence impact the inverse of FIM and the MSEbounds. That is, the decision on each and every bit transmittedby the individual targets impact the performance all radarparameters for all targets. Even for a single target, the FIMfor all parameters and data does not simplify and all datadecision errors impact the performance of the radar. Hence,some level of orthogonality in resource allocation is a must forgetting good performance in terms of bit error rate (BER) andparameter estimation error, where previously computed datarate bounds and parameter estimation bounds remain valid.

IV. JRC TRANSMITTER AND RECEIVER DESIGN

Following the analysis in Sec. III, one needs to minimize theTAFs to achieve the CRB. Minimization of TAFs is equivalentto maximization of the average norm E[‖xk ◦ tk‖2], andminimization of the following three inner products:(a) E[(xi,m ◦ tm)TR−1n (xi,n ◦ tn)],(b) E[(xi,m ◦ tm ◦ tm)TR−1n (xi,n ◦ tn ◦ tn)], and(c) E[(xi,m ◦ tm ◦Φm)TR−1n (xi,n ◦ tn) ◦Φm],simultaneously, for m,n = 1, . . . ,K and n 6= m, wherethe expectation is taken over all possible values of ranges,velocities, and azimuth angles.6

For L� 1, the above problem can be considered as a func-tional optimization, carried out through methods in calculusof variations [51]. Minimizing the inner product between timederivatives of the pulses corresponding to two different targetscan be formulated as an optimization problem, as discussed inthe next subsection. For the ease of presentation, we first usea continuous-time waveform, and relax that requirement later.

A. Optimization of Transmit Waveform

Consider a function f(t, y(t), y(t)) defined over an intervala ≤ t ≤ b, where y(t) , dy(t)

dt . Succinctly representedas f(t, y, y) for convenience, let the second order partialderivatives with respect to t, y, and, y be continuous. Then,the functional defined as

F (y, y) =

∫ b

a

f(t, y, y)dt (22)

attains its extrema when∂F

∂y=

d

dt

(∂F

∂y

). (23)

In the context of TAFs, we define the following cost functionF , as the normalized inner product

F (y, y) ,1

E

∫ Ts2

−Ts2y(t)y(t+ δ)dt, δ > 0, (24)

where E ,∫ Ts

2

−Ts2y(t)y(t)dt. This is equivalent to evaluating

the normalized auto-correlation of the first derivative of y(t).

6Recall that the conditions (a)-(c) follow our analysis with Gaussian pulses.However, they are also valid for all continuously differentiable functions withfinite support in time and frequency, e.g. nth order spline, half-sine, and raisedcosine pulses. For more details, refer to Section VI.

7

Sl. Envelope Type Energy-Normalized y(t) Scaling Factors

1 Gaussian A exp

(−[π tβ

]2)A =

(2β2

π

) 14

√erf(πTSβ√2

), −Ts

2≤ t ≤ Ts

2, e.g. β = 3

4TS

2 Cubic spline A

23− |t′|2 +

|t′|32

, 0 ≤ |t′| < 1(2−|t′|)3

6, 1 ≤ |t′| < 2

0 , |t′| ≥ 2

t′ = 4 tTs

, A =√

4Ts

3 Half-sine wave A sin(πtTs

)A =

√2Ts

Table I. Some well-known continuously differentiable transmit pulse envelopes.

Consider an iterative algorithm that finds an extremum ofthis function F . The smallest δ that minimizes F such thatF ∗(δ) ≤ ε, for some ε > 0, is considered as the smallestpossible ambiguity value for that ε. That is, the derivative ofy(t) should be similar to a Kronecker delta impulse function,such that F (y, y) < ε for an arbitrary δ > 0. Without lossof generality, the scale factor 1/E can be ignored during theoptimization stage.

For illustration, consider a simple case where y(t) is one ofthe continuously differentiable baseband waveforms given inTable I, to start the optimization procedure. Here, y(t) is thetransmit pulse, which modulates a carrier signal such as fre-quency modulated continuous wave (FMCW) or a single-tonecarrier, along with the amplitude scaling as per the code used.Note that these envelope waveforms have a time support of Tsseconds and frequency support of B Hz. Let y(t) evaluatedat the nth step of the algorithm be denoted as yn(t), and thealgorithm stops when ‖yn+1(t) − yn(t)‖2 < ε, where ‖ · ‖represents the Euclidean norm in the L2 space (or the Paley-Wiener space) for finite energy functions. Then, the followingtheorem specifies the condition under which the cost functionin (24) converges. To summarize, if the perturbation on yn(t)is assumed to be a time-scaled, time-shifted and amplitude-scaled version of y(t), such that the other parameters meetthe above mentioned conditions, then the algorithm convergesto an optimal waveform which minimizes the cost in (24).

Theorem 1. Let ‖yn+1(t)− yn(t)‖ ≤ λy(αt+ β). Then, thecondition for the convergence of an iterative algorithm is(

ytyt

+ytyt

)2

+ 4

(ytyt

)1−

(1− α2

yt+δ

)(

1− α2

yt

) ≥ 0, (25)

and choices of λ > 0, α > 0, β ∈ R, β 6= 0, and δ > 0determine the speed of convergence.

Proof. See Appendix B.

Discussion: Recall that the actual transmitted pulse will be aproduct of the envelope signal and a carrier. Towards wave-form optimization, the product signal will only help improvethe correlation properties of the waveform further. This will bedetailed in Sec. VI, where the correlation properties betweenan FMCW-modulated optimized waveform and that of onlythe optimized waveform are compared. Further, recall thatthe TAFs are functions of the waveform samples, where theinner product between the samples at two different instants –

dictated by the array geometry – is used as the cost function.As discussed in Appendix C, the sampling time-instants varywith the angle of arrival and the array geometry. The derivativeof the transmit waveform pulse will be sampled according tothe angle of arrival for each target and the array geometry.7

Having a large number of antennas yields a sufficient ofnumber of samples to represent the entire waveform. However,for short arrays, one needs to optimize the correlation forthose specific time instants averaged over all possible anglesof arrival. The sampling instants of the received waveformis illustrated for two array geometries in Appendix C. Sucha discrete functional variation optimization is similar to theone presented above. Section VI details this procedure vianumerical examples. The discrete CoV can be performed fora given array geometry and the transmit waveform can beoptimized by averaging the cost over all possible angles ofarrival. Note that a waveform can also be optimized even ifone restricts the array field of view (FoV) to a limited set ofangles or sectors.

B. Inter-Target Interference Suppression

From the waveform optimization procedure described inthe previous subsection, it is clear that the all the threeTAFs can be reduced when the ranges, velocities and anglesof the targets do not overlap. To quantify the impact ofthese reductions on the CRB of parameters, first consider therange estimation. In the case of a pulsed radar, the receivedwaveform is correlated with the transmitted pulse and thetime-lag at which correlation peak occurs is considered as theround-trip delay of that target. Therefore, any improvement inthe auto-correlation of the transmitted waveform, will directlyimpact the energy leakage of one target onto the another.Even in the case of an FMCW radar, the transmitted FMCWwaveform is mixed with the received waveform to obtain thetone frequency corresponding to the range of the target. Thebeat frequency in the mixer output is proportional to the delayin the reflected signal due to the nature of the FMCW signal.However, this is true only if the signal is not modulated.When the FMCW signal is modulated by another envelopesignal, the correlation between the baseband equivalents ofthe transmitted and received pulse is calculated to detect thedelays. This gives a better suppression of interference leakingfrom one target to another. In particular, the low pass filter after

7Here, the transmit and receive waveforms are assumed to be same forconvenience. However, the waveforms will be different in a multi-pathscenario, and the correlation of the corresponding derivatives have to beevaluated.

8

the mixing only gives a sinc(·) roll-off in comparison with thelower side-lobes obtained from a good auto-correlation in thetransmitted waveform.

Let Bs,R denote the side-lobe level obtained by the abovetransmit waveform optimization in the range dimension. Thatis, two objects separated by a small distance – equivalent toround trip delay of 1/2B – is recognizable by an FMCWsignal with bandwidth B, will be suppressed by Bs,R. In otherwords, we achieve a filtering of Bs,R across all ranges, aslong as the two objects are not in close proximity. Next, evenwhile processing the Doppler domain signal, a suppression isobserved due to the sinc(·) roll-off, even with a simple FFTprocessing. This suppression can be improved further by using,e.g. a Hamming or a Blackman-Harris window. Let Bs,D denotethis amount of suppression in the Doppler dimension. Later,even with a simple FFT processing for angle estimation, asuppression is observed due to the array vector correlationbetween two angles. Let Bs,A denote this suppression dueto the array vector correlation. This array vector correlationis described in detail for two separate array geometries inAppendix C, with no array tapering weights. Next, we willquantify the impact of these three suppression terms on theparameter estimation.

If two targets are closer in range, but not in Doppler andangle, then they can be separated out due to the filtering gaindue to Bs,D and Bs,A. That is, even if the two targets haveoverlapping spectrum in the range FFT, a total interferencesuppression of Bs,D +Bs,A is achievable. For notational conve-nience, we represent the gain from these three filters as theproduct of three Dsinc(·) terms, i.e. Dirichlet kernels. That is,the inter-target interference is reduced by,

Bs,tot ≈ Dsinc(∆r) Dsinc(∆v) Dsinc(∆θ), where

Dsinc(∆r) ,1

NR,FFT

{sin (π∆nNR,FFT)

sin (π∆n)

},

Dsinc(∆v) ,1

ND,FFT

{sin (π∆kND,FFT)

sin (π∆k)

},

∆n=

⌊2∆rBNR,FFT

cFrTs

⌋,∆k=

⌊∆fND,FFT

Fd

⌋,∆f=

2∆vfcc

,

Dsinc(∆θ) ,1

LaH(0)a(∆θ), (26)

where NR,FFT and ND,FFT denote the size of the FFTs in therange and Doppler domains, respectively, Fr and Fd denote thesampling rates in the range and Doppler domains, respectively,and a(·) denotes the array steering vector. We use Bk,j

s,tot todenote the suppression between the kth and jth targets, with∆

(k,j)r , |rk − rj |, ∆

(k,j)v , |vk − vj | and ∆

(k,j)θ = |θk − θj |.

C. Receiver Beamformer Design

Given that it is possible to design a filter to separate twoclosely placed targets in the Doppler and angle domains even iftheir reflected signals have overlapping spectrum in the rangedomain, it is desirable to first design a digital beamformer atthe receiver. Since the angles of targets are not known apriori,the FoV can be split into several sectors (beams) and weightsfor each beam to get an SINR improvement can be applied.

Among several methods that can be designed to find thebeamformer weights, we describe one next, where orthogonalweights can be constructed to cover the desired FoV, fora given array geometry. Adaptation on digital beamformerweights can be designed to give preference to an existingtarget direction, and to cover the other uncovered angles bythe existing targets. Depending on L, the receiver beams canoverlap for redundancy, and resilience to target movement.

The procedure is as follows. Following the beam-spaceweight matrix computation method given in [52], we proposeto select the orthogonal weights for the beamformer as thedominant eigenvectors of the covariance matrix given by

Q =

∫ θ2

θ1

w(θ)a(θ)aH(θ)dθ, (27)

where θ1 and θ2 denotes the boundary angles of the FoV andw(θ) denotes a weight function. In case of M < L beams aredesired, then M dominant eigenvectors in Q can be used. Onesuch design is illustrated later in Sec. VI.

D. Joint Radar and Communication

As mentioned earlier in Sec. II, we consider a JRC systemin which the radar transmits data in the odd frames, and thetargets communicate back in even frames. This simple protocolcan avoid overlapping transmissions between two targets inclose proximity. If the targets are distant, then they can beallowed to respond in all even frames. For simplicity, let theradar and the targets use the same codebook and coding rate.A suitable modulation and coding scheme can be used fordata transfer. A practical Gaussian spherical code constructionfor Gaussian channel is described in Appendix D, which maybe used to modulate the Gaussian pulses (or an optimallydesigned waveform) as described in the above section. Whenthe targets transmit their modulated pulses, the followingprocedure is implemented to decode the bits and estimate thetarget parameters.Step 1: First, digital beamforming is done on the received

data pulses, and the weights computed in Sec. IV-Cis used for a suppression of inter-target interferencein the angle domain. Note that this allows spectrumoverlapping, which was one of the constraints in [45],which limited the maximum data rate for the targets.

Step 2: The output of the beamformer is used to correlate withthe baseband equivalent of the transmitter waveform.

Step 3: The correlation peaks identify the target distances.The correlator peak outputs corresponding to all suchtransmitted pulses are summed up to enhance the SNRand resilience in range detection. This gives inter-target interference suppression in the range domain.

Step 4: The correlator peak output for all targets are passedto the FFT in Doppler dimension to estimate thevelocity, and the corresponding frequency is correctedon the correlator peak outputs. This gives inter-targetinterference suppression in the doppler domain.

Step 5: The resultant Doppler peak locations – for everyrange correlator peak – across all antennas are sent tothe angle estimation block, which further refines the

9

digital beam angles from which the range detectionwas obtained.

Step 6: The refined angles are used to apply beam weightson the range domain data, followed by the Dopplerfrequency correction to obtain the baseband signalcorresponding to the data transmitted by the individualtargets.

Step 7: Conventional data demodulation and error correctionare performed, including the matched filtering for thetransmit pulse shape.

Thus, one can perform both the Radar target parameterestimation and data transmission using the same Transmitwaveform, except that data is overriding on top of the transmitpulse envelope. Any incoherent sums used to improve SNR forrange and Doppler frequency estimation remains unaffected bythe code sent by the targets.

V. BOUNDS ON ACHIEVABLE DATA RATES

A. Mutual Information of the Forward Channel

Consider the forward channel between the transmitter andkth target. Let E[a2i ] = σ2

x = Pavg. The reflected signalwk,i(t), i = 1, . . . , N , from the target can be written as

wk,i(t)=aiαkγk

∫ tk

0

gk(t′)x([t−iTs −τk−t′]Tk)dt′

+ jk,i(t) + ni(t), (28)

where jk,i(t) is the interference for kth target due to a delib-erate jammer. For the ease of analysis, we assume that jk,i(t)is a Gaussian random process with zero mean and varianceσ2

j,k. However, the following analysis can be extended to otherpopular statistical models for clutter and jammer, e.g. Gammaor Rayleigh, by suitable modifications. Now, the signal-to-interference and noise ratio (SINR) due to the reflected signalfrom the kth target in the presence of other surrounding targetsand jammers can be written as

ρk =αkγkσ

2xGk

σ2n,k + σ2

j,k + σ2x

∑j 6=k

λ2Gjαjγj16π2(rk−rj)2

, (29)

where only the targets in close proximity to kth target areconsidered along with the corresponding free space pathloss,with λ denoting the wavelength of the carrier. Moreover,

Gk ,∫ B

−B

∣∣∣∣Xi

(ω

Tk

)Gk(ω)

∣∣∣∣2 dω (30)

denotes a factor that determines the variance of the reflectedsignal due to target channel response [53], and σ2

n,k denotesthe noise variance at the kth target’s receiver. Note that ρkis calculated based on the reflected signal power due to thefact that it is proportional to the RCS of the target, and RCScaptures the maximum antenna aperture that is available at thetarget. As an illustrative example, consider a target with nointerference from jammers, i.e. σ2

j = 0, Gk = 1, γk = κkAk,

αk = 14πr2k

and ρTk =σ2x

σ2n,k

, where Ak denotes the surface areaof the target, 0 ≤ κk ≤ 1 denotes the aperture efficiency. Then,

the SINR at the kth target on the forward channel simplifiesto

ρ(f)k =

ρTkκkAk

4πr2k

(1 +

σ2j,k

σ2n,k

+ ρTk

∑j 6=k

λ2Gjαjγj16π2(rk−rj)2

) , (31)

which is equivalent to the SNR of the signal at the kth target.Hence, a bound on the mutual information in the forwardchannel between the transmitter and kth target, I(X,Wk), canbe computed in a straightforward manner using the linearmodel and AWGN channel model as shown in Fig. 2 [50].That is,

I(X,Wk) ≥ N

2log

1+αkρTγkGk

1 +σ2

j,k

σ2n,k

+ ρTk

∑j 6=k

λ2Gjαjγj16π2(rk−rj)2

= log

(1 + ρ(f)

k

)N2

. (32)

Therefore, (32) gives a lower bound on the data rate that canbe communicated from the radar transmitter to kth target.

B. Mutual Information of the Reverse Channel

Similar to the derivation in Sec. V-A, we can derive alower bound on the mutual information for the reverse channelbetween each passive target k and the receiver as

I(X;Zk)≥ N2

log

1+α2

kρTγkGk

1 + ρT

{∑j 6=k

Bk,js,totα2

jγjGj

}

=log(

1 + ρ(r)k

)N2

, (33)

where ρ(r)k is the SINR at the receiver from the reverse channel,

and Bk,js,tot denotes the interference suppression obtained by the

three dimensional filtering, given in (26).8 The expression in(33) yields a lower bound on the achievable data rate betweenthe kth passive target and the radar source. Recall that thetarget is assumed to be a passive reflector, and only changes itsreflection coefficients to perform modulation on the reflectedsignal from the targets. For this reason, the strength of thereflected signal is taken as the received signal power in thereverse channel. Note the similarity between the expressions in(32) and (33), except for the additional α2

k in the numerator forthe reverse-channel to account for the additional pathloss andBk,j

s,tot term in the denominator which provides the 3-D filteringowing to the separation of the targets in 3-D space namely,range, velocity and angle of arrival.

VI. RESULTS AND DISCUSSION

In this section, we illustrate the utility of various designmethods discussed in the previous sections, using examplesthrough numerical techniques.

8We ignore the interference due to jammers to derive (33), under theassumptions that (a) There are no jammers near the radar receiver, (b) thejamming signals are significantly attenuated at the radar receiver, (c) thejamming signals get filtered out by the beamforming procedure.

10

-5 0 5Time in sec 10-5

-200

0

200

Am

plitu

de

Original EnvOptimized Env

-5 0 5Time in sec 10-5

-1

0

1

Der

ivat

ive

109

Original EnvOptimized Env

0 2 4 6Frequency in Hz 106

-150

-100

-50

0

Spec

trum

in d

B

Original EnvOptimized Env

-50 0 50 100Correlation Lags

-6

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-2

0

Nor

mal

ized

Cor

rela

tion

in d

B

Original EnvOptimized Env

(a)

-5 0 5Time in sec 10-5

-200

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0

100

200

Am

plitu

de

Original EnvOptimized Env

-5 0 5Time in sec 10-5

-1

-0.5

0

0.5

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ivat

ive

109

Original EnvOptimized Env

0 5 10Frequency in Hz 106

-150

-100

-50

Spec

trum

in d

B Original EnvOptimized Env

-100 -50 0 50 100Correlation Lags

-60

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0

Nor

mal

ized

Cor

rela

tion

in d

B

Original EnvOptimized Env

(b)

-3 -2 -1 0 1 2 3Correlation Lags (in sec) 10-6

-35

-30

-25

-20

-15

-10

-5

0

Nor

mal

ized

Cor

rela

tion

in d

B

-5 0 5

10-7

-30

-20

-10

0

FMCW onlyOptimized Env onlyCombined

(c)Figure 3: (a) Properties of the optimized Gaussian pulse waveform with 1 MHz bandwidth constraint, (b) Properties of the optimized Gaussian pulse waveformwith 10 MHz bandwidth constraint, and (c) Autocorrelation of the optimized envelope and FMCW modulated with the optimized envelope.

A. Optimization of Transmit Waveform

In this subsection, we demonstrate the optimal waveformdesign obtained by the CoV-based optimization. To begin with,the waveform was chosen to be Gaussian with Ts = 100 µs.In general, the choice can be any waveform listed in Table I.The cost function chosen was the sum of correlation lags from2 to 20 units. Reduction of the cost function was carried outover 200 iterations. Figures 3(a)-3(c) show the comparison ofvarious properties of the designed optimal envelope Note thatthe optimal envelope converges to a series of impulse func-tions. Depending on the bandwidth expansion allowed in thenew envelope function, the correlation lags were suppressedfurther. That is, the correlation suppression varies directly withthe envelope bandwidth. This bandwidth expansion constraintwas added as part of the iterations explained Section IV-A.Figure 3(c) shows correlation lags between 100 ns and 1 µs,which are lower than that observed over the unmodulatedFMCW carrier. In this example, the FMCW carrier has abandwidth of 10 MHz, which can resolve targets separated by100 ns. Thus, the 100 ns limitation in the optimal waveformcomes from the carrier rather than from the envelope. Thisdemonstrates that the optimizing the TAF functions reducesthe interference between two targets, as discussed in Sec. III.

B. Receive Beamforming

Consider an L = 12 element uniform linear array (ULA) ofantennas. We intend to design the digital beamformer weightswhose field of view (FoV) is ±45 degrees. Figures showsthe performance of the designed beamformer. Note that thestandard steering vectors are not orthogonal and there is nodirect way to incorporate the weight function for selecting aparticular sector or FoV. Figure 4 shows the array response fora set of 8 weights after applying Taylor window array tapering.By design, the array weights are orthogonal – before applyingthe array taper, and one can design upto L such vectors fromthe eigenvectors of the covariance matrix for the given arraygeometry. Note that the side-lobes and main beam width canbe altered using different array tapering values. These weightswere obtained for θ1 = −50 degrees and θ2 = 48 degrees.This asymmetry is deliberately created to produce complex

-100 -80 -60 -40 -20 0 20 40 60 80 100Azimuth Angle (degrees)

-80

-60

-40

-20

0

20

Dir

ectiv

ity (

dBi)

Azimuth Cut (elevation angle = 0.0°)

Weights 1Weights 2Weights 3Weights 4Weights 5Weights 6Weights 7Weights 8

Figure 4: Beamformer output for various weights designed with Taylorwindow weighting.

eigenvectors. As observed, the sidelobes are suppressed bymore than 30 dB in the Taylor window tapered weights case.

C. Parameter Estimation in JRC

A JRC system was implemented using the Radar toolboxin Matlab, with the parameters given in Table II. Two targets,one moving slow and one moving fast, both moving awayfrom the radar, were chosen. The (32, 8) Gaussian codebookdesigned in Appendix D was used for transmission. It can beseen that in both cases, the performance of the radar is almostidentical. The error in the measured position is mainly due tothe finite angle resolution due to the 12 element ULA used andsimple FFT processing was used to detect all three parameters,range, velocity and angle of arrival. The received signal afterpulse compression was sampled at 40 MHz sampling rate with4096 point range FFT and 512 point Doppler and angle FFTstaken for computing the radar parameters. The target detectionwas done in 3 stages, first two using order statistics-basedrange and Doppler detection, and the final based on a 2Dcentroid detection in the range-Doppler domain. Thresholdswere chosen to achieve a constant false-alarm rate.

In the case where data transmission was done, the data wasreceived with zero bit error rate for both targets since SINR ishigh (> 4 dB even at the farthest radial distance of 43 m). Justto compare the effect of data transmission on the Radar perfor-

11

Sl. Parameter Value Unit1 No. of transmit antennas 1 -2 No. of receive antennas 12 -3 Array geometry ULA -4 Antenna spacing 0.0078 m5 Antenna height 8 m6 Antenna grazing angle 6.38 degrees7 Antenna beamwidths (−45, 45) Azimuth deg

(−6, 6) Elevation deg8 FMCW Carrier bandwidth 100 MHz9 Num. of chirps per frame 512 -10 Chirp time, Ts 100 µs11 Envelope bandwidth 100 KHz12 Carrier frequency 24 GHz13 Forward error correction code (32, 8) -14 Target data rate 2500 bits/s15 Transmit pulse envelope Gaussian -16 RCS of targets 1 m2

17 Target velocities (0.5, 1) m/s(vx, vy) (5,−10) m/s

18 Target start positions (25, 25) m(rx, ry) (25, 35) m

Table II. Parameters for the simulation of a JRC system.

Figure 5: 24 GHz FMCW radar performance with and without data transmis-sion using Gaussian envelope.

mance, no clutter was added and target RCS was kept constanti.e., no random fluctuations such as Swerling model was used.But practical radar receiver impairments such as receiver noise,automatic gain control, and ADC quantization were added inthis study. Note that no tracking and smoothing algorithmssuch as Kalman filtering were applied on the data. The rawmeasured target position parameters are plotted in Figure 5after adjusting for the antenna height. This demonstrates that ifone can design radar transmit waveforms such that the sidelobeleakage of one target onto another (Refer to Eqn. (26)) isminimized, then we can achieve joint Radar communicationwith no loss in performance of parameter estimation as well asdata transmission system simultaneously. Moreover, one canachieve the performance bounds given in Eqns. (33) and (13)-(15). Rigorous proof of achievability is not discussed here, andwill be taken up as a future work.

D. Data Rate Bounds in JRC

To demonstrate the utility of the data rate expressions for theforward and reverse channel (Eqns. (32) and (33)), numericalsimulation was performed to compute the data rate bounds inthe forward and reverse channel for the above 24 GHz radar(but with 1 antenna at both ends and no error correction codeapplied) where ρT and ρTk are assumed to be 100 dB. Weassumed two static, point reflection targets with 1 sq.meterRCS area and no jammer. The two targets and the radar makean right angle triangle with different acute angle subtended atthe radar depending on the distance between the two targets.Moreover it is assumed that, γk = Gk = 1 and αk = 1

4πr2kfor all targets. The data rate is plotted as a function ofdistance from the radar. Note that, the distance of target 2is always larger than that of target 1 by a fixed amount forone simulation.

Figure 6 shows the data rate bounds for both channels for 3different distances between the targets. It can be observed thatthe forward channel rate I(X,Wk) for the target 1 reduceswith an increase in range. This reduction is sharp at smallrange compared to the linear reduction at high range values.This is largely due to the interference from target 2 causingadditional loss in the received SINR at low range. Note that thedata rate for target 1 is higher when target 2 is farther away.On the other hand, data rate for target 2 first increases andthen starts to decrease, due to the reduction in the interferencefrom target 1 at close ranges. Further, it is observed thatthe data rates increase as the separation between the targetsincrease. Both targets converge to nearly equal data rates astheir distance from the radar increases. However, a minordifference in rates exists due to the difference in their distancesfrom radar as well as distance between them. Looking at thereverse channel rate I(X,Zk), we observe that the rate fortarget 1 decreases with distance and is lower than I(X,Wk).Data rate for target 2 improves with range, and decreaseswith an increase in separation between them. Due to non-availability of multiple antennas at the receiver, no spatialfiltering can be designed except for SBrange. Moreover, notethat the drastic reduction in I(X,Zk) with distance, due tohigher (round trip) pathloss on the reverse channel comparedto the forward channel case.

VII. CONCLUSION AND FUTURE WORK

In this paper, a novel analytical approach using a linearchannel target response model was employed to derive lowerbounds for data rates and parameter estimation errors. Thisframework is valid in the context of a wideband JRC withsemi-passive targets, for any given array geometry. The radartransceiver was assumed to not only detect the targets, butalso to exchange information with them. The targets usecontrolled backscattering of the transmitted signal for datatransfer, and receive the data using active components. It wasshown that the proposed inter-target interference suppressionimproves the performance of the JRC system on the reversechannel. Moreover, it was observed that the forward channeldata communication does not impact the parameter estimationbounds. Furthermore, the data bounds were shown to be

12

0 10 20 30 40 50

2

3

456

I(X

;Wk)

in b

its/s

ec104

T1@R+ R=1m(F)T1@R+ R=3m(F)T1@R+ R=5m(F)T2@R+ R=1m(F)T2@R+ R=3m(F)T2@R+ R=5m(F)

0 10 20 30 40 50Range(R) in m

101

102

103

104

I(X

;Zk)

in b

its/s

ec

T1@R+ R=1m(R)T1@R+ R=3m(R)T1@R+ R=5m(R)T2@R+ R=1m(R)T2@R+ R=3m(R)T2@R+ R=5m(R)

Figure 6: Data rate bounds for the forward and reverse channels in a 24 GHzJRC radar.

achievable if and only if the TAFs are optimized by properselection of the transmit waveforms. Using CoV, an optimaltransmit waveform was designed to minimize the range TAF.By optimizing three different TAFs, it was shown that one canachieve a desired performance for each of the parameters. Theproof of achievability of these bounds are a part of the futurework. Additionally, design of practical codes to achieve thesebounds is an interesting research area.

APPENDIX

A. Target Ambiguity Functions

The TAFs are generalized versions of the conventionalambiguity functions, which arise naturally in the computationof CRBs. This requires computation of second order partialderivatives of logarithm of the joint PDF function. For theradar parameter estimation, the partial derivative with respectto the parameters can be computed using

∂ log fz(zi)

∂rm=

[∂ log fz(zi)

∂ (zi − aiXiΓi)

]T∂ (zi − aiXiΓi)

∂rm. (34)

Upon simplification, we get

∂ log fz(zi)

∂rm=−1

2

[2 (zi−aiXiΓi)

TR−1n

] ∂ (zi − aiXiΓi)

∂rm.

(35)

Further, it can be shown that(∂ Xi

∂ rm

)=

4π2T2mγm

∑j ζmjδ(t− tmj)β2 c

[xm ◦ tm] , (36)

(∂ Γi∂ rm

)=

0, . . . , 0, γm∑j

ζmjδ(t− tmj)

{−2e ε2 r−2e−1m }, 0, . . . , 0]T, (37)

∂ log fz(zi)

∂ rm

(a)≈ ai

4π2T2m γm

∑j ζmjδ(t− tmj) ε2r−2em

β2 c

(zi − aiXiΓi)T

R−1n [xm ◦ tm]

∂2 log fz(zi)

∂ rm∂ rn= ai

4π2T2m γm

∑j αmjδ(t− tmj)β2 c

∂

∂ rn

{(zi − aiXiΓi)

TR−1n [xm ◦ tm]

}, (38)

∂

∂ rn

{(zi − aiXiΓi)

TR−1n [xm ◦ tm]

}= (zi − aiXiΓi)

TR−1n

∂

∂rn[xm ◦ tm]︸ ︷︷ ︸=0

+

∂

∂rn(zi − aiXiΓi)

TR−1n [xm ◦ tm] , (39)

where the approximation denoted by (a) is obtained by ignor-ing the derivative of Γi with respect to rm, since the first termdominates the second. The final expression for ∂2 log fz(zi)

∂ rm∂ rnis

given in (40) at the top of next page. It can be observed that−E

[∂2 log fz(zi)∂ rm∂ rn

]= 0, since E[γmγn] = 0.

1) TAF in Range: Now, consider the function,

Ar(rm, rn) , E(

[xn ◦ tn]T

R−1n [xm ◦ tm]), (41)

which measures the average weighted inner product betweenthe two derived vectors, [xn ◦ tn] and [xm ◦ tm] where theterm xn ◦tn is proportional to the derivative of xn . Here, theexpectation is taken over all possible differences in the rangeof two targets except for rm = rn. That is, the expectation canbe computed as shown in (42), where f(rm, rn) is the jointPDF between two target ranges. Typically, one can evaluatethe integral for uniform distribution for the two target ranges.When Rn is a diagonal matrix, this function simplifies to

Ar(rm, rn) =1

σ2n

E(

[xn ◦ tn]T

[xm ◦ tm]), (43)

which measures the ambiguity or dissimilarity between thetime weighted columns of Xi. We refer this function as targetambiguity function (TAF) which is different from the ambi-guity function known in radar literature. TAF measures thesimilarity of the received vectors due to each target rather thanthe similarity between the ideal and time-frequency shiftedtransmit waveform. Moreover, TAF operates on the derivativeof the vector signal received from each target rather than thereceived signal vector. Even if the reflection from the targetsare correlated E[γmγn] 6= 0, one can design waveform andarray geometry which ensures TAF to be very small betweentwo targets, its parameters can be estimated more accurately,as observed in the sequel.

From (40), the expected value of second order partialderivative for range can be written as in (13), which can bethe inverse of the MSE if off-diagonal elements of the FIMare zero.9 It can be noticed that MSE is inversely proportional

9In general, the FIM is block diagonal in nature, where the 3x3 blockmatrices corresponding to all parameters for a given target can influence eachother on the MSE. However, the parameters are one target does not influencethe MSE of another. Also note that inverse of block diagonal matrix is a blockdiagonal matrix with the individual inverse of the blocks.

13

∂2 log fz(zi)

∂ rm∂ rn= −a2i

16π4T4m γmγnζmζnε

4r−2em r−2en

β4 c2[xn ◦ tn]

TR−1n [xm ◦ tm] . (40)

Ar(rm, rn) =

∫ rmax

rm=rmin

∫ rmax

rn=rmin,rn 6=rm[xn ◦ tn]

TR−1n [xm ◦ tm] fRm,Rn(rm, rn)drndrm, (42)

to the SNR, RCS parameter and norm of the derivative ofthe signal vector from that target. Hence, one should designa transmitter waveform which maximizes the average norm,E(‖xm ◦ tm‖2

)rather than the waveform with best self-

correlation property. At the same time, one also wishes tominimize E

((xm ◦ tm)

T(xn ◦ tn)

), to reduce the impact

due to cross terms in the FIM.2) TAF in Velocity: Similar to the previous case, one

can find the partial derivatives with respect to the velocityparameter as shown below. Note that,

∂ x(Tm[t− iTp − 2τm − ϕl])∂ vm

= −2Tm[t− iTp − 2τm − ϕl](

2c

(c− vm)2· π

2

β2

)x(Tm[t− iTp − 2τm − ϕl]) ◦ tm ◦ tm, (44)

using the fact that dTmdv = 2c

(c−vm)2 . We can write the secondorder derivative with respect to the velocity parameter as givenin (14). Thus, for minimizing the MSE for velocity estimation,one should design waveform which maximizes the norm of thesecond order derivative of the transmitter waveform. Moreover,define the TAF for velocity as

Av(vm, vn) = E([

xn ◦ t2n]T

R−1n[xn ◦ t2n

]). (45)

As in the case of range, the above TAF is computed over allpossible velocity values except when vm = vn.

3) TAF in Azimuth: Evaluation of the partial derivativeswith respect to the azimuth angle gives the TAF in angledomain. From the definition of ϕl , we can write

ϕl =cosφm cos θmxl+sinφm cos θmyl+sin θmzl

c, (46)

where (xl, yl, zl) are the Cartesian coordinates of the lth

receiver antenna, φm is azimuth angle of the mth target and θmis the corresponding elevation angle of the same target. Now,one can write the second order partial derivative as given in(15), and the TAF in angle can be written as

Aφ(φm, φn)=E(

[xn◦tn◦Φn]T

R−1n [xm◦tm◦Φm]), (47)

where Φm is defined as the vector of partial derivatives of ϕlwith respect to the angle φm for all l = 1, . . . , L. That is,

Φm ={sin θm (yl cosφm − xl sinφm) + zl cos θm}

c. (48)

Thus, this ambiguity function brings in the role of the arraygeometry to improve the MSE bound by minimizing the TAF.

4) Cross Terms in FIM: Using the above expressions, onecan write the expression for the mixed second order derivativesfor pairs of terms such as (range, velocity) or (range, angle)and (velocity, angle). Since these terms are not zero, the FIMhas the structure of a block diagonal matrix, where each blockbelongs to one target. That is, under independent scatteringassumption, one target does not influence the parameter es-timation of the other target provided we design waveformswhich minimizes the TAFs. Hence, to achieve the boundsgiven in Sec. III, we need to minimize all inner products givenin (49), so that the non-diagonal terms in FIM are reduced.

5) Generalized Target Ambiguity Function: Although wehave listed the ambiguity functions that arose from the partialderivatives taken with respect to different parameters, onecan combine all to define a generalized TAF from whichother TAFs can be derived as special cases. For instance, thegeneralized TAF given in (50) includes variations in range,velocity and array geometry.

B. Proof of Theorem 1

Consider the cost in (24) without scaling factor 1/E, whosepartial derivative with respect to y can be written as:

∂F (y, y)

∂y=

∫ Ts2

−Ts2

[y(t)

∂y(t+ δ)

∂y+∂y(t)

∂yy(t+ δ)

]dt, (51)

where ∂y(t)∂f denotes the variation in the temporal slope of y(t)

due to the functional variation in y. Similarly,

∂F (y, y)

∂y=

∫ Ts2

−Ts2[y(t) + y(t+ δ)] dt = 0, (52)

for y(t) whose end values are equal. The right hand side of(23) can be written as

d

dt

(∂F (y, y)

∂y

)= 0. (53)

Upon equating (51) and (53), we get the conditions foroptimality as∫ Ts

2

−Ts2

[y(t)

∂y(t+δ)

∂y

]dt=−

∫ Ts2

−Ts2

[∂y(t)

∂yy(t+δ)

]dt. (54)

Consider the first variation of the function as

yn+1(t) = yn(t)± λ y(α t+ β), (55)

where λ > 0, α > 0 and β ∈[−Ts2 ,

Ts2

)are constants.

The number of time shifts β are finite, since the number of

14

E[∂2 log fz(zi)

∂rmvm

]= −16ρTπ

4T3mE[γ2m]E[‖ζm‖2]ε4r−4em

β4(c− vm)2(xm ◦ tm ◦ tm)

T(xm ◦ tm)

E[∂2 log fz(zi)

∂ rmφm

]= −8ρTπ

4T3mE[γ2m]E[‖ζm‖2]ε4r−4em

β4 c(xm ◦ tm ◦Φm)

T(xm ◦ tm)

E[∂2 log fz(zi)

∂ vmφm

]= −8ρTπ

4T2mE[γ2m]E[‖ζm‖2] cε4r−4em

β4 (c− vm)2(xm ◦ tm ◦Φm)

T(xm ◦ tm ◦ tm) (49)

A(r1, r2, v1, v2, φ1, φ2)=E [xm(r1, v1, φ1) ◦ tm(r1, v1, φ1)◦Φm(φ1)]T

R−1n [xm(r2, v2, φ2)◦tm(r2, v2, φ2)◦Φm(φ2)]. (50)

dimensions of the considered functions – which is 2BWTs –in L2 are finite. Therefore, β will be changed in steps 1

2BWto accommodate the Nyquist sampling rate. Similarly, choiceof α also is limited within the range

[1

2BWTs, 1)

. For variouschoices of β and amplitude scale λ, y(t) is modified as givenin (55). Now, the optimality condition can be evaluated asfollows. Let z(t) = y(α t− β). Then,

y1(t) = y(t)

(1 +

λ z(t)

y(t)

),

∂yt(t)

∂y(t)=

(1 +

λ z

y

)+ y

(−α λ z

y2

)(∂z

∂y

)=

(1 +

λ z

y

)− α2

(λ z

yy

), (56)

where the time variable is dropped for brevity only positivesign is considered in (55), without loss of generality. Moreover,we have used the fact that

∂z

∂y=∂y(αt− β)

∂y= α

∂t

∂y=

(α

y

).

By substituting (56) in (54), and using some algebra, one canwrite the following where the time dependence is notated asa subscript.

ytyt+δ

(1− α2

yt+δ

)=yt+δyt

(1− α2

yt

). (57)

That is, the above condition is satisfed if the slope of thesignal does not change drastically for small delta, i.e. if y(t)is continuous, since yt ≈ yt+δ and yt ≈ yt+δ and theapproximation error can be evaluated using the Taylor seriesexpansion. By substituting the first order approximations forthe two ratios yt+δ

ytand yt+δ

ytin (57), we get a quadratic

equation in δ. Evaluating the condition for roots of thisequation to be real, gives the required result.

C. Array Vector Correlation

Here, we study the correlation between two array steeringvectors for two different array geometries at various steeringangles. Consider a uniform linear array (ULA) and a uniformcircular array (UCA) with 16 elements. The parameters listedin Table III were used for realizing the two array geometries.

Sl. Parameter ULA UCA

1 Element spacing λc2

π λc2

2 Array diameter 7.5 λc 8 λc3 Origin at the center at the center4 Orientation along X−axis along X − Y plane

Table III. Parameters for ULA and UCA with λc = 12.5 mm and L = 16.

Figure 7(a) shows the time delays across antenna elementswith respect to the origin for ULA and UCA respectively,for various azimuth angles while the elevation angle of thesource is kept at 45◦. The plots also show the correlationamong the array steering vectors. Note that UCA shows ahigher correlation for angles outside the main beam direction,compared to the ULA. However, the main beam width forUCA is smaller than that of ULA, for a given L. These timedelay variations also influence the TAFs, through the samplingtime instants for the received pulse before correlating it withthe transmit pulse. For various angles, the time delays in UCAare not concentrated around zero as opposed to ULA. Thiscauses wider peaks in the TAFs. That is, for azimuth anglesclose to 90◦ degrees, all elements receive the data nearly atthe same time and the time differences are nearly zero. Hence,UCA will be preferred array geometry for obtaining goodSBangle for the given number of antenna elements.

D. Code Construction Procedure

It is well-known that Gaussian random source achieves thecapacity in a Gaussian channel. Hence, the code constructionfor the data only transmission can adopt any of the wellknown methods such as the lattice Gaussian coding [54], [55].We elaborate a practical code construction along the lines ofShannon’s spherical code idea. The N -dimensional Gaussianvector with components, from i.i.d. Gaussian random variableswith zero mean and unit variance, can be scaled by σx toobtain the desired codeword with average transmitter powerP = σ2

x. A Gaussian spherical code with desired numberof codewords, e.g. 2r codewords or r bits per codeword,can be constructed by partitioning the surface area of an N -dimensional sphere with unit radius. This can be done bywell known methods such as K-means algorithm [56]. Now,use the normalized centroids of the 2r regions as the desiredcodewords for data transmission. Such a code is known toachieve the capacity of an AWGN channel for large N [55].

15

0 20 40 60 80 100 120 140 160 180-0.2

0

0.2

0

0.5

1

0

0.5

1T1T5T9T13T16Correlation

0 20 40 60 80 100 120 140 160 180Azimuth Angle (deg)

-0.2

0

0.2

Tim

e de

lay

w.r

.t ce

nter

(ns

ec)

0

0.5

1

Arr

ay V

ecto

r C

orre

latio

n

0

0.5

1

ULA

UCA

(a)

1 1.5 2Distance(codewords)

0

0.05

0.1

PD

F

-0.5 0 0.5Component amplitude

0

0.05

PD

F

0

0.5

PD

F

-1 -0.5 0Log

10(Norm)

10-16

0 20 40Codeword dimensions(x,y)

-2

0

2

Cor

rela

tion(

x,y)

10-3

(b)

2 4 6 8 10SNR (dB)

10-6

10-4

10-2

Sym

bol/B

it E

rror

Rat

e

SER:Gaussian(32,8)::SimBER:Gaussian(32,8)::SimBER:Uncoded BPSK::Theory

6 dB

(c)Figure 7: (a) Time delay across antenna elements with elevation angle 45◦ for ULA and UCA with L = 12, (b) Code book Statistics for (32, 8) Gaussianspherical code, and (c) Bit and symbol error rate performance of Gaussian spherical code (32, 8) in an AWGN channel.

Figure 7(b) shows the properties of one of such codesdesigned for length N = 32, code rate r = 1/4. K-meansalgorithm was used to design 256 size codebook with 32dimensions per codeword. 105 random Gaussian vectors wereused as input to the K-means algorithm after they were normal-ized to have unit norm. About 200 iterations were performedso that the update error between two iterations is less than10−5. It can be noticed that the all the codewords have unitnorm, and the distance between them is between 1 and 2.The correlation between the various dimensions is nearlyzero, in the order of 10−3. The PDF of the components ofthe codewords closely approximate the Gaussian distribution.Figure 7(c) demonstrates the bit error rate and symbol errorrate performances in an AWGN channel with a maximumlikelihood decoder, where a coding gain of 6 dB is observed.

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