Joint Design of Overlaid Communication Systems and Pulsed Radars

# Joint Design of Overlaid Communication Systems and Pulsed Radars

Le Zheng,  Marco Lops,
Xiaodong Wang,  and Emanuele Grossi,
Le Zheng and Xiaodong Wang are with the Electrical Engineering Department, Columbia University, New York, USA, 10027, e-mail: le.zheng.cn@gmail.com, wangx@ee.columbia.edu. Marco Lops and Emanule Grossi are with the DIEI, Universita degli Studi di Cassino e del Lazio Meridionale, Cassino 03043, Italy (e-mail: lops@unicas.it, e.grossi@unicas.it).
###### Abstract

The focus of this paper is on co-existence between a communication system and a pulsed radar sharing the same bandwidth. Based on the fact that the interference generated by the radar onto the communication receiver is intermittent and depends on the density of scattering objects (such as, e.g., targets), we first show that the communication system is equivalent to a set of independent parallel channels, whereby pre-coding on each channel can be introduced as a new degree of freedom. We introduce a new figure of merit, named the compound rate, which is a convex combination of rates with and without interference, to be optimized under constraints concerning the signal-to-interference-plus-noise ratio (including signal-dependent interference due to clutter) experienced by the radar and obviously the powers emitted by the two systems: the degrees of freedom are the radar waveform and the afore-mentioned encoding matrix for the communication symbols. We provide closed-form solutions for the optimum transmit policies for both systems under two basic models for the scattering produced by the radar onto the communication receiver, and account for possible correlation of the signal-independent fraction of the interference impinging on the radar. We also discuss the region of the achievable communication rates with and without interference. A thorough performance assessment shows the potentials and the limitations of the proposed co-existing architecture.

Coexistence, Compound rate, Pulsed radar, Spectrum sharing, Waveform design.

## I Introduction

Co-existence between radar and communication systems over overlapping (if not coincident) bandwidths has been a primary investigation field in recent years and has been put forward as a challenging topic at both theoretical and implementation stages [1, 2, 3, 4]. The prevailing approach so far—with some exception—has been to guarantee the detection and estimation performance of the radar by designing waveforms producing a tolerable level of interference on the communication system.

The paper outline is as follows. Sec. II is devoted to the signal model and the problem formulation, while Sec. III focuses on the constrained optimization of the aforementioned compound rate under white and colored radar noise. Sec. IV is devoted to the performance assessment and result validation, while conclusions and hints for future developments form the object of Sec. V.

## Ii System model and problem formulation

In this paper, the joint radar-communication system consists of an active, mono-static, pulsed radar and a single-user communication system: Fig. 1 outlines a scheme of the considered architecture. We assume that the radar and communication systems share the same bandwidth. The radar is allowed to transmit an amplitude-modulated pulse train with Pulse Repetition Time (PRT) , each pulse having the same duration , , as the symbol duration in the communication system, so that the situation of the overlay is the one outlined in Fig. 2.

Suppose that the amplitude of the -th radar pulse is , so that the length- pulse train emitted by the radar is

 ξ(t)=N∑n=1snϕ(t−(n−1)T) (1)

where satisfies the Nyquist criterion with respect to , whereby the bandwidth is , and range cells are defined. As to , we assume that it is chosen in such a way that no cell migration takes place in the time interval , i.e., that the targets do not change resolution cell for the duration of the pulse train. The signal transmitted by the communication system is

 χ(t)=∞∑ℓ=−∞xℓϕ(t−ℓTc) (2)

where denotes the -th transmitted symbol.

If a target is present in the -th range cell, , it back-scatters towards the radar antenna with a given coupling coefficient , and hits it with time delay , thus generating echoes at time instants : we assume that the targets of interest are coherent (e.g., follow a Swerling 0, I or III model), since no signal design could take place for scintillating targets (i.e., Swerling II or IV [17]). Letting and be the delay and the channel gain between the communication transmitter and the radar receiver, respectively, the radar should process the return from the -th range [18] cell to discriminate between the two hypotheses

 (3)

where is the measurement noise of the radar receiver, and is the scattering coefficient from all the other unwanted reflectors present (such as clutter and/or environmental reverberation) in the same range cell.111We have not accounted for the Doppler shift of the target, which boils down to considering it zero or to focusing on a specific Doppler resolution cell (e.g., the one with largest interference from the unwanted reflectors, in light of a worst-case design paradigm). Also, we are implicitly assuming point-like scatterers, which is quite reasonable in the narrowband low resolution scenario we are considering. This widely used signal model [18] is simple but allows to capture the main performance tradeoffs. The discrete-time representation of the observations pertaining to the -th range cell (corresponding to a delay ) after illumination through the pulse train can be obtained by projecting the previous observations onto the orthonormal functions , i.e.,

 rk,n =⟨r(t),ϕ(t−(n−1)T−kTc)⟩ ={asn+gxk−νCR+(n−1)K+csn+wk,n,under H1gxk−νCR+(n−1)K+csn+wk,n,under H0 (4)

where is the inner product, denoting complex conjugation, and . Note that this dicretization process can be equivalently performed by standard filtering through a filter matched to the pulse waveform and subsequent sampling at the inverse of the bandwidth , as it is commonly done in radar receivers. Regrouping the returns pertaining to each range cell in -dimensional vectors, we obtain, for the -th range cell,

 rk={as+gxk−νCR+cs+wk,under H1gxk−νCR+cs+wk,under H0 (5)

where , , , for integer, and , denoting transpose. We assume that the distribution of is , where is the all-zeros -dimensional vector, is the covariance matrix, on whose diagonal elements is the unique value , representing the noise power, and denotes the complex circularly symmetric Gaussian distribution. Notice that here the time scale is the PRT, so the vector includes data spaced apart. This model implicitly assumes that the radar, although undertaking clutter reduction functions, may still be left with some signal-dependent interference, whose intensity is encapsulated in the mean square value of the random coefficient ; moreover, this clutter suppression functions might introduce some correlation between the noise samples, encapsulated in the matrix , which is assumed from now on known, e.g., as a consequence of accurate estimation conducted offline through secondary data.

 z(t) =hξ(t−νCTc)+M∑m=1N∑n=1fkm,nsn ×ϕ(t−νRCTc−kmTc−(n−1)T)+v(t) (6)

where both and are unknown quantities,333Here we assume that the interferers’ Doppler shift with respect to the communication system receiver is zero, i.e. that they are either stationary or in tangential motion. The subsequent results, however, hold true whenever the relevant figure of merit is Doppler-independent, as it happens for one of the two major situations considered in this paper, i.e., totally incoherent scattering, which has a particularly important meaning (see Sec. II-B). Once again, accounting for arbitrary interference Doppler shift would be compatible with the present framework, once a worst-case philosophy is embraced (see also [23]). is the reflection coefficient of the -th interfering object444At the communication system side, we do not distinguish between target and clutter returns, for it is not concerned with target detection: any source of reflection causes interference and is therefore generally referred to as an interfering object. in the -th PRT, and is the measurement noise of the communication receiver. As to the reflection coefficients they are typically random and will be commented upon later on in the paper. This model does not rule out, in principle, the case that one of the interferers is a direct path from the radar transmitter to the communication receiver, and the subsequent derivations would hold true in this case too. However, the most damaging sources of interference onto the communication system are those produced by objects—either targets or reverberation—whose angles and ranges are unknown, since the direct path might in principle be canceled through such techniques beam-forming [24, 25]. By adopting the orthonormal basis , the projections of the observations received in the interval can be re-arranged in the form

 zk,n =⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩hxk−νC+νRC+(n−1)K+fk,nsn+vk,n,if k∈{k1,…,kM}hxk−νC+νRC+(n−1)K+vk,n,if k∈{1,…,K}∖{k1,…,kM}. (7)

for , where . The previous equation highlights that co-existence induces a form of range gating on the communication system, whereby, regrouping the above observations into vectors of dimension , whose entries represent samples, spaced apart, pertaining to a given range cell, we obtain the model for the communication signal:

 zk={hxk−νC+νRC+Sfk+vk, if k∈{k1,…,kM}hxk−νC+νRC+vk, if k∈{1,…,K}∖{k1,…,kM} (8)

where , , denoting the diagonal matrix whose diagonal entries are the input elements, , and . Here we assume that , with denoting the -dimensional identity matrix; also, we assume that555This assumption will be justified shortly. : in other words, the radar produces a signal-dependent interference onto the communication system, the nature of the reflecting object being encapsulated in the covariance matrix , on whose diagonal elements is a unique value, say, representing the intensity of the scattered power.

In principle, the communication system alone would achieve capacity by random coding through independent Gaussian codewords. In a co-existing architecture, a new degree of freedom is introduced, i.e., the covariance matrix , , where denotes conjugate transpose. This form of “rake encoding” corresponds to introducing correlation between coordinates spaced apart; on a different point of view, it amounts to generating a white -dimensional matrix, inducing the correlation between the elements of each column, and undertaking depth- interleaving (i.e., transmitting the elements along the rows). Fig. 3 shows an implementation of this encoding scheme. Also notice that the subscripts in (5) and in (8) become now irrelevant, whereby they will be omitted in subsequent derivations.

In next Section, we introduce the key performance measure that will be used for joint optimization of the communication and radar systems, while, in Sec. II-B, we present the optimization problem tackled in this paper.

### Ii-a Performance measures

For the radar system, the SINR is used as the figure of merit, which is expressed as

 \sf SINR(Rx,s) =Tr((σ2gRx+σ2cssH+M)−1σ2assH) =σ2asH(σ2gRx+σ2cssH+M)−1s (9)

where , , and are the variances of , and , respectively.666In what follows, the dependency of SINR on will be omitted whenever possible, so as to make the notation lighter. The SINR is a key performance measure [26, 5, 7] and is also closely related to another pivotal figure of merit used for detection optimization purposes: the pair of Kullback-Leibler divergences between the densities of the observations under the two alternative hypotheses [18, 27]. Indeed, the two divergences can be interpreted, in the light of the Chernoff-Stein Lemma [28], as error exponents of miss and false alarm probabilities in a Neyman-Pearson theoretic environment, and, in the framework of sequential decision rules [29], they offer guarantees on the average sample number necessary to make decisions. Also, the divergences have already been validated as valid alternatives to the usual probabilities of detection and false-alarm for waveform design purposes [30]. For the observation model in (5), when is deterministic and , maximizing the SINR is equivalent to maximizing the Kullback-Leibler divergences, as it is shown in Appendix -A, which reinforces the choice of adopting the SINR as a figure of merit in our framework.

For the communication system, assuming that rake-type random coding with Gaussian codewords is used, the rate of the -th channel bounces between that of an additive white Gaussian noise channel, i.e.,

 R0(Rx)=1Nlogdet(IN+|h|2σ2vRx)[bits/channel use] (10)

and that of an interference channel, where the interference has covariance matrix . In principle, such an interference might not be Gaussian (e.g., if the reflectors represent clutter); however, on top of the fact that we are considering low-resolution systems, wherein the Gaussian assumption might be justified, at the design stage and inasmuch as the communication system performance is considered the Gaussian model has strong theoretical motivations (e.g., in the light of mini-max principle [28, p. 298]), whereby we assume it outright, and the rate takes on the form

 R1,k(Rx,s) =1Nlogdet(IN+|h|2σ2vRx ×(IN+1σ2vSRf,kSH)−1). (11)

Also, we let be the “worst case” covariance777In this paper no form of cognition is assumed, whereby the basic philosophy must necessarily be that of the “worst case.” of the reflectors (more on this infra), and define the common value of its diagonal elements; this implies that , with

 R1(Rx,s) =1Nlogdet(IN+|h|2σ2vRx ×(IN+1σ2vSRfSH)−1). (12)

The objective function we propose is the convex combination of the rates in (II-A) and (10), hereinafter referred to as compound rate (CR), defined as888In what follows, the dependency of CR on will be omitted whenever possible, so as to make the notation lighter.

 \sf CR(Rx,s) =βR1+(1−β)R0 =βNlogdet(IN+|h|2σ2vRx ×(IN+1σ2vSRfSH)−1) +1−βNlogdet(IN+|h|2σ2vRx) (13)

where .

The above choice deserves some further comments. Denote as the indicator of the presence () or absence () of radar interference on the -th channel. If there is no “preferential” range where the reflectors are located, we may assume that are independent and identically distributed, with . Thus, in the considered scenario, the presence of a co-existing radar system is accounted for by modeling the communication system as mutually independent parallel channels, each of them being interfered with probability . In this framework, when and , the quantity can be interpreted as the conditional MI, given the channel state, between the input and the output of the -th channel,999In principle the input would be , but since , with a slight notational abuse, we use here as channel input. i.e., ; moreover differs from the input-output MI, , by less than . Consider indeed the identity

 I(xk,ζk;zk) =I(xk;zk)+I(ζk;zk|xk) =I(ζk;zk)+I(xk;zk|ζk). (14)

Since

 I(xk;zk|ζk) =αI(xk;zk|ζk=1) +(1−α)I(xk;zk|ζk=0) =N\sf CR (15)

the CR is in fact a conditional MI. Furthermore, we have

 I(xk ;zk)−N\sf CR =I(ζk;zk)−I(ζk;zk|xk), =H(ζk)−H(ζk|zk)−(H(ζk|xk)−H(ζk|xk,zk)), =−H(ζk|zk)+H(ζk|xk,zk) =−I(ζk;xk|zk) (16)

where denotes entropy and the independence between and has been exploited. As a consequence CR is not achievable through the proposed encoding scheme. However, since , we have that

 \sf CR−1N≤1NI(xk;zk)≤\sf CR (17)

which shows that the MI per channel use differs from the CR by less than . We hasten to underline here that, contrary to CR, represents the maximum achievable transmission rate, provided the pulse number is large enough. Notice that, lacking prior information on , the chosen value of (for optimization purposes) should depend on a coarse information (or forecast) on how crowded the scene is.

### Ii-B Problem Formulation

The degrees of freedom available for optimization are the covariance matrix of the communication system and the radar waveform . The objective function is the CR in (13), while a constraint is imposed on the minimum required SINR at the radar receiver [26], denoted , and on the maximum powers of the radar and of the communication system, denoted and , respectively. Concerning , it is usually set considering a reference target, i.e., a target set at a given distance (typically, the one corresponding to the -th range cell) and following a given fluctuation model with a given average Radar Cross-Section (RCS) [17]. The joint radar and communication waveform optimization problem could thus be set up in the following form:

 maxRx,s \sf CR(Rx,s) (18) s.t. \sf SINR(Rx,s)≥ρmin,1N∥s∥2≤Pr 1NTr(Rx)≤Pc,Rx⪰0

where denotes positive semi-definiteness.

Before presenting the solution to this problem, it is worthwhile giving the following comments.

###### Observation 1

The system model considered here relies on the assumption that the radar and the communication system sharing the same bandwidth are narrow-band: this implies that the channel is flat-fading, and that each reflector—whether it is a target the radar wants to detect or an interferer—scattering the radar signal towards the communication receiver produces a single resolvable path. Notice, however, that, should this not be the case, the mathematical setup of the design problem would require only updating the definition of the CR for the communication system. Indeed, an interferer producing resolvable paths would spread its reverberation across a number of different range cells (corresponding to delays that are multiple of ) with independent scattering coefficients[31]. As a consequence, from the point of view of the communication system, this would translate into a denser environment (i.e., larger values of ), while having no effect on the SINR, which is in fact evaluated focusing on a single range cell.

###### Observation 2

The complex vector in (5) and (8) represents the slow-time code sequence of the radar, in that the coefficients encode pulses spaced PRT apart. However, a similar discrete-time data model can be set up for fast-time coding. In that case, the transmitted pulse has duration and is composed of sub-pulses with bandwidth ; the coefficient , instead, represents the amplitude of the -th sub-pulse, as in [5, 7]; similarly, the codewords at the communication systems contain symbols spaced apart. However, the main advantage of designing the radar slow-time code is that we are not concerned with the stringent requirements—in terms of range resolution, peak-to-sidelobe levels of the correlation function, and, more generally, ambiguity function [32]—posed by fast-time coding. We underline here that the model lends itself to account for joint fast-time/slow-time coding, wherein a train of sophisticated pulses, each composed of encoded sub-pulses, is amplitude-modulated. The discrete-time model would be similar, with having a Kronecker product structure with entries, and the interference density at the communication system being -times larger. This problem is more challenging, since the constraints on the fast-time code must be included, and will be the subject of our future work.

## Iii Waveform optimization

Determining a general and closed-form solution to (18) for arbitrary appears unwieldy, but a deep insight into the consequences of having the two systems co-exist can be given by considering two important limiting situations, i.e.,

1. Coherent interference, such as, e.g., coherent targets yielding the rank-1 matrix , denoting the all-ones matrix;

2. Incoherent interference, such as scintillating scattering objects with low coherence time, yielding .

Here we consider, for both situations above, the case that the noise impinging on the radar is white. The case of colored noise is handled in Sec. III-B, where the relationship between these two cases is also inspected. Finally, in Sec. III-D, the scope is enlarged beyond the optimization problem in (18) by focusing the attention on the regions of achievable communication rate pairs.

### Iii-a Coherent Interference

When , the communication rate in (II-A) can be rewritten as

 R1(Rx,s) =1Nlogdet⎛⎜⎝IN+|h|2σ2vRx⎛⎝IN+σ2fσ2vssH⎞⎠−1⎞⎟⎠ =1N[logdet(IN+|h|2σ2vRx)+logdet(IN −(IN+|h|2σ2vRx)−1|h|2σ2vRxσ2fσ2vssH1+σ2fσ2v∥s∥2⎞⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥⎦ =1N[logdet(IN+|h|2σ2vRx) −log⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1+σ2fσ2v∥s∥21+σ2fσ2vsH(IN+|h|2σ2vRx)−1s⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥⎦ (19)

where the last equality follows from the fact that , with , whereby, plugging (19) into (13) and (18), the optimization problem can be reformulated as

 maxs,Rx {logdet(IN+|h|2σ2vRx) (20) s.t. σ2asH(σ2gRx+σ2cssH+σ2wIN)−1s≥ρmin 1N∥s∥2≤Pr,1NTr(Rx)≤Pc,Rx⪰0.

For the sake of completeness, here we also discuss the situation when is optimized under given and is optimized under fixed , on the understanding that the main result is the joint design. The proofs can be found in Appendix -B.

#### Iii-A1 Fixed communication codebook

In this case, the communication is given,101010This situation has a purely theoretical interest and is intended to show the difference between joint and single optimization, since in fact a non-coexisting communication system should use . with . The radar system is overlaid, and its waveform must be properly designed by solving (20) with fixed . This problem admits a solution only if

 ρmin≤σ2aNPrσ2gγN+σ2cNPr+σ2w (21)

where is the smallest eigenvalue of , in which case, the optimal radar waveform is

 s∗= ⎷ρmin(σ2gγN+σ2w)σ2a−σ2cρminuN (22)

where is the eigenvector of corresponding to . This situation matches the intuition that the radar transmits, with minimum power compatible with the constraint, in the least-interfered direction of the signal space where.

In this case, the radar waveform is given, with

 ⎧⎪⎨⎪⎩1N∥s∥2≤Prρmin≤σ2a∥s∥2σ2c∥s∥2+σ2w (23)

so that the radar constraints are satisfied when no interfering system is present. Once a communication system is overlaid, the covariance matrix of its codewords should solve (20) with fixed. The solution is

 R∗x=U∗diag(NPc−γ∗NN−1,…,NPc−γ∗NN−1,γ∗N)(U∗)H (24)

with any unitary matrix whose last column is , while the expression of is reported in Appendix -B, Eq. (54). This strategy boils down to splitting the power between the interference-free eigenvectors—where noise whiteness explains the uniform allocation—and the direction of the radar signal. The fraction of power allocated to the interfered direction is dictated by , which depends on the system parameters and constraints.

#### Iii-A3 Joint optimization

The problem is formulated by (20), and admits a solution only if

 ρmin≤σ2aNPrσ2cNPr+σ2w. (25)

In this case, the optimal covariance of the communication system is

 R∗x=U∗diag(NPc−γ∗NN−1,…,NPc−γ∗NN−1,γ∗N)(U∗)H (26)

where is any unitary matrix and is reported in Eq. (60) of Appendix -B, while the optimal radar waveform is

 s∗= ⎷ρmin(σ2gγ∗N+σ2w)σ2a−σ2cρminu∗N (27)

with denoting the last column of . The previous equation clearly shows that, under joint optimization, the structure of the solution in (26) and (27) is similar to (24) and (22).

### Iii-B Incoherent interference

When , the communication rate in (II-A) can be rewritten as

 R1(Rx,s) =1Nlogdet(IN+|h|2Rx(σ2vIN+σ2fSSH)−1) =1Nlogdet(IN+|h|2σ2vRx (28)

whereby, plugging (28) into (13) and (18), the problem becomes

 maxRx,s {(1−β)logdet(IN+|h|2σ2vRx)+βlogdet(IN (29) +|h|2σ2vRx∗diag⎛⎝{(1+σ2fσ2v|si|2)−1}Ni=1⎞⎠⎞⎠} s.t. σ2asH(σ2gRx+σ2cssH+σ2wIN)−1s≥ρmin 1N∥s∥2≤Pr,1NTr(Rx)≤Pc,Rx⪰0.

This situation represents the case that the communication system is affected by scintillating interferers. Also, as shown in [23], it can model the case where, lacking any prior information as to the Doppler frequencies of the objects producing interference on the communication receiver, the Doppler shifts—normalized to —are modeled as uniformly distributed on an interval of amplitude 1.

As shown in Appendix -C, Problem (29) admits a solution only if

 ρmin≤σ2aNPrσ2cNPr+σ2w (30)

in which case, the optimal covariance matrix of the communication system and radar waveform are

 R∗x =diag(NPc−γ∗NN−1,…,NPc−γ∗NN−1,γ∗N) (31a) s∗ = ⎷ρmin(σ2gγ∗N+σ2w)σ2a−σ2cρmin[0⋯01]T (31b)

respectively, where is reported in Eq. (64a) of Appendix -C.

This solution is similar to the one for the coherent scattering in (27) and (26), and results in the same optimized CR. However, the degree of freedom in the choice of the eigenvector matrix is now lost, and this may lead to practical implementation problems. Indeed, the radar waveform in (31b) might not be feasible, since all of the energy is concentrated in a single pulse, and this can break practical limitations on the Peak-to-Average Power Ratio (PAPR). In this case, one could include in the design the additional constraint

 maxn∈{1,…,N}|sn|2≤δ1N∥s∥2 (32)

where is the maximum allowed PAPR. Unfortunately, no closed-form solution to this optimization problem, appears to be available. Nevertheless, one can always approximate (31) as

 R∗x =Udiag(NPc−γ∗NN−1,…,NPc−γ∗NN−1,γ∗N)UH (33a) s∗ = ⎷ρmin(σ2gγ∗N+σ2w)σ2a−σ2cρminu (33b)

where

 u=[√N−δN(N−1)⋯√N−δN(N−1)√δN]T (34)

and is any unitary matrix with as its last column. A discussion of the impact of a PAPR constraint, whether solution (33) is adopted or exact solution is numerically determined, is deferred to Sec. IV

The results of the previous sections are based on the assumption that the radar is affected by white noise. In practical systems, the overall interference contains also a fraction of correlated noise, and in fact radar waveform optimization in colored noise is of great interest in radar community [30, 33]. Under coherent interference and colored noise case, Problem (18) is reformulated as

 maxs,Rx {logdet(IN+|h|2σ2vRx) (35) −βlog⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1+σ2fσ2v∥s∥21+σ2fσ2vsH(IN+|h|2σ2vRx)−1s⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎫⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪⎭ s.t. σ2asH(σ2gRx+σ2cssH+M)−1s≥ρmin 1N∥s∥2≤Pr,1NTr(Rx)≤Pc,Rx⪰0.

As shown in Appendix -D, this problem admits a solution only if

 ρmin≤σ2aNPrσ2cNPr+ϕN (36)

where is the smallest eigenvalue of . In this case, the optimal covariance of the communication system is

 R∗x=U∗diag(NPc−γ∗NN−1,…,NPc−γ∗NN−1,γ∗N)(U∗)H (37)

where is any unitary matrix whose last column is , the latter denoting the eigenvector of corresponding to , and is reported in Eq. (66a) of Appendix -D, while the optimal radar waveform is

 s∗= ⎷ρmin(σ2gγ∗N+ϕN)σ2a−σ2cρminvN. (38)

No closed-form solution is instead available in the case of incoherent interference, and system optimization should rely upon numerical methods.

### Iii-D Achievable communication rates

An important role is played by the region of communication rate pairs, , achievable under the constraints of Problem (18), i.e.,

 S(M) ={(R0(Rx,s),R1(Rx,s)): σ2asH(σ2gRx+σ2cssH+M)−1s≥ρmin, 1NTr(Rx)≤Pc,1N∥s∥2≤Pr,Rx⪰0}. (39)

Knowledge of this region allows determining the optimal transmission policy for any merit function of the form ; since is generally increasing in and , the solution to the optimization problem would be the point on the border of the achievable region that touches the level set corresponding to the largest . Following the proof in Appendix -E, we have following

###### Lemma 1

When , if , then there exists a point , such that .

According to the lemma, for fixed noise power in radar, white Gaussian noise is the worst case for waveform optimization in the presence of coherent scattering.

## Iv Analysis

We consider a communication system that, when operating in a non-coexisting mode, may rely on a received Signal-to-Noise Ratio (SNR) per symbol, , of  dB, so that its maximum rate is simply the capacity, i.e., bits/channel use, and we set . This is the maximum (un-attainable) rate and is the yardstick we compare our results to. As to the radar system, we assume it is designed so that, when operating in non-coexisting mode, under white noise and no clutter, it detects a reference target, located at the -th range cell, and whose RCS is exponentially distributed with average value  m, with probability of false alarm () and probability of detection () equal to and , respectively. Since, for such a Swerling I target,  [34], this conditions corresponds to requiring a cumulated SNR  dB. Similar to , we assume , the variance of the coupling coefficient from the communication transmitter to the radar receiver, equal to . Concerning the interference, we define, at the communication system, the Interference-to-Noise Ratio (INR) as and, at the radar, the Signal-to-Clutter Ratio (SCR) as .

At first, we study the impact, on the communication system, of the constraint forced on the minimum SINR received by the radar. The results, referring to the cases and (representing scattering environments with different densities) are represented in Fig. 4, where the optimum CR is plotted versus for different SCR’s when ,  dB, , and the radar noise is white. We underline here that, in the considered scenario, the limiting performance of the jointly optimal design under coherent and incoherent scattering is the same, whereby the curves corresponding to the optimum CR is unique. Not surprisingly, larger values of turn out to be detrimental in terms of CR; the effect of SCR, instead, is rather dramatic, because it limits the feasibility region of the optimum design, mainly because the SINR constraint cannot be met.

The advantages of a joint optimized design over a non-cooperative approach are outlined in Fig. 5; here we have reported the optimal joint design, the disjoint design, where both systems optimize the respective performance measure ignoring coexistence,111111In this case, the communication system uses the correlation matrix , while the radar, after having estimated the overall disturbance, employes an unmodulated pulse train with the minimum power satisfying the SINR constraint. and the orthogonal design, where the two systems transmit in orthogonal spaces; the other parameters are ,  dB,  dB, , and white radar noise. The results clearly indicate the disjoint design is nearly optimal for small values of , while being catastrophic at larger values of . Also, while for coherent scattering, the conservative approach of transmitting into orthogonal subspaces guarantees a performance level very close to that of the joint design, this is no longer the case for incoherent scattering, since the reflector scintillations spreads the interference at the communication systems over all of the directions of the signal space. Similar trends are observed for higher values of , even though we do not show this curves for space limitation.

Next, we analyze the effect of an additional PAPR constraint: as we have seen in Sec. III, this does not affect the design in the presence of coherent interference, where the eigenvector matrix can be freely chosen, but it may influence the design in the presence of incoherent interference. In the latter case, we resort to numerical methods to derive the solution, and we compare it with the “naïf” solution in (33). The results are shown in Figs. 6, where the optimized CR is reported versus for two values of and , when  dB,  dB, and , assuming white radar noise. In order to reduce the computational burden, we have set . It can be seen that the solution in (33) results in a CR almost coincident with the optimal one, even for the tightest PAPR constraint, , corresponding to a constant amplitude pulse train. We therefore use the solution in (33) to study the trends in CR performance at higher values of . In Fig. 7 the optimized CR is reported versus for and , when ,  dB,  dB, and , assuming white radar noise. Obviously, more stringent constraints on the PAPR (i.e., smaller values of ) result in larger losses in terms of CR with respect to the case where PAPR is unconstrained (). However, the sensitivity of the performance to the PAPR constraint is modest as far as is large enough to allow a substantial reduction of the amount of the interference along the dimensions that the optimum solution would guarantee interference-free.