optimal transmit beamforming for integrated sensing and

1

Optimal Transmit Beamforming for Integrated

Sensing and Communication

Haocheng Hua, Jie Xu, and Tony Xiao Han

Abstract

This paper studies the transmit beamforming in a downlink integrated sensing and communication

(ISAC) system, where a base station (BS) equipped with a uniform linear array (ULA) sends com-

bined information-bearing and dedicated radar signals to simultaneously perform downlink multiuser

communication and radar target sensing. Under this setup, we maximize the radar sensing performance

(in terms of minimizing the beampattern matching errors or maximizing the minimum beampattern

gains), subject to the communication users’ minimum individual signal-to-interference-plus-noise ratio

(SINR) requirements and the BS’s maximum transmit power constraints. In particular, we consider

two types of communication receivers, namely Type-I and Type-II receivers, which do not have and

do have the capability of cancelling the interference from the a-priori known dedicated radar signals,

respectively. Under both Type-I and Type-II communication receivers, the beampattern matching and

minimum beampattern gain maximization problems are non-convex and thus difficult to be optimally

solved in general. Fortunately, via applying the semidefinite relaxation (SDR) technique, we obtain

the globally optimal solutions by rigorously proving the tightness of SDR for both Type-I and Type-

II receivers under the two design criteria. It is shown that at the optimality, dedicated radar signals

are generally not required with Type-I communication receivers under some specific conditions, while

dedicated radar signals are always needed to enhance the performance with Type-II communication

receivers. Numerical results show that the minimum beampattern gain maximization leads to significantly

higher beampattern gains at the worst-case sensing angles, and also has a much lower computational

complexity than the beampattern matching design. It is also shown that by exploiting the capability of

Part of this paper has been submitted to IEEE Global Communications Conference (GLOBECOM), Madrid, Spain, December

7-11, 2021 [1].

H. Hua and J. Xu are with the Future Network of Intelligence Institute (FNii) and the School of Science and Engi-

neering (SSE), The Chinese University of Hong Kong (Shenzhen), Shenzhen, China (e-mail: [email protected],

[email protected]). J. Xu is the corresponding author.

T. X. Han is with the 2012 lab, Huawei, Shenzhen 518129, China (e-mail: [email protected]).

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canceling the interference caused by the dedicated radar signals, the case with Type-II receivers results

in better sensing performance than that with Type-I receivers and other conventional designs.

Index Terms

Integrated sensing and communication (ISAC), transmit beamforming, semidefinite relaxation (SDR),

multiple antennas, uniform linear array (ULA).

I. INTRODUCTION

Future wireless networks are envisaged to support emerging applications such as auto-driving,

virtual/augmented reality (AR/VR), smart home, unmanned aerial vehicles (UAVs), and fac-

tory automation [2], which require ultra reliable and low latency sensing, communication, and

computation. This thus calls for a paradigm shift in wireless networks, from the conventional

communications only design, to a new design with sensing-communication-computation inte-

gration. Towards this end, integrated sensing and communication (ISAC)1 [3]–[17], has recently

attracted growing research interests in both academia and industry, in which the radar sensing

capabilities are integrated into wireless networks (e.g., 6G cellular [18] and WiFi 7 [19], [20])

for a joint communications and sensing design. On one hand, with such integration, ISAC can

significantly enhance the spectrum utilization efficiency and cost efficiency by exploiting the

dual use of radio signals and infrastructures for both radar sensing and wireless communication

[18]. On the other hand, due to the advancements of millimeter wave (mmWave)/terahertz

(THz), wideband transmission (on the order of 1 GHz), and massive antennas in wireless

communications, ISAC can reuse the communication signals to provide high sensing resolution

(in both range and angular) and sensing accuracy (in detection and estimation) to meet the

stringent sensing requirements of emerging applications. In the industry, Huawei and Nokia

envisioned ISAC as one of the key technologies for 6G [18], [21], and IEEE 802.11 formed

the WLAN Sensing Task Group IEEE 802.11bf in September 2020, with the objective of

incorporating wireless sensing as a new feature for next-generation WiFi systems (e.g., Wi-Fi

7) [19], [20].

While ISAC is a relatively new research topic that emerged recently in communications

society, the interplay between wireless communications and radar sensing has been extensively

1In the literature, the ISAC is also referred to as joint radar communications (JRC) [3], joint communication and radar (JCR)

[4], dual-functional radar communications (DFRC) [5], [6], radar-communications (RadCom) [7], and joint communication and

radar/radio sensing (JCAS) [2].

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investigated in the literature, some examples of which include the communication and radar

spectrum sharing (CRSS) [11], [22]–[32] and the radar-centric communication integration [5],

[22], [33]–[41]. In particular, CRSS features the spectrum sharing between separate radar sensing

and communication systems, for which a large amount of research efforts have been put on

managing the inter-system interference via, e.g., interference null-space projection [24], [25],

opportunistic spectrum sharing [26], and transmit beamforming optimization [11], [27]–[31].

However, CRSS generally has limited performance in sensing and communication due to the

limited coordination and information exchange between them [2]. In another line of research

called radar-centric communication integration, the radar signals are revised to convey informa-

tion, thus incorporating communication capabilities in conventional radar systems. For instance,

the authors in [33]–[35] incorporated continuous phase modulation (CPM) into the conventional

linear frequency modulation (LFM) radar waveform to convey information, [40], [41] embedded

the information bits via controlling the amplitude and phase of radar side-lobes and waveforms,

and [36]–[39] modified the radar waveforms via index modulation to represent information by

the indexes of antennas, frequencies, or codes instead of directly modifying the radar waveforms.

However, the radar-centric communication integration has limited achievable data rates as the

radar signaling is highly suboptimal for data transmission in general.

Different from the CRSS and radar-centric communication integration, the ISAC shares the

same wireless infrastructures for radar sensing and communication, and optimizes the transceivers

design for both objectives, thus further enhancing both sensing and communication performance.

For example, the authors in [7]–[10] exploited the orthogonal frequency division multiplexing

(OFDM) waveform for probing and sensing. In particular, thanks to its great success in wireless

communications [42], [43] and radar sensing [44], [45], the multi-antenna or multiple-input

multiple-output (MIMO) technique has played a key role in ISAC by exploiting the spatial

degrees of freedom (DoF) via transmit beamforming to perform multi-target sensing and multi-

user communication simultaneously (see, e.g., [6], [11]–[17]). The benefit of MIMO may become

even more considerable in 5G and 6G, due to the employment of massive MIMO [46] and

mmWave/THz [47]. For instance, the authors in [13], [14] considered that one transmitter needs

to simultaneously communicate with one receiver and sense one target, in which two transmit

beamformers are designed towards the communication receiver and sensing target, respectively.

Then, [6], [11], [12], [15]–[17] considered the case with multiple receivers and multiple sensing

targets. To be more specific, [6] optimized the beamformers to minimize the cross interference

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between radar sensing and communication at the base station (BS), while matching the desired

transmit beampattern for sensing. [11], [12] further addressed the beampattern matching prob-

lem subject to the signal-to-interference-plus-noise ratio (SINR) constraints at communication

receivers and the per-antenna transmit power constraints at the BS. Most recently, [15] studied

the joint information multicasting and radar sensing, and [17] investigated the full-duplex hybrid

beamforming for ISAC.

How to design the transmit signals/waveforms and optimize the associated transmit beam-

forming is an essential issue to be dealt with in ISAC with multiple antennas. In [11], the

authors considered a conventional design by directly reusing the communication waveforms

for sensing and solved the SINR-constrained beampattern matching problems sub-optimally by

the techniques of semidefinite relaxation (SDR) together with Gaussian randomization [48].

Such conventional designs, however, may only have limited DoF in transmitted signals and

thus could result in transmit beampattern distortion, especially when the number of users is

smaller than that of transmit antennas [12]. To deal with this issue, [12] proposed to enhance the

sensing capability by sending dedicated radar signals combined with communication signals. By

treating the interference from radar signals as noise at communication receivers, [12] obtained the

globally optimal beamforming solution to the SINR-constrained beampattern matching problem

by showing the tightness of SDR. Despite the research progress in [12], it is still unknown

whether the dedicated radar signals are beneficial under optimized beamforming designs, by

considering more general scenarios with different design criteria and receiver structures (e.g.,

with or without interference cancellation), as well as specific channel conditions. This thus

motivates the investigation in this work.

This paper studies the transmit beamforming design in a downlink ISAC system, in which a

single BS equipped with an uniform linear array (ULA) sends combined information-bearing and

dedicated radar signals to perform downlink multiuser communication and radar targets sensing

simultanesously. Different from prior works, we consider two types of communication receivers

without and with the capability of canceling the interference from the a-priori known dedicated

radar signals, which are referred to as Type-I and Type-II receivers, respectively. Furthermore,

we consider two design criteria for radar target sensing, namely the beampattern matching and

the minimum beampattern gain maximization, respectively. Under such setups, our main results

are listed as follows.

• First, our objective is to design the joint information and radar transmit beamforming to

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minimize the beampattern matching error or maximize the minimum beampattern gain

at the sensing angles of interest, subject to the minimum SINR constraints at individual

communication receivers and the transmit power constraint at the BS. The formulated

beampattern matching and minimum beampattern gain maximization problems with both

Type-I and Type-II receivers are non-convex and thus difficult to be optimally solved in

general.

• Next, by applying the SDR technique [48], we obtain the globally optimal solutions to all

these four problems by rigorously proving that their SDRs are always tight. At the optimal

solution under both design criteria, we have rigorously proved that with Type-I receivers,

there is no need to add dedicated radar signals when the wireless communication channels

are line-of-sight (LOS); while with Type-II receivers, dedicated radar signals are always

beneficial in enhancing the sensing and communication performance.

• Numerical results are provided to validate the performance of the proposed designs. It is

shown that the minimum beamforming gain maximization design yields enhanced worst-

case radar sensing performance at the interested angles, and also enjoys significantly lower

computational complexity in terms of execution time than the beampattern matching design.

In addition, it is shown that the case with Type-II receivers leads to significantly enhanced

radar sensing performance as compared to that with Type-I receivers, thanks to the employ-

ment of dedicated radar signals at the BS together with the radar interference cancellation

at the receivers. Futhermore, it is observed that under Type-I receivers with high SINR

requirements, dedicated radar signal is not needed even when the wireless channels follow

Rayleigh fading.

The remainder of this paper is organized as follows. Section II presents the system model.

Sections III and IV study the transmit beamforming designs for SINR-constrained beampattern

matching and minimum beampattern gain maximization problems, respectively. Section V pro-

vides numerical results to validate the performance of our proposed designs. Section VI concludes

this paper.

Notations: Boldface letters refer to vectors (lower case) or matrices (upper case). For a square

matrix S, tr(S) and S−1 denote its trace and inverse, respectively, while S � 0, S � 0, S ≺ 0

and S � 0 mean that S is positive semidefinite, negative semidefinite, negative definite, and

non-positive semidefinite, respectively. For an arbitrary-size matrix M , rank(M ), M †, MH ,

and MT denote its rank, pseudoinverse, conjugate transpose, and transpose, respectively, and

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Figure 1. Illustration of the considered downlink ISAC system.

M ik denotes the element in the ith row and kth column of M . I and 0 denote an identity matrix

and an all-zero matrix, respectively, with appropriate dimensions. The distribution of a circularly

symmetric complex Gaussian (CSCG) random vector with mean vector x and covariance matrix

Σ is denoted by CN (x,Σ); and ∼ stands for “distributed as”. Cx×y denotes the space of x× y

complex matrices. R denotes the set of real numbers. E(·) denotes the statistical expectation.

‖x‖ denotes the Euclidean norm of a complex vector x. |z| and z∗ denote the magnitude of a

complex number z and its complex conjugate, respectively. For two real vectors x and y, x ≥ y

means that x is greater than or equal to y in a component-wise manner.

II. SYSTEM MODEL

We consider a downlink ISAC system as shown in Fig. 1, in which a BS sends wireless

signals to perform radar sensing towards potential targets and downlink communication with K

users simultaneously. The BS is equipped with a ULA with N > 1 antennas, and each user is

equipped with a single antenna.

In order to perform sensing and communication at the same time, we suppose that the BS uses

transmit beamforming to send information-bearing signals sk’s for the K users, together with a

dedicated radar signal s0 ∈ CN×1. Let K , {1, ..., K} denote the set of communication users.

The transmitted information signals {sk}Kk=1 are assumed to be independent random variables,

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with zero mean and unit variance, while the radar signal s0 has zero mean and covariance

matrix Rd = E[s0sH0 ] � 0 and is independent from {sk}Kk=1. Let tk ∈ CN×1 denote the transmit

beamforming vector for user k ∈ K. By combining sk’s and s0, the transmit signal x ∈ CN×1

by the BS is expressed as

x =K∑k=1

tksk + s0. (1)

Notice that different from the single-beam transmission for information-bearing signals, we

consider the general multi-beam transmission for the dedicated radar signal s0 (i.e., Rd is

with general rank). In particular, suppose that rank(Rd) = Ns. This then corresponds to a

set of Ns radar signal beams, which can be obtained via the eigenvalue decomposition (EVD)

of Rd. In this case, the sum transmit power at the BS is given as E[‖∑K

k=1 tksk + s0‖2]

=∑Kk=1 ‖tk‖2 + tr(Rd). Suppose that the maximum sum transmit power budget at the BS is P0.

Then we have the following power constraint2

K∑k=1

‖tk‖2 + tr(Rd) ≤ P0. (2)

First, we consider the radar target sensing, where the transmit beampattern is the key per-

formance metric of our interest. The transmit beampattern generally depicts the transmit signal

power distribution with respect to the sensing angle θ ranging from [−π2, π2]. In our considered

ISAC system with both radar and information signals, the resultant transmit beampattern gain

is expressed as [49]

P(θ) = E

[|aH(θ)(

K∑k=1

tksk + s0)|2]

= aH(θ)

(K∑k=1

tktHk + Rd

)a(θ), (3)

where

a(θ) = [1, ej2πdλsin θ, ..., ej2π

dλ(N−1) sin θ]T (4)

denotes the steering vector at angle θ, with λ and d denoting the carrier wavelength and the

spacing between two adjacent antennas, respectively. In practice, the transmit beampattern is

designed based on the radar target sensing requirements. For instance, if we need to perform

the detection task without knowing the direction of potential targets, a uniformly distributed

beampattern is desired. By contrast, if we roughly know the directions of the targets, e.g.,

2Notice that there are other types of power constraints that have been adopted in the previous literatures such as per-antenna

power constraints [11], [12]. In this paper, we focus on the sum-power constraint as it has been widely adopted in wireless

communication networks.

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for target tracking, we only need to maximize the beampattern gains towards these interested

potential directions [49].

Next, we consider the information reception at communication users. By letting hi denote the

channel vector from the BS to each user i ∈ K, the received signal at user i is then given as

yi = hHi x + ni = hHi (K∑k=1

tksk + s0) + ni, (5)

where ni ∼ CN (0, σ2i ) denotes the additive white Gaussian noise (AWGN) at the receiver of

user i. It is observed in (5) that each user i suffers from the interference induced by both the

information signals {sk}k 6=i, as well as the dedicated radar signal s0. Notice that s0 is pre-

determined sequences that can be a priori known by both the BS and the users. Therefore, we

consider the following two different types of communication users, namely Type-I and Type-II

receivers, which do not have and do have the capability of cancelling the resultant interference

by the radar signals s0, respectively.

• Type-I receivers (e.g., legacy users) do not have the capability of canceling the interference

of the radar signals before decoding its desirable information signal. In this case, the SINR

at receiver i ∈ K is

γ(I)i ({ti},Rd) =

∣∣hHi ti∣∣2∑k∈K,k 6=i

∣∣hHi tk∣∣2 + hHi Rdhi + σ2i

,∀i ∈ K. (6)

• Type-II receivers are dedicatedly designed for the ISAC system to have the capability of

perfectly canceling the interference caused by the dedicated sensing signal s0. In this case,

the SINR at receiver i ∈ K is

γ(II)i ({ti}) =

∣∣hHi ti∣∣2∑k∈K,k 6=i

∣∣hHi tk∣∣2 + σ2i

,∀i ∈ K. (7)

For the purpose of illustration, Figs. 2(a) and 2(b) show the ISAC system designs with Type-I

and Type-II receivers, respectively. For comparison, Fig. 2(c) shows the conventional design

without s0 [11]. Note that our consideration with both Type-I and Type-II receivers is new as

compared to [12], in which only Type-I receivers are considered3.

It is assumed that the BS perfectly knows all the channel vectors hi’s and the desirable beam-

3Notice that there is one parallel work [16] that also considered both Type-I and Type-II receivers for ISAC with rate-splitting

multiple access. However, this paper is significantly different from [16] in terms of the design optimization problems (SINR-

constrained beampattern matching and minimum beampattern gain in this paper versus the weighted sum-rate and beampattern

matching error in [16]) and solution approaches (the SDR for obtaining optimal solutions in this paper versus the alternating

direction method of multipliers (ADMM) for obtaining generally suboptimal solutions in [16]).

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Figure 2. Illustration of different ISAC transceivers designs: (a) Type-I receivers with dedicated radar signals at the BS; (b)

Type-II receivers with dedicated radar signals at the BS; (c) the conventional design without dedicated radar signal at the BS

[11].

pattern or interested sensing directions, while each user perfectly knows its own channel vector.

This assumption is made to characterize the fundamental performance upper bounds and gain

essential design insights for practical scenarios when such information is not perfectly known.

Under this setup, our objective is to design the joint sensing and communication beamforming

to maximize the radar sensing performance while ensuring the communication quality of service

(QoS) requirements at each user, as will be illustrated in the next two sections in more detail.

III. SINR-CONSTRAINED BEAMPATTERN MATCHING

In this section, we jointly optimize the information beamforming vectors tk,∀k ∈ K, and the

dedicated radar signal covariance matrix Rd, to minimize the matching error between the overall

transmit beampattern and a given desired beampattern, subject to the sum power constraint at

the BS and a set of minimum SINR constraints at communication users.

A. Problem Formulation

Let {P (θm)}Mm=1 denote a pre-designed beampattern, which specifies the desired transmit

power distribution at the M angles {θm}Mm=1 in the space. For target detection tasks, {P (θm)}Mm=1

can be uniformly distributed [49], while for target tracking, it can be non-zero constant at

interested angles with potential targets and zero elsewhere [11]. Let Γi ≥ 0 denote the minimum

SINR requirement at each receiver i ∈ K. Our objective is to minimize the beampattern

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matching error, for which the problem is formulated as follows with Type-I and Type-II receivers,

respectively.

(P1) : min{tk},Rd,α

M∑m=1

∣∣∣∣∣αP (θm)− aH(θm)(K∑k=1

tktHk + Rd)a(θm)

∣∣∣∣∣2

(8a)

s.t. γ(I)i ({ti},Rd) ≥ Γi, ∀i ∈ K (8b)

K∑k=1

||tk||2 + tr(Rd) = P0 (8c)

Rd � 0. (8d)

(P2) : min{tk},Rd,α

M∑m=1

∣∣∣∣∣αP (θm)− aH(θm)(K∑k=1

tktHk + Rd)a(θm)

∣∣∣∣∣2

(9a)

s.t. γ(II)i ({ti}) ≥ Γi, ∀i ∈ K (9b)

K∑k=1

||tk||2 + tr(Rd) = P0 (9c)

Rd � 0. (9d)

Here, α denotes a scaling factor and a (θm) denotes the steering vector at direction θm as in

(4). In problems (P1) and (P2), we consider the equality power constraints in (8c) and (9c) in

order for the BS to use up all the transmit power to maximize the radar performance for target

tracking or detection [45]. Notice that both problems (P1) and (P2) are non-convex and thus are

difficult to be optimally solved in general. Fortunately, we will use the SDR technique [48] to

obtain the optimal solutions, as will be shown next.

Remark 1. Notice that our considered problem (P1) is different from the beampattern matching

problem with Type-I receivers in [12] in the following two aspects. First, this paper considers

the beampattern matching error as the objective function, while [12] additionally considered

the mean-squared cross correlation pattern, thus leading to different solution structures (see

Proposition 4). Second, this paper considers the sum-power constraint at the BS (versus the

per-antenna constraints in [12]).

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B. Proposed Solution via SDR

In this subsection, we propose to use the SDR technique to solve problems (P1) and (P2).

Towards this end, we introduce new auxilary varaibles T k = tktHk ,∀k ∈ K, where T k � 0 and

rank(T k) = 1. In this case, problems (P1) and (P2) are equivalently reformulated as (P1.1) and

(P2.1) in the following, respectively.

(P1.1) : min{Tk},Rd,α

f1({Tk},Rd, α) =M∑m=1

∣∣∣∣∣αP (θm)− aH(θm)(K∑k=1

T k + Rd)a(θm)

∣∣∣∣∣2

(10)

s.t.tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− tr

(hih

Hi Rd

)− σ2

i ≥ 0,∀i ∈ K

K∑k=1

tr(T k) + tr(Rd) = P0

Rd � 0,T k � 0, rank(T k) = 1,∀k ∈ K

(P2.1) : min{Tk},Rd,α

f1({Tk},Rd, α) =M∑m=1

∣∣∣∣∣αP (θm)− aH(θm)(K∑k=1

T k + Rd)a(θm)

∣∣∣∣∣2

(11)

s.t.tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− σ2

i ≥ 0,∀i ∈ K

K∑k=1

tr(T k) + tr(Rd) = P0

Rd � 0,T k � 0, rank(T k) = 1,∀k ∈ K

However, problems (P1.1) and (P2.1) are still non-convex due to the rank-one constraints on

T k’s. To deal with this issue, we relax the rank-one constraints and accordingly get the SDRs

of problems (P1.1) and (P2.1), denoted as (SDR1) and (SDR2), respectively. Notice that both

(SDR1) and (SDR2) are convex and thus can be solved optimally by convex optimization solvers

such as CVX [50]. Suppose that the ranks of the obtained communication covariance matrices

T k’s in the optimal solutions to (SDR1) and (SDR2) are all equal to 1, then they are also

optimal for (P1.1) and (P2.1), respectively. Otherwise, we should further perform the Gaussian

randomization [48] to construct rank-one solutions that are generally sub-optimal. Fortunately,

the following propositions show that there always exist optimal rank-one solutions to (SDR1)

and (SDR2), and thus the Gaussian randomization is not needed in general.

Proposition 1. There always exists a globally optimal solution to problem (SDR1), denoted as

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{{T k}, Rd, α}, such that

rank(T k) = 1,∀k ∈ K.

Proof. Notice that problem (P1.1) has a similar structure as the joint information and radar

beamforming problem in [12], (30), with slight modifications in the objective function (by

eliminating the cross correlation pattern term) and the power constraints (sum-power constraint

versus per-antenna power constraints in [12]). In this case, Proposition 1 can be similarly proved

as in [12, Theorem 1], for which the details are omitted for brevity.

Proposition 2. There always exists a globally optimal solution to problem (SDR2), denoted as

{{T k}, Rd, α}, such that

rank(T k) = 1,∀k ∈ K.

Proof. See Appendix A.

Remark 2. By comparing problems (P1.1) and (P2.1), it is observed that every feasible solution

to problem (P1.1) is also feasible to problem (P2.1) but not vice versa. Therefore, it is evident

that problem (P2.1) always yields an equal or smaller beampattern matching error than (P1.1)

due to the enlarged feasible region. Based on this observation together with the fact that problems

(P1.1) and (P2.1) are identical under Rd = 0, it follows that dedicated radar signals are always

beneficial in enhancing the joint radar sensing and communication performance under Type-II

receivers with the capability of radar interference cancellation (as will be shown in Section V).

C. Properties of the Optimal Beamforming Solution to Problem (P1)

While Remark 2 shows that dedicated radar signals are beneficial for problem (P2) with Type-

II receivers, it still remains unknown whether they are beneficial for problem (P1) with Type-I

receivers. To answer this question for gaining insights, this subsection analyzes the properties of

the optimal transmit beamforming solution to problem (P1) or (P1.1) or (SDR1). To facilitate the

analysis, we introduce problem (P3) as the beampattern matching problem in the case without

dedicated radar signals, which can be expressed similarly as for (P1) and (P2) by setting Rd = 0.

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In this case, by introducing T k = tktHk , with T k � 0 and rank(T k) ≤ 1, ∀k ∈ K, we can

equivalently re-express problem (P3) as

(P3.1) : min{Tk},α

f3({Tk}, α) =M∑m=1

∣∣∣∣∣αP (θm)− aH(θm)(K∑k=1

T k)a(θm)

∣∣∣∣∣2

(12)

s.t.tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− σ2

i ≥ 0, ∀i ∈ K

K∑k=1

tr(T k) = P0

T k � 0, rank(T k) = 1,∀k ∈ K.

By removing the rank-one constraints on T k’s, we have the SDR of (P3.1) as (SDR3). Then we

have the following proposition.

Proposition 3. Suppose that {{T k}, Rd, α} is an optimal solution to problem (SDR1). Then

there always exist an optimal solution to (SDR3), denoted by {{T k}, α} with

Tk = Tk + βkRd,K∑k=1

βk = 1, βk ≥ 0, (13)

α = α, (14)

whereK∑k=1

Tk + Rd =K∑k=1

Tk. (15)

As a result, {{T k}, α} in (SDR3) achieves the same optimal beampatterns (and the same optimal

value) as that achieved by {{T k}, Rd, α} in (SDR1).

Proof. See Appendix B.

Based on Proposition 3 together with the SDR tightness between (P1.1)/(P1) and (SDR1) in

Proposition 1, it follows that problem (P1) actually achieves the same beampattern (and thus

the same objective value) as that by (SDR3). Nevertheless, the SDR tightness between (P3) and

(SDR3) may not hold in general, due to the limited DoF of the transmitted signals, especially

when the number of users becomes small [12]. Therefore, problem (P1) with dedicated radar

signals and Type-I receivers (or equivalently (SDR1)) can generally achieve better performance

than (P3) without dedicated radar signals, in terms of lower beampattern matching errors.

Nevertheless, under specific conditions (see Proposition 4), the SDR of (P3) is indeed tight

(i.e., (P3) and (SDR3) achieve the same optimal values), and thus (P1) and (P3) achieve the

14

same optimal matching error. In other words, the dedicated radar signals are not needed in this

special case.

Proposition 4. When the wireless channels from the BS to the communication users are LOS

(i.e., hi is given in the form of hi = [1, ejφi , ..., ej(N−1)φi ]T with φi = 2π dλ

sin(θi), where λ and

d denotes the carrier wavelength and the spacing between two adjacent antennas, respectively),

the SDR of problem (P3.1) (or equivalent (P3)) is tight.

Proof. See Appendix C.

IV. SINR-CONSTRAINED MINIMUM BEAMPATTERN GAIN MAXIMIZATION

In this section, we propose an alternative radar sensing design criterion termed minimum

beampattern gain maximization, based on which our objective is to maximize the minimum

beampattern gain illuminated at desirable directions of potential sensing targets, subject to SINR

constraints for individual communication users. We will show that this design leads to a better

sensing beampattern at lower computation complexity than the beampattern matching design in

Section III.

A. Problem Formulation

In this design, our objective is to maximize the minimum beampattern gain at a given set

of interested angles Θ , {θ1, θ2, ..., θQ}, where Q denotes the number of quantized angles of

interest. For instance, for target detection tasks without knowing any prior information about

targets’ potential locations, {θq}Qq=1 may correspond to the uniformly quantized angles ranging

from −π2

to π2. For target tracking tasks, the interested angles are specified based on the

targets’ potential locations (similarly as the angles with non-zero entries in the pre-designed

beampattern P(θm) in Section III). In this case, the SINR-constrained minimum beampattern

gain maximization problems are formulated as (P4) and (P5) in the following, by considering

15

Type-I and Type-II communication receivers, respectively.

(P4) : maxt,Rd,{tk}

t (16a)

s.t. aH (θ)

(Rd +

K∑k=1

tktHk

)a (θ) ≥ t, ∀θ ∈ Θ (16b)

γ(I)i ({ti},Rd) ≥ Γi, ∀i ∈ K (16c)

K∑k=1

||tk||2 + tr(Rd) ≤ P0 (16d)

Rd � 0. (16e)

(P5) : maxt,Rd,{tk}

t (17a)

s.t. aH (θ)

(Rd +

K∑k=1

tktHk

)a (θ) ≥ t, ∀θ ∈ Θ (17b)

γ(II)i ({ti}) ≥ Γi, ∀i ∈ K (17c)

K∑k=1

||tk||2 + tr(Rd) ≤ P0 (17d)

Rd � 0. (17e)

In (16d) and (17d), we consider the inequality power constraints, and it can be easily shown that

the optimality of problems (P4) and (P5) should be attained when they are met with equality.

Notice that problems (P4) and (P5) are both non-convex. In the following, we will adopt the

SDR technique to obtain their optimal solutions.

B. Proposed Solution via SDR

Similarly as in Section III, we define T k = tktHk with T k � 0 and rank(T k) ≤ 1, ∀k ∈

K. Accordingly, problems (P4) and (P5) can be reformulated as (P4.1) and (P5.1) as follows,

16

respectively.

(P4.1) : maxt,Rd,{Tk}

t (18)

s.t. aH (θ)

(Rd +

K∑k=1

T k

)a (θ) ≥ t,∀θ ∈ Θ

tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− tr

(hih

Hi Rd

)− σ2

i ≥ 0,∀i ∈ K

tr

(K∑k=1

T k + Rd

)≤ P0

Rd � 0,T k � 0, rank (T k) = 1, ∀k ∈ K

(P5.1) : maxt,Rd,{Tk}

t (19)

s.t. aH (θ)

(Rd +

K∑k=1

T k

)a (θ) ≥ t,∀θ ∈ Θ

tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− σ2

i ≥ 0,∀i ∈ K

tr

(K∑k=1

T k + Rd

)≤ P0

Rd � 0,T k � 0, rank (T k) = 1, ∀k ∈ K

Remark 3. Similar as Remark 1, it is evident that (P5)/(P5.1) with Type-II communication

receivers always yields an equal or greater minimum beampattern gains at interested angles than

(P4)/(P4.1) with Type-I receivers, due to the enlarged feasible region.

By removing the rank-one constraints, we have the SDRs of (P4.1) and (P5.1) as (SDR4) and

(SDR5), respectively, which are both separable semidefinite programming (SSDP) problems that

can be solved efficiently by CVX. Then we can show that the two SDRs are indeed tight in the

following proposition.

Proposition 5. There always exists globally optimal solutions to problems (SDR4) and (SDR5),

which are denoted as {{T k}, Rd, α}and {{T k}, Rd, α}, respectively, such that

rank(T k) = 1,∀k ∈ K,

rank(T k) = 1,∀k ∈ K.

17

Proof. Similar as the proof for Propositions 1 and 2, suppose that {{T ∗k},R∗d, α∗}({{T k}, Rd, α})

is one optimal solution to SDR4 (SDR5), which does not necessarily meet the rank-one con-

straints. As shown in Appendix A, we can always construct an alternative optimal solution

{{T k}, Rd, α} ({{T k}, Rd, α}) meeting the rank-one constraints, such that the same minimum

beampattern gains can be achieved by {{T k}, Rd, α} ({{T k}, Rd, α}). As the detailed proof

procedure is similar as that in Appendix A, the details are skipped for brevity.

Following Proposition 5, it is evident that we can always obtain rank-one optimal solutions

to (P4.1) and (P5.1) by solving (SDR4) and (SDR5). As a result, the optimal solutions to (P4)

and (P5) can be obtained accordingly.

Besides the SDR tightness shown in Proposition 5, we can also show the interesting properties

on the optimal beamforming solutions to problem (P4) with Type-I receivers. To facilitate

the illustration, we introduce the minimum beampattern gain maximization problem without

radar signals, denoted by (P6), which corresponds to problem (P4)/(P5) by setting Rd = 0.

Furthermore, we can reformulate (P6) as (P6.1) by introducing T k = tktHk with T k � 0 and

rank(T k) ≤ 1, ∀k ∈ K as follows.

(P6.1) : maxt,{Tk}

t, (20)

s.t. aH (θ)

(K∑k=1

T k

)a (θ) ≥ t,∀θ ∈ Θ

tr(hih

Hi Ti

)Γi

−∑

k 6=i,k∈K

tr(hih

Hi Tk

)− σ2

i ≥ 0,∀i ∈ K

tr

(K∑k=1

T k

)= P0

T k � 0, rank (T k) = 1, ∀k ∈ K

By removing the rank-one constraints, the SDR of (P6.1)/(P6) is denoted as (SDR6). Then

we have the following two propositions, for which the detailed proofs are similar to those for

Propositions 3 and 4, and thus are omitted for brevity.

Proposition 6. Suppose that {{T k}, Rd, α} is an optimal solution to problem (SDR4). Then

18

there always exist an optimal solution to (SDR6), denoted by {{T k}, α} with

Tk = Tk + βkRd,

K∑k=1

βk = 1, βk ≥ 0, (21)

α = α, (22)

whereK∑k=1

Tk + Rd =K∑k=1

Tk. (23)

As a result, {{T k}, α} in (SDR6) achieves the same optimal beampatterns (and the same optimal

value) as that achieved by {{T k}, Rd, α} in (SDR4).

Proposition 7. When the wireless channels from the BS to the communication users are LOS

(i.e., hi is given in the form of hi = [1, ejφi , ..., ej(N−1)φi ]T with φi = 2π dλ

sin(θi)), the SDR of

problem (P6.1) (or equivalent (P6)) is tight.

Proposition 6 shows that the optimal beampatterns achieved by (SDR4) and (SDR6) are

identical. Based on Proposition 6 together with the SDR tightness between (P4.1)/(P4) and

(SDR4) in Proposition 5, it follows that problem (P4) actually achieves the same beampattern

(and thus the same objective value) as that by (SDR4). Nevertheless, the SDR tightness between

(P6) and (SDR6) may not hold in general. Therefore, problem (P4) with dedicated radar signals

and Type-I receivers (or equivalently (SDR4)) can generally achieve better performance than (P6)

without dedicated radar signals, in terms of larger minimum beampattern gain at Θ. Nevertheless,

when the wireless channels are LOS, the SDR of (P6) is indeed tight (i.e., (P6) and (SDR6)

achieve the same optimal values), and thus (P4) and (P6) achieve the same minimum beampattern

gains. In other words, the dedicated radar signals are not needed in this case.

C. Complexity Analysis

In this subsection, we analyze the computational complexity for solving problem (SDR1)

/(SDR2) versus that for (SDR4)/(SDR5) in order to show the benefit of the newly proposed mini-

mum beampattern gain maximization design. Notice that problems (SDR1) and (SDR2) are both

quadratic semidefinite programs (QSDPs). Given a solution accuracy ε, the worst-case complexity

to solve the QSDP with the primal-dual interior-point method is O(K6.5N6.5 log(1/ε)) [12]. By

contrast, problems (SDR4) and (SDR5) are SSDPs. It follows from [48] that given a solution

accuracy ε, the worst-case complexity to solve the SSDP with the primal-dual interior-point

19

method is O(max{KN+N+1, Q+K+1}4(KN+N+1)1/2 log(1/ε)) ≈ O(K4.5N4.5 log(1/ε)),

which is significantly less than that for solving (SDR1) and (SDR2). This shows the benefit of

the proposed minimum beampattern gain maximization over the beampattern matching design,

in terms of the computational complexity.

V. NUMERICAL RESULTS

In this section, we provide numerical results to validate the performance of our proposed

transmit beamforming designs. We set P0 = 0.1 Watt (W) or 20 dBm and σ2i = −70 dBm.

Without loss of generality, we set Γi = Γ,∀i ∈ K. A passloss of 80 dB is assumed between the

BS and each communication receiver.

A. SINR-Constrained Beampattern Matching

This subsection considers the beampattern matching, in which we specify M = 101 discrete

sensing angles ranging from [−π2

: π100

: −π2]. Figs. 3(a) and 3(b) show the beampattern matching

error versus the SINR threshold Γ in the case with Rayleigh fading channels and LOS channels,

respectively, in which we set N = 8 and K = 5 and the results are obtained by averaging over

50 random channel realizations. Besides the proposed joint beamforming design with Type-I and

Type-II receivers via solving problems (P1)/(SDR1), and (P2)/(SDR2), for comparison, we also

consider the performance upper bound with radar sensing only (i.e., beampattern matching in

(P1)/(P2) with the SINR constraints omitted) and the benchmark design without radar signal via

solving (SDR3) (with Rd = 0). In addition, we further consider the two-stage design approach

without radar signals in [11] for comparison, where in the first stage, the optimal radar transmit

beamforming without SINR constraints is obtained to minimize the beampattern matching error,

while in the second stage, the information beamforming vectors are optimized via SDR to

minimize its difference with the obtained optimal radar transmit beamforming subject to SINR

constraints4.

In Fig. 3, it is observed that the proposed joint design with Type-II receivers achieves much

lower matching error than the other designs under both channel conditions, especially when the

SINR threshold Γ becomes high. This validates the necessity of adding dedicated radar signals

4Notice that in [11], Gaussian randomization is employed to obtain a feasible but sub-optimal rank-one information

beamforming solution. Here, we directly adopt the high-rank solution before the Gaussian randomization for the purpose of

comparison only, which generally achieves better performance or a lower average matching error than that in [11].

20

-6 -3 0 3 6 9 12 15 18 21 24

SINR Threshold (dB)

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Ave

rag

e M

atc

hin

g e

rro

r

Two-stage design w/o Rd

Benchmark w/o Rd via (SDR3)

Joint design w/ Type I receiver

Joint design w/ Type II receiver

Radar sensing only

(a) The case with Rayleigh fading channels

-6 -3 0 3 6 9 12 15 18 21 24

SINR Threshold (dB)

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Ave

rag

e M

atc

hin

g e

rro

r



Joint design w/ Type I receiver


Radar sensing only

(b) The case with LOS channels

Figure 3. Average beampattern matching error versus the SINR threshold Γ.

under Type-II receivers. It is also observed that the performance achieved by the proposed

design with Type-I receiver (via solving (P1)/(SDR1)) is exactly same as that by (SDR3), which

is consistent with Proposition 3. Besides, under Rayleigh fading channels, it is observed in the

simulations that the solution to (SDR1) satisfies that Rd = 0 and rank(T i) = 1,∀i ∈ K when

Γ ≥ 6dB and thus there is no need to add dedicated radar signal in this scenario. This is similar to

the scenario with LOS channels, where it follows from Propositions 3 and 4 that there is no need

to add dedicated radar signal over the whole SINR regimes with Type-I receivers. Furthermore, it

is observed that our proposed joint designs with both Type-I and Type-II receivers achieve much

smaller average beampattern matching error than the two-stage design without radar signals over

the whole SINR range.

-80 -60 -40 -20 0 20 40 60 80

Angle (Degree)

0

0.05

0.1

0.15

0.2

0.25

0.3

Tra

nsm

it B

ea

mp

att

ern



Joint design w/ Type-I receiver

Joint design w/ Type-II receiver

Figure 4. Obtained transmit beampattern via the beampattern matching under Rayleigh fading channels with Γ = 20dB.

21

Fig. 4 shows the obtained transmit sensing beampattern under Γ = 20 dB by considering

Rayleigh fading channels. It is observed that our proposed designs with both Type-I and Type-II

receivers allocate the transmit power into desired directions specified by the desired transmit

beampattern, while the two-stage separate design leads to more undesired power leakage into

uninterested anlges. Furthermore, the proposed joint design with Type-II receivers is observed to

outperform that with Type-I receiver, with higher power in interested regions and lower leakage

in uninterested angles. The benchmark scheme without Rd (via solving SDR3) is observed to

have exactly the same beampattern as the joint design with Type-I receivers. This is consistent

with Proposition 3.

B. SINR-Constrained Minimum Beampattern Gain Maximization

In this subsection, we consider the designs with minimum beampattern gain maximization.

Instead of considering the specified M point angles in Section V-A, here we only need to

choose a subset of them (with positive desired beampattern gains) as the angles of interest

Θ = {θ1, θ2, ..., θQ}. In particular, in the simulation, Θ is composed of the angles with non-zero

desired beampattern gains in Fig. 4.

-80 -60 -40 -20 0 20 40 60 80

Angle (Degree)

0

0.05

0.1

0.15

0.2

0.25

Tra

nsm

it B

ea

mp

att

ern

Max-min design w/ Type I

Max-min design w/ Type II

Max-min design w/o Rd via (SDR6)

Figure 5. Obtained transmit beampattern via the minimum beampattern gain maximization by considering Rayleigh fading

channels with Γ = 20dB.

Fig. 5 shows the obtained transmit beampattern via the minimum beampattern gain maxi-

mization. It is observed that with Type-II receivers, the transmit beampatter gain at all angles of

interest is higher than that with Type-I receivers, showing the benefit of dedicated radar signal

when the radar interference cancellation is implemented. Futhermore, the transmit beampattern

22

achieved by (SDR6) without Rd is observed to be same as that with Type-I receivers (via

(P4)/(SDR4)). This is consistent with Proposition 6.

-6 -3 0 3 6 9 12 15 18 21 24

SINR Threshold (dB)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Min

Be

am

pa

tte

rn G

ain

at


Max-min design w/ Type I receiver

Max-min design w/ Type II receiver

Max-min radar sensing only




Joint radar sensing only

Joint design w/o Rd via (SDR3)


-6 -3 0 3 6 9 12 15 18 21 24

SINR Threshold (dB)

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Min

Be

am

pa

tte

rn G

ain

at




Max-min radar sensing only




Joint radar sensing only

Joint design w/o Rd via (SDR3)


Figure 6. The minimum beampattern gain versus SINR threshold Γ.

Figs. 6(a) and 6(b) show the obtained minimum beampattern gain in Θ versus the SINR

threshold Γ by considering the Rayleigh fading and LOS channels, respectively, where N = 8

and K = 5. The results are obtained by averaging over 50 random channel realizations. It

is observed that the minimum beampattern gain maximization designs achieve much higher

worst-case beampattern gains at angles of interest than the beampattern matching designs. In

particular, under both design criteria, the case with Type-II receivers is observed to have much

better performance than other schemes, especially when the SINR threshold becomes high (e.g.,

Γ ≥ 10dB), thanks to its capability of cancelling the interference caused by radar signals.

Furthermore, under the Rayleigh fading channels in Fig. 6(a), it is observed that similar as

the case under SINR-constrained beampattern matching criterion, the performance achieved via

solving (P4)/(SDR4) is exactly same as that by (SDR6), which is consistent with Proposition 6.

Besides, under Rayleigh fading channels, it is observed in the simulations that the solution to

(SDR4) always satisfies that Rd = 0 and rank(T i) = 1,∀i ∈ K when Γ ≥ 3 dB and thus there

is no need to add dedicated radar signals in this case. This is similar to the scenario with Type-I

receivers under LOS channels, where it follows from Propositions 6 and 7 that there is no need

to add dedicated radar signal over the whole SINR regimes.

Fig. 7 shows the obtained transmit beampattern by the beampattern matching and minimum

beampattern gain maximization designs under Type-II receivers. It is observed that both designs

allocate more power at angles of interest while reducing the leakage to other angles, as compared

23

-80 -60 -40 -20 0 20 40 60 80

Angle (Degree)

0

0.05

0.1

0.15

0.2

0.25

Tra

nsm

it B

ea

mp

att

ern




Figure 7. Obtained transmit beampattern by the two different design criteria, by considering the Rayleigh fading channels with

Γ = 20dB.

with the two-stage separate design approach. Furthermore, it is observed that the minimum

beampattern gain maximization achieves more balanced beampattern gains over the angles of

interest than the beampattern matching design.

1 2 3 4 5 6

Number of Users (K)

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Min

Be

am

pa

tte

rn G

ain

at







1 2 3 4 5 6

Number of Users (K)

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Min

Be

am

pa

tte

rn G

ain

at







Figure 8. Minimum beampattern gain at angles of interest Θ obtained by five different methods with Γ = 20dB.

Fig. 8 shows the minimum beampattern gain at Θ versus the number of communication users

K, where we set the SINR threshold as Γ = 20 dB and the number of antennas as N = 8.

The results in Figs. 8(a) and 8(b) are obtained by averaging over 50 Rayleigh and LOS channel

realizations, respectively. As number of users increases, the minimum beampattern gain at Θ

are observed to decrease. The proposed minimum beampattern gain maximization design with

Type-II receivers is observed to outperform all other appraoches under both Rayleigh fading and

24

LOS channels.

Finally, Fig. 9 compares the computaional complexity (in terms of execution time) between the

minimum beampattern gain maximization (by solving (SDR4)) versus the beampattern matching

(by solving (SDR1)), which is obtained by averaging over 50 random realizations. It is observed

that as the number of antennas N increases, the execution time of solving (SDR1) grows much

faster than that of solving (SDR4), indicating the benefit of the minimum beampattern gain

maximization in terms of the implementation complexity. This is consistent with the analysis in

Section IV-C.

8 12 16 20 24 28 32

Number of antennas (N)

5

10

15

20

25

30

35

40

45

Avera

ge E

xecution T

ime (

s)

Problem (SDR1)

Problem (SDR4)

Figure 9. The execution time versus the number of antennas N at the BS.

VI. CONCLUSION

This paper considered a downlink ISAC system, in which one BS equipped with ULA jointly

designs the information and radar transmit beamforming to perform downlink multiuser com-

munication and target sensing simultaneously. By considering the unique property of dedicated

radar signals, we introduced two types of communication receivers (Type-I or Type-II) without

or with the capability of radar interference cancellation, and accordingly studied two design

criteria, namely the beampattern matching and the minimum beampattern gain maximization

subject to communication users SINR constraints. Though non-convex in general, we optimally

solved the two design problems under both types of receivers, by applying the technique of SDR.

It was shown that the minimum beampattern gain maximization with Type-II receivers achieves

the best joint sensing and communication performance with a reasonably low computational

25

complexity. It is our hope that this paper can provide useful guidelines in designing the joint

radar/information waveforms for practically implementing ISAC over wireless networks.

APPENDIX

A. Proof of Proposition 2

Suppose that {{T k}, Rd, α} denotes an optimal solution to (SDR2), where rank(T k) ≥ 1

holds in general for every k ∈ K. In this case, we construct a new solution {{T k}, Rd, α},

given by

α = α, (24a)

tk = (hHk T khk)−1/2T khk, T k = tkt

Hk , (24b)

Rd =K∑k=1

T k + Rd −K∑k=1

T k,∀k ∈ K. (24c)

It follows from (24b) that {T k} are all rank-one and positive semidefinite. According to (24a) and

(24c), the objective value achieved by {{T k}, Rd, α} remains the same and the corresponding

power constraint is met with equality.

Then, for any v ∈ CM , it holds that

vH(T k − T k

)v = vHT kv −

(hHk T khk

)−1 ∣∣∣vHk T khk

∣∣∣2 ,∀k ∈ K. (25)

According to the Cauchy-Schwarz inequality, we have(vHT kv

)(hHk T khk

)≥∣∣∣vHk T khk

∣∣∣2 ,∀k ∈ K. (26)

Thus, we have

vH(T k − T k

)v ≥ 0⇔ T k − T k � 0, ∀k ∈ K. (27)

From (24c), since Rd � 0 and the summation of a set of positive semidefinite matrices is also

positive semidefinite, it follows that Rd � 0.

Futhermore, it can be easily shown that

hHk T khk = hHk tktHk hk = hHk T khk,∀k ∈ K. (28)

Notice that the SINR constraints in (P2.1) can be reformulated as(1 +

1

Γi

)hHi Tihi − hHi

(K∑k=1

Tk

)hi − σ2

i ≥ 0,∀i ∈ K. (29)

26

Thus, we have(1 +

1

Γi

)hHi T ihi =

(1 +

1

Γi

)hHi T ihi

≥ hHi

(K∑k=1

Tk

)hi + σ2

i ≥ hHi

(K∑k=1

Tk

)hi + σ2

i . (30)

where the first equality follows from (28), and the first and the second inequalities follow from

(29) and (27), respectively. As a result, {T k} and Rd also ensure the SINR constraints at

communication users. By combining the results above, Proposition 2 is finally proved.

B. Proof of Proposition 3

First, we show that {{T k}, α} is indeed a feasible solution to (SDR3). Since {{T k}, Rd, α}

is the optimal solution to (SDR1), we have(1 +

1

Γi

)hHi T ihi =

(1 +

1

Γi

)hHi

(T i + βiRd

)hi

(a)

≥(

1 +1

Γi

)hHi T ihi

(b)

≥ hHi

(K∑k=1

Tk + Rd

)hi + σ2

i = hHi

(K∑k=1

Tk

)hi + σ2

i ,∀i ∈ K, (31)

where (a) follows due to the fact that βiRd � 0 and (b) holds as {{T k}, Rd, α} are feasible for

(SDR1). Besides, it follows that

tr

(K∑k=1

Tk + Rd

)= tr

(K∑k=1

Tk

)= P0. (32)

As a result, {{T k}, α} is feasible for (SDR3).

Furthermore, it can be shown that for any feasible solution {{T k}, α} to problem (SDR3),

f3({T k}, α) = f1({T k}, Rd, α) ≤ f1({T k}, 0, α) = f3({T k}, α), (33)

which shows the optimality of {{T k}, α} for (SDR3) and thus completes the proof.

C. Proof of Proposition 4

Let {T optk }Kk=1 denote the optimal solution of (SDR3). For each T opt

k , define ρk = rank(T optk ) ≥

1. Evidently, T optk can be decomposed as T opt

k =ρk∑=1

woptk`

(woptk`

)H, and thus the signal power

27

received from sk at receiver i through channel hi can be written as [51]

tr(T optk hih

Hi

)= tr

[ρk∑`=1

woptk`

(woptk`

)Hhih

Hi

]=

ρk∑`=1

tr[woptk`

(woptk`

)Hhih

Hi

]=

ρk∑`=1

tr[hHi w

optk`

(woptk`

)Hhi

]=

ρk∑`=1

∣∣hHi woptk`

∣∣2=

ρk∑`=1

∣∣∣v (φi)H wopt

k`

∣∣∣2 , (34)

where v (φi) = [1, ejφi , ..., ej(N−1)φi ]T . According to the Riesz-Fejer theorem [52], there exists a

vector woptk ∈ R× CN−1 that is independent of φi such that for all φi = 2π d

λsin(θi),

ρk∑`=1

∣∣∣v (φi)H wopt

k`

∣∣∣2 = |v (φi)H wopt

k |2 = tr(T opt

k hihHi ),

where T optk = wopt

k (woptk )H . Therefore, the new rank-one solution set {T opt

k }Kk=1 will not change

the SINR obtained by the original, possibly high-rank solution {T optk }Kk=1.

Secondly, we show that the aforementioned reconstructed rank-one solution meets the sum-

power constraint at the BS. Given a specific φ (index i is dropped here for simplicity), we

integrate out φ in (35) at both sides,1

2π

∫ π

−π

ρk∑`=1

∣∣∣v (φ)H woptk`

∣∣∣2 dφ =1

2π

∫ π

−π|v (φ)H wopt

k |2dφ, (35)

which is equivalent toρk∑`=1

(woptk` )H

1

2π

∫ π

−πv(φ)v(φ)Hdφwopt

kl = (woptk )H

1

2π

∫ π

−πv(φ)v(φ)Hdφwopt

k . (36)

Denote 12π

∫ π−π v(φ)v(φ)Hdφ as V ∈ CN×N . For the entry vk,m in the k-th row and m-th column

of V , k,m ∈ {1, 2, ..., N}, we have

vk,m =1

2π

∫ π

−πv(φ)kv(φ)Hmdφ =

1

2π

∫ π

−πej(k−1)φe−j(m−1)φdφ =

1

2π

∫ π

−πej(k−m)φdφ = δkm,

(37)where δkm = 1 for k = m and δkm = 0 for k 6= m. It thus immediately follows that

ρk∑`=1

||woptk` ||

2 = ||woptk ||

2 ⇔ tr

[ρk∑`=1

woptk`

(woptk`

)H]= tr

[woptk

(woptk

)H]⇔tr(T opt

k ) = tr(T optk )⇔

K∑k=1

tr(T optk ) =

K∑k=1

tr(T optk ). (38)

Thus, the set of {T optk } also meets the sum power constraints. Futhermore, by letting αopt = αopt,

28

we have

f3({T optk }

Kk=1, α

opt) =M∑m=1

∣∣∣∣∣αoptPd (θm)−K∑k=1

tr(T optk hmh

Hm)

∣∣∣∣∣2

=M∑m=1

∣∣∣∣∣αoptPd (θm)−K∑k=1

tr(T optk hmh

Hm)

∣∣∣∣∣2

= f3({T optk }

Kk=1, α

opt), (39)

where the second equality follows from the first part of the proof. This shows that the constructed

rank-one solution {T optk }Kk=1 also has the same objective function value as its original high-rank

counterpart {T optk }Kk=1. Combining the above results yields the proof.

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