optimal transmit beamforming for integrated sensing and
TRANSCRIPT
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
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
(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
Two-stage design w/o Rd
Benchmark w/o Rd via (SDR3)
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/o Rd via (SDR6)
Max-min design w/ Type I receiver
Max-min design w/ Type II receiver
Max-min radar sensing only
Two-stage design w/o Rd
Joint design w/ Type-I receiver
Joint design w/ Type-II receiver
Joint radar sensing only
Joint design w/o Rd via (SDR3)
(a) The case with Rayleigh fading channels
-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/o Rd via (SDR6)
Max-min design w/ Type I receiver
Max-min design w/ Type II receiver
Max-min radar sensing only
Two-stage design w/o Rd
Joint design w/ Type-I receiver
Joint design w/ Type-II receiver
Joint radar sensing only
Joint design w/o Rd via (SDR3)
(b) The case with LOS channels
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
Two-stage design w/o Rd
Joint design w/ Type II receiver
Max-min design w/ Type II receiver
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
Max-min design w/ Type I receiver
Max-min design w/ Type II receiver
Two-stage design w/o Rd
Joint design w/ Type-I receiver
Joint design w/ Type-II receiver
(a) The case with Rayleigh fading channels
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
Max-min design w/ Type I receiver
Max-min design w/ Type II receiver
Two-stage design w/o Rd
Joint design w/ Type-I receiver
Joint design w/ Type-II receiver
(b) The case with LOS channels
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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