# Joint Transmit Beamforming for Multiuser MIMO Communications and Radar

Xiang Liu, Tianyao Huang, Nir Shlezinger, Yimin Liu, Jie Zhou, and Yonina C.
Eldar  This work was supported by the National Natural Science Foundation of
China under Grant 61801258 and 61571260, Futurewei Technologies, and the Air
Force Office of Scientific Research under grant No. FA9550-18-1-0208. Part of
the work [1] were presented at 2019 IEEE International Conference on Signal,
Information and Data Processing. X. Liu, T. Huang and Y. Liu are with the
Department of Electronic Engineering, Tsinghua University, Beijing, China
(e-mail<EMAIL_ADDRESS>{huangtianyao;
yiminliu}@tsinghua.edu.cn). J. Zhou is with the Institute of Electronic
Engineering, China Academy of Engineering Physics, Mianyang, China (e-mail:
zhoujie_iee@yeah.net). N. Shlezinger and Y. C. Eldar are with the Faculty of
Mathematics and Computer Science, Weizmann Institute of Science, Rehovot,
Israel (e-mail: {nir.shlezinger; yonina}@weizmann.ac.il).

###### Abstract

Future wireless communication systems are expected to explore spectral bands
typically used by radar systems, in order to overcome spectrum congestion of
traditional communication bands. Since in many applications radar and
communication share the same platform, spectrum sharing can be facilitated by
joint design as dual function radar-communications system. In this paper, we
propose a joint transmit beamforming model for a dual-function multiple-input-
multiple-output (MIMO) radar and multiuser MIMO communication transmitter
sharing the spectrum and an antenna array. The proposed dual-function system
transmits the weighted sum of independent radar waveform and communication
symbols, forming multiple beams towards the radar targets and the
communication receivers, respectively. The design of the weighting
coefficients is formulated as an optimization problem whose objective is the
performance of the MIMO radar transmit beamforming, while guaranteeing that
the signal-to-interference-plus-noise ratio (SINR) at each communication user
is higher than a given threshold. Despite the non-convexity of the proposed
optimization problem, it can be relaxed into a convex one, which can be solved
in polynomial time, and we prove that the relaxation is tight. Then, we
propose a reduced complexity design based on zero-forcing the inter-user
interference and radar interference. Unlike previous works, which focused on
the transmission of communication symbols to synthesize a radar transmit beam
pattern, our method provides more degrees of freedom for MIMO radar and is
thus able to obtain improved radar performance, as demonstrated in our
simulation study. Furthermore, the proposed dual-function scheme approaches
the radar performance of the radar-only scheme, i.e., without spectrum
sharing, under reasonable communication quality constraints.

###### Index Terms:

Spectrum sharing, dual-function radar communication, MIMO radar, multiuser
MIMO, transmit beamforming

##  I Introduction

The increasing demands on wireless communications networks give rise to a
growing need for spectrum sharing between radar and communication systems.
Nowadays, military radars utilize numerous spectrum bands below 10 GHz, like
S-band (2-4 GHz) and C-band (4-8 GHz), while spectrum congestion is becoming a
serious problem which limits the throughput of wireless communications
operating in neighbouring bands. To tackle this congestion, it has been
recently proposed to allow wireless communications to share spectrum with
radar systems, allowing both functionalities to simultaneously operate over
the same wide frequency bands [2, 3, 4, 5].

The common strategy to allow individual radar and communication systems to
share spectrum with controllable mutual interference is to facilitate co-
existence by some level of cooperation [6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18]. These techniques include opportunistic spectrum access [6, 7],
transmit interference nulling [8, 9], adaptive receive interference
cancellation [10, 11, 12, 13] and optimization based beamforming design [14,
15, 16, 17, 18] to mitigate the mutual interference. This approach typically
requires the individual radar and communication systems to either be
configured using some centralized entity, or alternatively, to exchange
information, such as knowledge of the interference channel and radar waveform
parameters, significantly increasing the complexity of realizing such systems
[19].

The difficulty associated with coordinating spectrum sharing radar and
communication systems is notably reduced when these functionalities operate on
the same device. In fact, various emerging technologies, such as automotive
vehicles [20], implement both radar sensing and data transmission from the
same platform, as illustrated in Fig. 1. In such cases, spectrum sharing can
be realized by jointly designing a dual-function radar-communications (DFRC)
system [21, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 19, 42]. One clear advantage of DFRC methods over
individual co-existing systems is that the functionalities share radio
frequency (RF) front-end and aperture, thus reducing the cost and weight of
hardware [43]. Moreover, radar and communication are naturally combined in a
DFRC system, and no additional cost is required for cooperation. Nonetheless,
DFRC design has several associated challenges. From a hardware perspective,
the requirements of radar and communications may be quite distinct in terms
of, e.g., power amplifiers operation mode [24, 19]. From the algorithmic side,
properly combining radar and communications is a challenging task, and a broad
range of strategies for doing so have been proposed in the literature, see,
e.g., the detailed survey in [20].

Figure 1: A dual function system in which communication and radar share the
transmit platform.

Early works on DFRC systems consider single-antenna devices. One way to
implement such spectrum-sharing dual function signaling is by utilizing
orthogonal individual signals for radar and communications, as proposed in
[21] which studied time-division based DFRC systems. Alternatively, one can
achieve both functions simultaneously by employing an appropriate integrated
waveform, which can be utilized for both target detection and information
transmission. For instance, the probing capabilities of orthogonal frequency
division multiplexing (OFDM) waveforms, which are widely used for
communication signaling, were studied in [22, 23, 24, 25, 26]. The combination
of linear frequency modulation (LFM), which is a traditional radar waveform,
with continuous phase modulation (CPM) to realize a dual-function signal
capable of conveying information was studied [27, 28, 29, 30]. However, these
schemes inherently result in performance loss for either the radar or the
communication [19]. For instance, LFM-CPM usually exhibits higher side-lobes
than standard LFM [27]. Moreover, a common problem emerging in these single-
antenna schemes is that radar systems with integrated waveforms usually form a
single directional beam, which illuminates the radar target inside the beam.
Therefore, single-antenna schemes are not able to illuminate multiple targets
and communicate with multiple users simultaneously as in Fig. 1. That leads to
notable degradation in signal-to-noise-ratio (SNR) when the communication
receivers are not physically located within the radar main lobe.

Recognizing this limitation of single-antenna schemes, recent works on DFRC
methods employ multiple-input-multiple-output (MIMO) systems, which provide
higher degrees of spatial freedom, and can simultaneously synthesize multiple
beams towards several communication users and radar targets. These studies can
be divided into two categories: information embedding [31, 32, 33, 34, 35, 36,
37, 39] and transmit beamforming [19, 40, 41, 42]. In information embedding
systems, radar is typically considered as the primary function, and the
communication message is encoded into the MIMO radar waveform. For example,
the works [34, 35] proposed to embed communication bits by controlling the
amplitude and phase of radar spatial side-lobes. The works in [31, 32, 33]
proposed to convey the message in the form of index modulation, via the
selection of active radar transmit antennas and the allocation of radar
waveforms across active antennas. Embedding data bits in the parameters of a
radar waveform, e.g., phase, antenna index, and frequency, can yield
communications in the form of phase modulation [36], spatial modulation [37],
and carrier frequency modulation [39]. However, such methods carry a very
limited number of communication symbols per radar pulse, usually yielding low
information rate of the same order of radar pulse repetition frequency [19].

The second approach for implementing MIMO DFRC systems is based on transmit
beamforming. Here, the spatial degrees of freedom is exploited to synthesize
multiple beams towards several communication users and radar targets. As
opposed to information embedding strategies, transmit beamforming enables each
function to use its individual waveform, potentially supporting higher data
rates and guaranteed radar performance by utilizing conventional dedicated
signals for each functionality [19]. In this approach, the main design goal is
to properly beamform both the radar and communication signals such that each
can operate reliably.

In [40, 41], the array probing signal is designed to synthesize radar and
communication waveform towards different directions. The method in [40, 41]
considers waveform synthesis at the main beam direction but cannot suppress
the azimuth side-lobe of the radar transmit beam pattern. The work [19]
extended the method of [40, 41] to designing the array probing signal to match
the radar transmit beam pattern and minimize the interference power at
multiple users. However, the methods in [40, 41, 19] only minimize the
interference power, and do not consider the signal-to-interference-plus-noise
ratio (SINR) at each user, which is the quantity dictating the communications
rate. In [42], the authors studied transmit beamforming in DFRC systems with
multiple receivers, i.e., multiuser setup, in which the communication waveform
is utilized as a radar transmit waveform. In such a dual-function system, the
available degrees of freedom (DoF) for the MIMO radar waveform, which affects
the resulting radar beam pattern, is equal to the number of communication
users. Since the DoF of conventional MIMO radar, i.e., without communication
functionality, is at most the number of transmit antennas, the resulting MIMO
radar cannot utilize its full DoF when the number of users is smaller than the
number of antennas, potentially leading to significant distortion of radar
beam pattern. Furthermore, the resulting problem is a non-convex optimization
problem, which is solved by sub-optimal methods.

In this paper, we design a transmit beamforming based MIMO DFRC system. As was
done in [42], we design our transmit beamforming to optimize both the radar
transmit beam pattern and the SINR at the communication users. Unlike [42],
our proposed joint transmitter utilizes jointly precoded individual
communication and radar waveforms, allowing to extend the MIMO radar waveform
DoF to its maximal value, i.e., the number of antennas. In fact, the
previously proposed formulation of [42] can be regarded as a special case of
the proposed one by nullifying the dedicated radar waveform. Furthermore,
while we utilize individual signals for radar and communications, we exploit
the fact that both signals are transmitted from the same device, which also
accommodates the radar receiver. Consequently, the radar receiver has complete
knowledge of the transmitted communication waveform, which is utilized for
target detection in addition to the dedicated radar signal. This approach
contributes to the power-efficiency of the DFRC system, further exploiting the
inherent advantages of joint design over co-existing separate radar and
communication systems.

We formulate the design of the resulting precoding method as a non-convex
optimization problem, which we tackle using two different methods: First, we
show that it can be relaxed into an equivalent semidefinite problem, where the
latter can be solved using conventional optimization tools, and prove that the
relaxation is tight. To circumvent the computational burden of recovering the
optimal precoders from the relaxed formulation, we propose an additional
design approach based on zero-forcing the interference. Our numerical results
demonstrate that, due to the increased DoF of the MIMO radar waveform, our
approach obtains improved radar transmit beam pattern compared to [42], under
the same SINR constraints at the communication users. Furthermore, we
demonstrate that under high SINR constraints our reduced complexity zero-
forcing technique is capable of achieving comparable performance to that of
the optimal beamforming scheme, whose compuataion is substantially more
complex, indicating the potential of our approach in designing reliable DFRC
beamforming at controllable complexity.

The rest of this paper is organized as follows. Section II proposes the system
model. Section III introduces the performance metrics of MIMO radar and
multiuser MIMO communication, respectively. Section IV establishes the
optimization model for joint beamforming, and proposes two methods for
designing precoders based on that formulation. Simulation results are
presented in Section V. Finally, Section VI provides concluding remarks.

_Notations:_ In this paper, $(\cdot)^{H}$, $(\cdot)^{c}$ and $(\cdot)^{T}$
denote Hermitian transpose, conjugate and transpose, respectively. Vectors are
denoted by bold lower class letters and matrices are denoted by bold upper
class letters. For a matrix $\bm{A}$, the $(i,j)$-th elements of $\bm{A}$ is
denoted by $\left[\bm{A}\right]_{i,j}$, and $\left[\bm{A}\right]_{1:j}$
denotes the sub-matrix containing the first $j$ columns of $\bm{A}$. We let
$\bm{I}_{n}$ and $\bm{0}_{m\times n}$ denote $n$-dimensional identity matrix
and $m\times n$ zero matrix, respectively. We use $\mathbf{E}(\cdot)$ for the
stochastic expectation. For an integer $n>0$, the set consisting of all
$n$-dimensional complex positive semidefinite matrices is denoted by
$\mathcal{S}_{n}^{+}$.

##  II Signal Model

Consider an antenna array shared by a colocated monostatic MIMO radar system
and a multiuser MIMO communication transmitter as depicted in Fig. 1. In our
work, both functionalities operate simultaneously by joint beamforming. The
system diagram of our joint beamforming transmitter is shown in Fig. 2,
demonstrating that the transmited signal is a weighted sum of communication
symbols and radar waveform. We consider an antenna array of $M$ elements, and
let the discrete-time transmit signal of this array at time index $n$ be given
by

$\bm{x}[n]=\bm{W}_{r}\bm{s}[n]+\bm{W}_{c}\bm{c}[n],\ n=0,\ldots,N-1.$ (1)

Here, the $M\times 1$ vector $\bm{s}[n]=[s_{1}[n],\ldots,s_{M}(n)]^{T}$
includes $M$ individual radar waveforms, and the $M\times M$ matrix
$\bm{W}_{r}$ is the beamforming matrix (or precoder) for radar waveform.
Similarly, $\bm{c}[n]=[c_{1}[n],\ldots,c_{K}[n]]^{T}$ is a $K\times 1$ vector
including $K$ parallel communication symbol streams to be communicated to $K$
users, respectively, while the $M\times K$ matrix $\bm{W}_{c}$ is the
communication precoder. To achieve alias-free signal sampling and symbol
transmission, the communication symbol duration or the radar code duration,
denoted by $t_{s}$, should satisfy $t_{s}\geq 1/(2B)$ [44], where $B$ is the
baseband bandwidth of the transmit platform. The maximal available radar delay
resolution is bounded by the symbol duration, and is given by $1/(2B)$. The
maximal available symbol rate is $2B$. We note that the scheme proposed in
[42], which beamformed the communications symbols to be utilized for probing,
can be regarded as a special case of our system by letting the radar waveform
be zero, namely transmitting only communication symbols, as depicted in Fig.
2.

Figure 2: (a) The joint transmitter jointly precodes both communication
symbols and radar waveform; (b) The joint transmitter only utilizes precoded
communication symbols.

Our goal is to design the matrices $W_{c}$ and $W_{r}$ in (1). Without loss of
generality, we make the following assumptions: 1) Both radar and communication
signals are zero-mean, temporally-white and wide-sense stationary stochastic
process; 2) The communication symbols are uncorrelated with radar waveform,
i.e.,

$\mathbf{E}\left(\bm{s}[n]\bm{c}^{H}[n]\right)=\mathbf{E}\left(\bm{s}[n]\right)\mathbf{E}(\bm{c}^{H}[n])=\bm{0}_{M\times
K};$ (2)

3) Communication symbols intended to different users are uncorrelated, namely,

$\mathbf{E}\left(\bm{c}[n]\bm{c}^{H}[n]\right)=\bm{I}_{K};$ (3)

4) The individual radar waveform are generated by pseudo random coding [45,
46, 47, 48, 49], and thus are uncorrelated with each other, resulting in

$\mathbf{E}\left(\bm{s}[n]\bm{s}^{H}[n]\right)=\bm{I}_{M}.$ (4)

Here, both signals are normalized to have unit power, and their real power is
encapsulated in their corresponding precoders $\bm{W}_{r}$ and $\bm{W}_{c}$.

Since we focus on DFRC schemes in which individual uncorrelated waveforms are
used for radar and communications, radar interference can induce a notable
degradation in the ability of the communication receivers to recover the
transmitted symbols. In particular, to achieve radar detection, the transmit
power of the dual function platform is usually much higher than that of a
typical communication transmitter, because the echos reflected from the
targets are attenuated with a two-way propagation loss. However, radar
transmit beams are designed to be highly directional, and thus radar
interference is dominant only when the radar beam is steered towards the
communication receivers. In particular, when the communication receivers lie
in the same direction as that of the radar targets, they are expected to
observe high radar interference. Nevertheless, the transmitted communication
waveform is completely known at the radar receiver and thus its reflected
signal can also be utilized for target detection, indicating that this
challenge can be overcome by forming a high-gain communication beam to
simultaneously cover the targets and communication receiver for jointly radar
sensing and data transmission, as in the shared integrated waveform DFRC
design [22, 23, 24, 25, 26, 27, 28, 29, 30]. Under such a waveform reuse
design, the “radar signal” at communication received is not interference but
the expected communication signal.

In order to implement joint transmit beamforming, the precoders $\bm{W}_{c}$
and $\bm{W}_{r}$ are to be jointly designed in consideration of the system
performance. The performance metrics of MIMO radar and multiuser MIMO
communication are detailed in Section III. In practice, the precoders should
satisfy some constraints representing the transmit hardware. Here, we require
that the transmit waveform satisfies a per-antenna power constraint, namely,
that the transmit power of each antenna is identical. The per-antenna power
constraint settles with the common practice that radar waveforms should be
transmitted with their maximal available power [50], and has also been applied
in multi-antenna communication systems [51, 52, 53]. We note that the per-
antenna power constraints can be extended to represent other power-related
limitations, such as total power constraints, according to the hardware
requirements.

To formulate the power constraint, define the covariance of transmit waveform
as

$\bm{R}=\mathbf{E}\left(\bm{x}[n]\bm{x}^{H}[n]\right).$ (5)

Substituting (1)-(4) into (5) yields the covariance $\bm{R}$ as

$\bm{R}=\bm{W}_{r}\bm{W}_{r}^{H}+\bm{W}_{c}\bm{W}_{c}^{H}.$ (6)

The per-antenna power constraint implies that for each $\ m=1,\ldots,M$ it
holds that

$\left[\bm{R}\right]_{m,m}=\left[\bm{W}_{r}\bm{W}_{r}^{H}+\bm{W}_{c}\bm{W}_{c}^{H}\right]_{m,m}=P_{t}/M,$
(7)

where $P_{t}$ is the total transmit power. Under this constraint, we discuss
the radar and communication metrics for precoder design in the following
section.

##  III Performance metrics of radar and communication

Based on the signal model of joint transmit beamforming, we aim to design the
precoders in light of the following guidelines: For MIMO radar, the precoder
is designed to synthesize transmit beams towards radar targets of interests;
For multiuser MIMO communication, the precoder is designed to guarantee the
receiving SINR at communication users. These performance metrics of MIMO radar
and multiuser MIMO communication are properly formulated in Subsections III-A
and III-B, respectively.

###  III-A MIMO Radar Performance

The main purpose of MIMO radar beamforming is to direct the transmit beam
towards several given directions, so that one can obtain more information of
the targets illuminated by these beams. These directions are typically known
to the transmitter: When radar works in tracking mode, the beam direction is
inferred from the direction of the targets acquired at previous observations;
When radar works in searching mode, the beam direction is given by the center
of angular sector-of-interest. Consequently, to formulate the performance
metric associated with MIMO radar beamforming, we first express the
transmitted signal at each direction, and then develop a loss function
evaluating the transmit beam pattern. Combining the loss function and the per-
antenna power constraint, we achieve an optimization problem which accounts
for the radar performance.

In DFRC systems, the communication signals can also be used for sensing, since
the radar receiver has complete knowledge of the transmitted communication
waveform. In this way, the communication signal is not regarded as
interference at the radar receiver. Under the assumption that the transmit
waveform is narrow-band and the propagation path is line of sight (LoS), the
baseband signal at direction $\theta$ can be expressed as

$y[n;\theta]=\bm{a}^{H}(\theta)\bm{x}[n],$ (8)

where $\bm{a}(\theta)$ is the array steering vector of direction $\theta$.
When the waveform is reflected from a point target located at angular
direction $\theta$, the received signal can be written as

$\bm{r}[n]=\beta\bm{a}^{c}(\theta)\bm{a}^{H}(\theta)\bm{x}[n-n^{\prime}]+\bm{v}[n],$
(9)

where $\beta$ is the complex amplitude proportional to the radar-cross
sections (RCS) of the target, $n^{\prime}$ represents the discrete time delay,
and $\bm{v}[n]$ is additive zero-mean temporally-white noise with covariance
$\bm{R}_{v}$.

Following the guidelines for MIMO radar probing signal design stated in [50],
the desired goals of MIMO radar transmit beamforming include:

  1. 1)

Optimize the transmit power at given directions, or generally match a desired
beam pattern;

  2. 2)

Decrease the cross correlation pattern among signals at several given
directions, which is essential for the performance of adaptive MIMO radar
techniques.

Here, the transmit power (beam pattern) at angular direction $\theta$ is

$\displaystyle
P(\theta;\bm{R})=\mathbf{E}\left(\left|y[n;\theta]\right|^{2}\right)$
$\displaystyle=\mathbf{E}\left(\bm{a}^{H}(\theta)\bm{x}[n]\bm{x}^{H}[n]\bm{a}(\theta)\right)=\bm{a}^{H}(\theta)\bm{R}\bm{a}(\theta),$
(10)

and the cross correlation pattern between direction $\theta_{1}$ and
$\theta_{2}$ is defined as

$\displaystyle
P_{c}(\theta_{1},\theta_{2};\bm{R})=\mathbf{E}\left(y^{*}[n;\theta_{1}]y[n;\theta_{2}]\right)$
$\displaystyle=\mathbf{E}\left(\bm{a}^{H}(\theta_{2})\bm{x}[n]\bm{x}^{H}[n]\bm{a}(\theta_{1})\right)=\bm{a}^{H}(\theta_{2})\bm{R}\bm{a}(\theta_{1}).$
(11)

From (10) and (11), both the transmit beam pattern and cross correlation
pattern are determined by the covariance $\bm{R}$. Then, properly beamforming
of MIMO radar waveforms is achieved by designing the covariance matrix
$\bm{R}$ [50, 54].

To this aim, we use the loss function proposed in [50, 54] to evaluate the
radar performance, which is the weighted sum of two parts: beam pattern error
and cross correlation. In particular, the first part is the mean square error
(MSE) between the obtained beam pattern and some desired beam pattern, given
by

$L_{r,1}(\bm{R},\alpha)=\frac{1}{L}\sum_{l=1}^{L}\left|\alpha
d(\theta_{l})-P(\theta_{l};\bm{R})\right|^{2},$ (12)

where $\alpha$ is a scaling factor, $d(\theta)$ is the given desired beam
pattern, and $\left\\{\theta_{l}\right\\}_{l=1}^{L}$ are sampled angle grids.
The second part is the mean-squared cross correlation pattern, expressed as

$L_{r,2}(\bm{R})=\frac{2}{P^{2}-P}\sum_{p=1}^{P-1}\sum_{q=p+1}^{P}\left|P_{c}(\overline{\theta}_{p},\overline{\theta}_{q};\bm{R})\right|^{2},$
(13)

where $\\{\overline{\theta}_{p}\\}_{p=1}^{P}$ are the given directions of the
targets. The summation in (13) is normalized by $\frac{2}{P^{2}-P}$, as there
exists $\frac{P^{2}-P}{2}$ pairs of distinct directions in the set
$\\{\overline{\theta}_{p}\\}$. The loss function of radar is then

$L_{r}(\bm{R},\alpha)=L_{r,1}(\bm{R},\alpha)+w_{c}L_{r,2}(\bm{R}),$ (14)

where $w_{c}$ is a weighting factor. As discussed in [50, 54], the loss
function $L_{r}(\bm{R},\alpha)$ can be written as a positive-semidefinite
quadratic function of $\bm{R}$ and $\alpha$.

Combining the loss functions in (12) and (13), the covariance of the
transmitted signal in the absence of communication constrains, i.e., in a
radar-only setup, can be designed in light of the overall radar objective
under per-antenna power constraints [50], i.e.

$\displaystyle\min_{\bm{R},\alpha}\ \ \ \ $ $\displaystyle\
L_{r}(\bm{R},\alpha)$ (15a) $\displaystyle\mathrm{subject\ to}\ $
$\displaystyle\ \bm{R}\in\mathcal{S}_{M}^{+},$ (15b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M.$ (15c)

We denote the optimal covariance of this problem by $\bm{R}_{0}$. Generally,
the performance requirement of multiuser MIMO communication, detailed in
Subsection III-B, cannot be satisfied if the covariance of transmit waveform
is $\bm{R}_{0}$. In other words, there is an inherent radar performance loss
due to spectrum sharing with communications compared to the radar-only case.
To address the communication performance of our DFRC system, we discuss the
communication metric in the next subsection.

###  III-B Multiuser MIMO Communication Performance

A common performance measure for multiuser broadcast communications is the
SINR, which is directly related to the achievable rate under reduced
complexity decoding [55, Ch. 8]. Consequently, we design the precoders of MIMO
transmission to optimize the users’ SINR. To this aim, we first present the
communication signal model, and derive the expression of SINR with respect to
the precoders (for radar and communication signals) and the channel matrix,
followed by a formulation of the combined objective which accounts for
communication performance.

Consider a down-link multiuser MIMO transmission scenario with $K<M$ single
antenna users observing the output of a frequency flat Gaussian noise channel.
The channel output at the $K$ users at time instance $n$, represented via the
$K\times 1$ vector ${\bm{r}}[n]$, is given by

$\bm{r}[n]=\bm{H}\bm{W}_{c}\bm{c}[n]+\bm{H}\bm{W}_{r}\bm{s}[n]+\bm{v}[n],$
(16)

where $\bm{H}$ is the $K\times M$ narrow-band channel matrix and $\bm{v}[n]$
is additive white Gaussian noise (AWGN) with covariance
$\sigma^{2}\bm{I}_{K}$.

In multiuser transmit beamforming, the precoder should be designed to
guarantee a certain level of SINR at the users. Here, it is assumed that the
transmit array knows the instantaneous downlink channel $\bm{H}$. This
knowledge can be obtained for example, by exploiting wireless channel
reciprocity when operating in time-division duplex mode, i.e., the downlink
channel is obtained via uplink channel estimation. Alternatively, in
frequency-division duplex mode, downlink channel can be obtained via channel
feedback from the users, see, e.g., [56]. Define the equivalent radar-to-user
channel and equivalent inter-user channel matrices as

$\displaystyle\bm{F}_{r}=\bm{H}\bm{W}_{r},$ (17a)
$\displaystyle\bm{F}_{c}=\bm{H}\bm{W}_{c},$ (17b)

respectively. Since the users are generally not able to cooperate with each
other, the off-diagonal elements of $\bm{F}_{c}$ lead to inter-user
interference, which should be mitigated by precoding. At the same time, since
the users generally do not have any prior information on radar waveform,
$\bm{F}_{r}$ leads to interference from radar. At the $k$-th user, the signal
power is

$\mathbf{E}\left(|[\bm{F}_{c}]_{k,k}c_{k}(t)|^{2}\right)=\left|[\bm{F}_{c}]_{k,k}\right|^{2},$
(18)

the power of inter-user interference is

$\mathbf{E}\Big{(}\sum_{i\neq
k}|[\bm{F}_{c}]_{k,i}c_{i}(t)^{2}\Big{)}=\sum_{i\neq
k}\left|[\bm{F}_{c}]_{k,i}\right|^{2},$ (19)

and the power of interference from radar is

$\mathbf{E}\Big{(}\sum_{i=1}^{M}|[\bm{F}_{r}]_{k,i}s_{i}(t)|^{2}\Big{)}=\sum_{i=1}^{M}\left|[\bm{F}_{r}]_{k,i}\right|^{2}.$
(20)

Therefore, the SINR at the $k$-th user is expressed as

$\gamma_{k}=\frac{|[\bm{F}_{c}]_{k,k}|^{2}}{\sum_{i\neq
k}|[\bm{F}_{c}]_{k,i}|^{2}+\sum_{i=1}^{M}|[\bm{F}_{r}]_{k,i}|^{2}+\sigma^{2}}.$
(21)

Two typical design criteria for multiuser beamforming are [57, 58]:

  * •

_Throughput_ : maximizing the sum rate

$C(\bm{\gamma})=\sum_{k=1}^{K}\log_{2}\left(1+\gamma_{k}\right),$ (22)

  * •

_Fairness_ : maximizing the minimal SINR, referred to henceforth as the
fairness SINR:

$F(\bm{\gamma})=\min\\{\gamma_{1},\ldots,\gamma_{K}\\},$ (23)

where $\bm{\gamma}=[\gamma_{1},\ldots,\gamma_{K}]^{T}$. In this work, we use
fairness SINR $F(\bm{\gamma})$ as the performance metric for multiuser
communication, and require it to be higher than a given threshold $\Gamma$,
guaranteeing a minimal level of communication quality of service at each user,
i.e.

$\gamma_{k}\geq\Gamma,\ k=1,\ldots,K.$ (24)

Moreover, the fairness beamforming is simpler in terms of computation
complexity and can be solved in polynomial time, while the optimal throughput
beamforming is NP hard [57]. We note that in the formulated joint beamforming
problem in Section IV, the fairness SINR requirement can be extended to having
$K$ individual SINR constraints [59], namely,

$\gamma_{k}\geq\Gamma_{k},\ k=1,\ldots,K,$ (25)

where $\Gamma_{k}$ is the SINR threshold at the $k$-th user.

##  IV Joint Transmit Beamforming

With the proposed MIMO radar and communication performance metrics, we now
turn to design a DFRC joint beamforming scheme. We begin by formulating the
joint transmit beamforming as an optimization problem with respect to the
precoding matrices in Subsection IV-A. To solve this problem, we propose a
semidefinite relaxation (SDR) based optimization scheme in IV-B, and a zero-
forcing (ZF) methods which cancels the inter-user interference and the radar
interference in Subsections IV-C, respectively.

###  IV-A Problem formulation

The goal of our joint DFRC beamforming is to optimize the radar beam pattern
under the transmit power and communication quality of service constraints. In
particular, we minimize the loss function on radar beam pattern defined in
(14), under the per-antenna power constraint (7) and the fairness SINR
constraint (24) for each downlink user.

Let $\bm{W}=[\bm{W}_{c},\bm{W}_{r}]$ be the overall precoding matrix. The
precoding matrix can be obtained by solving the following optimization problem

$\displaystyle\min_{\bm{W},\alpha}\ $ $\displaystyle\ L_{r}(\bm{R},\alpha)$
(26a) $\displaystyle\mathrm{subject\ to}\ $ $\displaystyle\
\bm{R}=\bm{W}\bm{W}^{H}\in\mathcal{S}_{M}^{+},$ (26b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (26c) $\displaystyle\
\gamma_{k}\geq\Gamma,\ k=1,\ldots,K,$ (26d)

where (26a)-(26c) come from (15) addressing the radar performance, and (26d)
follows from considering the fairness SINR requirement (24).

The selection of the threshold $\Gamma$ affects the trade-off between the
communication quality and radar performance. When $\Gamma=0$, (26d) always
holds, and the joint radar-communication beamforming problem (26) reduces to
the radar-only optimization (15). When $\Gamma>0$, compared with the radar-
only transmit beamforming problem in (15), the precoder $\bm{W}$, which
dictates the equivalent channels via (17), is restricted by the SINR
constraints in (26d). Therefore, there can be an inherent radar performance
loss induced by the need to meet the communication performance guarantees, as
compared to the radar-only case. If higher $\Gamma$ is set, higher signal
power and less interference is expected to be observed at the user side,
further restricting the precoding matrices. As a result, the performance loss
of MIMO radar becomes more significant if higher $\Gamma$ is set.

The optimization problem (26) is not convex because of the quadratic equality
constraint in (26b) and is thus difficult to solve. Nonetheless, we show in
Subsection IV-B that it can be recast using semidefinite relaxation (SDR) such
that the solution to the solvable relaxed problem is also the global optimizer
of the original non-convex (26), i.e., the relaxation is tight. To further
reduce the computation complexity, we propose a sub-optimal zero-forcing
beamforming strategy in Subsection IV-C, which is shown to be able to approach
the performance of the global solution to (26) in our numerical study
presented in Section V.

###  IV-B Joint Transmit Beamforming via SDR

In this subsection, we tackle the non-convex problem (26) using an SDR
strategy [60, 61]. To this aim, we first explicitly write the relationship
(26b) as a quadratic constraint with respect to each column of $\bm{W}$. Let
$\bm{w}_{i}$ denote the $i$-th column of $\bm{W}$, for $i=1,\ldots,M+K$. Then
(26b) becomes

$\bm{R}=\sum_{i=1}^{M+K}\bm{w}_{i}\bm{w}_{i}^{H}.$ (27)

Defining $\bm{R}_{i}=\bm{w}_{i}\bm{w}_{i}^{H}$, we have

$\bm{R}=\sum_{i=1}^{M+K}\bm{R}_{i},$ (28)

where we omit the rank-one constraints. The SINR constraints in (26d) can be
converted to linear constraints in the rank-one matrices $\\{\bm{R}_{i}\\}$.
Letting $\bm{h}_{k}^{H}$ denote the $k$-th row of $\bm{H}$, $k=1,\ldots,K$,
the entires of the equivalent channel matrices can be written as
$[\bm{F}_{c}]_{k,i}=\bm{h}_{k}^{H}\bm{w}_{i}$, and
$[\bm{F}_{r}]_{k,i}=\bm{h}_{k}^{H}\bm{w}_{i+K}$. Consequently, the SINR
constriant becomes

$\displaystyle\gamma_{k}$
$\displaystyle=\frac{\bm{h}_{k}^{H}\bm{w}_{k}\bm{w}_{k}^{H}\bm{h}_{k}}{\sum_{1\leq
i\leq M+K,i\neq
k}\bm{h}_{k}^{H}\bm{w}_{i}\bm{w}_{i}^{H}\bm{h}_{k}+\sigma^{2}}$
$\displaystyle=\frac{\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}}{\sum_{1\leq i\leq
M+K,i\neq k}\bm{h}_{k}^{H}\bm{R}_{i}\bm{h}_{k}+\sigma^{2}}$
$\displaystyle=\frac{\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}}{\bm{h}_{k}^{H}\bm{R}\bm{h}_{k}-\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}+\sigma^{2}}\geq\Gamma.$
(29)

We now cast (26) as an equivalent quadratic semidefinite programming (QSDP)
with rank-one constraints

$\displaystyle\min_{\bm{R}.\\{\bm{R}_{i}\\},\alpha}\ $ $\displaystyle\
L_{r}(\bm{R},\alpha)$ (30a) $\displaystyle\mathrm{subject\ to}\ $
$\displaystyle\ \bm{R}=\sum_{i=1}^{M+K}\bm{R}_{i}\in\mathcal{S}_{M}^{+},$
(30b) $\displaystyle\ \left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (30c)
$\displaystyle\ \bm{R}_{i}\in\mathcal{S}_{M}^{+},\
\mathrm{rank}(\bm{R}_{i})=1,\ i=1,\ldots,K+M,$ (30d) $\displaystyle\
\left(1+\Gamma^{-1}\right)$
$\displaystyle\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}\geq\bm{h}_{k}^{H}\bm{R}\bm{h}_{k}+\sigma^{2},\
k=1,\ldots,K,$ (30e)

where $\eqref{eq-3-ranke}$ is derived from (29). We observe that in problem
(30), the individual matrices $\\{\bm{R}_{i}\\}_{i\geq K+1}$ have no effect on
the SINR constraints and are only encapsulated in the overall covariance
matrix $\bm{R}$. Therefore, we can remove the variables
$\\{\bm{R}_{i}\\}_{i\geq K+1}$, and (30) is relaxed to

$\displaystyle\min_{\bm{R},\bm{R}_{1},\ldots,\bm{R}_{K},\alpha}\ $
$\displaystyle\ L_{r}(\bm{R},\alpha)$ (31a) $\displaystyle\mathrm{subject\
to}\ $ $\displaystyle\ \bm{R}\in\mathcal{S}_{M}^{+},\
\bm{R}-\sum_{k=1}^{K}\bm{R}_{k}\in\mathcal{S}_{M}^{+},$ (31b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (31c) $\displaystyle\
\bm{R}_{k}\in\mathcal{S}_{M}^{+},\ \mathrm{rank}(\bm{R}_{k})=1,\
k=1,\ldots,K,$ (31d) $\displaystyle\ \left(1+\Gamma^{-1}\right)$
$\displaystyle\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}\geq\bm{h}_{k}^{H}\bm{R}\bm{h}_{k}+\sigma^{2},\
k=1,\ldots,K.$ (31e)

The optimization problem (31) is still non-convex because of the rank-one
constraints. Omitting these constraints leads to the following relaxation:

$\displaystyle\min_{\bm{R},\bm{R}_{1},\ldots,\bm{R}_{K},\alpha}\ $
$\displaystyle\ L_{r}(\bm{R},\alpha)$ (32a) $\displaystyle\mathrm{subject\
to}\ $ $\displaystyle\ \bm{R}\in\mathcal{S}_{M}^{+},\
\bm{R}-\sum_{k=1}^{K}\bm{R}_{k}\in\mathcal{S}_{M}^{+},$ (32b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (32c) $\displaystyle\
\bm{R}_{k}\in\mathcal{S}_{M}^{+},\ k=1,\ldots,K,$ (32d) $\displaystyle\
\left(1+\Gamma^{-1}\right)$
$\displaystyle\bm{h}_{k}^{H}\bm{R}_{k}\bm{h}_{k}\geq\bm{h}_{k}^{H}\bm{R}\bm{h}_{k}+\sigma^{2},\
k=1,\ldots,K.$ (32e)

This relaxed optimization model (32) is a convex QSQP, because the target
function is a positive-semidefinite quadratic form and all the constraints are
either linear or semidefinite. The global optimum of (32) can be obtained in
polynomial time with convex optimization toolboxes [62, 63, 64, 65].

The relaxation used in SDR is tight if the optimal
$\bm{R}_{1},\ldots,\bm{R}_{K}$ for (32) are exactly rank-one, i.e., the
solution to the relaxed problem is also a solution to the original non-convex
problem. While such relaxations are not necessarily tight, the SDR used in
obtaining (32) from (31) is tight, as stated in the following theorem:

######  Theorem 1.

There exists a global optimum for (32), denoted by
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$, satisfying

$\mathrm{rank}(\tilde{\bm{R}}_{k})=1,\ k=1,\ldots,K.$

###### Proof.

See Appendix -A. ∎

We note that Theorem 1 only states that the rank-one global optimum exists.
Generally, the global optimum to (32) may not be unique and convex
optimization software may not give a rank-one solution. Once the optimal
solution $\hat{\bm{R}},\hat{\bm{R}}_{1},\ldots,\hat{\bm{R}}_{K}$ are obtained,
we use them to obtain the rank-one optimal solution
$\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ and the corresponding optimal
precoder $\tilde{\bm{w}}_{1},\ldots,\tilde{\bm{w}}_{K}$, as presented in
Appendix -A. First, we compute $\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$
and $\tilde{\bm{R}},\tilde{\bm{w}}_{1},\ldots,\tilde{\bm{w}}_{K}$ via

$\tilde{\bm{R}}=\hat{\bm{R}},\
\tilde{\bm{w}}_{k}=\big{(}\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\big{)}^{-1/2}\hat{\bm{R}}_{k}\bm{h}_{k},\
\tilde{\bm{R}}_{k}=\tilde{\bm{w}}_{k}\tilde{\bm{w}}_{k}^{H},$ (33)

for $k=1,\ldots,K$. According to the proof of Theorem 1,
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ is optimal to
(31) and hence is also optimal (32). To show that
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ is also optimal
to (30), we construct rank-one matrices $\\{\tilde{\bm{R}}_{i}\\}_{i\geq K+1}$
as $\tilde{\bm{R}}_{i}=\tilde{\bm{w}}_{i}\tilde{\bm{w}}_{i}^{H}$, where the
vectors $\tilde{{\bm{w}}}_{i}$ for $i>K$ are calculated by the Cholesky
decomposition [66]

$\bm{W}_{r}\bm{W}_{r}^{H}=\tilde{\bm{R}}-\sum_{k=1}^{K}\tilde{\bm{w}}_{k}\tilde{\bm{w}}_{k}^{H},$
(34)

where
$\bm{W}_{r}=\left[\tilde{\bm{w}}_{K+1},\ldots,\tilde{\bm{w}}_{K+M}\right]$ is
a lower triangular matrix. From (34), it can be verified that constraint (30b)
holds for $\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K+M}$.
Therefore, $\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K+M}$ is
a feasible solution to (30) and hence is also an optimal solution to (30).
Furthermore, the precoding matrix
$\tilde{\bm{W}}=\left[\tilde{\bm{w}}_{1},\ldots,\tilde{\bm{w}}_{K+M}\right]$
is a solution to (26).

We summarize the procedure to compute the precoding matrix $\bm{W}$ in
Algorithm 1. The main computational burden in Algorithm 1 stems from solving
the QSDP (32). Specifically, given a solution accuracy $\epsilon$, the worst
case complexity to solve the QSDP (32) with the primal-dual interior-point
algorithm in [67, 68] is $\mathcal{O}(K^{6.5}M^{6.5}\log(1/\epsilon))$.

1:

2: Total transmit power $P_{t}$;

3: Power of AWGN at users $\sigma^{2}$;

4: Expression of the MIMO radar loss function $L_{r}(\bm{R},\bm{\alpha})$;

5: Instantaneous downlink channel $\bm{H}$;

6: SINR threshold $\Gamma$.

7:

8: The overall precoding matrix $\bm{W}$.

9: Steps:

10:Compute the optimal value of
$\hat{\bm{R}},\hat{\bm{R}}_{1},\ldots,\hat{\bm{R}}_{K}$ by solving (32) with
convex optimization solvers.

11:Compute $\tilde{\bm{w}}_{1},\ldots,\tilde{\bm{w}}_{K}$ via (33).

12:Compute $\tilde{\bm{w}}_{K+1},\ldots,\tilde{\bm{w}}_{K+M}$ via (34).

13:Set the overall precoding matrix
$\tilde{\bm{W}}=\left[\tilde{\bm{w}}_{1},\ldots,\tilde{\bm{w}}_{K+M}\right]$.

Algorithm 1 Joint transmit beamforming via SDR

###  IV-C Joint Transmit Beamforming via ZF

The computational burden associated with obtaining the precoder via Algorithm
1 motivates seeking a reduced complexity sub-optimal beamforming strategy. In
this subsection, we focus on ZF beamforming. ZF methods facilitate obtaining
closed-form, tractable, and interpretable precoders [69, 58]. In addition to
its relative simplicity, from a communications perspective, ZF beamforming is
known to asymptotically approach the sum-capacity in broadcast channels [70],
indicating its potential to approach optimal performance in setups involving
multi-user communications.

We design the precoders to eliminate the inter-user interference and radar
interference, obtained by restricting $\bm{F}_{c}$ to a diagonal matrix and
$\bm{F}_{r}$ to a zero matrix, i.e.

$\bm{F}_{c}=\mathrm{diag}\left(\sqrt{p_{1}},\ldots,\sqrt{p_{K}}\right),\
\bm{F}_{r}=\bm{0}_{K\times M}.$ (35)

Here, $p_{k}$ is the signal power at the $k$-th user, for $1\leq k\leq K$.
Enforcing the interference to be canceled facilitates achieving high SINR
values at the users. In our numerical study in Section V we demonstrate that
the achievable performance under the additional ZF constraint approaches that
of the global solution to (26), obtained with increased computational burden
via Algorithm 1, when the SINR threshold is high.

In ZF beamforming, the SINR constraint (26d) is reformulated as
$\frac{1}{\Gamma}p_{k}\geq\sigma^{2}$, and the corresponding optimization
problem (26) becomes

$\displaystyle\min_{\bm{W},\alpha}\ $ $\displaystyle\ L_{r}(\bm{R},\alpha)$
(36a) $\displaystyle\mathrm{subject\ to}\ $ $\displaystyle\
\bm{R}=\bm{W}\bm{W}^{H}\in\mathcal{S}_{M}^{+},$ (36b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (36c) $\displaystyle\
\bm{HW}=\left[\mathrm{diag}(\sqrt{\bm{p}}),\bm{0}_{K\times M}\right],$ (36d)
$\displaystyle\ \frac{1}{\Gamma}{p}_{k}\geq\sigma^{2},\ k=1,\ldots,K.$ (36e)

The ZF beamforming optimization (36) is still non-convex. The following
theorem shows that it can be converted to convex problem.

######  Theorem 2.

Given a covariance matrix $\bm{R}\in\mathcal{S}_{n}^{+}$ and a full rank
$K\times(K+M)$ matrix $\bm{F}$, there exists a matrix $\bm{W}$ satisfying
(36b) and

$\bm{H}\bm{W}=\bm{F}$ (37)

if and only if

$\bm{H}\bm{R}\bm{H}^{H}=\bm{F}\bm{F}^{H}.$ (38)

###### Proof.

See Appendix -B. ∎

Theorem 2 indicates that constraints (36b) and (36d) are equivalent to

$\bm{H}\bm{R}\bm{H}^{H}=\mathrm{diag}\left(\bm{p}\right),$ (39)

by letting $\bm{F}=\left[\mathrm{diag}(\sqrt{\bm{p}}),\bm{0}_{K\times
M}\right]$. Using (39), the globally optimal $\bm{R}$ to (36) is found by

$\displaystyle\min_{\bm{R},\alpha}\ $ $\displaystyle\ L_{r}(\bm{R},\alpha)$
(40a) $\displaystyle\mathrm{subject\ to}\ $ $\displaystyle\
\bm{R}\in\mathcal{S}_{+}^{M},\ \bm{HRH}^{H}=\mathrm{diag}\left(\bm{p}\right),$
(40b) $\displaystyle\ \left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (40c)
$\displaystyle\ \frac{1}{\Gamma}{p}_{k}\geq\sigma^{2},\ k=1,\ldots,K.$ (40d)

Similar to (32), the optimization (40) is a convex QSDP, and the global
optimum of (40) can be obtained in polynomial time. As we show in the sequel,
the overall complexity of ZF beamforming is substantially lower than that of
recovering the global optimum via Algorithm 1.

The solution of (40), i.e., the matrix $\tilde{\bm{R}}$ and the vector
$\tilde{\bm{p}}$, are used to construct the optimal precoding matrix
$\tilde{\bm{W}}$, As detained in the proof of Theorem 2. Here, we briefly give
the final expressions. First, we recover an $M\times M$ matrix $\bm{L}_{r}$
which satisfies $\tilde{\bm{R}}=\bm{L}_{r}\bm{L}_{r}^{H}$. This can be
obtained using, e.g., Cholesky decomposition, though $\bm{L}_{r}$ does not
have to be triangular and any matrix satisfying
$\tilde{\bm{R}}=\bm{L}_{r}\bm{L}_{r}^{H}$ may be used to calculate
$\tilde{\bm{W}}$. Then, the resulting precoder $\tilde{\bm{R}}$ is

$\tilde{\bm{W}}=\bm{L}_{r}\bm{Q}_{h}^{H}\left[\bm{Q}_{f}^{T}\right]^{T}_{{1:M}},$
(41)

where $\bm{Q}_{h}$ and $\bm{Q}_{f}$ are obtained by applying row QR
decomposition to $\bm{H}\bm{L}_{r}$ and $\bm{F}$, respectively. Since
$\bm{F}=\left[\mathrm{diag}(\sqrt{\tilde{\bm{p}}}),\bm{0}_{K\times M}\right]$
is diagonal, it holds that $\bm{Q}_{f}=\bm{I}_{M+K}$, and thus (41) is
simplified to

$\tilde{\bm{W}}=\left[\bm{L}_{r}\bm{Q}_{h}^{H},\bm{0}_{M\times K}\right].$
(42)

According to the proof of Theorem 1, $\tilde{\bm{W}},\tilde{\bm{R}}$ is a
feasible solution to (36). Since $\tilde{\bm{R}}$ is the global optimum to
(36), $\tilde{\bm{W}}$ is also globally optimal to (36).

The resulting ZF beamforming method is summarized below as Algorithm 2. The
main computational burden in Algorithm 2 stems from solving the QSDP problem
(40), as is also the case in Algorithm 1. Given a solution accuracy
$\epsilon$, the worst case complexity to solve the QSDP problem (40) with the
primal-dual interior-point algorithm in [67, 68] is
$\mathcal{O}(M^{6.5}\log(1/\epsilon))$. Compared to the recovering the global
solution via the SDR-based Algorithm 1, the worst-case computation complexity
for ZF beamforming is lower by a factor of $K^{6.5}$. This computational
complexity reduction stems from the fact that the optimization problem (40)
involves only one semidefinite constraint, while the problem (32), from which
Algorithm 1 originates, involves $K+2=\mathcal{O}(K)$ such constraints.

1:

2: Total transmit power $P_{t}$;

3: Power of AWGN at users $\sigma^{2}$;

4: Expression of the MIMO radar loss function $L_{r}(\bm{R},\bm{\alpha})$;

5: Instantaneous downlink channel $\bm{H}$;

6: SINR threshold $\Gamma$.

7:

8: The overall precoding matrix $\tilde{\bm{W}}$.

9: Steps:

10:Compute the optimal $\tilde{\bm{R}}$, and $\tilde{\bm{p}}$ by solving
optimization problem (40) with convex optimization solvers.

11:Compute the Cholesky decomposition of $\tilde{\bm{R}}$ as
$\tilde{\bm{R}}=\bm{L}_{r}\bm{L}_{r}^{H}$.

12:Given $\bm{HL}_{r}$, calculate $\bm{Q}_{h}$ with the row QR decomposition
(59) shown later in Appendix -B.

13:Compute the overall precoding matrix $\tilde{\bm{W}}$ using (42).

Algorithm 2 Joint transmit beamforming via ZF

We next discuss how the selection of the SINR threshold $\Gamma$ affects
trade-off between communications and radar when using ZF beamforming. As noted
in the discussion following the original optimization problem (26), the radar
loss function here decreases as $\Gamma$ increases, i.e., the less restrictive
the communication constraints are, the better the radar functionality can
perform. However, there are two phenomenons which are explained in the sequel,
that are different under ZF beamforming compared to the original optimization
problem (26): 1) When $\Gamma$ approaches zero, the radar performance achieved
is generally different from the radar-only optimal performance; 2) The radar
loss function and the obtained fairness SINR remain constant if $\Gamma$ is
lower than some positive value.

To understand phenomenon 1), we specialize the ZF optimization problem (40)
for the case of $\Gamma=0$, resulting in

$\displaystyle\min_{\bm{R},\bm{p},\alpha}\ $ $\displaystyle\
L_{r}(\bm{R},\alpha)$ (43a) $\displaystyle\mathrm{subject\ to}\ $
$\displaystyle\ \bm{R}\in\mathcal{S}_{M}^{+},$ (43b) $\displaystyle\
\left[\bm{R}\right]_{m,m}=P_{t}/M,\ m=1,\ldots,M,$ (43c) $\displaystyle\
\bm{HRH}^{H}=\mathrm{diag}\left(\bm{p}\right).$ (43d)

Here, we note that this formulation is distinct from the radar-only
optimization problem (15), since, even when the SINR can take any value, we
still force the interference to be cancelled. This restriction is reflected in
the additional constraint (43d) imposed on $\bm{R}$, namely that
$\bm{H}\bm{R}\bm{H}^{H}$ should be a diagonal matrix. The optimal radar-only
covariance $\bm{R}_{0}$, which is not forced to satisfy this interference
cancelling constraint, generally does not satisfy it, i.e.
$\bm{H}\bm{R}_{0}\bm{H}^{H}$ is not a diagonal matrix. If $\bm{R}_{0}$ is not
a feasible solution of problem (43), the radar-only optimal performance cannot
be achieved.

In order to explain phenomenon 2), we again focus on the ZF optimization
specialized to the case of no SINR constraints in (43), and denote its
solution by $\\{\bm{R}_{\rm II},\alpha_{\rm II},\bm{p}_{\rm II}\\}$. Given
$\bm{p}_{\rm II}$, the resulting fairness SINR is given by

$\Gamma_{\rm II}=\min\\{\bm{p}_{\rm II}\\}/\sigma^{2}.$ (44)

In problem (40), if the given $\Gamma$ is not greater than $\Gamma_{\rm II}$,
i.e. $0\leq\Gamma\leq\Gamma_{\rm II}$, the constraint (40d) always holds and
can be regarded as being invariant to the actual solution $\\{\bm{R}_{\rm
II},\alpha_{\rm II},\bm{p}_{\rm II}\\}$. In this case, $\\{\bm{R}_{\rm
II},\alpha_{\rm II},\bm{p}_{\rm II}\\}$ is still a feasible solution for (40).
Thus, the minimized radar loss function is equal to $L_{r}(\bm{R}_{\rm
II},\alpha_{\rm II})$, and the obtained fairness SINR is equal to $\Gamma_{\rm
II}$, for ZF beamforming derived under every SINR constraint satisfying
$\Gamma\leq\Gamma_{\rm II}$.

Here we compare the two proposed beamforming methods. The key difference
between them is whether to completely eliminate the interference. As a
globally optimal method, the radar performance of SDR beamforming should be
better than that of the sub-optimal ZF beamforming under the same
communication requirement. However, the performance gap may become small with
reasonably large $\Gamma$, because the interference is expected to be
eliminated under strict constraint on SINR. In this case, ZF beamforming is
preferable since its corresponding QSDP problem has a much simpler form. To
explain the performance gap when the given $\Gamma$ is low, we note that it is
unnecessary to completely eliminate the interference, which restricts the
precoder in a null space. Thanks to the more degrees of freedom for designing
$\bm{W}$, SDR beamforming enjoys better radar performance. In addition, as
$\Gamma$ goes to zero, the radar performance of SDR beamforming goes to the
optimal radar-only performance, while the radar performance of ZF beamforming
cannot and stays constant when $\Gamma$ is lower than some value.

##  V Numerical results

In this section we numerically evaluate the proposed joint beamforming
methods, i.e. SDR beamforming (Algorithm 1) and ZF beamforming (Algorithm 2),
in a simulation study. We begin by analyzing the achievable radar beampattern
of the proposed schemes, compared to the DFRC method of [42] in Subsection
V-A. Then, we compare ZF and SDR beamforming in Subsection V-B, while the
comparison of SDR beamforming and the DFRC method of [42] in terms of their
inherent radar-communication tradeoffs is presented in Subsection V-C.

In the experiments reported in this section, we use the following settings:
The transmit array is a uniform linear array with half wavelength spaced
elements. The number of transmit antennas is $M=10$ and the total transmit
power $P_{t}=1$. For MIMO radar transmit beamforming, the ideal beam pattern
consists of three main beams, whose the directions are
$\overline{\theta}_{1}=-40^{\circ}$, $\overline{\theta}_{2}=0^{\circ}$ and
$\overline{\theta}_{3}=40^{\circ}$. The width of each ideal beam is
$\Delta=10^{\circ}$, and thus the desired beam pattern is

$d(\theta)=\left\\{\begin{array}[]{cl}1,&\overline{\theta}_{p}-\frac{\Delta}{2}\leq\theta\leq\overline{\theta}_{p}+\frac{\Delta}{2},\
p=1,2,3,\\\ 0,&\mathrm{otherwise}.\end{array}\right.$ (45)

In (12), the direction grids $\left\\{\theta_{l}\right\\}_{l=1}^{L}$ are
obtained by uniformly sampling the range of $-90^{\circ}$ to $90^{\circ}$ with
resolution of $0.1^{\circ}$. The radar loss in (14) accounts for both
objectives equally, namely, the weighting factor is set to $w_{c}=1$. The
multi-user communications channel obeys a Rayleigh fading model, i.e., the
entries of $\bm{H}$ are i.i.d. standard complex normal random variables, and
the channel output at each user is corrupted with an additive white Gaussian
noise of variance $\sigma^{2}=0.01$.

In our simulations we use SINR threshold values $\Gamma$ varying from
$4\mathrm{dB}$ to $24\mathrm{dB}$, and number of users simulated is $K=2,4,6$.
We simulate different $\Gamma$ and $K$ to test the impact of these parameters
on the performance of the proposed joint beamforming methods. For each value
of $\Gamma$ and $K$, the performance is averaged over $1000$ Monte Carlo
tests. The individual radar waveform and communication symbols comprising the
transmitted signal $\bm{x}[n]$ in (1) are generated as random quadrature-
phase-shift-keying modulated sequences, and the transmit signal block size set
to is $N=1024$.

The MATLAB CVX toolbox [71, 72] is used to solve the QSDP problems (32) and
(40). We compare our joint beamforming schemes with the DFRC beamforming
method proposed in [42], in which only communication symbols are precoded.
Specially, we use gradient projection method to solve the sum-square penalty
(SSP) problem under per-antenna constraint in [42]. In the sum-square penalty
problem in [42], the weighing factors are $\rho_{1}=1$, $\rho_{2}=2$ and the
given SINR at each user is equal to the SINR threshold $\Gamma$ in (26) .

###  V-A MIMO Radar Transmit Beam Pattern

First, we numerically evaluated the MIMO radar transmit beam patterns
$P(\theta;\bm{R})$ defined in (10) for SDR beamforming, ZF beamforming, and
the SSP approach [42]. The transmit beam patterns for $\Gamma=12$ dB are
depicted in Fig. 3 for $K=2$ and in Fig. 4 for $K=4$. The optimal radar-only
beam pattern, obtained from (15), are also evaluated for comparison.

Observing Fig. 3, we note that when $K=2$, the average beam pattern for SDR
beamforming and ZF beamforming approaches that of the optimal radar-only
beamforming, while the SSP beamformer of [42] only synthesizes two main beams
towards $0^{\circ}$ and $40^{\circ}$. The fact that the SSP beamformer is
unable to steer three main beams for $K=2$ stems from its decreased MIMO radar
DoF. In particular, as noted in (10), the MIMO transmit beam pattern is
determined by the covariance of transmit waveform, and thus the DoF for MIMO
radar transmit beamforming is given by the rank of covariance matrix. In the
SSP approach, only communication symbols are precoded and thus the DoF cannot
be larger than the number of users $K$, namely, the rank of covariance matrix
cannot exceed $K$. In our scheme however, both communication symbols and radar
waveform are precoded, and thus the DoF can be as high as its maximal value
$M$, i.e., the covariance can have full rank. Numerically solving (15) using
the CVX toolbox reveals that rank of the optimal radar-only covariance
$\bm{R}_{0}$ is $4$. In other words, the required DoF to achieve the optimal
performance of radar is $4$. As a result, if $K=2<4$, the SSP approach does
not have enough DoF to form three main beams as in the optimal radar beam
pattern, explaining the degraded beam pattern observed in Fig. 3. Our scheme
are capable of forming beam patterns which are close to the optimal radar beam
pattern, since the available DoF in our schemes is $M=10>4$. When $K=4$, the
SSP approach has enough DoF and is thus capable of forming a beam pattern
comparable to the optimal radar beam pattern, as shown in Fig. 4.

We also observe in Figs. 3-4 that the main-lobe power of the ZF beamforming is
lower than that of the SDR beamforming, implying an expected radar performance
loss for ZF beamforming compared to SDR beamforming. This follows since when
the SINR threshold $\Gamma$ is not very high, one can achieve the desired SINR
level without canceling the interference, allowing to further optimize the
radar beam pattern by proper optimization. In ZF beamforming, the interference
is completely eliminated regardless of the SINR threshold, namely
$\bm{HW}=\left[\mathrm{diag}(\sqrt{\bm{p}}),\bm{0}_{K\times M}\right]$ even if
$\Gamma$ is low. This additional constraint limits the DoF of $\bm{W}$ and
introduces the radar performance loss compared to the SDR beamforming.
Nevertheless, in order to fully compare ZF beamforming to SDR beamforming, one
must also account for the communication performance. In particular, ZF
beamforming can provide improved communication rates compared SDR beamforming
due to the fact that it completely eliminates the interference regardless of
the specified SINR threshold. Our numerical results detailed in the sequel
show that the obtained SINR of ZF beamforming may be much higher than
$\Gamma$, while the obtained SINR by the SDR beamforming is generally quite
close to $\Gamma$. To understand the inherent tradeoffs of the proposed
schemes, in the following subsection we compare SDR and ZF beamforing in terms
of both their radar and communication performance measures.

Figure 3: Transmit beam pattern of MIMO radar, for $\Gamma=12$ dB and $K=2$.
Figure 4: Transmit beam pattern of MIMO radar, for $\Gamma=12$ dB and $K=4$.

###  V-B Comparison Between ZF and SDR Beamforming

In this subsection, we compare the radar performance and communication
performance of SDR and ZF beamforming. Radar performance is evaluated using
the beam pattern MSE, defined as the MSE between the obtained MIMO radar
transmit beam pattern and the optimal radar-only beam pattern, and is written
as

$\mathrm{MSE}=\frac{1}{L}\sum_{l=1}^{L}\left|P(\theta_{l};\bm{R}_{0})-P(\theta_{l};\bm{R})\right|^{2},$
(46)

where $P(\theta_{l};\bm{R}_{0})$ is the optimal radar-only beam pattern with
$\bm{R}_{0}$ obtained from (15). Low beam pattern MSE indicates improved MIMO
radar transmit beamforming. The numerically compared beam pattern MSE values
versus the SINR threshold $\Gamma$ are depicted in Fig. 5. As expected, the
beam pattern MSE increases with the increment of $\Gamma$, implying that the
more restrictive SINR demands naturally come at the cost of radar performance.
The results in Fig. 5 validate three characters of the two proposed joint
beamforming schemes: 1) SDR beamforming achieves improved radar performance
compared the the sub-optimal ZF strategy; 2) The performance gap between the
two methods notably narrows at high SINR constraints, i.e., as $\Gamma$
increases; 3) When $\Gamma$ is lower than some value, radar performance of the
ZF beamforming stays constant, as discussed in Subsection IV-C. It is also
observed Fig. 5 that the more communication receivers the DFRC system has to
communicate with reliably, i.e., as $K$ increases, the higher the beam pattern
MSE is, again indicating the inherent tradeoff between radar and
communications in DFRC systems. In particular, it is observed that the impact
of $K$ on the beam pattern MSE is more significant than the impact of
$\Gamma$, namely, the demand to support an increased number of users is more
restrictive in terms of radar performance compared to the requirement to
provide improved SINR at each user.

The communication performance is evaluated in terms of the achievable sum rate
defined in (22). The resulting values are depicted in Fig. 6, where we observe
that ZF beamforming achieves higher communciation rate compared to SDR
beamforming, despite its performance loss for radar. This follows since, as
discussed in the previous subsection, ZF beamforming typically yields SINR
values higher than the imposed threshold $\Gamma$, as it nullifies the
interference regardless of the value of $\Gamma$. Conversely, SDR beamforming,
which aims at improving radar performance without imposing any structure on
the resulting interference, does so by tunning its SINR to be close to the
threshold $\Gamma$, allowing to further improve radar performance without
violating the SINR constraint. Hence, the achievable sum rate of SDR
beamforming demonstrates an approximate linear increase with the SINR
constraint $\Gamma$ in Fig. 6.

From Figs. 5 and 6, it is observed that performance of the two methods
coincides as $\Gamma$ increases. For large values of $\Gamma$, the
interference tends to be naturally eliminated by the SDR beamforming in order
to meet the SINR constraints. To demonstrate this property, we depict in Fig.
7 the interference-to-noise ratio at the first user versus SINR threshold for
SDR beamforming. Observing Fig. 7, we note that when $\Gamma$ is high enough,
the interference power becomes much dominant than the noise power, and thus
the interference can be effectively ignored. Therefore, under high SINR
conditions, it is reasonable to completely eliminate the interference, and ZF
beamforming is asymptotically optimal.

Figure 5: Beam pattern MSE versus SINR threshold $\Gamma$. Figure 6:
Achievable sum rate versus SINR threshold $\Gamma$. Figure 7: Radar-
interference-to-noise ratio versus SINR threshold $\Gamma$, SDR beamforming.

We note that optimization problems tackled by SDR beamforming (32) and ZF
beamforming (36) are not always feasible. As the total transmit power is fixed
to be $P_{t}$, the signal power at users should have an upper bound.
Correspondingly, the achievable feasible SINR should have an upper bound. If
the given $\Gamma$ is too high, the joint beamforming problem (32) and (36)
may become infeasible. To calculate the feasible probability under a given
$\Gamma$ and $K$, we ran multiple Monte Carlo tests, randomizing a new channel
realization in each test. The feasible probability is calculated by dividing
the number of feasible tests by the total number of tests.

The relationship between the feasible probability and $\Gamma$ is demonstrated
in Fig. 8, for $K=2,4,6$. It is observed in Fig. 8 that the feasible
probability is roughly the same for ZF and SDR beamforming, and that both
curves decrease as the number of users and SINR threshold increases. This
implies that our optimization approach may fail with very high SINR
restrictions, and thus for practical applications, the SINR threshold should
be carefully set. If the given threshold is too high, the two problems may be
infeasible and our method will fail to return any meaningful solution.
Nevertheless, this result shows that the feasibility can almost always be
ensured if $\Gamma$ is lower than some value under Rayleigh channel. We also
note that this infeasible situation can be avoided if one changes the SINR
constraints into a part of penalty functions, i.e. (32a) or (40a), as done in
the scheme in [42]. We leave the analysis of this modification for future
investigation.

Figure 8: Feasible probability the SDR beamforming (32) and ZF beamforming
(36) versus SINR threshold $\Gamma$.

###  V-C Comparing SDR beamforming with SSP DFRC Method

Finally, we compare our proposed SDR beamforming method to the SSP DFRC scheme
previously proposed in [42]. To that aim, we evaluate their tradeoff between
the communication performance, encapsulated in the achieved fairness SINR
defined in (23) and the radar beam pattern MSE defined in (46). The
numerically evaluated tradeoffs for number of users $K=2,4,6$ are depcited in
Fig. reffig:p8. As discussed in Subsection V-A, our scheme notably outperform
the SSP approach for $K=2$, as clearly demonstrated in 9. When $K=4,6$, our
SDR beamforming technique still outperforms the SSP approach, although the
gain is less notable compared to $K=2$. The fact that SDR beamforming
outperforms the SSP method of [42] even when the latter is capable of
exploiting the full MIMO radar DoF stems from the following reasons: 1) The
SSP problem is non-convex and the obtained solution may be a local optimum; 2)
In the SSP problem, the radar lost function, defined as
$\|\bm{R}-\bm{R}_{0}\|_{F}^{2}$, does not directly reflect the performance of
radiation beam pattern.

Figure 9: Beam pattern MSE versus SINR threshold $\Gamma$.

Since the beam pattern MSE is not the only performance measure for radar, we
also analyze the sensing capabilities at radar receiver. To extract the range
and angular profile of radar targets from the received radar signal, we first
perform range compression [73] to obtain the range profile, and then use the
least square (LS) Capon method [74, 75] to calculate the spatial spectrum in
each range resolution bin.

The first simulation is conducted to examine the range resolution and angular
resolution of the MIMO radar. In the simulation, there are five targets in the
field of view of radar. The coordinate of targets in radar polar coordinate
system is defined by the discrete time delay $n^{\prime}$ (or the range
resolution bin index) and the angular direction $\theta$ as defined in (9). In
our parameter setting, the coordinate of these targets are $(10,0^{\circ})$,
$(20,-40^{\circ})$, $(20,0^{\circ})$, $(20,40^{\circ})$ and $(30,0^{\circ})$,
respectively, and the complex amplitude $\beta$ in (9) for each target is $1$.
The radar receive signal is corrupted with Gaussian noise with covariance
$\bm{R}_{v}=\sigma_{r}^{2}\bm{I}$, where $\sigma_{r}^{2}=1$. The Capon spatial
spectrum at the $20$-th range resolution bin and the range profile at
direction $0^{\circ}$ in one test are demonstrated in Fig. 10, for $K=2$ and
$\Gamma=12$dB. In 10, the range profile and Capon spatial spectrum for the
radar-only case, the SSP approach and SDR beamforming are compared. From Fig.
10, it is observed that the range and angular resolution for SDR beamforming
is close to that for the radar-only case. The performance degradation of the
SSP approach resulting from the lack of radar DoF is significant, since the
SSP approach cannot form a notable peak around the coordinate
$(20,0^{\circ})$, see Figs. 10 and 10, and the amplitude estimation error at
the $20$-th range resolution bin is very large, see Fig. 10. When $K=2$, the
reflected signal from the three targets at the $20$-th range resolution bin in
the SSP approach are linearly dependent, and thus the cross correlation
defined in (11) cannot be suppressed effectively. Therefore, the performance
of adaptive MIMO radar processing technique for the SSP approach decreases
significantly.

Figure 10: Capon spatial spectrum at the $20$-th range resolution bin and the
range profile at direction $0^{\circ}$, for $K=2$ and $\Gamma=12$ dB. (a)
Range profile for the radar-only case. (b) Capon spatial spectrum for the
radar-only case. (c) Range profile for the SSP method. (d) Capon spatial
spectrum for the SSP method. (e) Range profile for the SDR method. (f) Capon
spatial spectrum for the SDR method.

The second simulation is conducted to evaluate the spatial processing
performance of MIMO radar, including the angle estimation accuracy and target
detection performance. We simulate three radar targets located at directions
$\overline{\theta}_{1}=-40^{\circ}$, $\overline{\theta}_{2}=0^{\circ}$, and
$\overline{\theta}_{3}=40^{\circ}$, respectively. These targets are in the
same range resolution bin and the complex amplitude of the targets are all
$1$. The targets’ reflected signal is corrupted with additive noise whose
covariance is $\bm{R}_{v}=\sigma_{r}^{2}\bm{I}$. The angle of the targets is
estimated by finding the peaks of the Capon spatial spectrum. The angle
estimation performance is evaluated by the root-mean-square-error (RMSE),
defined as

$\mathrm{RMSE}=\sqrt{\mathbf{E}\left\\{\frac{1}{3}\sum_{p=1}^{3}(\overline{\theta}_{p}-\hat{\theta}_{p})^{2}\right\\}},$
(47)

where $\overline{\theta}_{p}$ is the real angle and $\hat{\theta}_{p}$ is the
estimated angle for the $p$-th target, for $p=1,\ldots,3$. The generalized
likelihood ratio test proposed in [76] is applied to detect the target. To
demonstrate the target detection performance, we study the relationship
between the detection probability and the transmit SNR given by
$P_{t}N/\sigma_{r}^{2}$, under a fixed false alarm probability $P_{fa}$. To
calculate the detection probability, we ran $1000$ Monte Carlo tests to
produce randomized Gaussian noise for each channel realization, and thus the
total number of tests is $10^{6}$.

The numerically evaluated tradeoff between angle estimation RMSE and achieved
fairness SINR for SDR beamforming and the SSP DFRC system of [42] is depicted
in Fig. 11 for $K=2,4,6$. Here the RMSE of the SSP approach for $K=2$ is not
evaluated since it frequently fails to detect the targets near the true angle
direction of the targets. The angle estimation RMSE in radar-only case is also
displayed for comparison. Observing Fig. 11, we note that the angle estimation
RMSE tends to increase with the fairness SINR, again indicating that the
improved communication performance induces some loss on the radar performance.
If $K=2$, the angle estimation performance of SDR beamforming is almost
identical to the performance in radar-only case, indicating that the proposed
DFRC system achieves angle estimation performance close to that of the radar-
only scheme. The RMSE of angle estimation slightly increases if more
communication users are under service. It is also noted that under most
considered fairness SINR values, our proposed SDR beamforming achieves
improved angle estimation RMSE compared to the SSP method.

The numerically evaluated detection probability versus transmit SNR for SDR
beamforming, SSP DFRC system of [42] and the radar-only case is depicted in
Fig. 12, for $\Gamma=12$ dB and $P_{fa}=10^{-4}$. From [42], it is noted that
there exists detection performance loss for simultaneous multiuser information
transmission compared to the radar-only case. If $K=2$, the detection
performance of SDR beamforming notably outperforms that of the the SSP
approach, because the SSP approach usually cannot provide enough DoF to form
three beams to cover the three target. Hence, reflected signal from one of the
targets may experience notable SNR loss, significantly reducing the detection
probability. If $K=4,6$, although the detection performance of SDR beamforming
and the SSP approach is close, we note that the SDR beamforming can achieve
better communication quality. In particular, the SDR beamforming guarantees
the achieved fairness SINR is higher than $\Gamma$, while in the SSP approach
the SINR is considered as a penalty term in the penalty function and the
obtained SINR at users is generally less than $\Gamma$ in our simulation.

Figure 11: RMSE for angle estimation with LS-Capon method versus SINR
threshold $\Gamma$. Figure 12: Detection probability versus transmit SNR under
false alarm probability $P_{fa}=10^{-4}$, for $\Gamma=12$ dB.

##  VI Conclusion

In this paper, we proposed two joint beamforming approaches for MIMO radar and
multiuser MIMO communication sharing spectrum and transmit array. The
precoders of the individual radar waveform and communication symbols are
designed to optimize the performance of MIMO radar transmit beamforming while
meeting SINR constraints at communication users. To solve the proposed
optimization problem efficiently, we proposed an algorithm based on SDR, which
allows to accurately recover the optimal beamforming scheme. To reduce the
computational burden of SDR beamforming, we proposed a reduced complexity sub-
optimal strategy based on joint beamforming with zero-forced inter-user and
radar interference. Simulation results showed that the radar beam patterns and
angle estimation performance obtained by the proposed dual-function system is
comparable to those of the optimal radar-only scheme. We also observed the
advantage of our method over the previous dual-function work that precodes
only multiuser communication symbols from simulations in terms of their
inherent radar-communication tradeoffs. These performance gains are most
notable when the number of communication users is relatively small, and
particularly less than the MIMO radar DoF required to meet the desired
transmit beam pattern.

###  -A Proof of Theorem 1

Let $\hat{\bm{R}},\hat{\bm{R}}_{1},\ldots,\hat{\bm{R}}_{K}$ be an arbitrary
global optimum to (32). We prove the theorem by constructing
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ from
$\hat{\bm{R}},\hat{\bm{R}}_{1},\ldots,\hat{\bm{R}}_{K}$ with

$\tilde{\bm{R}}=\hat{\bm{R}},\
\tilde{\bm{w}}_{k}=\big{(}\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\big{)}^{-1/2}\hat{\bm{R}}_{k}\bm{h}_{k},\
\tilde{\bm{R}}_{k}=\tilde{\bm{w}}_{k}\tilde{\bm{w}}_{k}^{H},$ (48)

for $k=1,\ldots,K$. It is clear that
$\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ are positive semidefinite and
are rank-one.

We now show that $\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$
is also a global optimum to (32). Since the target function $L(\bm{R},\alpha)$
is determined by $\bm{R}$ and $\hat{\bm{R}}=\tilde{\bm{R}}$, we only need to
validate that $\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ is
a feasible solution to (32).

First, one can derive that

$\bm{h}_{k}^{H}\tilde{\bm{R}}_{k}\bm{h}_{k}=\bm{h}_{k}^{H}\tilde{\bm{w}}_{k}\tilde{\bm{w}}_{k}^{H}\bm{h}_{k}=\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}$
(49)

by substituting (48). Thus

$\displaystyle\left(1+\Gamma^{-1}\right)\bm{h}_{k}^{H}\tilde{\bm{R}}_{k}\bm{h}_{k}=\left(1+\Gamma^{-1}\right)\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}$
(50)
$\displaystyle\geq\bm{h}_{k}^{H}\hat{\bm{R}}\bm{h}_{k}+\sigma^{2}=\bm{h}_{k}^{H}\tilde{\bm{R}}\bm{h}_{k}+\sigma^{2},$

namely constraint (32e) holds for
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$.

Next, we show that
$\hat{\bm{R}}_{k}-\tilde{\bm{R}}_{k}\in\mathcal{S}_{M}^{+}$. For any
$\bm{v}\in\mathbb{C}^{M}$, it holds that

$\bm{v}^{H}\big{(}\hat{\bm{R}}_{k}-\tilde{\bm{R}}_{k}\big{)}\bm{v}=\bm{v}^{H}\hat{\bm{R}}_{k}\bm{v}-\big{(}\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\big{)}^{-1}\left|\bm{v}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\right|^{2}.$
(51)

According to the Cauchy-Schwarz inequality, one has

$\big{(}\bm{h}_{k}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\big{)}\big{(}\bm{v}^{H}\hat{\bm{R}}_{k}\bm{v}\big{)}\geq\left|\bm{v}^{H}\hat{\bm{R}}_{k}\bm{h}_{k}\right|^{2},$
(52)

so $\bm{v}^{H}\big{(}\hat{\bm{R}}_{k}-\tilde{\bm{R}}_{k}\big{)}\bm{v}\geq 0$
holds for any $\bm{v}\in\mathbb{C}^{M}$, i.e.
$\hat{\bm{R}}_{k}-\tilde{\bm{R}}_{k}\in\mathcal{S}_{M}^{+}$. It therefore
follows that

$\tilde{\bm{R}}-\sum_{k=1}^{K}\tilde{\bm{R}}_{k}=\hat{\bm{R}}-\sum_{k=1}^{K}\hat{\bm{R}}_{k}+\sum_{k=1}^{K}\big{(}\hat{\bm{R}}_{k}-\tilde{\bm{R}}_{k}\big{)}\in\mathcal{S}_{M}^{+},$

namely, the constraint (32b) holds for
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$. Finally, since
$\hat{\bm{R}}=\tilde{\bm{R}}$, (32c) also holds for $\tilde{\bm{R}}$.

With the derivation above, it is verified that
$\tilde{\bm{R}},\tilde{\bm{R}}_{1},\ldots,\tilde{\bm{R}}_{K}$ is a feasible
solution, and furthermore, it is also a global optimum to (32), completing the
proof. ∎

###  -B Proof of Theorem 2

When the conditions (36b) and (37) hold, it follows that

$\bm{HRH}^{H}=\bm{HWW}^{H}\bm{H}^{H}=\bm{F}\bm{F}^{H},$ (53)

i.e. (38) holds, proving the necessity.

Next, we prove that condition (38) is also sufficient. Assume that condition
(38) holds. We will then construct a $\bm{W}$ that satisfies (36b) and (37).
To this aim, we recall that the QR decomposition [66] of a $n\times m$ matrix
$\bm{B}$ with $n\geq m$ is defined as

$\bm{B}=\bm{P}^{\prime}_{a}\bm{U}_{a}=\bm{P}_{a}\left[\begin{array}[]{c}\bm{U}_{a}\\\
\bm{0}_{(n-m)\times m}\end{array}\right],$ (54)

where $\bm{U}_{a}$ is a $m\times m$ upper triangular matrix,
$\bm{P}^{\prime}_{a}$ is a $n\times m$ matrix with orthogonal unit columns,
and $\bm{P}_{a}$ is a $n\times n$ unitary matrix. Then, define the row QR
decomposition of a $m\times n$ matrix $\bm{A}=\bm{B}^{T}$ with $m\leq n$ as

$\bm{A}=\bm{L}_{a}\bm{Q}^{\prime}_{a}=\left[\bm{L}_{a},\bm{0}_{m\times(n-m)}\right]\bm{Q}_{a},$
(55)

where $\bm{L}_{a}=\bm{U}_{a}^{T}$ is a $m\times m$ lower triangular matrix,
$\bm{Q}^{\prime}_{a}=(\bm{P}_{a}^{\prime})^{T}$ is a $m\times n$ matrix with
orthogonal unit rows, and $\bm{Q}_{a}=\bm{P}_{a}^{T}$ is a $n\times n$ unitary
matrix, i.e.

$\bm{Q}_{a}\bm{Q}_{a}^{H}=\bm{Q}_{a}^{H}\bm{Q}_{a}=\bm{I}_{n}.$ (56)

We note that

$\displaystyle\bm{A}\bm{A}^{H}$
$\displaystyle=\left[\bm{L}_{a},\bm{0}_{m\times(n-m)}\right]\bm{Q}_{a}\bm{Q}_{a}^{H}\left[\begin{array}[]{c}\bm{L}_{a}^{H}\\\
\bm{0}_{(n-m)\times m}\end{array}\right]$ (57)
$\displaystyle=\bm{L}_{a}\bm{L}_{a}^{H}.$

Observing the left hand side of (38), we proceed by writing the Cholesky
decomposition of $\bm{R}$ as

$\bm{R}=\bm{L}_{r}\bm{L}_{r}^{H},$ (58)

and writing the row QR decomposition to $\bm{HL}_{r}$ as

$\bm{HL}_{r}=\left[\bm{L}_{h},\bm{0}_{K\times(M-K)}\right]\bm{Q}_{h},$ (59)

where $\bm{Q}_{h}$ is a $M\times M$ unitary matrix and $\bm{L}_{h}$ is a
$K\times K$ lower triangular matrix. Applying (57), we rewrite the left hand
side of (38) as

$\bm{H}\bm{R}\bm{H}^{H}=\bm{HL}_{r}\bm{L}_{r}^{H}\bm{H}^{H}=\bm{L}_{h}\bm{L}_{h}^{H}.$
(60)

Similarly, applying row QR decomposition to $\bm{F}$ yields

$\bm{F}=\left[\bm{L}_{f},\bm{0}_{K\times M}\right]\bm{Q}_{f},$ (61)

and then

$\bm{F}\bm{F}^{H}=\bm{L}_{f}\bm{L}_{f}^{H},$ (62)

according to (57). In (61), $\bm{Q}_{f}$ is a $(M+K)$-dimension unitary matrix
and $\bm{L}_{f}$ is a $K\times K$ lower triangular matrix.

Here, we note that both $\bm{H}\bm{R}\bm{H}^{H}$ and $\bm{F}\bm{F}^{H}$ are
positive definite given that $\bm{F}$ is a full rank $K\times(K+M)$ matrix ,
indicating that the diagonal elements of $\bm{L}_{h}$ and $\bm{L}_{f}$ are all
non-zero real numbers.

Since $\bm{L}_{h}$ and $\bm{L}_{f}$ are lower triangular matrices, we find
that (60) is the Cholesky decomposition of $\bm{H}\bm{R}\bm{H}^{H}$, and (62)
is the Cholesky decomposition of $\bm{F}\bm{F}^{H}$. Since
$\bm{H}\bm{R}\bm{H}^{H}=\bm{F}\bm{F}^{H}$ and the Cholesky decomposition of a
positive definite matrix is unique [66], we have that

$\bm{L}_{h}=\bm{L}_{f},$ (63)

if we require that the diagonal elements of $\bm{L}_{h}$ and $\bm{L}_{f}$ are
positive real numbers.

We can now construct the matrix $\bm{W}$ as $\bm{W}=\bm{L}_{r}\bm{Q}_{w}$ to
satisfy (36b), where $\bm{Q}_{w}$ is a $M\times(M+K)$ matrix obeying
$\bm{Q}_{w}\bm{Q}_{w}^{H}=\bm{I}_{M}$. Since we also require $\bm{W}$ to meet
(37), the matrix $\bm{Q}_{w}$ should satisfy that

$\bm{H}\bm{L}_{r}\bm{Q}_{w}=\bm{F}.$ (64)

To this aim, the matrix $\bm{Q}_{w}$ is constructed as
$\bm{Q}_{w}=\bm{Q}_{h}^{H}\hat{\bm{Q}}_{f}$, where the $M\times(M+K)$ matrix
$\hat{\bm{Q}}_{f}=\left[\bm{Q}_{f}^{T}\right]^{T}_{{1:M}}$ denotes the first
$M$ rows of $\bm{Q}_{f}$, and satisfies

$\hat{\bm{Q}}_{f}\hat{\bm{Q}}_{f}^{H}=\bm{I}_{M}$ (65)

according to (56). Thus, the matrix $\bm{W}$ is computed as

$\bm{W}=\bm{L}_{r}\bm{Q}_{h}^{H}\hat{\bm{Q}}_{f}.$ (66)

Using (66), we can calculate $\bm{W}$ from $\bm{R}$ and $\bm{F}$ with
$\bm{L}_{r}$, $\bm{Q}_{h}$ and $\bm{Q}_{f}$ obtained by applying matrix
decomposition, i.e. (58), (59) and (61), respectively.

To prove (36b) and (36d), we substitute (66) into these two equations,
yielding

$\bm{W}\bm{W}^{H}=\bm{L}_{r}\bm{Q}_{h}^{H}\hat{\bm{Q}}_{f}\hat{\bm{Q}}_{f}^{H}\bm{Q}_{h}\bm{L}_{r}^{H}\stackrel{{\scriptstyle(a)}}{{=}}\bm{L}_{r}\bm{L}_{r}^{H}\stackrel{{\scriptstyle(b)}}{{=}}\bm{R},$
(67)

and

$\displaystyle\bm{H}\bm{W}=\bm{H}\bm{L}_{r}\bm{Q}_{h}^{H}\hat{\bm{Q}}_{f}\stackrel{{\scriptstyle(c)}}{{=}}\left[\bm{L}_{h},\bm{0}_{K\times(M-K)}\right]\bm{Q}_{h}\bm{Q}_{h}^{H}\hat{\bm{Q}}_{f}$
(68)
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\left[\bm{L}_{h},\bm{0}_{K\times(M-K)}\right]\hat{\bm{Q}}_{f}\stackrel{{\scriptstyle(e)}}{{=}}\left[\bm{L}_{f},\bm{0}_{K\times
M}\right]\bm{Q}_{f}\stackrel{{\scriptstyle(f)}}{{=}}\bm{F},$

respectively, where $(a)$ follows from (65) and
$\bm{Q}_{h}^{H}\bm{Q}_{h}=\bm{I}_{M}$ (56); $(b)$ stems from (58); $(c)$ is
due to (59); $(d)$ applies again $\bm{Q}_{h}^{H}\bm{Q}_{h}=\bm{I}_{M}$; $(e)$
uses (63); and $(f)$ follows from (61). Therefore, the condition (38) is also
sufficient, completing the proof. ∎

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