# Optimal Precoding Design for Monostatic ISAC Systems: MSE Lower Bound and
DoF Completion

Yuanhao Cui, Fan Liu, Weijie Yuan,
Junsheng Mu, Xiaojun Jing, and Derrick Wing Kwan Ng Y. Cui, J. Mu, and X. Jing
are with the School of Information and Communication Engineering, Beijing
University of Posts and Communications (BUPT), Beijing 100876, China.
(email:{cuiyuanhao, mujs, jxiaojun}@bupt.edu.cn).F. Liu and W. Yuan is with
the Department of Electrical and Electronic Engineering, Southern University
of Science and Technology, Shenzhen 518055, China. (e-mail:
{liuf6,yuanwj}@sustech.edu.cn).D. W. K. Ng are with the School of Electrical
Engineering and Telecommunications, University of New South Wales, Sydney, NSW
2052, Australia. (e-mail: w.k.ng@unsw.edu.au).

###### Abstract

In this letter, we study the parameter estimation performance for monostatic
downlink integrated sensing and communications (ISAC) systems. In particular,
we analyze the mean squared error (MSE) lower bound for target sensing in the
downlink ISAC system that reveals the suboptimality in re-using the
conventional communication waveform for sensing. To realize a practical dual-
functional waveform, we propose a waveform augmentation strategy that imposes
an extra signal structure, namely the degrees-of-freedom (DoF) completion
method. The proposed approach is capable of improving the parameter estimation
performance of the ISAC system and achieving the derived MSE lower bound. To
improve the performance of the proposed strategy, we formulate an MSE
minimization problem to design the ISAC precoder, subject to the communication
users’ signal-interference-plus-noise-ratio (SINR) constraints. Despite the
non-convexity of the waveform design problem, we obtain its globally optimal
solution via semi-definite relaxation (SDR) and the proposed constructive
method. Simulation results validate the proposed DoF completion technology
could achieve the derived MSE lower bound and the effectiveness of the MSE-
based ISAC waveform design.

###### Index Terms:

Integrated Sensing and Communications, Multi-user Communications

##  I Introduction

Benefiting from the improved spectral-, energy- and hardware efficiency, and
the ability to deploy sensing functionality into the current communication
networks, the research interest for integrated sensing and communication
(ISAC) has arisen in the design of the sixth-generation (6G) systems [1].
Depending on the geographical configurations, current ISAC systems are
categorized into three classical configurations [2], i.e., 1) the monostatic
deployment that transmits communication signals and then captures the target
echoes via the co-located receiver, such as enabling a base station (BS) as a
sensor; 2) the bistatic deployment that collects the reflected and scattered
echoes from a separated receiver, such as various Wi-Fi sensing applications;
3) the distributed deployment that characterizes a target via signals
collected from widely distributed receivers. In particular, the monostatic
ISAC systems are appealing for practical implementation as it promises a
pilot-free signaling strategy [1].

To unlock the potential integration gain promised by ISAC signaling [3], one
of the major challenges is to design a fully unified waveform design via
jointly considering both the sensing and the communication performances. As
such, several ISAC signaling strategies have been proposed to strike a balance
between communications and sensing [4], e.g., embedding communication data
into a sensing waveform [5] and the direct use of standard-compatible
communication waveforms for sensing [6]. Yet, there are limited discussions
regarding the estimation error for sensing functionality in ISAC systems,
which is also a fundamental performance metric [7].

In this letter, we focus on the downlink monostatic ISAC system where an ISAC
BS serves multiple users and simultaneously sensing the surrounding
environment. In particular, we analyze the parameter estimation performance of
the re-used communication waveform, by formulating a transmit waveform design
problem that minimizes the MSE for the target sensing. To this end, a MSE
lower bound is derived that paves the way for establishing a bound-achieving
waveform augmentation strategy, namely degrees-of-freedom (DoF) completion,
that facilitates the transmit waveform to achieve the MSE lower bound. The
simulation results validate the performance of proposed DoF completion
strategy and the effectiveness of the MSE-based ISAC waveform design. We
highlight our contributions as following:

  * •

A MSE lower bound is derived for the monostatic downlink ISAC system, which
indicates that re-using conventional communication waveforms without further
processing could not achieve the lower bound even if the signal-interference-
plus-noise-ratio (SINR) asymptotically high.

  * •

A waveform augmentation strategy, i.e., DoF completion, is systematically
introduced. Comparing to other pioneer work [8, 9], we confirm that the
proposed strategy can achieve the derived MSE lower bound, and then, apply it
to a newly proposed MSE-based ISAC waveform design problem. Finally, we obtain
its globally optimal solution via semidefinite relaxation (SDR) by
constructively showing that the adopted SDR is tight.

Notations: Matrices are denoted by bold uppercase letters, vectors are
represented by bold lowercase letters; $\text{Tr}(\cdot)$ and
$\text{vec}(\cdot)$ denote the trace and the vectorization operations,
$(\cdot)^{T}$, $(\cdot)^{H}$, $\text{rank}(\cdot)$, and $(\cdot)^{-1}$ stand
for the transpose, the Hermitian transpose, the rank, and the inverse of the
input matrices, respectively. $\otimes$, and $\mathbb{E}(\cdot)$ denotes the
Kronecker product and the expectation operation, respectively. $\mathbf{I}$
represents the identity matrix.

Figure 1: Illustration of the monostatic downlink ISAC system with $K$
communication users and a target.

##  II System Model

###  II-1 Communication Model

We consider an integrated sensing and communication BS equipped with $N_{T}$
transmitting antennas and $N_{R}$ receiving antennas, as shown in Fig. 1. The
BS transmits data to $K$ single-antenna communication users and simultaneously
extracting environmental information from reflected echoes such that $K\leq
N_{T}$. The channel matrix from the BS to $\mathcal{K}=\\{1,\cdots,K\\}$ users
is represented by $\mathbf{H}\in\mathbb{C}^{K\times
N_{T}}=[\mathbf{h}_{1},\cdots,\mathbf{h}_{K}]^{H}$, with each
$\mathbf{h}_{k}\in\mathbb{C}^{N_{T}\times 1}$ denoting the channel vector from
the BS to user $k\in\mathcal{K}$. We further assume that the communication
channel experiences block fading, in which the data symbol
$s_{k,l}\in\mathbb{C}$ transmitted to user $k$ over time slot
$l\in\mathcal{L}=\\{1,\cdots,L\\}$ is precoded by a spatial precoding vector
$\mathbf{p}_{k}\in\mathbb{C}^{K\times 1}$, where $L$ is the channel coherence
time interval with $K\leq N_{T}\leq L$ [4, 9]. Taking the inter-user
interference into account, the received signal at the $k$-th user is given by

$y_{k,l}=\mathbf{h}_{k}^{H}\mathbf{p}_{k}s_{k,l}+\sum_{j\in\mathcal{K}/k}\mathbf{h}_{k}^{H}\mathbf{p}_{j}s_{j,l}+n_{k,l},\forall
k\in\mathcal{K},l\in\mathcal{L},$ (1)

where $n_{k,l}\sim\mathcal{CN}(0,\delta_{\text{C}}^{2})$ is the additive
complex Gaussian white noise at user $k$. Without loss of generality, we
assume that the channel knowledge is perfectly known at the BS. Then, the
achievable rate at user $k$ can be written as

$\mathcal{C}_{k}(\mathbf{P})=\log_{2}(1+\gamma_{k}(\mathbf{P})),$ (2)

where $\mathbf{P}\in\mathbb{C}^{N_{T}\times
K}=[\mathbf{p}_{1},\cdots,\mathbf{p}_{K}]$ denotes the overall precoding
matrix at the BS transmitter and $\gamma_{k}(\mathbf{P})$ is the received SINR
at user $k$. By stacking the data symbols transmitted at slot $l$ into a
vector $\mathbf{s}_{l}=[s_{1,l},\cdots,s_{K,l}]^{T}$, we assume that the
transmitted data symbols are independent with unitary power, i.e.,
$\mathbb{E}(\mathbf{s}_{l}\mathbf{s}_{l}^{H})=1$, such that the instantaneous
per-user SINR can be given by

$\gamma_{k}(\mathbf{P})=\frac{|\mathbf{h}_{k}^{H}\mathbf{p}_{k}|^{2}}{\sum_{j\in\mathcal{K}/k}|\mathbf{h}_{k}^{H}\mathbf{p}_{j}|^{2}+\delta_{\text{C}}^{2}},\forall
k\in\mathcal{K}.$ (3)

###  II-2 Sensing Model

The BS re-uses the transmitted waveform for sensing the surrounding
environment. The corresponding sensing waveform
$\mathbf{X}=\mathbf{P}\mathbf{S}$ is an $N_{T}\times L$ dimensional matrix
that jointly governed by the precoder and transmitted data symbol matrices,
where $\mathbf{S}=[\mathbf{s}_{1},\cdots,\mathbf{s}_{L}]$ is a $K\times L$
dimensional data matrix with orthogonal data streams
$\frac{1}{L}\mathbf{S}\mathbf{S}^{H}=\mathbf{I}$. Thus, the sample covariance
matrix of $\mathbf{X}$ can be written as

$\mathbf{R}_{\text{X}}=\frac{1}{L}\mathbf{X}\mathbf{X}^{H}=\frac{1}{L}\mathbf{P}\mathbf{S}\mathbf{S}^{H}\mathbf{P}^{H}=\mathbf{P}\mathbf{P}^{H}=\mathbf{R}_{\text{P}}.$
(4)

Then, the reflected sensing echo
$\mathbf{Y}_{\text{S}}\in\mathbb{C}^{N_{R}\times L}$ received at the BS can be
written in a matrix form as

$\mathbf{Y}_{\text{S}}=\mathbf{G}\mathbf{X}+\mathbf{N}_{\text{S}},$ (5)

where $\mathbf{G}\in\mathbb{C}^{N_{R}\times N_{T}}$ is the target response
matrix and $\mathbf{N}_{\text{S}}\in\mathbb{C}^{N_{R}\times L}$ is the noise
matrix. For sensing, we aim for estimating all target responses, i.e.,
elements of matrix $\mathbf{G}$. For the ease of presentation, we adopt the
following operator,
$\text{vec}(\mathbf{A}\mathbf{B})=(\mathbf{B}^{T}\otimes\mathbf{I})\text{vec}(\mathbf{A})$,
to vectorize (5) as

$\mathbf{y}_{\text{S}}=(\mathbf{X}^{T}\otimes\mathbf{I})\mathbf{g}+\mathbf{n}_{\text{S}}=\bar{\mathbf{X}}\mathbf{g}+\mathbf{n}_{\text{S}},$
(6)

where $\mathbf{g}=\text{vec}(\mathbf{G})$,
$\mathbf{n}_{\text{S}}=\text{vec}(\mathbf{N}_{\text{S}})$, and
$\bar{\mathbf{X}}=\mathbf{X}^{T}\otimes\mathbf{I}$ denotes the $N_{T}N_{R}$
dimensional vectorized target response matrix, noise vector, and the
equivalent sensing waveform matrix, respectively. Typically, the target
information is unknown to the sensing system. Hence, we assume that
$\mathbf{g}$ is a complex Gaussian random vector
$\mathbf{g}\sim\mathcal{CN}(\mathbf{0},\mathbf{R}_{\text{g}})$, and is
independent of the noise vector
$\mathbf{n}_{\text{S}}\sim\mathcal{CN}(\mathbf{0},\delta_{\text{S}}^{2}\mathbf{I})$.

###  II-3 Target Model

Here, we consider an extended target model, where a sensory target (e.g. a
human being or a vehicle) is located near to the BS, such that the sensing
waveforms are reflected by various target components (scatters). Therefore, in
a rich scattering environment, $\mathbf{R}_{\text{g}}$ is typically of full
rank. Recalling that the transmitting and the receiving antennas employed for
sensing are colocated at the BS, thus, $\mathbf{R}_{\text{g}}$ can be modeled
as [10]

$\mathbf{R}_{\text{g}}=\sum_{m}\alpha_{m}^{2}(\mathbf{a}(\theta_{m})\mathbf{a}^{H}(\theta_{m}))\otimes(\mathbf{b}(\theta_{m})\mathbf{b}^{H}(\theta_{m})),$
(7)

where the scalar $\alpha_{m}$ is the radar cross section (RCS) of scatter $m$,
$\mathbf{a}(\theta_{m})$ and $\mathbf{b}(\theta_{m})$ are the corresponding
transmit and receive steering vectors to echo’s direction of arrival
$\theta_{m}$, respectively. Since we are only interested in the MSE-based
waveform design at the transmitter side, the receive steering vectors are
assumed to orthogonal with each other. Therefore, by introducing the transmit
covariance matrix
$\mathbf{R}_{\text{g}_{\text{T}}}=\sum_{m}\alpha_{m}^{2}\mathbf{a}(\theta_{m})\mathbf{a}^{H}(\theta_{m})$
and the receive covariance matrix
$\mathbf{R}_{\text{g}_{\text{R}}}=\mathbf{I}_{\text{N}_{\text{R}}}$, we can
simplify $\mathbf{R}_{\text{g}}$ as

$\mathbf{R}_{\text{g}}=\mathbf{R}_{\text{g}_{\text{T}}}\otimes\mathbf{I}_{\text{N}_{\text{R}}}.$
(8)

###  II-4 MSE-Based Transmit Waveform Design

From the sensing perspective, we consider an estimation-oriented waveform
design that minimizes the MSE for target sensing. Since the received signal
$\mathbf{y}_{\text{S}}$ and the target responses $\mathbf{g}$ are jointly
Gaussian distributed, the MMSE estimator is indeed a linear MMSE (LMMSE)
estimator. Recall the fact that the sensing waveform $\bar{\mathbf{X}}$ is
known to the BS. In particular, the MMSE estimator is expressed as

$\mathbf{g}_{\text{MMSE}}=(\bar{\mathbf{X}}^{H}\bar{\mathbf{X}}+\delta_{\text{S}}^{2}\mathbf{I})^{-1}\bar{\mathbf{X}}^{H}\mathbf{y}_{\text{S}}.$
(9)

Therefore, the MSE matrix $\mathbf{E}$ is

$\displaystyle\mathbf{E}=\mathbb{E}\\{\|\mathbf{g}-\mathbf{g}_{\text{MMSE}}\|^{2}_{2}\\}{=}\text{Tr}\\{(\delta_{\text{S}}^{-2}\bar{\mathbf{X}}^{H}\bar{\mathbf{X}}+\mathbf{R}_{\text{g}}^{-1})^{-1}\\}.$
(10)

To bridge the connection between the sensing waveform design and communication
precoder, we further observe that
$\bar{\mathbf{X}}^{H}\bar{\mathbf{X}}=(\mathbf{X}\mathbf{X}^{H})\otimes\mathbf{I}_{\text{N}_{\text{R}}}=\mathbf{R}_{\text{P}}\otimes\mathbf{I}_{\text{N}_{\text{R}}}.$
By taking (8) into account, the to-be-designed precoding matrix $\mathbf{P}$
appears in the MMSE sensing waveform design criteria. Exploiting the identity
$\text{Tr}(\mathbf{A}\otimes\mathbf{I}_{\text{N}_{\text{R}}})=N_{R}\text{Tr}(\mathbf{A})$,
the MMSE transmit waveform design problem is given by

$\begin{split}\mathbf{E}(\mathbf{R}_{\text{P}})&=\text{Tr}\\{((\delta_{\text{S}}^{-2}\mathbf{R}_{\text{P}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})\otimes\mathbf{I}_{\text{N}_{\text{R}}})^{-1}\\}\\\
&=N_{R}\text{Tr}\\{(\delta_{\text{S}}^{-2}\mathbf{R}_{\text{P}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}.\end{split}$
(11)

Therefore, when the transmitted communication signal is re-used for sensing,
the waveform design problem that minimizes the target estimation error under
the maximum power budget can be formulated as

$\displaystyle\underset{\mathbf{R}_{\text{P}}}{\text{minimize}}\quad$
$\displaystyle\text{Tr}\\{(\delta_{\text{S}}^{-2}\mathbf{R}_{\text{P}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}$
s.t $\displaystyle\text{Tr}\\{\mathbf{R}_{\text{P}}\\}\leq
P_{T},\mathbf{R}_{\text{P}}\succeq\mathbf{0}.$ (12)

Obviously, the optimization problem (12) is convex w.r.t
$\mathbf{R}_{\text{P}}$. Its optimal point can be found by the well-known
Karush–Kuhn–Tucker (KKT) conditions [11]. Yet, in the next section, our
analysis on the optimal solutions of (10) will unveil the parameter estimation
performance of the downlink waveform in the monostatic ISAC system.

##  III A MSE Lower Bound

In this section, we analyze the parameter estimation performance by examining
the above MSE problem. Here, we start with a MSE lower bound inspired from
(12).

Theorem 1: Let $\mathbf{R}_{\text{P}}$ be the covariance matrix of the BS
precoder and $\mathbf{R}_{\text{g}_{\text{T}}}$ be the target response matrix,
which are both $N_{T}\times N_{T}$ positive-definite full-rank Hermitian
matrices. Let $\sigma_{i,\text{P}}$ and $\sigma_{i,\text{g}_{\text{T}}}$ be
the $i$-th large eigenvalue of $\mathbf{R}_{\text{P}}$ and
$\mathbf{R}_{\text{g}_{\text{T}}}$, respectively. Then, the MSE is lower-
bounded by

$\mathbf{E}(\mathbf{R}_{\text{P}})\geq\sum_{i=1}^{N_{T}}(\delta_{\text{S}}^{-2}\sigma_{i,{\text{P}}}+\sigma_{i,\text{g}_{\text{T}}}^{-1})^{-1}.$
(13)

The equality holds if and only if the eigenvector matrix
$\mathbf{U}_{\text{P}}$ = $\mathbf{U}_{\text{g}_{\text{T}}}$, where
$\mathbf{U}_{\text{P}}$ and $\mathbf{U}_{\text{g}_{\text{T}}}$ consist of
eigenvectors corresponding to descending ordered eigenvalues of
$\mathbf{R}_{\text{P}}$ and $\mathbf{R}_{\text{g}_{\text{T}}}$, respectively.

###### Proof.

It can be proved by following the similar procedure as in [12] and is omitted
here due to the page limitation. ∎

The MSE matrix evaluates the parameter estimation performance of sensing
functionality. Typically, if $\mathbf{R}_{\text{g}_{\text{T}}}$ is known,
Theorem 1 can be employed to find the optimal eigenvalues
$\\{{\sigma}^{*}_{i,{\text{P}}}\\}_{i=1}^{N_{T}}$ that minimize the MSE. In
(12), with the maximum power budget $P_{T}$ at BS, the optimal eigenvalues is
the water-filling solution given by

$\sigma^{*}_{i,{\text{P}}}=\left[\delta_{\text{S}}(\frac{1}{\sqrt{\lambda}}-\frac{\delta_{\text{S}}}{\sigma_{i,\text{g}_{\text{T}}}})\right]^{+},\quad\sum_{i=1}^{N_{T}}\sigma^{*}_{i,{\text{P}}}=P_{T}.$
(14)

Consequently, the optimal precoder can be designed as
$\mathbf{P}=\mathbf{U}_{\text{P}}\text{diag}\\{{\sigma^{*}}_{i,{\text{P}}}^{\frac{1}{2}},\cdots,{\sigma^{*}}_{N_{T},{\text{P}}}^{\frac{1}{2}}\\}$.
If the to-be-designed $\mathbf{P}$ is unconstrained, it is obvious that the
optimal solution is determined by $\mathbf{R}_{\text{g}_{\text{T}}}$’s
eigenvalues and the corresponding eigenvectors. However, in a practical
downlink monostatic ISAC sensing system, the transmitted waveform matrix is
often rank-deficient.

Typically, in the communication BS, the precoder $\mathbf{P}$ delivers $K$
data streams, stacking as $\mathbf{S}$, to its corresponding users via $N_{T}$
transmitting antennas. If we re-use the communication waveform $\mathbf{X}$
for sensing, the rank of the transmitted sensing waveform is given by

$\text{rank}(\mathbf{R}_{\text{X}})\overset{\eqref{X2P}}{=}\text{rank}(\mathbf{R}_{\text{P}})\leq\min\\{K,N_{T}\\}.$
(15)

By examining (13) and (15), we further observe that (13) still holds when
$K=N_{T}$. However, when $K<N_{T}$, there are $N_{T}-K$ eigenvalues
$\\{{\sigma}_{i,{\text{P}}}\\}_{i=K+1}^{N_{T}}$ of $\mathbf{R}_{\text{P}}$
that are zero, inspiring the following corollary.

Corollary 1: Let $\mathbf{R}_{\text{P}}$ be the rank-deficient covariance
matrix of the BS precoder, where $\text{rank}(\mathbf{R}_{\text{P}})=K$ and
$K<N_{T}$. Then, the MSE is given by

$\mathbf{E}(\mathbf{R}_{\text{P}})\geq\sum_{k=1}^{K}(\delta_{\text{S}}^{-2}\sigma_{k,{\text{P}}}+\sigma_{k,\text{g}_{\text{T}}}^{-1})^{-1}+\sum_{i=K+1}^{N_{T}}\sigma_{i,\text{g}_{\text{T}}}.$
(16)

The equality holds if and only if the eigenvector matrix
$\mathbf{U}_{\text{P}}$ = $\mathbf{U}_{\text{g}_{\text{T}}}$, where
$\mathbf{U}_{\text{P}}$ and $\mathbf{U}_{\text{g}_{\text{T}}}$ consist of
eigenvectors corresponding to descending ordered eigenvalues of
$\mathbf{R}_{\text{P}}$ and $\mathbf{R}_{\text{g}_{\text{T}}}$, respectively.

###### Proof.

The proof is given as

$\begin{split}\mathbf{E}(\mathbf{R}_{\text{P}})=&\text{Tr}\\{(\delta_{\text{S}}^{-2}\mathbf{R}_{\text{P}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}\geq\sum_{i=1}^{N_{T}}(\delta_{\text{S}}^{-2}\sigma_{i,{\text{P}}}+\sigma_{i,\text{g}_{\text{T}}}^{-1})^{-1}\\\
=&\sum_{k=1}^{K}(\delta_{\text{S}}^{-2}\sigma_{k,{\text{P}}}+\sigma_{k,\text{g}_{\text{T}}}^{-1})^{-1}+\sum_{i=K+1}^{N_{T}}(\delta_{\text{S}}^{-2}\sigma_{i,{\text{P}}}+\sigma_{i,\text{g}_{\text{T}}}^{-1})^{-1}\\\
=&\sum_{k=1}^{K}(\delta_{\text{S}}^{-2}\sigma_{k,{\text{P}}}+\sigma_{k,\text{g}_{\text{T}}}^{-1})^{-1}+\sum_{i=K+1}^{N_{T}}\sigma_{i,\text{g}_{\text{T}}}.\end{split}$

∎

Corollary 1 indicates that in the downlink monostatic ISAC system, if the
transmitted communication waveform is re-used for sensing, the MSE matrix is
governed by the rank of the transmitted waveform, which is indeed, controlled
by the number of served communication users. Moreover, the estimation
performance attains its MSE bound when the user number $K$ is equal to the
antenna numbers $N_{T}$. On the other hand, when the transmitted signal power
is unlimited, which implies that all $\sigma_{i,{\text{P}}}$ of
$\mathbf{R}_{\text{P}}$ are sufficiently large, (13) and (16) can then be
characterized as

$\liminf_{\\{\sigma_{i,{\text{P}}}\\}_{i=1}^{N_{T}}\rightarrow\infty}\mathbf{E}(\mathbf{R}_{\text{P}})=0,$
(17)

and

$\liminf_{\\{\sigma_{i,{\text{P}}}\\}_{i=1}^{K}\rightarrow\infty}\mathbf{E}(\mathbf{R}_{\text{P}})=\sum_{i=K+1}^{N_{T}}\sigma_{i,\text{g}_{\text{T}}},$
(18)

respectively, it can be seen that the $\mathbf{E}(\mathbf{R}_{\text{P}})$ is
lower bounded by $0$ when $\mathbf{R}_{\text{X}}$ is full rank, however,
$\mathbf{E}(\mathbf{R}_{\text{P}})$ will always be positive because the
eigenvalues of the target echoes
$\sum_{i=K+1}^{N_{T}}\sigma_{i,\text{g}_{\text{T}}}$ are always positive. In
other words, the sensing MSE of the downlink monostatic ISAC system could not
always achieve the lower bound $0$ for $K<N_{T}$, if the transmitted
communication waveform is re-used without any further processing.

##  IV DoF completion : A Scheme to Achieve MSE Lower Bound

In this section, we introduce a waveform augmentation strategy to improve the
sensing performance of the downlink monostatic ISAC system, namely DoF
completion. The main idea is to complete the transmitted signal from a rank-
deficient matrix to a full rank one, by embedding an additional signal
structure into the transmitted communication signals. Therefore, the resultant
waveform dedicated for sensing contains not only the re-used communication
signal, but also an augmented waveform matrix to extend the spatial DoFs. To
this end, we express the augmented waveform $\hat{\mathbf{X}}$ as

$\hat{\mathbf{X}}=[\mathbf{P}\quad\mathbf{P}_{\text{A}}]\begin{bmatrix}\mathbf{S}\\\
\mathbf{S}_{\text{A}}\end{bmatrix}\in\mathbb{C}^{N_{T}\times L},$ (19)

where $\mathbf{P}_{\text{A}}$ and $\mathbf{S}_{\text{A}}$ are the
$N_{T}\times(N_{T}-K)$ and $(N_{T}-K)\times L$ additional sensing precoding
matrix and data matrix, respectively. We assume that the additional data
matrix is a unitary matrix such that
$\mathbf{S}_{\text{A}}\mathbf{S}^{H}_{\text{A}}=\mathbf{I}_{N_{T}-K}.$
Therefore, the covariance matrix of $\hat{\mathbf{X}}$ can be written as

$\mathbf{R}_{\hat{\text{X}}}=[\mathbf{P}\quad\mathbf{P}_{\text{A}}]\begin{bmatrix}\mathbf{P}^{H}\\\
\mathbf{P}_{\text{A}}^{H}\end{bmatrix}=\mathbf{R}_{\text{P}}+\mathbf{R}_{\text{P}_{\text{A}}}=\mathbf{R}_{\hat{\text{P}}},$
(20)

where $\mathbf{R}_{\text{P}_{\text{A}}}\in\mathbb{C}^{N_{T}\times N_{T}}$ is
the covariance matrix dedicated to the additional precoder
$\mathbf{P}_{\text{A}}$ and $\mathbf{R}_{\hat{\text{P}}}$ denotes the
covariance matrix of the augmented precoder matrix.

Hence, when the augmented signal waveform is employed, the resulting MSE
matrix that measures the sensing performance can then be written as

$\mathbf{E}(\mathbf{R}_{\hat{\text{P}}})=N_{R}\text{Tr}\\{(\delta_{\text{S}}^{-2}\mathbf{R}_{\hat{\text{P}}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}.$
(21)

It is also worth noting that the to-be-designed matrix is now
$\mathbf{R}_{\hat{\text{P}}}$ instead of the previsou $\mathbf{R}_{\text{P}}$
for DoF completion. Recalling Corollary 1, the MSE matrix
$\mathbf{E}(\mathbf{R}_{\hat{\text{P}}})$ is also lower bounded by the
eigenvalues of $\mathbf{R}_{\hat{\text{P}}}$ and
$\mathbf{R}_{\text{g}_{\text{T}}}^{-1}$ as

$\mathbf{E}(\mathbf{R}_{\hat{\text{P}}})\geq\sum_{i=1}^{N_{T}}(\delta_{\text{S}}^{-2}\sigma_{i,{\hat{\text{P}}}}+\sigma_{i,\text{g}_{\text{T}}}^{-1})^{-1}.$
(22)

Consequently, the proposed strategy is able to achieve the MSE lower bound of
0 when the transmitted signal power is unlimited, i.e.,

$\liminf_{\\{\sigma_{i,{\hat{\text{P}}}}\\}_{i=1}^{N_{T}}\rightarrow\infty}\mathbf{E}(\mathbf{R}_{\hat{\text{P}}})=0.$
(23)

Remark: In general, the idea behind DoF completion strategy is to generate a
transmit waveform that is full-rank. In the radar context, a full-rank
waveform matrix is able to guarantee the feasibility of unbiased estimation
[8], as well as to realize omni-directional beampattern without additional
mechanical rotation systems or phased-array beam scanning. In the above
analysis, we proved that the full-rank waveform could achieve the MSE lower
bound of 0 for a sufficient large transmit power, which implies that the
parameter estimation errors from target echoes can be minimized. However, the
above strategy still brings several drawbacks. On the one hand, the emission
of the additional signal structure will always take additional costs of the
transmission power. Intuitively, these additional power is dedicated for
sensing but may not necessarily be beneficial for the communication
functionality. On the other hand, while the additional signal structure from
the DoF completion strategy can be removed by the successive interference
cancellation (SIC) technology, this requires high computation capability at
the communication users’ side. Therefore, we introduce the following MMSE
waveform design approach to deal with the dual-functional waveform design
problem.

##  V MMSE Waveform Designs via DoF completion

In this section, we apply the DoF completion strategy into a monostatic
downlink ISAC waveform design problem. Our objective function is to minimize
the MSE of the sensing performance, under the per-user SINR constraint for
communication functionality. Note that this is a typical waveform design
problem in the monostatic downlink ISAC scenario. Therefore, the waveform
design can be formulated as

$\displaystyle\underset{\hat{\text{P}}}{\text{minimize}}\quad$
$\displaystyle\text{Tr}\\{(\delta_{\text{S}}^{-2}\mathbf{R}_{\hat{\text{P}}}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}$
(24) s.t $\displaystyle\gamma_{k}(\text{P})\geq\gamma_{0},\forall
k\in\mathcal{K},$ (25)
$\displaystyle\text{Tr}\\{\mathbf{R}_{\hat{\text{P}}}\\}\leq
P_{T},\mathbf{R}_{\hat{\text{P}}}\succeq\mathbf{0}.$ (26)

where the received SINR of each communication user is required to be larger
than the threshold $\gamma_{0}$ given by (25) and (26) limits the overall
transmit power at the BS. When (25) is removed or $\gamma_{0}=0$, the waveform
design concerns only the target estimation performance that is equivalent to
(12). Due to the SINR constraint (25) for communication functionality, the
overall problem is non-convex. Fortunately, (25) can be reformulated by
employing SDR technology, $\forall k\in\mathcal{K}$,

$\mathbf{Q}_{k}=\mathbf{p}_{k}\mathbf{p}_{k}^{H}\Longleftrightarrow\mathbf{Q}_{k}\succeq\mathbf{0},\text{rank}(\mathbf{Q}_{k})=1,$
(27)

and $\mathbf{R}_{\text{P}}=\sum_{k\in\mathcal{K}}\mathbf{Q}_{k}$. We then
recast constraint (25) as

$\text{tr}(\mathbf{H}_{k}\mathbf{Q}_{k})\geq\gamma_{0}(\sum_{j\in\mathcal{K}/k}\text{tr}(\mathbf{H}_{k}\mathbf{Q}_{k})+\delta_{C}^{2}),\forall
k,\text{rank}(\mathbf{Q}_{k})=1,$ (28)

where $\mathbf{H}_{k}=\mathbf{h}_{k}\mathbf{h}_{k}^{H}$. Similarly, by
defining
$\\{\mathbf{Q}_{k}\\}_{k=K+1}^{N_{T}}=\sum_{k=K+1}^{N_{T}}\mathbf{p}_{k}\mathbf{p}_{k}^{H}$,
we could find
$\mathbf{R}_{\text{P}_{\text{A}}}=\sum_{k=K+1}^{N_{T}}\mathbf{Q}_{k}$. Then,
the optimization problem can be reformulated as

$\displaystyle\underset{\\{\mathbf{Q}_{k}\\}_{k=1}^{N_{T}}}{\text{minimize}}$
$\displaystyle\text{Tr}\\{(\delta_{\text{S}}^{-2}\sum_{k=1}^{N_{T}}\mathbf{Q}_{k}+\mathbf{R}_{\text{g}_{\text{T}}}^{-1})^{-1}\\}$
(29) s.t
$\displaystyle\frac{\text{tr}(\mathbf{H}_{k}\mathbf{Q}_{k})}{\sum_{j\in\mathcal{K}/k}\text{tr}(\mathbf{H}_{k}\mathbf{Q}_{k})+\delta_{C}^{2}}\geq\gamma_{0},\forall
k\in\mathcal{K}$
$\displaystyle\text{Tr}\\{\sum_{k=1}^{N_{T}}\mathbf{Q}_{k}\\}\leq
P_{T},\text{rank}(\mathbf{Q}_{k})=1.$

By dropping the rank-one constraint $\text{rank}(\mathbf{Q}_{k})=1$, i.e., the
relaxed problem is a convex one that can be solved by the well-known CVX
toolbox. However, in this case, the resulting optimal solution is a relaxed
solution of (29), which is typically not guaranteed to yield a rank-one
optimal point. By employing Theorem 4 in [8], we could construct a tight rank-
one solution. Denoting the relaxed optimal solution as
$\\{\hat{\mathbf{Q}}_{k}\\}_{k=1}^{N_{T}}$ with convariance matrix
$\hat{\mathbf{R}}_{\hat{\text{P}}}$ , the rank-one beamformers
$\mathbf{p}_{k}$ in $\mathbf{P}$ can be constructed by

$\mathbf{p}_{k}=(\mathbf{h}_{k}^{H}\hat{\mathbf{Q}}_{k}\mathbf{h}_{k})^{-1/2}\hat{\mathbf{Q}}_{k}\mathbf{h}_{k},\forall
k\in\mathcal{K},$ (30)

and $\mathbf{P}_{A}$ can always be obtained from

$\mathbf{P}_{\text{A}}\mathbf{P}_{\text{A}}^{H}=\hat{\mathbf{R}}_{\hat{\text{P}}}-\sum_{k\in\mathcal{K}}\hat{\mathbf{Q}}_{k}.$
(31)

Note (30) and (31) yield an optimal solution of (24). Therefore, the above
MMSE waveform design problem is solved optimally.

##  VI Simulations

In this section, we provide numerical results to verify the advantages of our
proposed DoF completion strategy, as well as to show the performance of the
MSE-based sensing waveform design. Without loss of generality, the ISAC BS is
equipped with $N_{R}=10$ receive antennas, under the maximun power budget of
$P_{T}=40$ dBm. The transmission frame length $L=30$, the carrier frequency is
set as $2.4$ GHz, the noise powers are
$\delta_{\text{C}}^{2}=\delta_{\text{S}}^{2}=-100$ dBm [13]. The communication
channel experiences Rayleigh fading, where each entry of the channel matrix
follows the standard complex Gaussian distribution. The path loss exponent is
$2.2$ according to the 3GPP path loss model. Besides, $K$ users are randomly
and uniformly distributed in a circle centered at (40 m, 0 m) with a radius of
10 m. Moreover, we assume that the entries of the target response matrix
$\mathbf{G}$ follows Swerling 2 model with Gaussian distributed complex
amplitude [8, 14].

Figure 2: The sensing estimation performances versus the number of
communication users with and without our proposed DoF completion strategy at
$N_{T}=16$ and $N_{T}=21$.

In Fig. 2, we compare the estimation performance of the sensing functionality
in terms of MSE versus with the growth of $K$ with and without our proposed
DoF completion strategy. The number of antennas are set to be $N_{T}=16$ and
$N_{T}=21$, respectively. Without DoF completion strategy, the increase of $K$
leads to a lower MSE. This observation is due to the fact that the rank of the
transmit waveform grows with the number of users that facilitates the design
to achieve the MSE lower bound in (16). On the contrary, for the schemes
without DoF completion, the resulting MSE performance is not affected by the
number of users, which is indeed the MSE lower bound. More interestingly, when
$K=N_{T}=16$, the MSE performance without DoF completion also achieves the
lower bound. This observation validates the correctness of our MSE analysis in
Section IV, where the performance gap between the MSE lower bound and
performance of optimal MSE solution vanishes at $K=N_{T}$.

Figure 3: The sensing estimation performances versus the minimum required SINR
to the communication users with and without the proposed DoF completion
strategy.

In Fig. 3, we evaluate the performance of SINR constrained MMSE waveform
design for the downlink monostatic ISAC system (29). The Cramer-Rao Bound
(CRB) based waveform design in [8] acts as a baseline. We observe that without
the DoF completion strategy, the MMSE waveform design can achieve the CRB at
$K=N_{T}=16$. More interestingly, by employing the propsoed DoF completion
strategy, the performance of MMSE waveform design is no longer affected by the
number of served communication users. The comparison between the MMSE waveform
designs with and without DoF completion strategy shows the performance gain of
the proposed strategy.

##  VII Conclusion

In this letter, the estimation performance of downlink monostatic ISAC system
was investigated. We derived a parameter estimation lower bound that
facilitated the derivation of the downlink monostatic ISAC waveform, and then
proposed a waveform augmentation strategy to improve the parameter estimation
performance. The proposed strategy was adopted in the MMSE ISAC waveform
design constrained by the per-user SINR of communication functionality. The
MSE lower bound and the effectiveness of the proposed waveform design were
revealed by the simulations.

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