# Joint Waveform and Passive Beamformer Design in Multi-IRS Aided Radar

Zahra Esmaeilbeig1, Arian Eamaz1, Kumar Vijay Mishra, and Mojtaba Soltanalian
$\leavevmode\nobreak\ {}^{1}$Equal contribution. This work was sponsored in
part by the National Science Foundation Grant ECCS-1809225, and in part by the
Army Research Office, accomplished under Grant Number W911NF-22-1-0263. The
views and conclusions contained in this document are those of the authors and
should not be interpreted as representing the official policies, either
expressed or implied, of the Army Research Office or the U.S. Government. The
U.S. Government is authorized to reproduce and distribute reprints for
Government purposes notwithstanding any copyright notation herein.Z. E., A. E.
and M. S. are with the Department of Electrical and Computer Engineering,
University of Illinois Chicago, Chicago, IL 60607, USA (e-mails:
<EMAIL_ADDRESS>aeamaz2@uic.edu_).K. V. M. is with the United States DEVCOM
Army Research Laboratory, Adelphi, MD 20783 USA.

###### Abstract

Intelligent reflecting surface (IRS) technology has recently attracted a
significant interest in non-light-of-sight radar remote sensing. Prior works
have largely focused on designing single IRS beamformers for this problem. For
the first time in the literature, this paper considers multi-IRS-aided
multiple-input multiple-output (MIMO) radar and jointly designs the transmit
unimodular waveforms and optimal IRS beamformers. To this end, we derive the
Cramér-Rao lower bound (CRLB) of target direction-of-arrival (DoA) as a
performance metric. Unimodular transmit sequences are the preferred waveforms
from a hardware perspective. We show that, through suitable transformations,
the joint design problem can be reformulated as two unimodular quadratic
programs (UQP). To deal with the NP-hard nature of both UQPs, we propose
unimodular waveform and beamforming design for multi-IRS radar (UBeR)
algorithm that takes advantage of the low-cost power method-like iterations.
Numerical experiments illustrate that the MIMO waveforms and phase shifts
obtained from our UBeR algorithm are effective in improving the CRLB of DoA
estimation.

###### Index Terms:

Beamforming, IRS-aided radar, non-line-of-sight sensing, unimodular sequences,
waveform design.

##  I Introduction

An intelligent reflecting surface (IRS) is composed of a large array of
scattering meta-material elements, which reflect the incoming signal after
introducing a pre-determined phase shift [1]. Recently, the benefits of IRS
have been investigated for future wireless communications [2, 3, 4]
applications, including multi-beam design [5], energy transfer [6], secure
parameter estimation [7] and joint sensing-communications [8, 9]. In this
paper, we focus on the IRS-aided radar, where combined processing of line-of-
sight (LoS) and non-LoS (NLoS) paths has shown improvement in target
estimation and detection [10, 11, 12, 13, 14] through an optimal design of IRS
phase shifts.

Target detection via multiple-input multiple-output (MIMO) IRS-aided radar was
studied extensively in [10, 11]. In our earlier works on target estimation
[12, 15], we derived the optimal IRS phase shifts based on the mean-squared-
error of the best linear unbiased estimator (BLUE) for complex target
reflection factor [12] and the Cramér-Rao lower bound (CRLB) of Doppler
estimation for moving targets [15]. Recent studies [13, 16] focused on
optimization of IRS beamforming based on CRLB of DoA estimation for a single
IRS-aided radar. More recent works [15, 17] demonstrate the benefits of
deploying multiple IRS platforms instead of a single IRS.

Similar to a conventional radar [18], a judicious design of transmit waveforms
improves the performance of IRS-aided radar. Whereas designing radar probing
signals is a well-studied problem [18, 19, 20, 21, 22], it is relatively
unexamined for IRS-aided radar. In this context, transmit sequences that
mitigate the non-linearities of amplifiers and yield a uniform power
transmission over time are of particular interest. Unimodular sequences with
the minimum peak-to-average power ratio exhibit these properties and have been
studied in previous non-IRS works for radar applications [21]. In this paper,
we jointly design unimodular sequences and IRS beamformers.

Multipath propagation through multiple IRS platforms increases the spatial
diversity of the radar system [23]. To this end, we investigate the benefits
of multipath processing for multi-IRS-aided target estimation. We first derive
the CRLB of target parameters for a multi-IRS-aided radar. Then, we formulate
the unimodular waveform design problem based on the CRLB minimization for IRS-
aided radar as a unimodular quadratic program (UQP). The unimodularity
constraint makes the UQP an NP-hard problem. In general, UQP may be relaxed
via a semi-definite program (SDP) formulation but the latter has a high
computational complexity as well[24, 25]. Inspired by the power method that
has the advantage of simple matrix-vector multiplications, [26, 22] proposed
_power method like iterations_ (PMLI) algorithm to approximate UQP solutions
leading to a low-cost algorithm. We formulate the IRS beamforming design as a
_unimodular quartic programming_ (UQ2P). Prior works [27, 19] on unimodular
waveform design with good correlation properties also lead to UQ2Ps, for which
they employ a more costly majorization-minimization technique. On the
contrary, we use a quartic to bi-quadratic transformation to solve UQ2P by
splitting it into two quadratic subproblems. Our _u_ nimodular waveform and
_be_ amforming design for multi-IRS _r_ adar (UBeR) algorithm is based on the
cyclic application of PMLI and provides the optimized CRLB.

Throughout this paper, we use bold lowercase and bold uppercase letters for
vectors and matrices, respectively. We represent a vector
$\mathbf{x}\in\mathbb{C}^{N}$ in terms of its elements $\\{x_{i}\\}$ as
$\mathbf{x}=[x_{i}]^{N}_{i=1}$. The $mn$-th element of the matrix $\mathbf{B}$
is $\left[\mathbf{B}\right]_{mn}$. The sets of complex and real numbers are
$\mathbb{C}$ and $\mathbb{R}$, respectively; $(\cdot)^{\top}$,
$(\cdot)^{\ast}$and $(\cdot)^{\mathrm{H}}$ are the vector/matrix transpose,
conjugate and the Hermitian transpose, respectively; trace of a matrix is
$\operatorname{Tr}(.)$; the function $\textrm{diag}(.)$ returns the diagonal
elements of the input matrix; and $\textrm{Diag}(.)$ produces a
diagonal/block-diagonal matrix with the same diagonal entries/blocks as its
vector/matrices argument. The Hadamard (element-wise) and Kronecker products
are $\odot$ and $\otimes$, respectively. The vectorized form of a matrix
$\mathbf{B}$ is written as $\mathrm{vec}\left(\mathbf{B}\right)$. The
$s$-dimensional all-ones vector, all-zeros vector, and the identity matrix of
size $s\times s$ are $\mathbf{1}_{s}$, $\mathbf{0}_{N}$, and $\mathbf{I}_{s}$,
respectively. The minimum eigenvalue of $\mathbf{B}$ is denoted by
$\lambda_{min}(\mathbf{B})$. The real, imaginary, and angle/phase components
of a complex number are $\operatorname{Re}\left(\cdot\right)$,
$\operatorname{Im}\left(\cdot\right)$, and
$\operatorname{arg}\left(\cdot\right)$, respectively.
$\mathrm{vec}_{{}_{K,L}}^{-1}\left(\mathbf{c}\right)$ reshapes the input
vector $\mathbf{c}\in\mathbb{C}^{KL}$ into a matrix
$\mathbf{C}\in\mathbb{C}^{K\times L}$ such that
$\mathrm{vec}\left(\mathbf{C}\right)=c$. Also, $\mathbf{0}_{N}$ is the all
zero vector of size $N$.

##  II Multi-IRS-Aided Radar System Model

Consider a colocated MIMO radar with $N_{t}$ transmit and $N_{r}$ receive
antennas, each arranged as uniform arrays (ULA) with inter-element spacing
$d$. The $M$ IRS platforms indexed as IRS1, IRS2,…,IRSM, are implemented at
stationary and known locations, each equipped with $N_{m}$ reflecting elements
arranged as ULA, with element spacing of $d_{m}$ between the
antennas/reflecting elements of IRSm. The continuous-time signal transmitted
from the $n$-th antenna at time instant $t$ is $x_{n}(t)$. Denote the
$N_{t}\times 1$ vector of all transmit signals as
$\bm{x}(t)=[x_{{}_{i}}(t)]_{i=1}^{N_{t}}\in\Omega^{N_{t}}$, where the set of
complex unimodular sequences is

$\Omega^{n}=\left\\{\mathbf{s}\in\mathbb{C}^{n}|\mathbf{s}=[e^{\textrm{j}\omega_{i}}]^{n}_{i=1},\omega_{i}\in[0,2\pi]\right\\}.$
(1)

The steering vectors of radar transmitter, receiver and the m-th IRS are,
respectively,
$\mathbf{a}_{t}(\theta)=[1,e^{\textrm{j}\frac{2\pi}{\lambda}dsin\theta},\ldots,e^{\textrm{j}\frac{2\pi}{\lambda}d(N_{t}-1)sin\theta}]^{\top}$,
$\mathbf{a}_{r}(\bm{\theta})=[1,e^{\textrm{j}\frac{2\pi}{\lambda}dsin\theta},\ldots,e^{\textrm{j}\frac{2\pi}{\lambda}d(N_{r}-1)sin\theta}]^{\top}$,
and
$\mathbf{b}_{m}(\bm{\theta})=[1,e^{\textrm{j}\frac{2\pi}{\lambda}d_{m}sin\theta},\ldots,e^{\textrm{j}\frac{2\pi}{\lambda}d_{m}(N_{m}-1)sin\theta}]^{\top}$,
where $\lambda$, is the carrier wavelength and $d$ and $d_{m}$ are usually
assumed to be half the carrier wavelength. Each reflecting element of IRSm
reflects the incident signal with a phase shift and amplitude change that is
configured via a smart controller [28]. We denote the phase shift vector of
IRSm by
$\mathbf{v}_{m}=[e^{\textrm{j}\phi_{{}_{m,1}}},\ldots,e^{\textrm{j}\phi_{{}_{m,N_{m}}}}]^{\top}\in\mathbb{C}^{N_{m}}$,
where $\phi_{{}_{m,k}}\in[0,2\pi]$ is the phase shift associated with the
$k$-th passive element of IRSm.

Denote the angle between the radar-target, radar–IRSm, and target-IRSm by
$\theta_{tr}$, $\theta_{ri,m}$, and $\theta_{ti,m}$, respectively. Denote
target-radar channel by
$\mathbf{H}_{tr}=\mathbf{a}_{r}(\theta_{tr})\in\mathbb{C}^{N_{r}\times 1}$;
and radar-target by
$\mathbf{H}_{rt}=\mathbf{a}_{t}(\theta_{tr})^{\top}\in\mathbb{C}^{1\times
N_{t}}$. The LoS or radar-target-radar channel matrix is
$\mathbf{H}_{rtr}=\mathbf{a}_{r}{(\theta_{tr})}\mathbf{a}_{t}{(\theta_{tr})}^{\top}\in\mathbb{C}^{N_{r}\times
N_{T}}$. Analogously, for the multi-IRS aided radar the NLoS channel matrices
associated with IRSm are defined as
$\mathbf{H}_{ri,m}=\mathbf{b}_{m}(\theta_{ri,m})\mathbf{a}_{t}^{\top}(\theta_{ri,m})\in\mathbb{C}^{N_{m}\times
N_{T}}$ for radar-IRSm;
$\mathbf{H}_{it,m}=\mathbf{b}_{m}^{\top}(\theta_{ti,m})\in\mathbb{C}^{1\times
N_{m}}$ for IRSm-target;
$\mathbf{H}_{ti,m}=\mathbf{b}_{m}(\theta_{ti,m})\in\mathbb{C}^{N_{m}\times 1}$
for target-IRSm; and
$\mathbf{H}_{ir,m}=\mathbf{a}_{r}(\theta_{ri,m})\mathbf{b}^{\top}_{m}(\theta_{ri,m})\in\mathbb{C}^{N_{r}\times
N_{m}}$ for IRSm-radar paths.

The received signal back-scattered from a single target is the superimposition
of echoes from both LoS and NLoS paths as

$\displaystyle\mathbf{y}(t)$
$\displaystyle=\alpha_{{}_{rtr}}\mathbf{H}_{rtr}\mathbf{x}(t-\tau_{rtr})$
$\displaystyle+\sum_{m=1}^{M}\alpha_{{}_{ritr,m}}\mathbf{H}_{tr}\mathbf{H}_{it,m}\bm{\Phi}_{m}\mathbf{H}_{ri,m}\mathbf{x}(t-\tau_{ritr,m})$
$\displaystyle+\sum_{m=1}^{M}\alpha_{{}_{rtir,m}}\mathbf{H}_{ir,m}\bm{\Phi}_{m}\mathbf{H}_{ti,m}\mathbf{H}_{rt}\mathbf{x}(t-\tau_{rtir,m})$
$\displaystyle+\sum_{m=1}^{M}\alpha_{{}_{ritir,m}}\mathbf{H}_{ir,m}\bm{\Phi}_{m}\mathbf{H}_{ti,m}\mathbf{H}_{it,m}\bm{\Phi}_{m}\mathbf{H}_{ri,m}$
$\displaystyle\mathbf{x}(t-\tau_{ritir,m})+\mathbf{w}(t),\;\in\mathbb{C}^{N_{r}},$
(2)

where $\bm{\Phi}_{m}=\mathrm{Diag}\left(\mathbf{v}_{m}\right)$,
$\alpha_{{}_{{(\cdot),m}}}$ is the complex reflectivity which depends on the
target back-scattering coefficient and the atmospheric attenuation, and
$\mathbf{w}(t)\sim\mathcal{CN}(\bm{0},\sigma^{2}\mathbf{I}_{N_{t}})$ denotes a
stationary (homoscedastic) additive white Gaussian noise (AWGN). In general,
the received signal may also have an additional inter-IRS interference that
should be included while accounting for the SNR. When there is some blockage
or obstruction between the radar and target, we have $\alpha_{rtr}\simeq 0$,
$\alpha_{ritr,m}\simeq 0$ and $\alpha_{rtir,m}\simeq 0$. We replace
$\alpha_{{}_{ritir,m}}$ by $\alpha_{{}_{m}}$ for notation brevity. The
received signal becomes

$\displaystyle\mathbf{y}(t)$
$\displaystyle=\sum_{m=1}^{M}\alpha_{{}_{m}}\mathbf{H}_{ir,m}\bm{\Phi}_{m}\mathbf{H}_{ti,k}\mathbf{H}_{it,m}\bm{\Phi}_{m}\mathbf{H}_{ri,m}$
$\displaystyle\hskip 56.9055pt\bm{x}(t-\tau_{ritir,m})+\mathbf{w}(t).$ (3)

Our goal is to design a radar system for inspecting a range cell located at
distance $d_{tr}$ with respect to (w.r.t.) the radar transmitter/receiver for
a potential target. Assume that the relative time gaps between any two
multipath signals are very small in comparison to the actual roundtrip delays,
i.e., $\tau_{ritir,m}\approx\tau_{0}=\frac{2d_{tr}}{c}$ for
$m\in\\{1,\ldots,M\\}$, where $c$ is the speed of light. We collect $N$ slow-
time samples at the rate $1/T_{s}$ from the signal, at $t=nT_{s}$,
$n=0,\ldots,N-1$. Hence, corresponding to the range-cell of interest, the
received signal vector is

$\mathbf{y}[n]=\sum_{m=1}^{M}\alpha_{{}_{m}}\mathbf{H}_{m}\mathbf{x}[n]+\mathbf{w}[n],\quad\mathbf{y}[n]\in\mathbb{C}^{N_{r}\times
1},$ (4)

where $\mathbf{x}[n]=\mathbf{x}(\tau_{0}+nT_{s})\in\mathbb{C}^{N_{t}\times
1}$, $\mathbf{y}[n]=[y_{{}_{i}}[n]]_{i=1}^{N_{r}}$, and we define
$\mathbf{H}_{m}=\mathbf{H}_{ir,m}\bm{\Phi}_{m}\mathbf{H}_{ti,m}\mathbf{H}_{it,m}\bm{\Phi}_{m}\mathbf{H}_{ri,m}\in\mathbb{C}^{N_{r}\times
N_{t}}$. The delay $\tau_{0}$ is aligned on-the-grid so that
$n_{0}=\tau_{0}/T_{s}$ is an integer [29].

Collecting all discrete-time samples for $N_{r}$ receiver antennas, the
received signal is the $N_{r}\times N$ matrix

$\mathbf{Y}=[\mathbf{y}[0],\ldots,\mathbf{y}[N-1]]=\sum_{m=1}^{M}\alpha_{{}_{m}}\mathbf{H}_{m}\mathbf{X}+\mathbf{W},$
(5)

where
$\mathbf{X}=[\mathbf{x}[0],\ldots,\mathbf{x}[N-1]]\in\mathbb{C}^{N_{t}\times
N}$and $\mathbf{W}$ $=[\mathbf{w}[0],\ldots,$
$\mathbf{w}[N-1]]\in\mathbb{C}^{N_{r}\times N}$. Vectorizing as
$\mathbf{y}=\mathrm{vec}\left(\mathbf{Y}\right)$ yields

$\mathbf{y}=\sum_{m=1}^{M}\alpha_{{}_{m}}\mathrm{vec}\left(\mathbf{H}_{m}\mathbf{X}\right)+\mathrm{vec}\left(\mathbf{W}\right)=\tilde{\mathbf{X}}\tilde{\mathbf{H}}\bm{\alpha}+\tilde{\mathbf{w}},$
(6)

where $\tilde{\mathbf{X}}=\mathbf{X}^{\top}\otimes\mathbf{I}_{{}_{N_{r}}}$,
$\tilde{\mathbf{H}}=[\tilde{\mathbf{H}}_{{}_{1}},\ldots,\tilde{\mathbf{H}}_{{}_{M}}]$,
$\tilde{\mathbf{H}}_{m}=\mathrm{vec}\left(\mathbf{H}_{m}\right)$ for
$m\in\\{1,\ldots,M\\}$,
$\tilde{\mathbf{w}}=\mathrm{vec}\left(\mathbf{W}\right)$ and
$\bm{\alpha}=[\alpha_{m}]_{m=1}^{M}$. Given that $\mathbf{w}(t)$ is AWGN in
(II), it is easily observed that
$\mathbf{y}\sim\mathcal{CN}(\bm{\mu},\mathbf{R})$, where
$\bm{\mu}=\tilde{\mathbf{X}}\tilde{\mathbf{H}}\bm{\alpha}$ and
$\mathbf{R}=\sigma^{2}\mathbf{I}_{{}_{N_{r}N}}$. Note that, since
$\mathbf{w}(n)$ is a stationary process and i.i.d. with $\sigma^{2}$ variance,
through vectorization and stacking all ensembles as one vector, the resulting
process is still stationary and i.i.d with the same variance.

Our goal is to show the effectiveness of placing $M$ IRS platforms in
estimating the DoA of the target in the LoS path, i.e. $\theta_{tr}$. For
simplicity, we consider a two-dimensional (2-D) scenario, where the radar, IRS
platforms and the target are in the same plane. Our analysis can be easily
extended to 3-D scenarios. The following remark states that the estimation of
DoAs in the NLoS paths, $\theta_{ti,m}$, for $m\in\\{1,\ldots,M\\}$ is
equivalent to an estimation of $\theta_{tr}$.

######  Remark 1.

Estimation of the vector of target DoAs, $\bm{\zeta}=[\theta_{ti,1},$
$\ldots,\theta_{ti,M}]^{\top}$ is equivalent to estimating scalar DoA
parameter, $\theta_{tr}$. This follows because, given the locations of radar,
IRS platforms and potential target range in the 2-D plane, we have
$\bm{\zeta}=[\theta_{tr}+\theta_{1},\ldots,\theta_{tr}+\theta_{{}_{M}}]^{\top}$,
where $\theta_{m}$ for $m\in\\{1,\ldots,M\\}$ are known.

##  III UQP-Based CRLB Optimization

For an unbiased estimator of a parameter $\theta_{tr}$ ($\theta$, hereafter),
the variance of $\hat{\theta}$ is lower bounded as
$\mathbf{E}\\{{(\hat{\theta}-\theta)(\hat{\theta}-\theta)^{H}}\\}\geq\textrm{CRLB}(\theta)$
[30]. Theorem 1 below unveils the Fisher information
$F_{\theta}=\left(\textrm{CRLB}(\theta)\right)^{-1}$.

######  Theorem 1.

Consider for the multi-IRS-aided radar, the receive signal model in (6). The
Fisher information of LoS DoA $\theta$ is

$F_{\theta}=\frac{2}{\sigma^{2}}\operatorname{Re}\left(\bm{\alpha}^{\mathrm{H}}\dot{\tilde{\mathbf{H}}}^{\mathrm{H}}\tilde{\mathbf{X}}^{\mathrm{H}}\tilde{\mathbf{X}}\dot{\tilde{\mathbf{H}}}\bm{\alpha}\right),$
(7)

where
$\dot{\tilde{\mathbf{H}}}=\left[\dot{\tilde{\mathbf{H}}}_{{}_{1}},\ldots,\dot{\tilde{\mathbf{H}}}_{{}_{M}}\right]$,
$\dot{\tilde{\mathbf{H}}}_{m}=\mathrm{vec}\left(\dot{\mathbf{H}}_{{}_{m}}\right)$
and
$\dot{\mathbf{H}}_{m}=\frac{\partial\mathbf{H}_{{}_{m}}}{\partial\theta}=b_{m}\mathbf{H}_{ir,m}\bm{\Phi}_{m}(\mathbf{b}_{m}(\theta_{ti,m})(\mathbf{d}\odot\mathbf{b}_{m}(\theta_{ti,m}))^{\top}$
$+(\mathbf{d}\odot\mathbf{b}_{m}(\theta_{ti,m}))\mathbf{b}_{m}(\theta_{ti,m})^{\top})\bm{\Phi}_{m}\mathbf{H}_{ri,m}$,
with NLoS DoAs being $\theta_{ti,m}=\theta+\theta_{m}$,
$b_{m}=\mathrm{j}\frac{2\pi d_{m}}{\lambda}\cos(\theta_{m})$ and
$\mathbf{d}=[0,\ldots,N_{m}-1]^{\top}$.

###### Proof.

Given the observations
$\mathbf{y}\sim\mathcal{CN}(\bm{\mu}(\theta),\mathbf{R})$, using Slepian-Bangs
formula [30, Chapter 3C], the Fisher information is

$\displaystyle
F_{\theta}=\mathrm{Tr}\left(\mathbf{R}^{-1}\frac{\partial\mathbf{R}}{\partial\theta}\mathbf{R}^{-1}\frac{\partial\mathbf{R}}{\partial\theta}\right)+2\operatorname{Re}\left(\frac{\partial\bm{\mu}(\theta)}{\partial\theta}^{H}\mathbf{R}^{-1}\frac{\partial\bm{\mu}(\theta)}{\partial\theta}\right).$
(8)

From (6), $\bm{\mu}(\theta)=\tilde{\mathbf{X}}\tilde{\mathbf{H}}\bm{\alpha}$.
Also, given the above mentioned definitions, we have
$\frac{\partial\mu(\theta)}{\partial\theta}=\tilde{\mathbf{X}}\dot{\tilde{\mathbf{H}}}\bm{\alpha}$.
Substituting this in (8) and using $\mathbf{R}=\sigma^{2}\mathbf{I}$, one
arrives at (7). ∎

######  Remark 2.

In the absence of the IRS, ceteris paribus, the LoS Fisher information is
$F_{\theta}=\frac{2|\alpha_{rtr}|^{2}}{\sigma^{2}}\|\tilde{\mathbf{X}}\dot{\tilde{\mathbf{H}}}_{rtr}\|_{2}^{2}$,
where
$\dot{\tilde{\mathbf{H}}}_{rtr}=\mathrm{vec}\left(\dot{\mathbf{H}}_{{}_{rtr}}\right)$,
$\dot{\mathbf{H}}_{rtr}=\frac{\partial\mathbf{H}_{{}_{rtr}}}{\partial\theta}=\mathrm{j}\frac{2\pi
d}{\lambda}\cos(\theta)\left(\left(\mathbf{d}^{\prime}\odot\mathbf{a}_{r}{(\theta)}\right)\mathbf{a}_{t}{(\theta)}^{\top}\right.$
$\left.+\mathbf{a}_{r}{(\theta)}(\mathbf{d}^{\prime}\odot\mathbf{a}_{t}{(\theta)}^{\top})\right)$,
$N_{r}=N_{t}$, and $\mathbf{d}^{\prime}=[0,\ldots,N_{r}-1]^{\top}$.

To design the unimodular waveform $\mathbf{X}$, the following proposition
casts $F_{\theta}$ in standard quadratic form.

######  Proposition 1.

The Fisher information $F_{\theta}$ of LoS DoA is

$\displaystyle
F_{\theta}\left(\mathbf{X}\right)=\mathrm{vec}\left(\mathbf{X}\right)^{\mathrm{H}}(\mathbf{I}_{N}\otimes\mathbf{B})^{\mathrm{H}}(\mathbf{I}_{N}\otimes\mathbf{B})\mathrm{vec}\left(\mathbf{X}\right),$
(9)

where
$\mathbf{B}=\frac{\sqrt{2}}{\sigma}\mathrm{vec}_{{}_{{}_{N_{r},N_{t}}}}^{-1}\left(\dot{\tilde{\mathbf{H}}}\alpha\right)\in\mathbb{C}^{N_{r}\times
N_{t}}$.

###### Proof.

Given $\tilde{\mathbf{X}}=\mathbf{X}^{\top}\otimes\mathbf{I}_{{}_{N_{r}}}$,
rewrite Fisher information in (7) as

$F_{\theta}=\frac{2}{\sigma^{2}}\operatorname{Re}\left\\{\left(\left(\mathbf{X}^{\top}\otimes\mathbf{I}_{N_{r}}\right)\dot{\tilde{\mathbf{H}}}\alpha\right)^{\mathrm{H}}\left(\left(\mathbf{X}^{\top}\otimes\mathbf{I}_{N_{r}}\right)\dot{\tilde{\mathbf{H}}}\alpha\right)\right\\}.$
(10)

Since the argument of real operator is a real number, we can put it out of the
real operator. Using the identity
$\left(\mathbf{X}^{\top}\otimes\mathbf{I}_{N_{r}}\right)\dot{\tilde{\mathbf{H}}}\alpha=\left(\mathbf{I}_{N}\otimes\operatorname{vec}_{{}_{N_{r},N_{t}}}^{-1}\left(\dot{\tilde{\mathbf{H}}}\alpha\right)\right)\mathrm{vec}\left(\mathbf{X}\right)$
in (10), we immediately get (9). ∎

Using the expression in (9), we recast the unimodular waveform design
objective as a unimodular quadratic objective that leads to a UQP. To proceed
with IRS beamformer design, define,
$\dot{\tilde{\mathbf{H}}}_{{}_{m}}=\mathbf{D}_{m}\mathrm{vec}\left(\mathbf{V}_{m}\right)$,
$\mathbf{D}=\mathrm{Diag}\left(\mathbf{D}_{1},\ldots,\mathbf{D}_{m}\right)$,

$\mathbf{D}_{m}=\left(\mathbf{C}_{m}^{\top}\mathrm{diag}\left(\mathbf{d}\right)\otimes\mathbf{C}_{m}^{\top}\right)+\left(\mathbf{C}_{m}^{\top}\otimes\mathbf{C}_{m}^{\top}\mathrm{diag}\left(\mathbf{d}\right)\right)$
(11)

and
$\mathbf{C}_{m}=\mathrm{Diag}\left(\mathbf{b}_{m}(\theta_{ti,m})\right)\mathbf{H}_{ri,m}$,
where the unimodular phase shifts for IRSm are given by
$\mathbf{v}_{m}=\mathrm{diag}\left(\bm{\Phi}_{m}\right)$ or
$\mathbf{V}_{m}=\mathrm{vec}\left(\mathbf{v}_{m}\mathbf{v}_{m}^{\top}\right)$.
In order to obtain (11), we imposed the reciprocity,
$\mathbf{H}_{ir,m}=\mathbf{H}_{ri,m}^{\top}$ for a radar with collocated
antennas and $N_{r}=N_{t}$. For the IRS beamforming, the Fisher information
$F_{\theta}$ w.r.t. phase shifts is recast in the following proposition.

######  Proposition 2.

The Fisher information is quartic in phase shifts:

$F_{\theta}(\upnu)=\upnu^{\mathrm{H}}\mathbf{Q}^{\mathrm{H}}_{1}(\upnu)\mathbf{T}\mathbf{Q}_{1}(\upnu)\upnu=\upnu^{\mathrm{H}}\mathbf{Q}_{2}^{\mathrm{H}}(\upnu)\mathbf{T}\mathbf{Q}_{2}(\upnu)\upnu,$
(12)

where
$\upnu=\left[\mathbf{v}^{\top}_{1},\mathbf{v}^{\top}_{2},\cdots,\mathbf{v}^{\top}_{M}\right]^{\top}\in\mathbb{C}^{MN_{m}}$,
$\mathbf{T}=\mathbf{D}^{H}\mathbf{P}^{\mathrm{H}}\mathbf{Z}^{\ast}\mathbf{P}\mathbf{D}$,
$\mathbf{Q}_{1}(\upnu)=\mathrm{Diag}\left([\mathbf{v}_{1}\otimes\mathbf{I}_{N_{m}},\ldots,\mathbf{v}_{{}_{M}}\otimes\mathbf{I}_{N_{m}}]\right)$,
$\mathbf{Q}_{2}(\upnu)=\mathrm{Diag}\left([\right.$
$\left.\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{1},\ldots,\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{{}_{M}}]\right)$,
$\mathbf{Z}=(\mathbf{I}_{N_{r}N_{t}}\otimes\bm{\alpha}^{*}\bm{\alpha}^{\top})^{\top}(\tilde{\mathbf{X}}^{\top}\tilde{\mathbf{X}}^{*}\otimes\mathbf{I}_{M})$,
and $\mathbf{P}$ is the commutation matrix, i.e.,
$\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\top}\right)=\mathbf{P}\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}\right)$.

###### Proof.

The Fisher information is
$F_{\theta}=\mathrm{Tr}\left(\dot{\tilde{\mathbf{H}}}\bm{\alpha}\bm{\alpha}^{\mathrm{H}}\dot{\tilde{\mathbf{H}}}^{\mathrm{H}}\tilde{\mathbf{X}}^{\mathrm{H}}\tilde{\mathbf{X}}\right)=\mathrm{vec}\left(\bm{\alpha}^{*}\bm{\alpha}^{\top}\dot{\tilde{\mathbf{H}}}^{\top}\right)^{\top}\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\mathrm{H}}\tilde{\mathbf{X}}^{\mathrm{H}}\tilde{\mathbf{X}}\right)=\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\top}\right)^{\top}\mathbf{Z}\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\mathrm{H}}\right)$.
Since $F_{\theta}$ is real, we have
$F_{\theta}\left(\mathbf{z}\right)=\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\top}\right)^{\mathrm{H}}\mathbf{Z}^{\ast}\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}^{\top}\right)=\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}\right)^{\mathrm{H}}\mathbf{P}^{\mathrm{H}}\mathbf{Z}^{\ast}\mathbf{P}\mathrm{vec}\left(\dot{\tilde{\mathbf{H}}}\right)=\mathbf{z}^{\mathrm{H}}\mathbf{D}^{H}\mathbf{P}^{\mathrm{H}}\mathbf{Z}^{\ast}\mathbf{P}\mathbf{D}\mathbf{z}$,
where
$\mathbf{z}=\mathrm{vec}\left([\mathbf{V}_{1},\ldots,\mathbf{V}_{M}]\right)=[\mathrm{vec}\left(\mathbf{v}_{m}\mathbf{v}_{m}^{\top}\right)^{\top},\ldots,\mathrm{vec}\left(\mathbf{v}_{m}\mathbf{v}_{m}^{\top}\right)^{\top}]^{\top}$.
Applying the identity

$\operatorname{vec}\left(\mathbf{v}_{m}\mathbf{v}^{\top}_{m}\right)=\left(\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{m}\right)\mathbf{v}_{m}=\left(\mathbf{v}_{m}\otimes\mathbf{I}_{N_{m}}\right)\mathbf{v}_{m},$
(13)

yields $\mathbf{z}=\mathbf{Q}_{1}(\upnu)\upnu=\mathbf{Q}_{2}(\upnu)\upnu$.
This completes the proof. ∎

To jointly optimize $\mathbf{v}_{m}=\mathrm{diag}\left(\bm{\Phi}_{m}\right)$
and $\mathbf{X}$, we solve

$\displaystyle\underset{\mathbf{X}\in\Omega^{N_{t}\times
N},\upnu\in\Omega^{MN_{m}}}{\textrm{maximize}}\quad
F_{\theta}(\upnu,\mathbf{X}),$ (14)

which leads to the CRLB minimization. Note that this problem is UQP w.r.t.
$\mathbf{X}$ but quartic or UQ2P w.r.t. the phase shifts $\upnu$.

##  IV UBeR Algorithm

We resort to a task-specific _alternating optimization_ (AO) or _cyclic
algorithm_ [31, 32, 22], wherein we optimize (14) for $\mathbf{X}$ and $\upnu$
cyclically. To tackle each subproblem, we adopt power method-like iterations
(PMLI) [26], which is a computationally efficient procedure to tackle the UQP.
The PMLI resembles the well-studied _power method_ for computing the dominant
eigenvalue/vector pairs of matrices [26]. Given a matrix $\mathbf{G}$, the
following problem is a UQP[26]:

$\mathcal{P}_{1}:\leavevmode\nobreak\
\underset{\mathbf{s}\in\Omega^{n}}{\textrm{maximize}}\quad\mathbf{s}^{\mathrm{H}}\mathbf{G}\mathbf{s}.$
(15)

If $\mathbf{G}$ is positive semidefinite, the PMLI iterations

$\mathbf{s}^{(t+1)}=e^{\textrm{j}\operatorname{arg}\left(\mathbf{G}\mathbf{s}^{(t)}\right)},$
(16)

lead to a monotonically increasing objective value for the UQP.

Unimodular Waveform Design: From Proposition 1, the Fisher information
$F_{\theta}$ for the unimodular waveform $\mathbf{X}$ is the unimodular
quadratic objective in (9). Let $\bm{s}=\mathrm{vec}\left(\mathbf{X}\right)$,
and
$\mathbf{G}=(\mathbf{I}_{N}\otimes\mathbf{B})^{\mathrm{H}}(\mathbf{I}_{N}\otimes\mathbf{B})$.
Therefore, the uniform $\mathbf{X}$ is obtained from $\mathcal{P}_{1}$ via the
iterations ($t\geq 0$):

$\mathrm{vec}\left(\mathbf{X}^{(t+1)}\right)=e^{\textrm{j}\operatorname{arg}\left((\mathbf{I}_{N}\otimes\mathbf{B})^{\mathrm{H}}(\mathbf{I}_{N}\otimes\mathbf{B})\mathrm{vec}\left(\mathbf{X}^{(t)}\right)\right)}.$
(17)

If $\mathbf{G}$ is not positive semidefinte, at each iteration we use the
diagonal loading technique, i.e.,
$\tilde{\bm{G}}\leftarrow\mathbf{G}+\lambda_{m}\mathbf{I}$, where the loading
parameter $\lambda_{m}\geq-\lambda_{min}(\mathbf{G})$. Note that diagonal
loading with $\lambda_{m}\mathbf{I}$ has no effect on the solution of (17)
because
$\mathbf{s}^{\mathrm{H}}\tilde{\mathbf{G}}(\bm{s})\mathbf{s}=\lambda_{m}N_{{}_{t}}N+\mathbf{s}^{\mathrm{H}}\mathbf{G}\mathbf{s}$.

IRS Beamforming Design: For the phase shifts optimization, we find an
alternative bi-quadratic formulation to the quartic
$F_{\theta}\left(\upnu\right)$. Define

$g\left(\upnu_{1},\upnu_{2}\right)=\frac{1}{2}\left(\upnu^{\mathrm{H}}_{1}\mathbf{G}_{1}\left(\upnu_{2}\right)\upnu_{1}+\upnu^{\mathrm{H}}_{2}\mathbf{G}_{1}\left(\upnu_{1}\right)\upnu_{2}\right),$
(18)

where
$\mathbf{G}_{1}(\upnu)=\mathbf{Q}^{\mathrm{H}}_{1}(\upnu)\mathbf{T}\mathbf{Q}_{1}(\upnu)$.
The function $g(.,.)$ is symmetric, i.e.,
$g\left(\upnu_{1},\upnu_{2}\right)=g\left(\upnu_{2},\upnu_{1}\right)$ and from
proposition 2, $F_{\theta}(\upnu)=g(\upnu,\upnu)$. According to (13), one can
readily verify that
$\mathbf{Q}_{1}(\upnu_{1})\upnu_{2}=\mathbf{Q}_{2}(\upnu_{2})\upnu_{1}$. As a
result, $g\left(\upnu_{1},\upnu_{2}\right)$ is rewritten as

$g\left(\upnu_{1},\upnu_{2}\right)=\mathbf{v}^{\mathrm{H}}_{1}\mathbf{E}\left(\upnu_{2}\right)\upnu_{1},$
(19)

where
$\mathbf{E}\left(\upnu_{2}\right)=\frac{\left(\mathbf{G}_{1}\left(\upnu_{2}\right)+\mathbf{G}_{2}\left(\upnu_{2}\right)\right)}{2}$,
and
$\mathbf{G}_{2}(\upnu)=\mathbf{Q}^{\mathrm{H}}_{2}(\upnu)\mathbf{T}\mathbf{Q}_{2}(\upnu)$.
Fixing either $\upnu_{1}$ or $\upnu_{2}$ and minimizing
$g\left(\upnu_{1},\upnu_{2}\right)$ w.r.t. the other variable requires solving
the following UQP:

$\displaystyle\underset{\upnu_{j}\in\Omega^{MN_{m}}}{\textrm{minimize}}\quad\upnu_{j}^{\mathrm{H}}\tilde{\mathbf{E}}(\upnu_{i})\upnu_{j},\quad
i\neq j\in\left\\{1,2\right\\},$ (20)

where we used the diagonal loading,
$\tilde{\mathbf{E}}(\upnu_{i})\leftarrow\lambda_{M}\mathbf{I}-\mathbf{E}(\upnu_{i})$,
with $\lambda_{M}$ being the maximum eigenvalue of $\mathbf{E}(\upnu_{i})$.
Note that diagonal loading has no effect on the solution because
$\upnu_{j}^{\mathrm{H}}\tilde{\mathbf{E}}(\upnu_{i})\upnu_{j}=\lambda_{M}MN_{m}-\upnu_{j}^{\mathrm{H}}\mathbf{E}(\upnu_{i})\upnu_{j}$.
Moreover, $\tilde{\mathbf{E}}(\upnu_{i})$ is positive semidefinite, to satisfy
the requirement of PMLI.

To guarantee that the maximization of $g\left(\upnu_{1},\upnu_{2}\right)$
w.r.t. $\upnu_{1}$ and $\upnu_{2}$ also maximizes
$F_{\theta}\left(\upnu\right)$, a regularization would be helpful. Therefore,
we add the norm-$2$ error between $\upnu_{1}$ and $\upnu_{2}$ as a _penalty_
function to (20), we obtain

$\underset{\upnu_{j}\in\Omega^{MN_{m}}}{\textrm{minimize}}\quad\upnu_{j}^{\mathrm{H}}\tilde{\mathbf{E}}(\upnu_{i})\upnu_{j}+\eta||\upnu_{i}-\upnu_{j}||_{2}^{2},\quad
i\neq j\in\left\\{1,2\right\\},$ (21)

where $\eta$ is Lagrangian multiplier. Rewrite the objective of (21) as

$\begin{pmatrix}\upnu_{j}\\\
1\end{pmatrix}^{\mathrm{H}}\underbrace{\begin{pmatrix}\tilde{\mathbf{E}}(\upnu_{i})&-\eta\upnu_{i}\\\
-\eta\upnu_{i}^{\mathrm{H}}&2\eta
MN_{m}\end{pmatrix}}_{\mathcal{E}(\upnu_{i})}\begin{pmatrix}\upnu_{j}\\\
1\end{pmatrix}.$ (22)

Then, UQP for (14) w.r.t. $\upnu$ becomes

$\mathcal{P}_{2}:\leavevmode\nobreak\
\underset{\upnu_{j}\in\Omega^{MN_{m}}}{\textrm{maximize}}\quad\bar{\upnu}_{j}^{\mathrm{H}}\underbrace{\begin{pmatrix}\hat{\lambda}_{M}\mathbf{I}-\tilde{\mathbf{E}}(\upnu_{i})&\eta\upnu_{i}\\\
\eta\upnu_{i}^{\mathrm{H}}&\hat{\lambda}_{M}-2\eta
MN_{m}\end{pmatrix}}_{=\hat{\mathbf{E}}(\upnu_{i})=\hat{\lambda}_{M}\mathbf{I}-\mathcal{E}(\upnu_{i})}\bar{\upnu}_{j},$
(23)

where $\hat{\lambda}_{M}$ is the maximum eigenvalue of
$\mathcal{E}(\upnu_{i})$, and $\bar{\upnu}_{j}=\left[\upnu^{\top}_{j}\quad
1\right]^{\top}$.

To tackle the UQ2P for maximizing $F_{\theta}$, we solve the _bi-quadratic_
program (23) using PMLI in (16). Algorithm 1 summarizes the proposed steps.
The PMLI in UBeR have previously been shown to be convergent in terms of both
the optimization objective and variable [22, 21].

Algorithm 1 _U_ nimodular waveform and _be_ amforming design for multi-IRS _r_
adar (UBeR).

1:Input: Initialization values $\mathbf{X}^{(0)}$ and $\upnu_{1}^{(0)}$ and
$\upnu_{2}^{(0)}$, Lagrangian multiplier $\eta$, total number of iterations
$\Gamma_{1}$ and $\Gamma_{2}$ for problems $\mathcal{P}_{1}$ and
$\mathcal{P}_{2}$, respectively.

2:Output: Optimized phase shifts $\upnu^{*}$, unimodular waveform
$\mathbf{X}^{*}$.

3:Obtain $F_{\theta}\left(\mathbf{X}\right)$ and
$F_{\theta}\left(\upnu\right)$ from (9) and (12), respectively.

4:$\mathbf{B}\leftarrow\mathrm{vec}^{-1}\left(\dot{\tilde{\mathbf{H}}}\bm{\alpha}\right)\in\mathbb{C}^{N_{r}\times
N_{t}}$,

5:$\mathbf{G}\leftarrow\left(\mathbf{I}_{N}\otimes\mathbf{B}\right)^{\mathrm{H}}\left(\mathbf{I}_{N}\otimes\mathbf{B}\right)$.

6:for $t_{1}=0:\Gamma_{1}$ do $\triangleright$ Update the unimodular waveform

7: for $t_{2}=0:\Gamma_{2}$ do $\triangleright$ Bi-quadratic programming via
PMLI

8: $\upnu^{(t_{2}+1)}_{1}\leftarrow
e^{\textrm{j}\operatorname{arg}\left(\begin{bmatrix}\mathbf{I}_{{}_{MN_{m}}}\mathbf{0}_{{}_{MN_{m}}}\end{bmatrix}\hat{\mathbf{E}}\left(\upnu^{(t_{2})}_{2},\mathbf{X}^{(t_{1})}\right)\bar{\upnu}^{(t_{2})}_{1}\right)},$

9: $\upnu^{(t_{2}+1)}_{2}\leftarrow
e^{\textrm{j}\operatorname{arg}\left(\begin{bmatrix}\mathbf{I}_{{}_{MN_{m}}}\mathbf{0}_{{}_{MN_{m}}}\end{bmatrix}\hat{\mathbf{E}}\left(\upnu^{(t_{2}+1)}_{1},\mathbf{X}^{(t_{1})}\right)\bar{\upnu}^{(t_{2})}_{2}\right)}.$

10: $\upnu^{(t_{1}+1)}\leftarrow$ $\upnu^{(\Gamma_{2})}_{1}$ or
$\upnu^{(\Gamma_{2})}_{2}$.

11: $\mathrm{vec}\left(\mathbf{X}^{(t_{1}+1)}\right)\leftarrow
e^{\textrm{j}\operatorname{arg}\left(\mathbf{G}\left(\upnu^{(t_{1}+1)}\right)\mathrm{vec}\left(\mathbf{X}^{(t_{1})}\right)\right)}$.

12:return
$\left\\{\upnu^{\star},\mathbf{X}^{\star}\right\\}\leftarrow\left\\{\upnu^{(\Gamma_{1})},\mathbf{X}^{(\Gamma_{1})}\right\\}$.

##  V Numerical Experiments

Figure 1: The optimized CRLB of DoA versus (a) $\sigma^{2}$ for fixed number
of iterations $\Gamma_{1}=50$, and (b) $\Gamma_{1}$ for fixed
$\sigma^{2}=0.1$. In both experiments, $\Gamma_{2}$ was set to $20$.

We consider a radar, equipped with $N_{r}=N_{t}=8$ antennas for
transmitter/receiver, positioned in the 2-D Cartesian plane at [$0$ m, $0$ m],
sensing a target at [$5000$ m, $5000$ m]. We placed three IRS platforms with
$N_{m}=8$ reflecting elements arranged as ULA with the first elements located
at [$500$ m, $500$ m], [$500$ m, $-800$ m], and [$300$ m, $1300$ m]. The IRS
platforms were deployed at far ranges w.r.t. the radar. Usually, distant
targets tend to have obstructed or very weak LoS signal. In these cases, the
received signal from the NLoS paths, i.e., the signal propagated through IRS
platforms, is helpful in boosting the reflected LoS echo strength.

For a point-like target, given the radar, target and IRS positions, the
corresponding radar–IRSm and target–IRSm angles $\theta_{ir,m}$ and
$\theta_{ti,m}$, for $m\in\\{1,\ldots,M\\}$, are obtained through geometric
computations. The complex reflectivity coefficients $\\{\alpha_{m}\\}$, which
correspond to a Swerling 0 target model are generated from a
$\mathcal{CN}(0,1)$. In Algorithm 1, we set $\Gamma_{1}=50$ and
$\Gamma_{2}=20$ for all iterations. Throughout all our experiments the
Lagrangian multiplier $\eta$ is tuned to $0.1$. Initially, all IRS platforms
are set to impose zero phase shift $\upnu_{i}^{(0)}=\bm{0}_{{}_{MN_{m}}}$ for
$i\in\\{1,2\\}$. The samples in $\mathbf{X}^{(0)}$ are generated from a normal
distribution. Fig. 1a illustrates that the multiple IRS-aided radar
outperforms the single-IRS aided radar. Further, Fig. 1b indicates that
iterations of Algorithm 1 result in a monotonically decreasing CRLB.

##  VI Summary

Waveform design for IRS-aided radar is relatively unexplored in prior works.
In this context, this paper studies a new set of waveform design problems.
Numerical experiments demonstrate that the deployment of multiple IRS
platforms leads to a better achievable estimation performance compared to non-
IRS and single-IRS systems. Some other issues that should be accounted for in
the future include inter-IRS interference and quantization of phases.

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