#  Moving Target Detection via Multi-IRS-Aided OFDM Radar

Zahra Esmaeilbeig, Arian Eamaz, Kumar Vijay Mishra and Mojtaba Soltanalian
Zahra Esmaeilbeig, Arian Eamaz and Mojtaba Soltanalian are with the ECE
Departement, University of Illinois at Chicago, Chicago, IL 60607 USA. Email:
{zesmae2, aeamaz2<EMAIL_ADDRESS>Vijay Mishra is with the United States
DEVCOM Army Research Laboratory, Adelphi, MD 20783 USA. E-mail:
<EMAIL_ADDRESS>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.

###### Abstract

An intelligent reflecting surface (IRS) consists of passive reflective
elements capable of altering impinging waveforms. The IRS-aided radar systems
have recently been shown to improve detection and estimation performance by
exploiting the target information collected via non-line-of-sight paths.
However, the waveform design problem for an IRS-aided radar has remained
relatively unexplored. In this paper, we consider a multi-IRS-aided orthogonal
frequency-division multiplexing (OFDM) radar and study the theoretically
achievable accuracy of target detection. In addition, we jointly design the
OFDM signal and IRS phase-shifts to optimize the target detection performance
via an alternating optimization approach. To this end, we formulate the IRS
phase-shift design problem as a unimodular _bi-quadratic_ program which is
tackled by a computationally cost-effective approach based on power-method-
like iterations. Numerical experiments illustrate that our proposed joint
design of IRS phase-shifts and the OFDM code improves the detection
performance in comparison with conventional OFDM radar.

###### Index Terms:

Intelligent reflecting surfaces, non-line-of-sight sensing, OFDM, unimodular
bi-quadratic programming, waveform design.

##  I Introduction

Intelligent reflective surface (IRS) is an emerging technological advancement
for next-generation wireless systems. An IRS comprises meta-material units
that enable smart and programmable radio environments by introducing
predetermined phase-shifts to the impinging signal [1]. The IRS-aided wireless
communications are shown to provide range extension to users with obstructed
direct links [2], enhance physical layer security [3, 4], facilitate unmanned
air vehicle (UAV) communications [5], and shaping the wireless channel through
multi-beam design [6]. Recent works have also introduced IRS to integrated
communications and sensing systems [7, 8, 3, 9].

Recently, following the advances in [10, 7], IRS-aided sensing for non-line-
of-sight (NLoS) target estimation has been investigated in [10, 11, 12]. In
[13], the phase-shift matrix of the IRS was optimized for collocated MIMO
radar to improve the estimation and detection performance. Target detection
was also considered in cases where the radar is aided by a single IRS [14] or
multiple IRS platforms [15, 16]. The deployment of multiple IRS platforms is
necessary to overcome line-of-sight (LoS) blockage or obstruction in cases
where the NLoS path formed by a single IRS is unable to provide the desired
coverage. To this end, [17] jointly designed the radar transmitter and IRS
beamformers for a multi-IRS-aided radar. Similar to a conventional radar [18],
a judicious design of transmit waveforms improves the performance of IRS-aided
radar.

In general, radar waveform design is a well-investigated problem [18, 19, 20].
However, it is relatively unexamined for IRS-aided scenarios. Among prior
works, [21] designed a transmit radar code with constant-modulus for a
narrowband IRS-aided radar. However, wideband signaling compensates for signal
fading resulting from multipath propagation [22]. Therefore, very recent works
[8, 23, 24] investigate wideband waveforms such as orthogonal frequency-
division multiplexing (OFDM) signaling to improve detection with IRS-aided
radar.

In this paper, we focus on designing a wideband radar waveform for multi-IRS-
aided radar jointly with the IRS phase-shifts. In particular, we formulate the
detection problem as a hypothesis test to decide the presence of a target in a
particular range cell. Then, we jointly design the OFDM signal and the IRS
phase-shifts to enhance the receiver operating characteristics (RoC)
associated with moving target detection. We adopt noncentrality parameter of
the asymptotic distribution of the generalized likelihood ratio test (GLRT)
statistic [22] as the performance metric for target detection. We demonstrate
that maximizing the noncentrality parameter with respect to the system
parameters such as the transmit waveform and phase-shifts of IRS, yields
improvement in the probability of detection. Contrary to prior works, wherein
only IRS phase-shifts were optimized in an IRS-aided radar [10, 16, 15], we
show that jointly optimal waveform and phase-shifts increase the probability
of detection. Further, our IRS-aided radar outperforms the multipath OFDM
radar [22] with specular reflection in the exactly identical paths between the
target and radar.

The remainder of this paper is organized as follows. In the next section, we
describe the signal model for the multi-IRS-aided OFDM radar. The moving
target detector based on GLRT is introduced in Section III. We present our
joint waveform and IRS phase-shift design in Section IV. We validate our model
and methods via numerical experiments in Section V and conclude in Section VI.

Throughout this paper, we use bold lowercase and bold uppercase letters for
vectors and matrices, respectively. The $(m,n)$-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 Hermitian transpose, respectively; the 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 $\mathrm{vec}\left(\mathbf{B}\right)$ and the block
diagonal vectorization [25] is denoted by $\operatorname{vecb}(\mathbf{B})$.
The $s$-dimensional all-ones vector and the identity matrix of size $s\times
s$ are $\mathbf{1}_{s}$, 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.
$\mathrm{vec}_{{}_{K,L}}^{-1}\left(\mathbf{b}\right)$ reshapes the input
vector $\mathbf{b}\in\mathbb{C}^{KL}$ into a matrix
$\mathbf{B}\in\mathbb{C}^{K\times L}$ such that
$\mathrm{vec}\left(\mathbf{B}\right)=\mathbf{b}$. Also, $\mathbf{0}_{N}$ is
the all-zero vector of size $N$. The generalized inversion of a matrix
$\mathbf{B}$ is $\left(\mathbf{B}\right)^{-}$. We use
$\textrm{Pr}\left(.\right)$ to denote the probability.

##  II System Model

Consider a multi-IRS-aided radar with transmitter and receiver located at
$\uprho_{{}_{r}}=[0,0]^{T}$ in the two-dimensional (2-D) Cartesian coordinate
system. The radar transmits an OFDM signal with the bandwidth $B$ Hz
consisting of $L$ subcarriers as

$s_{{}_{\textrm{OFDM}}}(t)=\sum_{l=0}^{L-1}a_{l}e^{\mathrm{j}2\pi
f_{l}t},\quad 0\leq t\leq T,$ (1)

where $a_{l}$ is the waveform code, $f_{l}=f_{c}+l\Delta_{f}$ denotes the
$l$-th subcarrier frequency and the subcarrier spacing is chosen as
$\Delta_{f}=B/(L+1)=1/T$ to guarantee the orthogonality of the subcarriers.
The vector $\boldsymbol{a}=[a_{1},\ldots,a_{L}]^{\top}$ collects the OFDM
coefficients of all subcarriers for which we have
$\|\boldsymbol{a}\|_{2}^{2}=1$. The pulses in (1) are transmitted with the
pulse repetition interval (PRI) $T_{{\textrm{PRI}}}$.

The $M$ IRS platforms denoted as IRS1, IRS2,…, IRSM are installed at
stationary known locations (Fig. 1). Each IRS is a uniform linear array (ULA)
with $N_{m}$ reflecting elements, and with an inter-element spacing of $d$.
The first element of IRSm is located at a known coordinate
$\uprho_{{}_{i}}^{(m)}=[x_{i}^{(m)},y_{i}^{(m)}]^{T}$. The space-frequency
steering vector of the $m$-th IRS is
$\mathbf{b}_{m}(\boldsymbol{\theta},f_{l})=\left[1,e^{\textrm{j}\frac{2\pi
f_{l}}{c}dsin\theta},\ldots,e^{\textrm{j}\frac{2\pi
f_{l}}{c}d(N_{m}-1)sin\theta}\right]^{\top}$, where $c$ is the speed of light,
$f_{l}$ is the subcarrier frequency, $d$ is the half-wavelength Nyquist
spacing and $\theta$ is the direction of the impinging wavefront at the ULA.
Each reflecting element of IRSm reflects the incident signal with a phase-
shift change that is configured via a smart controller [1]. 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. Clearly, $\mathbf{v}_{m}$ is a unimodular
vector chosen from the set $\Omega^{N_{m}}$, where
$\Omega=\left\\{s\in\mathbb{C}|\mathbf{s}=e^{\textrm{j}\omega_{i}},\omega_{i}\in[0,2\pi)\right\\}$.

Assume a target at $\uprho_{{}_{t}}=[x_{t},y_{t}]^{\top}$ moving with velocity
$\upnu_{t}=[\upnu_{x},\upnu_{y}]^{\top}$. In the LoS path, the target is
characterized by its Doppler shift and time delay given by, respectively,

$\displaystyle\nu_{0}$
$\displaystyle=\frac{1}{c}\frac{2\upnu_{t}^{\top}(\uprho_{{}_{t}}-\uprho_{{}_{r}})}{d_{tr}},$
(2)

and

$\displaystyle\tau_{0}$ $\displaystyle=\frac{2d_{tr}}{c},$ (3)

where $d_{tr}=\|\uprho_{{}_{t}}-\uprho_{{}_{r}}\|_{2}$ is the distance between
the radar and target. The IRS deployment also yields $M$ non-line-of-sight
(NLoS) paths from the target to the radar. The Doppler shift and time delay in
the radar-IRSm-target-IRSm-radar path are, respectively,

$\displaystyle\nu_{m}$
$\displaystyle=\frac{1}{c}\frac{2\upnu_{t}^{\top}(\uprho_{{}_{t}}-\uprho^{(m)}_{{}_{i}})}{d^{(m)}_{it}},$
(4)

and

$\displaystyle\tau_{m}$ $\displaystyle=2\frac{d^{(m)}_{ri}+d^{(m)}_{it}}{c},$
(5)

for $m=1,\ldots,M$, where
$d^{(m)}_{ri}=\|\uprho^{(m)}_{{}_{i}}-\uprho_{{}_{r}}\|_{2}$ and
$d^{(m)}_{it}=\|\uprho_{{}_{t}}-\uprho^{(m)}_{{}_{i}}\|_{2}$ are the radar-
IRSm and target-IRSm distances, respectively.

We make the following assumptions about the IRS-aided OFDM radar and target
parameters:

A1

“Bandwidth-invariant Doppler”: The bandwidth of OFDM signal is much smaller
than the carrier frequency, i.e., $B\ll f_{c}$. Hence, the phase-shifts
arising from the Doppler effect are identical over all subcarriers.

A2

“Slow Target”: The Doppler frequency of the target does not change during one
coherent processing interval (CPI) i.e.
$\nu_{m}<<\frac{1}{NT_{{}_{{\textrm{PRI}}}}B}$. Therefore, the following
piecewise-constant approximation holds
$\nu_{m}t\approx\nu_{m}nT_{{}_{{\textrm{PRI}}}}$ , for
$t\in[nT_{{}_{{\textrm{PRI}}}},(n+1)T_{{}_{{\textrm{PRI}}}}]$.

A3

“Narrow surveillance area”: The radar is deployed in a region, where the range
of the target is much greater than the width or cross-range extent of the
surveillance area. The relative time gaps between any two signals received
from NLoS paths are very small in comparison to the LoS round trip delays,
i.e., $\tau_{m}\approx\tau_{0}=\frac{2d_{tr}}{c}$ for $m\in\\{1,\ldots,M\\}$.

A4

“Frequency-invariant IRS phase-shift”: The IRS platforms impose the same
phaseshifts over all subcarrier frequencies and therefore the IRS phase-shift
matrix is not indexed over different frequencies, i.e.,
$\boldsymbol{\Phi}_{m}(f_{l})=\boldsymbol{\Phi}_{m}$, for
$l\in\\{0,\ldots,L-1\\}$ and $m\in\\{1,\ldots,M\\}$.

A5

“Inter-IRS interference”: The mutual interference between various IRS
platforms is negligible. In other words, the interference caused by
reflections in the radar-IRSm-target-IRS${}_{m^{\prime}}$-radar path for
$m\neq m^{\prime}$ is insignificant because IRS is a passive reflector and the
reflections in non-beamformed directions are weaker.

Figure 1: A simplified illustration of various NLoS or virtual LoS links
provided by multiple IRS platforms mounted on urban infrastructure between the
radar and the hidden moving target.

Define the NLoS channel along the $l$-th subcarrier and $m$-th path as

$h_{{}_{lm}}=\mathbf{b}(\theta_{ir,m},f_{l})^{\top}\boldsymbol{\Phi}_{m}\mathbf{b}(\theta_{ti,m},f_{l})\mathbf{b}(\theta_{it,m},f_{l})^{\top}\boldsymbol{\Phi}_{m}\mathbf{b}(\theta_{ri,m},f_{l}),$
(6)

for $m>1$ and $h_{l0}$ is the LoS channel [22]. We define
$\boldsymbol{\Phi}_{m}=\mathrm{Diag}\left(\mathbf{v}_{m}\right)$ as
diagonalization of the 1-D phase-shift vector of IRSm. Assume the LoS path
between radar and target is obstructed, i.e., $h_{l0}\approx 0$. The signal
reflected from a Swerling-0 [26] target with $\alpha_{{}_{lm}}$ as the complex
reflectivity/amplitude along the $l$-th subchannel and $m$-th path is a
delayed, modulated and scaled version of the transmit signal in (1) as

$\displaystyle y_{l}(t)$
$\displaystyle=\sum_{m=0}^{M}a_{l}h_{{}_{lm}}\alpha_{lm}e^{\mathrm{j}2\pi
l\Delta_{f}(1+\nu_{m})(t-\tau_{m})}$ $\displaystyle\times e^{-\mathrm{j}2\pi
f_{c}(1+\nu_{m})\tau_{m}}e^{\mathrm{j}2\pi f_{c}\nu_{m}t}e^{\mathrm{j}2\pi
f_{c}t}+w_{l}(t),$ (7)

where the signal independent interference (noise) for the $l$-th subcarrier is
denoted by $w_{l}(t)$.

We collect $N$ samples from the signal, at
$t=\tau_{0}+nT_{{}_{{\textrm{PRI}}}}$, $n=0,\ldots,N-1$. By applying
$\nu_{m}t\approx\nu_{m}nT_{{}_{{\textrm{PRI}}}}$ (A2) and
$\tau_{m}\approx\tau_{0}$ (A3) to (II), the discrete-time received signal
corresponding to the range-cell of interest is

$\displaystyle y_{l}[n]$
$\displaystyle=\sum_{m=0}^{M}a_{l}h_{{}_{lm}}\alpha_{{}_{lm}}p_{{}_{l}}(n,\nu_{m})+w_{l}[n],$
(8)

where, $p_{{}_{l}}(n,\nu_{m})=e^{-\mathrm{j}2\pi
f_{l}\tau_{0}}e^{\mathrm{j}2\pi f_{l}\nu_{m}nT_{{}_{{\textrm{PRI}}}}}$
contains the unknown target delay and Doppler.

We stack measurements from all L subchannels to obtain the $L\times 1$ vector

$\mathbf{y}[n]=\mathrm{Diag}\left(\boldsymbol{a}\right)\mathbf{X}\mathbf{p}(n,\boldsymbol{\nu})+\mathbf{w}[n],$
(9)

where the Doppler steering vector is
$\mathbf{p}(n,\boldsymbol{\nu})=[\mathbf{p}_{{}_{0}}(n,\boldsymbol{\nu})^{\top},\ldots,\mathbf{p}_{{}_{L-1}}(n,\boldsymbol{\nu})^{\top}]^{\top}$,
with
$\mathbf{p}_{{}_{l}}(n,\boldsymbol{\nu})=[p_{{}_{l}}(n,\nu_{{}_{0}}),\ldots,p_{{}_{l}}(n,\nu_{{}_{M}})]^{\top}$,
and the $L\times 1$ noise vectors is
$\mathbf{w}[n]=[w_{0}[n],\ldots,w_{L-1}[n]]^{\top}$. Stacking all $N$ temporal
measurements, the $L\times N$ OFDM received signal matrix is

$\mathbf{Y}_{{}_{\textrm{OFDM}}}=\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})+\mathbf{N},$
(10)

where $\mathbf{A}=\mathrm{Diag}\left(\boldsymbol{a}\right)$,
$\mathbf{N}=[\mathbf{w}[0],\ldots,\mathbf{w}[N-1]]^{\top}$ and the Doppler
information of the target is collected in

$\mathbf{P}(\boldsymbol{\nu})=[\mathbf{p}(0,\boldsymbol{\nu}),\ldots,\mathbf{p}(N-1,\boldsymbol{\nu})],$
(11)

and

$\displaystyle\mathbf{X}$ $\displaystyle=\mathbf{D}\odot\mathbf{H},$ (12)
$\displaystyle\mathbf{D}$
$\displaystyle=\mathrm{Diag}\left(\boldsymbol{\alpha}_{{}_{0}}^{\top},\ldots,\boldsymbol{\alpha}_{{}_{L-1}}^{\top}\right),$
(13) $\displaystyle\boldsymbol{\alpha}_{{}_{l}}$
$\displaystyle=[\alpha_{{}_{l1}},\ldots,\alpha_{{}_{lM}}]^{\top},$ (14)
$\displaystyle\mathbf{H}$
$\displaystyle=\mathrm{Diag}\left(\mathbf{h}_{{}_{0}}^{\top},\ldots,\mathbf{h}_{{}_{L-1}}^{\top}\right),$
(15) $\displaystyle\mathbf{h}_{l}$
$\displaystyle=[h_{{}_{l1}},\ldots,h_{{}_{lM}}]^{\top},$ (16)

We assume the noise is from complex zero-mean Gaussian distribution and
correlated with a positive-definite covariance $\boldsymbol{\Sigma}$. The
columns of $\mathbf{N}$ are assumed to be (independent and identically
distributed) i.i.d. Then, OFDM measurements are distributed as

$\mathbf{Y}_{{}_{\textrm{OFDM}}}\sim\mathcal{CN}(\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu}),\mathbf{I}_{N}\otimes\boldsymbol{\Sigma})$
(17)

where $\mathbf{I}_{N}\otimes\boldsymbol{\Sigma}$ is the covariance of the
temporally white noise. Our goal is to design a waveform that maximizes the
detection of a moving target located at a given range.

##  III Target Detection

In order to decide whether a target is present in a particular known range-
cell, we perform binary hypothesis testing between $\mathcal{H}_{0}$ (target-
free hypothesis) and $\mathcal{H}_{1}$ (target-present hypothesis), that is

$\displaystyle\mathcal{H}_{0}:$
$\displaystyle\quad\mathbf{Y}_{{}_{\textrm{OFDM}}}=\mathbf{N},$ (18)
$\displaystyle\mathcal{H}_{1}:$
$\displaystyle\quad\mathbf{Y}_{{}_{\textrm{OFDM}}}=\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})+\mathbf{N}.$
(19)

The likelihood ratio is [27, 22]

$\mathcal{L}\left(\mathbf{Y}_{\text{OFDM}};\boldsymbol{\nu}\right)=\frac{f_{\mathcal{H}_{1}}\left(\mathbf{Y}_{\text{OFDM}};\boldsymbol{\nu},\mathbf{X},\boldsymbol{\Sigma}_{1}\right)}{f_{\mathcal{H}_{0}}\left(\mathbf{Y}_{\text{OFDM
}};\boldsymbol{\Sigma}_{0}\right)},$ (20)

where $f_{\mathcal{H}_{0}}$ and $f_{\mathcal{H}_{1}}$ are the likelihood
functions under $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively and
$\boldsymbol{\nu}$ is the Doppler frequency under test. Since the
$\boldsymbol{\Sigma}$ and target parameters are unknown, we employ a
generalized likelihood ratio test (GLRT) by replacing the unknowns with their
maximum likelihood estimates (MLEs) in the
$\mathcal{L}\left(\mathbf{Y}_{\text{OFDM}};\boldsymbol{\nu}\right)$ to obtain
the GLRT for our detection problem (18) as

$\mathcal{T}_{{}_{\text{GLR}}}=\frac{f_{\mathcal{H}_{1}}\left(\mathbf{Y}_{\text{OFDM}};\boldsymbol{\nu},\widehat{\mathbf{X}},\widehat{\boldsymbol{\Sigma}}_{1}\right)}{f_{\mathcal{H}_{0}}\left(\mathbf{Y}_{\text{OFDM
}};\widehat{\boldsymbol{\Sigma}}_{0}\right)}\quad\underset{\mathcal{H}_{2}}{\overset{\mathcal{H}_{1}}{\gtrless}}\gamma,$
(21)

where $\widehat{\boldsymbol{\Sigma}}_{0}$ and
$\widehat{\boldsymbol{\Sigma}}_{1}$ are the MLEs of
$\boldsymbol{\boldsymbol{\Sigma}}$ under $\mathcal{H}_{0}$ and
$\mathcal{H}_{1},\widehat{\mathbf{X}}$ is the MLE of $\mathbf{X}$ under
$\mathcal{H}_{1}$, and $\gamma$ is the detection threshold [27, 22]. The MLEs
of unknown covariance matrices are

$\displaystyle\widehat{\boldsymbol{\Sigma}}_{1}$
$\displaystyle=\frac{1}{N}\left(\mathbf{Y}_{\text{OFDM}}-\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\right)^{\mathrm{H}}\left(\mathbf{Y}_{\text{OFDM}}-\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\right),$
(22) $\displaystyle\widehat{\boldsymbol{\Sigma}}_{0}$
$\displaystyle=\frac{1}{N}\mathbf{Y}_{\text{OFDM}}^{\mathrm{H}}\mathbf{Y}_{\text{OFDM}}.$
(23)

Therefore, In the Gaussian noise scenario, the GLR becomes

$\mathcal{T}_{{}_{\text{GLR}}}=\frac{\operatorname{det}\left(\mathbf{Y}_{\text{OFDM}}^{\mathrm{H}}\mathbf{Y}_{\text{OFDM}}\right)}{\operatorname{det}\left(\left(\mathbf{Y}_{\text{OFDM}}-\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\right)^{\mathrm{H}}\left(\mathbf{Y}_{\text{OFDM}}-\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\right)\right)}.$
(24)

It follows from the Wilk’s theroem [28] that, as $N\rightarrow\infty$, the
GLRT statistic under $\mathcal{H}_{0}$ asymptotically garners a complex chi-
square (central) distribution, i.e.,

$N\ln\mathcal{T}_{{}_{{\textrm{GLR}}}}\sim\mathbb{C}\chi_{{}_{rL}}^{2},$ (25)

and under $\mathcal{H}_{1}$,

$N\ln\mathcal{T}_{{}_{{\textrm{GLR}}}}\sim\mathbb{C}\chi_{{}_{rL}}^{{}^{2}}(\delta),$
(26)

where $r=\operatorname{rank}\left(\mathbf{P}(\boldsymbol{\nu})\right)$, $rL$
is degrees of freedom, and $\delta$ is the noncentrality parameter of chi-
square distribution obtained in [22] as

$\delta=\mathrm{Tr}\left(\boldsymbol{\Sigma}^{-1}\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\mathbf{A}^{\mathrm{H}}\right).$
(27)

We have the probability of false alarm ($P_{{}_{\textrm{FA}}}$) and
probability of detection ($P_{{}_{\textrm{D}}}$) as

$\displaystyle P_{{}_{\textrm{FA}}}=\textrm{Pr}$
$\displaystyle\left(\mathcal{T}_{{}_{{\textrm{GLR}}}}>\gamma|\mathcal{H}_{0}\right)=Q_{{}_{rL}}(\gamma)$
$\displaystyle P_{{}_{\textrm{D}}}=\textrm{Pr}$
$\displaystyle\left(\mathcal{T}_{{}_{{\textrm{GLR}}}}>\gamma|\mathcal{H}_{1}\right)=Q^{M}_{\frac{rL}{2}}(\sqrt{\delta},\gamma)=Q^{M}_{{}_{\frac{rL}{2}}}\left(\sqrt{\delta},Q_{{}_{rL}}^{-1}(P_{{}_{\textrm{FA}}})\right)$
(28)

where $1-Q_{{}_{rL}}(.)$ is the cumulative distribution function (CDF) of chi-
squared distribution and $1-Q^{M}_{{}_{\frac{rL}{2}}}(.,.)$ is the Marcum
Q-function accounting for the CDF of non-central chi-squared distribution [28,
29]. The Marcum Q-function is strictly increasing in $\delta$. Thus, to
maximize $P_{\textrm{D}}$, we maximize $\delta=f(\boldsymbol{a},\mathbf{v})$
with respect to parameters such as the OFDM coefficients $\boldsymbol{a}$ and
vector of all IRS phase-shifts,
$\mathbf{v}=\left[\mathbf{v}^{\top}_{1},\mathbf{v}^{\top}_{2},\cdots,\mathbf{v}^{\top}_{M}\right]^{\top}\in\mathbb{C}^{MN_{m}}$.
We define the _signal-to-noise ratio_ (SNR) matrix
$\mathbf{A}\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\mathbf{A}^{\mathrm{H}}$
and its trace as in (27), as the _SNR metric_[27, 22].

##  IV Joint Waveform and IRS Phase-Shift Design

As discussed earlier in Section III, the probability of detection for a given
probability of false alarm is a monotonically increasing function of the
_SNR_. Hence, the design problem may also be formulated to maximize the SNR
with respect to the system parameters. The joint waveform and IRS phase-shift
design problem is

$\displaystyle\underset{\mathbf{v},\boldsymbol{a}}{\textrm{maximize}}\quad
f(\boldsymbol{a},\mathbf{v})$ $\displaystyle\textrm{subject
to}\quad\boldsymbol{a}^{H}\boldsymbol{a}=1.$ (29)

We resort to a task-specific cyclic algorithm, wherein we cyclically optimize
(IV) for $\boldsymbol{a}$ and $\mathbf{v}$ [31, 20]. To ensure the
unimodularity constraint of IRS phase-shift, we need a projection-based
optimization method.

###  IV-A OFDM Waveform Design

Problem (IV) with respect to $\boldsymbol{a}$ is recast as

$\displaystyle\mathcal{P}_{1}:\;\;\;$
$\displaystyle\underset{\boldsymbol{a}}{\textrm{maximize}}\quad\boldsymbol{a}^{\mathrm{H}}\left[\left(\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\right)^{\top}\odot\boldsymbol{\Sigma}^{-1}\right]\boldsymbol{a}$
$\displaystyle\textrm{subject to}\quad\boldsymbol{a}^{H}\boldsymbol{a}=1.$
(30)

The optimal OFDM coefficient $\boldsymbol{a}$ is the eigenvector corresponding
to the dominant eigenvalue of
$\left(\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\right)^{\top}\odot\boldsymbol{\Sigma}^{-1}$,
evaluated based on the _power method_[32] at each iteration $s$ as follows:

$\boldsymbol{a}^{(s+1)}=\frac{\left[\left(\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\right)^{\top}\odot\boldsymbol{\Sigma}^{-1}\right]\boldsymbol{a}^{(s)}}{\left\|\left[\left(\mathbf{X}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{X}^{\mathrm{H}}\right)^{\top}\odot\boldsymbol{\Sigma}^{-1}\right]\boldsymbol{a}^{(s)}\right\|_{2}},\;s\geq
0.$ (31)

###  IV-B IRS Beamforming Design

To design the phase-shift parameters, we maximize the SNR metric with respect
to $\mathbf{v}$:

$\displaystyle\mathcal{P}_{2}:\;\;\;\;\underset{\mathbf{v}}{\textrm{maximize}}$
$\displaystyle\quad f(\boldsymbol{a},\mathbf{v}).$ (32)

In what follows, we show that $\mathcal{P}_{2}$ can be written as a unimodular
quadratic programming (UQP) problem. To tackle $\mathcal{P}_{2}$ with respect
to $\mathbf{v}$, we adopt a computationally efficient procedure of the _power-
method-like iterations (PMLI)_ algorithm [33]. This method closely resembles
the widely used power method for computing the dominant eigenvalue/vector
pairs of matrices.

The UQP is defined as

$\underset{\mathbf{s}\in\Omega^{n}}{\textrm{maximize}}\quad\mathbf{s}^{\mathrm{H}}\mathbf{G}\mathbf{s}.$
(33)

The sequence of unimodular vectors at the $t$-th PMLI iteration is

$\mathbf{s}^{(t+1)}=e^{\textrm{j}\operatorname{arg}\left(\mathbf{G}\mathbf{s}^{(t)}\right)},$
(34)

leads to a monotonically increasing objective value for the UQP, when
$\mathbf{G}$ is a positive semidefinite matrix [33, 19].

The following Lemma 1 states the required transformations of (32) to
facilitate the application of the PMLI approach.

######  Lemma 1.

Define
$\mathbf{C}=\left[\begin{array}[]{c|c|c}\boldsymbol{\Upsilon}_{1}&\cdots&\boldsymbol{\Upsilon}_{L}\end{array}\right]$,$\boldsymbol{\Upsilon}_{l}=\sum^{LM}_{i=1}\mathbf{C}_{i}(\boldsymbol{\aleph}_{l}^{\top}\mathbf{E}_{i}^{\top}\otimes\mathbf{I})$,
$\mathbf{C}_{i}=\mathbf{e}_{i}\otimes\mathbf{I}_{{}_{L}}$, $\mathbf{e}_{i}$ as
a $LM\times 1$ vector whose $i$-th element is unity and remaining elements are
zero, $\boldsymbol{\aleph}_{l}$ as a $L\times L$ matrix with
$[\boldsymbol{\aleph}]_{{}_{ll}}=1$ and zero everywhere else, and
$\mathbf{E}_{i}=\textrm{vec}^{-1}_{{}_{(M,L)}}\left(\mathbf{e}_{i}\right)$.
Then,

$\mathrm{vec}\left(\mathbf{H}\right)=\mathbf{C}\mathbf{h},$ (35)

###### Proof.

Following the definitions in Lemma. 1, we have

$\displaystyle\mathrm{vec}\left(\mathbf{H}\right)$
$\displaystyle=\mathrm{vec}\left(\mathrm{Diag}\left(\mathbf{h}_{1}^{\top},\ldots,\mathbf{h}_{L}^{\top}\right)\right)=\mathrm{vec}\left(\sum_{l=1}^{L}\boldsymbol{\aleph}_{l}\otimes\mathbf{h}_{l}^{\top}\right)$
$\displaystyle=\sum_{l=1}^{L}\mathrm{vec}\left(\boldsymbol{\aleph}_{l}\otimes\mathbf{h}_{l}^{\top}\right)=\sum_{l=1}^{L}\sum^{LM}_{i=1}\mathbf{C}_{i}\left(\boldsymbol{\aleph}_{l}\otimes\mathbf{h}_{l}^{\top}\right)\mathbf{e}_{i}$
$\displaystyle=\sum_{l=1}^{L}\sum^{LM}_{i=1}\mathbf{C}_{i}\left(\boldsymbol{\aleph}_{l}\otimes\mathbf{h}_{l}^{\top}\right)\mathrm{vec}\left(\mathbf{E}_{i}\right)$
$\displaystyle=\sum_{l=1}^{L}\sum^{LM}_{i=1}\mathbf{C}_{i}\mathrm{vec}\left(\mathbf{h}_{l}^{\top}\mathbf{E}_{i}\boldsymbol{\aleph}_{l}\right)=\sum_{l=1}^{L}\sum^{LM}_{i=1}\mathbf{C}_{i}(\boldsymbol{\aleph}_{l}^{\top}\mathbf{E}_{i}^{\top}\otimes\mathbf{I})\mathbf{h}_{l}$
$\displaystyle=\sum_{l=1}^{L}\boldsymbol{\Upsilon}_{l}\mathbf{h}_{l}=\mathbf{C}\mathbf{h}.$
(36)

∎

Subsequently, we will use Lemma. 1 to propose a quadratic form with respect to
$\mathbf{h}$ for the SNR metric.

######  Proposition 1.

Denote
$\mathbf{h}=[\mathbf{h}_{1}^{\top},\ldots,\mathbf{h}_{L}^{\top}]^{\top}$,
$\mathbf{W}=\mathbf{C}^{\mathrm{H}}\mathbf{U}^{\mathrm{H}}\mathcal{A}\mathbf{U}\mathbf{C}$
and
$\mathbf{U}=\mathrm{Diag}\left(\mathrm{vec}\left(\mathbf{A}\mathbf{D}\right)\right)$.
Then, the SNR metric becomes

$f(\boldsymbol{a},\mathbf{v})=\mathbf{h}^{\mathrm{H}}\mathbf{W}\mathbf{h}.$
(37)

###### Proof.

Assume $\mathbf{B}=\mathbf{A}\mathbf{X}$. We recast the objective in (32) as

$\displaystyle f(\boldsymbol{a},\mathbf{v})$
$\displaystyle=\mathrm{Tr}\left(\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\mathbf{B}^{\mathrm{H}}\boldsymbol{\Sigma}^{-1}\mathbf{B}\mathbf{P}(\boldsymbol{\nu})\right)$
$\displaystyle=\mathrm{Tr}\left(\mathbf{B}^{\mathrm{H}}\boldsymbol{\Sigma}^{-1}\mathbf{B}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\right)$
$\displaystyle=\mathrm{vec}\left(\mathbf{B}^{\ast}\right)^{\top}\mathrm{vec}\left(\boldsymbol{\Sigma}^{-1}\mathbf{B}\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\right)$
$\displaystyle=\mathrm{vec}\left(\mathbf{B}\right)^{\mathrm{H}}\left(\left(\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\right)^{\top}\otimes\boldsymbol{\Sigma}^{-1}\right)\mathrm{vec}\left(\mathbf{B}\right)$
$\displaystyle=\mathrm{vec}\left(\mathbf{B}\right)^{\mathrm{H}}\mathcal{A}\mathrm{vec}\left(\mathbf{B}\right),$
(38)

where
$\mathcal{A}=\left(\mathbf{P}(\boldsymbol{\nu})\mathbf{P}^{\mathrm{H}}(\boldsymbol{\nu})\right)^{\top}\otimes\boldsymbol{\Sigma}^{-1}$.
We have
$\mathrm{vec}\left(\mathbf{B}\right)=\mathrm{vec}\left(\mathbf{A}\mathbf{X}\right)=\mathrm{vec}\left(\mathrm{Diag}\left(\boldsymbol{a}\right)\left(\mathbf{D}\odot\mathbf{H}\right)\right)=\mathrm{vec}\left(\mathbf{H}\right)\odot\mathrm{vec}\left(\mathbf{A}\mathbf{D}\right)=\mathbf{C}\mathbf{h}\odot\mathrm{vec}\left(\mathbf{A}\mathbf{D}\right)$
and using Lemma. 1,
$\mathrm{vec}\left(\mathbf{B}\right)=\mathbf{U}\mathbf{C}\mathbf{h}$.
Substituting this in (IV-B) completes the proof. ∎

We now reformulate the SNR metric as a quartic function in the optimization
parameter $\mathbf{v}$.

######  Proposition 2.

The SNR metric is quartic in phase-shifts, i.e.

$\displaystyle f(\boldsymbol{a},\mathbf{v})$
$\displaystyle=\mathbf{v}^{\mathrm{H}}\mathbf{Q}_{1}(\mathbf{v})^{\mathrm{H}}\mathbf{W}\mathbf{Q}_{1}(\mathbf{v})\mathbf{v},$
$\displaystyle=\mathbf{v}^{\mathrm{H}}\mathbf{Q}_{2}(\mathbf{v})^{\mathrm{H}}\mathbf{W}\mathbf{Q}_{2}(\mathbf{v})\mathbf{v},$
(39)

where

$\displaystyle\mathbf{Q}_{1}(\mathbf{v})$
$\displaystyle=\left[\begin{array}[]{c|c|c}(\mathbf{S}_{1}\mathbf{Q}(\mathbf{v}))^{\top}&\cdots&(\mathbf{S}_{L}\mathbf{Q}(\mathbf{v}))^{\top}\end{array}\right]^{\top},$
(41) $\displaystyle\mathbf{Q}_{2}(\mathbf{v})$
$\displaystyle=\left[\begin{array}[]{c|c|c}(\mathbf{S}_{1}\mathbf{Q}^{\prime}(\mathbf{v}))^{\top}&\cdots&(\mathbf{S}_{L}\mathbf{Q}^{\prime}(\mathbf{v}))^{\top}\end{array}\right]^{\top},$
(43) $\displaystyle\mathbf{Q}(\mathbf{v})$
$\displaystyle=\mathrm{Diag}\left(\mathbf{v}_{1}\otimes\mathbf{I}_{N_{m}},\ldots,\mathbf{v}_{{}_{M}}\otimes\mathbf{I}_{N_{m}}\right),$
$\displaystyle\mathbf{Q}^{\prime}(\mathbf{v})$
$\displaystyle=\mathrm{Diag}\left(\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{1},\ldots,\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{{}_{M}}\right),$
$\displaystyle\mathbf{S}_{l}$
$\displaystyle=\mathrm{Diag}\left(\mathrm{vec}\left(\mathbf{S}_{l1}\right)^{\top},\ldots,\mathrm{vec}\left(\mathbf{S}_{lM}\right)^{\top}\right),$
$\displaystyle\mathbf{S}_{lm}$
$\displaystyle=\left(\mathbf{b}(\theta_{ir,m},f_{l})\odot\mathbf{b}(\theta_{ti,m},f_{l})\right)\left(\mathbf{b}(\theta_{ri,m},f_{l})\odot\mathbf{b}(\theta_{it,m},f_{l})\right)^{\top}.$
(44)

###### Proof.

Given $\boldsymbol{\Phi}_{m}=\mathrm{Diag}\left(\mathbf{v}_{m}\right)$, it is
straightforward to verify from (6) that we have
$h_{{}_{lm}}=\mathbf{v}_{m}^{\top}\mathbf{S}_{{}_{lm}}\mathbf{v}_{m}$ and

$\displaystyle\mathbf{h}_{l}$
$\displaystyle=[h_{{}_{l1}},\ldots,h_{{}_{lM}}]^{\top}=\left[\mathrm{Tr}\left(\mathbf{S}_{{}_{l1}}\mathbf{v}_{{}_{1}}\mathbf{v}_{{}_{1}}^{\top}\right),\ldots,\mathrm{Tr}\left(\mathbf{S}_{{}_{lM}}\mathbf{v}_{{}_{M}}\mathbf{v}_{{}_{M}}^{\top}\right)\right]^{\top},$
$\displaystyle=\mathrm{Diag}\left(\mathrm{vec}\left(\mathbf{S}_{{}_{l1}}\right)^{\top},\ldots,\mathrm{vec}\left(\mathbf{S}_{{}_{lM}}\right)^{\top}\right)$
$\displaystyle\qquad\times\left[\mathrm{vec}\left(\mathbf{v}_{{}_{1}}\mathbf{v}_{{}_{1}}^{\top}\right),\ldots,\mathrm{vec}\left(\mathbf{v}_{{}_{M}}\mathbf{v}_{{}_{M}}^{\top}\right)\right]^{\top}.$
(45)

Applying the identity [32]

$\operatorname{vec}\left(\mathbf{v}_{m}\mathbf{v}^{\top}_{m}\right)=\left(\mathbf{I}_{N_{m}}\otimes\mathbf{v}_{m}\right)\mathbf{v}_{m}=\left(\mathbf{v}_{m}\otimes\mathbf{I}_{N_{m}}\right)\mathbf{v}_{m},$
(46)

to (IV-B) produces

$\displaystyle\mathbf{h}_{{}_{l}}=\mathbf{S}_{{}_{l}}\mathbf{Q}(\mathbf{v})\mathbf{v}=\mathbf{S}_{{}_{l}}\mathbf{Q}^{\prime}(\mathbf{v})\mathbf{v},$
(47)

where $\mathbf{S}_{{}_{l}}$, $\mathbf{Q}(\mathbf{v})$ and
$\mathbf{Q}^{\prime}(\mathbf{v})$ are given in (41). Concatenating the vectors
in (47), we obtain

$\mathbf{h}=\mathbf{Q}_{1}(\mathbf{v})\mathbf{v}=\mathbf{Q}_{2}(\mathbf{v})\mathbf{v},$
(48)

where $\mathbf{Q}_{1}(\mathbf{v})$ and $\mathbf{Q}_{2}(\mathbf{v})$ are given
in (41). ∎

###  IV-C Proposed Algorithm

We cyclically tackle the SNR maximization via its bi-quadratic transformation
with respect to auxiliary variables $\mathbf{v}_{(1)}$ and $\mathbf{v}_{(2)}$.
In the sequel, $\mathbf{v}_{(1)},\mathbf{v}_{(2)}\in\mathbb{C}^{MN_{m}}$ are
vectors produced by symmetrization, representing the collection of phase-
shifts of all IRS platforms, whereas in the previous parts
$\mathbf{v}_{1},\mathbf{v}_{2}\in\mathbb{C}^{N_{m}}$ were the vector of phase-
shift of IRS1 and IRS2. In the following proposition, we recast the SNR metric
as a bi-quadratic function of $\mathbf{v}_{(1)}$ and $\mathbf{v}_{(2)}$.

######  Proposition 3.

The function

$\displaystyle g(\mathbf{v}_{{}_{(1)}},\mathbf{v}_{{}_{(2)}})$
$\displaystyle=\mathbf{v}_{{}_{(1)}}^{\mathrm{H}}\mathbf{E}(\mathbf{v}_{{}_{(2)}})\mathbf{v}_{{}_{(1)}},$
$\displaystyle=\mathbf{v}_{{}_{(2)}}^{\mathrm{H}}\mathbf{E}(\mathbf{v}_{{}_{{}_{(1)}}})\mathbf{v}_{{}_{(2)}},$
(49)

where

$\displaystyle\mathbf{E}(\mathbf{v})$
$\displaystyle=\frac{\mathbf{G}_{1}(\mathbf{v})+\mathbf{G}_{2}(\mathbf{v})}{2},$
$\displaystyle\mathbf{G}_{1}(\mathbf{v})$
$\displaystyle=\mathbf{Q}^{\mathrm{H}}_{1}(\mathbf{v})\mathbf{W}\mathbf{Q}_{1}(\mathbf{v}),$
$\displaystyle\mathbf{G}_{2}(\mathbf{v})$
$\displaystyle=\mathbf{Q}^{\mathrm{H}}_{2}(\mathbf{v})\mathbf{W}\mathbf{Q}_{2}(\mathbf{v}),$
(50)

is a bi-quadratic transformation of SNR metric in (2).

###### Proof.

To show symmetry, from (41) and (46), we observe that
$\mathbf{Q}_{1}(\mathbf{v}_{{}_{(i)}})\mathbf{v}_{{}_{(k)}}=\mathbf{Q}_{2}(\mathbf{v}_{{}_{(k)}})\mathbf{v}_{{}_{(i)}}$
for $i\neq k\in\left\\{1,2\right\\}$. Therefore,

$\displaystyle
g\left(\mathbf{v}_{{}_{(1)}},\mathbf{v}_{{}_{(2)}}\right)=\mathbf{v}_{{}_{(1)}}^{\mathrm{H}}\mathbf{E}\left(\mathbf{v}_{{}_{(2)}}\right)\mathbf{v}_{{}_{(1)}}$
$\displaystyle=\mathbf{v}_{{}_{(1)}}^{\mathrm{H}}\frac{\mathbf{G}_{1}\left(\mathbf{v}_{{}_{(2)}}\right)+\mathbf{G}_{2}\left(\mathbf{v}_{{}_{(2)}}\right)}{2}\mathbf{v}_{{}_{(1)}}$
$\displaystyle=\mathbf{v}_{{}_{(1)}}^{\mathrm{H}}\frac{\mathbf{Q}^{\mathrm{H}}_{1}\left(\mathbf{v}_{{}_{(2)}}\right)\mathbf{W}\mathbf{Q}_{1}\left(\mathbf{v}_{{}_{(2)}}\right)+\mathbf{Q}^{\mathrm{H}}_{2}\left(\mathbf{v}_{{}_{(2)}}\right)\mathbf{W}\mathbf{Q}_{2}\left(\mathbf{v}_{{}_{(2)}}\right)}{2}\mathbf{v}_{{}_{(1)}}$
$\displaystyle=\mathbf{v}_{{}_{(2)}}^{\mathrm{H}}\frac{\mathbf{Q}^{\mathrm{H}}_{2}\left(\mathbf{v}_{{}_{(1)}}\right)\mathbf{W}\mathbf{Q}_{2}\left(\mathbf{v}_{{}_{(1)}}\right)+\mathbf{Q}^{\mathrm{H}}_{1}\left(\mathbf{v}_{{}_{(1)}}\right)\mathbf{W}\mathbf{Q}_{1}\left(\mathbf{v}_{{}_{(1)}}\right)}{2}\mathbf{v}_{{}_{(2)}}$
$\displaystyle=\mathbf{v}_{{}_{(2)}}^{\mathrm{H}}\frac{\mathbf{G}_{2}\left(\mathbf{v}_{{}_{(1)}}\right)+\mathbf{G}_{1}\left(\mathbf{v}_{{}_{(1)}}\right)}{2}\mathbf{v}_{{}_{(2)}}$
$\displaystyle=\mathbf{v}_{{}_{(2)}}^{\mathrm{H}}\mathbf{E}\left(\mathbf{v}_{{}_{(1)}}\right)\mathbf{v}_{{}_{(2)}}=g\left(\mathbf{v}_{{}_{(2)}},\mathbf{v}_{{}_{(1)}}\right).$
(51)

By Substituting $\mathbf{v}_{{}_{(1)}}=\mathbf{v}_{{}_{(2)}}=\mathbf{v}$ and
(3) in (3), and comparing it with (2), one can verify that
$f(\boldsymbol{a},\mathbf{v})=g(\mathbf{v},\mathbf{v})$. ∎

Since $f(\boldsymbol{a},\mathbf{v})=g(\mathbf{v},\mathbf{v})$, we propose to
maximize $f(\boldsymbol{a},\mathbf{v})$ by alternately fixing
$\mathbf{v}_{(1)}$ or $\mathbf{v}_{(2)}$ and maximizing
$g\left(\mathbf{v}_{{}_{(1)}},\mathbf{v}_{{}_{(2)}}\right)$ with respect to
the other variable while enforcing
$\mathbf{v}_{{}_{(1)}}=\mathbf{v}_{{}_{(2)}}$ as a constraint. From
Proposition 3, fixing either $\mathbf{v}_{{}_{(1)}}$ or
$\mathbf{v}_{{}_{(2)}}$ and maximizing
$g\left(\mathbf{v}_{{}_{(1)}},\mathbf{v}_{{}_{(2)}}\right)$ with respect to
the other variable requires solving the following

$\displaystyle\underset{\mathbf{v}_{(k)}\in\Omega^{MN_{m}}}{\textrm{maximize}}\quad\mathbf{v}_{(k)}^{\mathrm{H}}\mathbf{E}\left(\mathbf{v}_{(i)}\right)\mathbf{v}_{(k)},\quad
i\neq k\in\left\\{1,2\right\\},$ (52)

######  Remark 1.

In UQP, the diagonal loading technique is used to ensure the positive
semidefiniteness of the matrix, without changing the optimal solution [12]. In
(52), the diagonal loading as
$\widetilde{\mathbf{E}}(\mathbf{v})\leftarrow\lambda_{m}\mathbf{I}-\mathbf{E}(\mathbf{v})$,
with $\lambda_{{}_{m}}$ being the maximum eigenvalue of
$\mathbf{E}(\mathbf{v})$, results is an equivalent problem.

Note that diagonal loading has no effect on the solution of (52) because
$\mathbf{v}^{\mathrm{H}}\widetilde{\mathbf{E}}(\mathbf{v})\mathbf{v}=\lambda_{{}_{m}}MN_{m}-\mathbf{v}^{\mathrm{H}}\mathbf{E}(\mathbf{v})\mathbf{v}$.
The equivalent problem to (52) is

$\displaystyle\underset{\mathbf{v}_{(k)}\in\Omega^{MN_{m}}}{\textrm{minimize}}\quad\mathbf{v}_{(k)}^{\mathrm{H}}\widetilde{\mathbf{E}}\left(\mathbf{v}_{(i)}\right)\mathbf{v}_{(k)},\quad
i\neq k\in\left\\{1,2\right\\}.$ (53)

The following theorem demonstrates that the IRS beamforming design problem
$\mathcal{P}_{2}$ is equivalent to a unimodular bi-quadratic programming
(UBQP) that we solve using the PMLI approach in (34).

######  Theorem 1.

The SNR maximization problem $\mathcal{P}_{2}$ with respect to phase-shifts is
equivalent to the following UBQP

$\displaystyle\underset{\mathbf{v}_{{}_{(k)}}\in\Omega^{MN_{m}}}{\textrm{maximize}}$
$\displaystyle\quad\begin{pmatrix}\mathbf{v}_{(k)}\\\
1\end{pmatrix}^{\mathrm{H}}\underbrace{\begin{pmatrix}\widehat{\lambda}_{m}\mathbf{I}-\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})&\eta\mathbf{v}_{{}_{(i)}}\\\
\eta\mathbf{v}_{{}_{(i)}}^{\mathrm{H}}&\widehat{\lambda}_{m}-2\eta
MN_{m}\end{pmatrix}}_{=\widehat{\mathbf{E}}\left(\mathbf{v}_{{}_{(i)}}\right)}\begin{pmatrix}\mathbf{v}_{(k)}\\\
1\end{pmatrix},$ (54)

where $i\neq k\in\left\\{1,2\right\\}$, and $\widehat{\lambda}_{m}$ is the
maximum eigenvalue of $\mathcal{E}(\mathbf{v}_{{}_{(i)}})$ as defined in
(IV-C).

###### Proof.

From Proposition 3 and Remark 1, we know that $\mathcal{P}_{2}$ is equivalent
to the following problems,

$\displaystyle\underset{\mathbf{v}_{(k)}\in\Omega^{MN_{m}}}{\textrm{minimize}}\quad\mathbf{v}_{(k)}^{\mathrm{H}}\widetilde{\mathbf{E}}(\mathbf{v}_{(i)})\mathbf{v}_{(k)},\quad
i\neq k\in\left\\{1,2\right\\},$ $\displaystyle\textrm{subject
to}\quad\mathbf{v}_{{}_{(i)}}=\mathbf{v}_{{}_{(k)}}.$ (55)

We add the $\ell_{2}$-norm penalty term between $\mathbf{v}_{{}_{(1)}}$ and
$\mathbf{v}_{{}_{(2)}}$ as a _penalty_ function to (IV-C), which yields

$\underset{\mathbf{v}_{{}_{(k)}}\in\Omega^{MN_{m}}}{\textrm{minimize}}\quad\mathbf{v}_{{}_{(k)}}^{\mathrm{H}}\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})\mathbf{v}_{{}_{(k)}}+\eta\|\mathbf{v}_{{}_{(i)}}-\mathbf{v}_{{}_{(k)}}\|_{2}^{2},\quad
i\neq k\in\left\\{1,2\right\\},$ (56)

where $\eta$ is a Lagrangian multiplier. The regularizer as well as the main
objective are quadratic in $\mathbf{v}_{{}_{(1)}}$ and
$\mathbf{v}_{{}_{(2)}}$. Consequently, we recast the objective of (56) as

$\displaystyle\mathbf{v}_{{}_{(k)}}^{\mathrm{H}}\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})\mathbf{v}_{{}_{(k)}}+\eta\|\mathbf{v}_{{}_{(i)}}-\mathbf{v}_{{}_{(k)}}\|_{2}^{2},$
$\displaystyle=\mathbf{v}_{{}_{(k)}}^{\mathrm{H}}\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})\mathbf{v}_{{}_{(k)}}-2\eta\operatorname{Re}\left(\mathbf{v}_{{}_{(k)}}^{\mathrm{H}}\mathbf{v}_{{}_{(i)}}\right)+2\eta
MN_{m},$ $\displaystyle=\begin{pmatrix}\mathbf{v}_{{}_{(k)}}\\\
1\end{pmatrix}^{\mathrm{H}}\begin{pmatrix}\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})&\eta\mathbf{v}_{{}_{(i)}}\\\
\eta\mathbf{v}_{{}_{(i)}}^{\mathrm{H}}&2\eta
MN_{m}\end{pmatrix}\begin{pmatrix}\mathbf{v}_{{}_{(k)}}\\\ 1\end{pmatrix},$
(57)

where in the first equality, we used
$\|\mathbf{v}_{{}_{(k)}}\|_{2}^{2}=MN_{m}$, due to unimodularity of
$\mathbf{v}_{{}_{(k)}}$. Substituting (IV-C) in (56) yields

$\displaystyle\underset{\mathbf{v}_{{}_{(k)}}\in\Omega^{MN_{m}}}{\textrm{minimize}}$
$\displaystyle\quad\begin{pmatrix}\mathbf{v}_{(k)}\\\
1\end{pmatrix}^{\mathrm{H}}\underbrace{\begin{pmatrix}\widetilde{\mathbf{E}}(\mathbf{v}_{{}_{(i)}})&\eta\mathbf{v}_{{}_{(i)}}\\\
\eta\mathbf{v}_{{}_{(i)}}^{\mathrm{H}}&2\eta
MN_{m}\end{pmatrix}}_{=\mathcal{E}(\mathbf{v}_{{}_{(i)}})}\begin{pmatrix}\mathbf{v}_{(k)}\\\
1\end{pmatrix},$ $\displaystyle\quad i\neq k\in\left\\{1,2\right\\}.$ (58)

We use diagonal loading as introduced in Remark 1 to obtain (54). ∎

Based on Theorem 1, the IRS phase assignment problem can be formulated as a
UBQP and therefore it may be tackled in an alternating manner over
$\bar{\mathbf{v}}_{{}_{(1)}}=[\mathbf{v}^{\top}_{{}_{(1)}}1]^{\top}$ and
$\bar{\mathbf{v}}_{{}_{(2)}}=[\mathbf{v}^{\top}_{{}_{(2)}}1]^{\top}$, by the
PMLI iterations in (34). The PMLI has been shown to be convergent in terms of
both the optimization objective and variable[20, 19]. Algorithm 1 summarizes
the steps for joint waveform and IRS phase-shift design.

Algorithm 1 Joint IRS phase-shift and OFDM waveform design

1:Input: Initialization values $\mathbf{v}^{(0)}_{1}$,$\mathbf{v}^{(0)}_{2}$,
$\boldsymbol{a}^{(0)}$, the Lagrangian multiplier $\eta$, total number of
iterations $\Gamma_{1}$ ($\Gamma_{2}$) for the problem $\mathcal{P}_{1}$
($\mathcal{P}_{2}$).

2:Output: Optimized phase-shifts $\mathbf{v}^{\star}$ and OFDM signal
coefficients $\boldsymbol{a}^{\star}$.

3:for $s=0:\Gamma_{1}-1$ do $\triangleright$ $\mathbf{v}^{(t)}_{(1)}$ and
$\mathbf{v}^{(t)}_{(2)}$ are the solutions at the $t$-th iteration.

4: for $t=0:\Gamma_{2}-1$ do

5: $\mathbf{v}^{(t+1)}_{(1)}\leftarrow
e^{\mathrm{j}\operatorname{arg}\left(\begin{bmatrix}\mathbf{I}_{{}_{MN_{m}}}\mathbf{0}_{{}_{MN_{m}}}\end{bmatrix}\left(\widehat{\lambda}_{m}\mathbf{I}-\mathcal{E}(\mathbf{v}^{(t+1)}_{(2)},\boldsymbol{a}^{(s)})\right)\bar{\mathbf{v}}^{(t)}_{(1)}\right)}$.

6: $\mathbf{v}^{(t+1)}_{(2)}\leftarrow
e^{\mathrm{j}\operatorname{arg}\left(\begin{bmatrix}\mathbf{I}_{{}_{MN_{m}}}\mathbf{0}_{{}_{MN_{m}}}\end{bmatrix}\left(\widehat{\lambda}_{m}\mathbf{I}-\mathcal{E}(\mathbf{v}^{(t)}_{(1)},\boldsymbol{a}^{(s)})\right)\bar{\mathbf{v}}^{(t)}_{(2)}\right)}$,

7: $\mathbf{v}^{(s)}\leftarrow\mathbf{v}^{(\Gamma_{2})}_{(1)}$ or
$\mathbf{v}^{(\Gamma_{2})}_{(2)}$.

8:
$\mathbf{h}^{(s)}\leftarrow\mathbf{Q}_{1}(\mathbf{v}^{(s)})\mathbf{v}^{(s)}$.

9: Update $\mathbf{X}^{(s)}$ according to (12)-(15).

10: Update $\boldsymbol{a}^{(s)}$ according to (31).

11:return
$\left\\{\boldsymbol{a}^{\star},\mathbf{v}^{\star}\right\\}\leftarrow\left\\{\boldsymbol{a}^{(\Gamma_{1})},\mathbf{v}^{(\Gamma_{1})}\right\\}$.

##  V Numerical analysis

We performed numerical experiments to analyze the performance of Algorithm 1
through the RoC of the proposed detector. The radar located at
$\uprho_{r}=[0,0]^{\top}$ was set to transmit $N=50$ pulses with pulse-width
$T=50$ ns, carrier frequency $f_{c}=1$ GHz, bandwidth $B=100$ MHz, and PRI
$T_{{}_{{\textrm{PRI}}}}=20\mu$s. The OFDM signal had $L=4$ subcarriers with
spacing $\Delta_{f}=1/T=20$ MHz. The IRS1 was located at
$\uprho_{i}^{(1)}=[0.1,0.1]^{\top}$km and IRS2 is located at
$\uprho_{i}^{(2)}=[-0.1,0.1]^{\top}$km. Each IRS comprises $N_{m}=8$ elements
arranged as a ULA. The target was located at $\uprho_{{}_{t}}=[0,5]^{\top}$ km
moving with $\upnu=[10,10]^{T}$m/s. The complex target reflectivity
coefficients $\alpha_{{}_{lm}}$ corresponding to a Swerling-0 target model
were drawn from $\mathcal{CN}(0,1)$.

We compared multi-IRS-aided OFDM radar with a multipath OFDM radar [22]. Fig.
2 illustrates the RoC obtained after $10^{3}$ Monte-Carlo trials for fixed
probability of false alarm $P_{{}_{\textrm{FA}}}$. In each trial, the
threshold $\gamma$ is set according to (III) for the desired
$P_{{}_{\textrm{FA}}}$. We observe that deploying $M=1$ IRS platforms with
phase-shifts obtained by Algorithm 1 improves $P_{{}_{\textrm{D}}}$ over the
non-IRS (LoS) OFDM radar. Moreover, for $M=2$, IRS-aided outperforms multipath
OFDM radar proposed in [22] with specular reflection $\\{h_{{}_{lm}}\\}=1$ in
the exactly identical two paths between the target and radar. Deploying
multiple IRS platforms in comparison with single IRS and non-IRS scenarios
provides additional degrees of freedom (DoFs) and improves performance. Also,
multi-IRS-aided radar outperforms single IRS because an optimal deployment of
more IRSs provides more NLoS paths and, hence, enhanced detection of NLoS
targets especially those that may not be accessible via only one IRS.

Figure 2: RoC of detection for LoS OFDM radar (single path), OFDM radar with 2
specular paths, $M=1$ IRS-aided OFDM radar, $M=2$ IRS-aided OFDM radar.

##  VI Summary

We investigated the moving target detection problem using a multi-IRS-aided
OFDM radar. The IRS phase-shifts along with the OFDM transmit signal
coefficients were designed by taking advantage of an alternating optimization
of the non-centrality parameter of the GLRT. We showed that maximizing the
non-centrality parameter improved the probability of detection. By means of
numerical investigations, we demonstrated that our proposed method enhances
the $P_{{}_{\textrm{D}}}$ over the non-IRS OFDM radar systems.

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