# Hybrid Beamforming for Intelligent Reflecting Surface Aided Millimeter Wave
MIMO Systems

Sung Hyuck Hong, Jaeyong Park, Sung-Jin Kim, and Junil Choi Sung Hyuck Hong
and Junil Choi are with the School of Electrical Engineering, Korea Advanced
Institute of Science and Technology, Daejeon 34141, South Korea (e-mail:
{shong16; junil}@kaist.ac.kr).Jaeyong Park and Sung-Jin Kim are with C&M
Standard Lab, Future Technology Center, LG Electronics Inc. (e-mail:
{jaeyong630.park; sj88.kim}@lge.com).

###### Abstract

As communication systems that employ millimeter wave (mmWave) frequency bands
must use large antenna arrays to overcome the severe propagation loss of
mmWave signals, hybrid beamforming has been considered as an integral
component of mmWave communications. Recently, intelligent reflecting surface
(IRS) has been proposed as an innovative technology that can significantly
improve the performance of mmWave communication systems through the use of
low-cost passive reflecting elements. In this paper, we study IRS-aided mmWave
multiple-input multiple-output (MIMO) systems with hybrid beamforming
architectures. We first exploit the sparse-scattering structure and large
dimension of mmWave channels to develop the joint design of IRS reflection
matrix and hybrid beamformer for narrowband MIMO systems. Then, we generalize
the proposed joint design to broadband MIMO systems with orthogonal frequency
division multiplexing (OFDM) modulation by leveraging the angular sparsity of
frequency-selective mmWave channels. Simulation results demonstrate that the
proposed joint designs can significantly enhance the spectral efficiency of
the systems of interest and achieve superior performance over the existing
designs.

###### Index Terms:

Millimeter wave (mmWave) communications, multiple-input multiple-output
(MIMO), hybrid beamforming, intelligent reflecting surface (IRS), frequency-
selective channels, orthogonal frequency division multiplexing (OFDM).

##  I Introduction

The explosive growth in wireless data traffic and resultant issue of bandwidth
shortage have increased the necessity of resorting to higher frequency bands
that are relatively uncongested [1]. Since large bandwidth can be utilized at
millimeter wave (mmWave) frequency bands to attain possibly up to gigabit-per-
second data rates, mmWave communications have been viewed as one of the key
technologies that can realize many new applications and services that the
fifth generation (5G) wireless networks aim to support [2]. However, the
propagation properties of mmWave signals, such as severe path loss and
atmospheric attenuation, must be carefully examined in order to fully reap the
expected benefits of mmWave communications [3, 4].

Massive multiple-input multiple-output (MIMO), the concept of utilizing a
large number of antennas at the transceivers, has been proposed as a promising
solution to combat the high path loss of mmWave signals and achieve
significant capacity gain through simultaneous transmission of multiple data
streams [4, 5]. Despite its numerous advantages, massive MIMO makes the
traditional fully-digital beamforming at baseband prohibitively expensive
since such processing requires dedicated radio frequency (RF) chain for each
antenna [6]. To address such issue, hybrid beamforming architectures that use
the combination of a low-dimensional digital baseband beamformer and a high-
dimensional analog beamformer have been proposed to increase the energy
efficiency of massive MIMO systems, significantly reducing the number of RF
chains at the cost of only slight performance degradation [7].

Many works have investigated mmWave MIMO communication systems with hybrid
beamforming architectures [8, 9, 10, 11, 12, 13, 14, 15]. In [8], the problem
of designing hybrid precoders and combiners that maximize spectral efficiency
was recast as a sparsity-constrained matrix reconstruction problem, which was
subsequently solved using an algorithm based on orthogonal matching pursuit.
Manifold optimization (MO)-based algorithms were proposed in [9] to tackle the
hybrid precoder design problem in both narrowband and broadband mmWave MIMO
systems. The authors in [10] presented a unified heuristic algorithm to design
beamformers for frequency-selective mmWave channels under fully-connected and
partially-connected hybrid architectures. Hybrid beamformer designs that aim
to minimize mean square error (MSE) of data streams were proposed in [11, 12,
13], while some works considered the use of finite-resolution phase shifters
in implementing analog beamformers [14, 15].

Recently, intelligent reflecting surface (IRS) has been proposed as a cost-
effective and innovative technology to smartly reconfigure the wireless
propagation environments in a real-time manner, thereby significantly
improving the performance of future communication systems [16]. Labeled as one
of the key components of sixth generation (6G) wireless networks, IRS is a
metasurface consisting of a large number of passive reflecting elements, each
of which induces an adjustable phase and/or amplitude shift to the incident
electromagnetic waves [17, 18]. Since mmWave signals are highly vulnerable to
blockages due to the narrow beamwidth, IRS is, along with massive MIMO and
hybrid beamforming, expected to play a pivotal role in enhancing the coverage
and spectral/energy efficiency of mmWave communication systems [18, 19].

As 6G wireless networks aim to provide even higher data rates and energy
efficiency than 5G networks in which mmWave frequency bands are heavily
employed, upcoming communication systems must integrate IRS into mmWave MIMO
systems with large antenna arrays and hybrid beamforming architectures to
successfully attain such performance targets [20, 21]. However, to reap the
full benefits of such integration, it is necessary to overcome the challenge
of jointly designing the IRS reflection matrix and hybrid beamformer, both of
which are subject to the hardware constraints that must be carefully
considered [16]. Among the limited number of studies on the IRS reflection
matrix design for point-to-point MIMO communications [22, 23, 24, 25, 26, 27],
only [26] and [27] considered mmWave MIMO systems with hybrid beamforming
architectures. In [26], a two-stage algorithm based on MO was developed to
design the IRS reflection matrix and hybrid beamformer for mmWave systems in
which the direct channel from the TX to RX is not present. In [27], the
authors constructed the IRS reflection matrix and hybrid beamformer for
broadband MIMO systems under frequency-selective channels. However, the hybrid
beamformer design proposed in [27] focused on minimizing bit error rate (BER)
and thus entails an inevitable loss in spectral efficiency. Furthermore, the
designs in [26] and [27] construct the hybrid beamformer without considering
the effect of the IRS reflection matrix on mmWave channels, although doing so
can lead to significant performance improvement and complexity reduction.

Motivated by these facts, we propose in this paper the joint designs of IRS
reflection matrix and hybrid beamformer for narrowband and broadband mmWave
MIMO systems. The proposed designs aim to maximize the spectral efficiency of
IRS-aided mmWave MIMO systems in which the reflected channel from the
transmitter (TX) to IRS to receiver (RX) and the direct channel from the TX to
RX coexist. To the best of our knowledge, no prior work on IRS-aided mmWave
MIMO systems with hybrid beamforming architectures has considered the presence
of the direct channel. Furthermore, by exploiting how the IRS reflection
matrix influences the structure of mmWave channels when constructing the
hybrid beamformer, the proposed joint designs achieve significant performance
gains over the existing benchmarks. Our main contributions are summarized as
follows:

  * •

We develop an IRS reflection matrix design that successfully establishes the
favorable communication environments for narrowband mmWave MIMO systems. Then,
to demonstrate the effectiveness of the design, we provide an asymptotic
analysis of the effect of the proposed IRS reflection matrix on the structure
of mmWave channels.

  * •

We propose a hybrid beamformer design that takes into account the inherent
structure of mmWave channels so as to attain the performance close to that of
fully-digital beamforming. By carefully examining how the channels are
adjusted according to the proposed IRS reflection matrix design, we construct
hybrid precoders and combiners that fully reap considerable spectral
efficiency gains offered by IRS.

  * •

We generalize the proposed joint design of IRS reflection matrix and hybrid
beamformer for narrowband MIMO systems to broadband MIMO systems with
orthogonal frequency division multiplexing (OFDM) modulation. Smartly
leveraging the sparsity of frequency-selective mmWave channels in the angular
domain, the generalized design can significantly enhance the spectral and
energy efficiency of IRS-aided MIMO-OFDM systems.

The rest of the paper is organized as follows. The system model and problem
formulation are described in Section II. The joint design of IRS reflection
matrix and hybrid beamformer for narrowband MIMO systems is proposed in
Section III. In Section IV, the proposed joint design is extended to broadband
MIMO-OFDM systems. The complexity analysis of the joint designs and simulation
results are presented in Section V and Section VI, respectively. Finally,
conclusions are drawn in Section VII.

Notation: Vectors and matrices are represented by lower and upper boldface
letters. The $m\times n$ matrix whose elements are all zero is represented by
${\boldsymbol{0}}^{m\times n}$, while ${\mathbb{C}}^{m\times n}$ denotes the
set of all $m\times n$ complex matrices. The transpose, conjugate transpose,
inverse, determinant, and rank of the matrix ${\mathbf{A}}$ are represented by
${\mathbf{A}}^{\mathrm{T}}$, ${\mathbf{A}}^{\mathrm{H}}$, ${\mathbf{A}}^{-1}$,
$\text{det}({\mathbf{A}})$, and $\text{rank}({\mathbf{A}})$, respectively. The
Frobenius norm of ${\mathbf{A}}$ is denoted by
$\lVert{\mathbf{A}}\rVert_{\mathrm{F}}$, while $\text{row}({\mathbf{A}})$ and
$\text{col}({\mathbf{A}})$ indicate the number of rows and columns of
${\mathbf{A}}$, respectively. The element in the $i$-th row and $j$-th column
of ${\mathbf{A}}$ is denoted by $[{\mathbf{A}}]_{i,j}$, whereas
$[{\mathbf{A}}]_{:,j}$ represents the $j$-th column vector of ${\mathbf{A}}$.
Similarly,
$[{\mathbf{A}}]_{:,1:j}\in\mathbb{C}^{\text{row}({\mathbf{A}})\times j}$ and
$[{\mathbf{A}}]_{1:j,1:j}\in\mathbb{C}^{j\times j}$ each denote the matrix
whose columns are given by the first $j$ columns of ${\mathbf{A}}$ and that
consisting of the first $j$ rows and columns of ${\mathbf{A}}$. The $j$-th
element of the vector ${\mathbf{x}}$ is denoted by $[{\mathbf{x}}]_{j}$, while
$\text{diag}({\mathbf{x}})$ represents the diagonal matrix that contains the
elements of ${\mathbf{x}}$ on its main diagonal. The $N\times N$ identity
matrix and the expectation operator are represented by ${\mathbf{I}}_{N}$ and
$\mathbb{E}[\cdot]$, respectively. The function $\text{max}(a,0)$ defined for
a real number $a$ is denoted by $(a)^{+}$, whereas $|{\cdot}|$ and
$\lVert\cdot\rVert_{2}$ each indicate the absolute value of a scalar and the
$\ell_{2}$-norm of a vector. The complex and real normal distributions with
mean $m$ and variance $\sigma^{2}$ are denoted by
${\mathcal{C}}{\mathcal{N}}(m,\sigma^{2})$ and ${\mathcal{N}}(m,\sigma^{2})$,
respectively.

Figure 1: Illustration of an IRS-aided mmWave MIMO system.

##  II System Model and Problem Formulation

In this paper, we consider an IRS-aided mmWave MIMO system with a hybrid
beamforming architecture. For the sake of simplicity, we first describe a
narrowband system in this section, and introduce a broadband system later in
Section IV. As shown in Fig. 1, an IRS consisting of $M$ passive reflecting
elements assists the communication between the TX with $N_{\text{t}}$ antennas
and the RX with $N_{\text{r}}$ antennas. Let
${\mathbf{H}}_{\text{TR}}\in{\mathbb{C}}^{N_{\text{r}}\times N_{\text{t}}}$
denote the direct channel from the TX to RX. Similarly, let
${\mathbf{H}}_{\text{TI}}\in{\mathbb{C}}^{M\times N_{\text{t}}}$ and
${\mathbf{H}}_{\text{IR}}\in{\mathbb{C}}^{N_{\text{r}}\times M}$ represent the
channel from the TX to IRS and that from the IRS to RX, respectively. With
each reflecting element of the IRS serving as a single point source that
scatters the received signal after applying to it a controllable phase shift,
the effect of IRS can be modeled by the diagonal matrix
${\boldsymbol{\Phi}}=\text{diag}([e^{j\theta_{1}},\dots,e^{j\theta_{M}}])\in\mathbb{C}^{M\times
M}$, where $\theta_{m}\in[0,2\pi)$ denotes the phase shift of $m$-th IRS
element, $m\in\\{1,\dots,M\\}$ [16, 17]. The total combined channel from the
TX to RX can then be expressed as
${\mathbf{H}}_{\text{tot}}={\mathbf{H}}_{\text{TR}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}$.

The TX sends $N_{\text{s}}$ data streams to the RX using the digital baseband
precoder
${\mathbf{F}}_{\text{BB}}\in{\mathbb{C}}^{N_{\text{t}}^{\text{RF}}\times
N_{\text{s}}}$ and analog precoder
${\mathbf{F}}_{\text{RF}}\in{\mathbb{C}}^{N_{\text{t}}\times
N_{\text{t}}^{\text{RF}}}$, where the number of RF chains at the TX is denoted
by $N_{\text{t}}^{\text{RF}}$ and is subject to the constraint
$N_{\text{s}}\leq N_{\text{t}}^{\text{RF}}\leq N_{\text{t}}$. We impose the
total power constraint $P_{\text{TX}}$ on the transmit power, i.e.,
$\lVert{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}\rVert_{\mathrm{F}}^{2}\leq
P_{\text{TX}}$. The RX uses the analog combiner
${\mathbf{W}}_{\text{RF}}\in\mathbb{C}^{N_{\text{r}}\times
N^{\text{RF}}_{\text{r}}}$ and digital baseband combiner
${\mathbf{W}}_{\text{BB}}\in\mathbb{C}^{N^{\text{RF}}_{\text{r}}\times
N_{\text{s}}}$ to process the received signal, where the number of RF chains
at the RX $N_{\text{r}}^{\text{RF}}$ is subject to the constraint
$N_{\text{s}}\leq N_{\text{r}}^{\text{RF}}\leq N_{\text{r}}$. As the analog
combiner ${\mathbf{W}}_{\text{RF}}$ and precoder ${\mathbf{F}}_{\text{RF}}$
are implemented with phase shifters, the constant modulus constraint is
imposed on each of their elements, i.e.,
$|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n$. The processed received signal is expressed as

$\displaystyle\tilde{{\mathbf{y}}}$
$\displaystyle={\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}{\mathbf{s}}+{\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{n}},$
(1)

where ${\mathbf{s}}\in{\mathbb{C}}^{N_{\text{s}}\times 1}$ is the symbol
vector satisfying
$\mathbb{E}[{\mathbf{s}}{\mathbf{s}}^{\mathrm{H}}]={\mathbf{I}}_{N_{\text{s}}}$,
and ${\mathbf{n}}\in{\mathbb{C}}^{N_{\text{r}}\times 1}$ is an additive white
Gaussian noise (AWGN) vector whose entries are independently and identically
distributed (i.i.d) with ${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2}_{n})$. To
accurately evaluate the effectiveness of the proposed IRS reflection matrix
and hybrid beamformer design, we assume in this paper that perfect channel
state information (CSI) is available at the TX and RX [8, 9, 10, 24, 25, 26,
27]. CSI acquisition at IRS-aided mmWave systems is currently a topic of
active research. Recently, several algorithms have been proposed in [28, 29,
30] to efficiently estimate mmWave channels in IRS-aided MIMO systems. The
achievable spectral efficiency when the transmitted symbols follow a Gaussian
distribution is given by

$\displaystyle R$
$\displaystyle=\log_{2}\text{det}({\mathbf{I}}_{N_{\text{s}}}+{\mathbf{R}}_{\bar{{\mathbf{n}}}}^{-1}{\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}{\mathbf{F}}^{\mathrm{H}}_{\text{BB}}{\mathbf{F}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}{\mathbf{W}}_{\text{BB}}),$
(2)

where
${\mathbf{R}}_{\bar{{\mathbf{n}}}}=\sigma^{2}_{n}{\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{W}}_{\text{RF}}{\mathbf{W}}_{\text{BB}}$
represents the covariance matrix of the noise
$\bar{{\mathbf{n}}}={\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{n}}$
after combining.

Throughout this paper, we adopt the widely used Saleh-Valenzuela model [8, 9,
11, 12, 13, 14, 26, 27] to represent mmWave channels. Each of the narrowband
mmWave channels is expressed as

$\displaystyle{\mathbf{H}}_{i}=\sum_{q=0}^{N_{\text{path}}^{i}-1}\alpha_{i,q}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{i,q},\theta^{\text{r}}_{i,q}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{i,q},\theta^{\text{t}}_{i,q})^{\mathrm{H}},$
(3)

where $i\in\\{\text{TR, TI, IR}\\}$ is the subscript for the channel matrices,
$N_{\text{path}}^{i}$ is the number of physical propagation paths in
${\mathbf{H}}_{i}$, and $\alpha_{i,q}$ is the complex gain of the $q$-th path
in ${\mathbf{H}}_{i}$. We assume that $\alpha_{i,q}$ are independently
distributed with
${\mathcal{C}}{\mathcal{N}}(0,\gamma_{i}^{2}10^{-0.1PL(d_{i})}),\forall
q\in\\{0,\dots,N_{\text{path}}^{i}-1\\}$, where
$\gamma_{i}=\sqrt{\text{row}({\mathbf{H}}_{i})\text{col}({\mathbf{H}}_{i})/N_{\text{path}}^{i}}$
is the normalization factor, and $PL(d_{i})$ represents the path loss that
depends on the distance $d_{i}$ between the two entities associated with
${\mathbf{H}}_{i}$ [31]. For example, $d_{\text{TR}}$ denotes the distance
between the TX and RX. Lastly, the normalized receive and transmit array
response vectors corresponding to the $q$-th path in ${\mathbf{H}}_{i}$ are
respectively denoted by
${\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{i,q},\theta^{\text{r}}_{i,q})\in\mathbb{C}^{\text{row}({\mathbf{H}}_{i})\times
1}$ and
${\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{i,q},\theta^{\text{t}}_{i,q})\in\mathbb{C}^{\text{col}({\mathbf{H}}_{i})\times
1}$, where $\phi^{\text{r}}_{i,q}(\theta^{\text{r}}_{i,q}$) and
$\phi^{\text{t}}_{i,q}(\theta^{\text{t}}_{i,q})$ each stand for the azimuth
(elevation) angles of arrival and departure (AoAs and AoDs) of the path.

We assume in this paper that uniform planar arrays (UPAs) are employed at the
TX, RX, and IRS. For the sake of clarity, set $i=\text{TR}$ and define
$s\in\\{0,\dots,N_{\text{path}}^{\text{TR}}-1\\}$. The transmit array response
vector
${\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{TR},s},\theta^{\text{t}}_{\text{TR},s})\in\mathbb{C}^{N_{\text{t}}\times
1}$ corresponding to the $s$-th path in ${\mathbf{H}}_{\text{TR}}$ is then
given by

$\displaystyle{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{TR},s},\theta^{\text{t}}_{\text{TR},s})=\frac{1}{\sqrt{N_{\text{t}}}}\Big{[}1,\dots,e^{j\frac{2\pi
d}{\lambda}(i_{h}\sin(\phi^{\text{t}}_{\text{TR},s})\sin(\theta^{\text{t}}_{\text{TR},s})+i_{v}\cos(\theta^{\text{t}}_{\text{TR},s}))},$
$\displaystyle\hskip 180.0pt\dots,e^{j\frac{2\pi
d}{\lambda}((N^{h}_{\text{t}}-1)\sin(\phi^{\text{t}}_{\text{TR},s})\sin(\theta^{\text{t}}_{\text{TR},s})+(N^{v}_{\text{t}}-1)\cos(\theta^{\text{t}}_{\text{TR},s}))}\Big{]}^{\mathrm{T}},$
(4)

where $\lambda$ is the signal wavelength, and $d$ is the spacing between the
antennas or IRS elements. The horizontal and vertical indices for the transmit
antennas are respectively denoted by $0\leq i_{h}<N^{h}_{\text{t}}$ and $0\leq
i_{v}<N^{v}_{\text{t}}$, where
$N_{\text{t}}=N^{h}_{\text{t}}N^{v}_{\text{t}}$. Other receive and transmit
array response vectors can be similarly defined.

In this paper, we aim to maximize the spectral efficiency in (2) by jointly
designing the IRS reflection matrix ${\boldsymbol{\Phi}}$, hybrid precoder
${\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}$, and hybrid combiner
${\mathbf{W}}_{\text{RF}}{\mathbf{W}}_{\text{BB}}$. The problem of maximizing
the spectral efficiency is formulated as

$\displaystyle\text{(P1)}\hskip
42.0pt\max_{{\mathbf{W}}_{\text{BB}},{\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}},{\mathbf{F}}_{\text{BB}},{\mathbf{F}}_{\text{RF}}}R$
(5a) $\displaystyle\hskip 10.0pt\text{subject to }\hskip
13.0pt{\boldsymbol{\Phi}}=\text{diag}([e^{j\theta_{1}},\dots,e^{j\theta_{M}}]),$
(5b) $\displaystyle\hskip
74.0pt\lVert{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}\rVert_{F}^{2}\leq
P_{\text{TX}},$ (5c) $\displaystyle\hskip
74.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (5d) $\displaystyle\hskip
74.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n.$ (5e)

It is challenging to obtain the solution of the optimization problem (P1)
since the constant modulus constraints (5b), (5d), (5e) on
${\boldsymbol{\Phi}}$, ${\mathbf{W}}_{\text{RF}}$, and
${\mathbf{F}}_{\text{RF}}$ are non-convex. Additionally, the objective
function $R$, which is coupled with the five matrix variables
${\\{{\mathbf{W}}_{\text{BB}},{\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}},{\mathbf{F}}_{\text{BB}},{\mathbf{F}}_{\text{RF}}\\}}$,
is neither convex nor concave and thus makes the problem (P1) intractable to
solve. To tackle these challenges, we reformulate the problem (P1) by
exploiting its structure in the next section.

##  III Joint Design of IRS Reflection Matrix and Hybrid Beamformer

In this section, we propose the joint design of IRS reflection matrix and
hybrid beamformer for narrowband mmWave MIMO systems described in Section II.
Effectively exploiting the angular sparsity and large dimension of mmWave MIMO
channels, the proposed design provides the systems of interest with
significant increases in spectral efficiency.

###  III-A Formulation of Effective Channel Design Problem

In this subsection, we transform (P1) into the effective channel design
problem, whose purpose is to properly construct the analog combiner
${\mathbf{W}}_{\text{RF}}$, IRS reflection matrix ${\boldsymbol{\Phi}}$, and
analog precoder ${\mathbf{F}}_{\text{RF}}$ so that the effective channel
${\mathbf{H}}_{\text{eff}}={\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}{\mathbf{F}}_{\text{RF}}$
is capable of supporting high spectral efficiency. We first re-express the
objective function $R$ of (P1) as

$\displaystyle R$
$\displaystyle=\log_{2}\text{det}\left({\mathbf{I}}_{N_{\text{s}}}+{\mathbf{R}}_{\bar{{\mathbf{n}}}}^{-1}{\mathbf{W}}^{\mathrm{H}}_{\text{BB}}{\mathbf{H}}_{\text{eff}}{\mathbf{F}}_{\text{BB}}{\mathbf{F}}_{\text{BB}}^{\mathrm{H}}{\mathbf{H}}_{\text{eff}}^{\mathrm{H}}{\mathbf{W}}_{\text{BB}}\right),$
(6)

where
${\mathbf{H}}_{\text{eff}}={\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}{\mathbf{F}}_{\text{RF}}$
represents the effective channel after analog precoding and combining are
applied. The expression in (6) shows that, when ${\mathbf{H}}_{\text{eff}}$ is
given, i.e., when ${\boldsymbol{\Phi}},{\mathbf{W}}_{\text{RF}},$ and
${\mathbf{F}}_{\text{RF}}$ are chosen to satisfy the constraints in (5b),
(5d), and (5e), the problem (P1) can be simplified as

$\displaystyle(\text{P2})\hskip
5.0pt\max_{{\mathbf{W}}_{\text{BB}},{\mathbf{F}}_{\text{BB}}}\log_{2}\text{det}({\mathbf{I}}_{N_{\text{s}}}+{\mathbf{R}}_{\bar{{\mathbf{n}}}}^{-1}{\mathbf{W}}_{\text{BB}}^{\mathrm{H}}{\mathbf{H}}_{\text{eff}}{\mathbf{F}}_{\text{BB}}{\mathbf{F}}_{\text{BB}}^{\mathrm{H}}{\mathbf{H}}_{\text{eff}}^{\mathrm{H}}{\mathbf{W}}_{\text{BB}})$
(7a) $\displaystyle\hskip 22.0pt\text{subject to }\hskip
35.0pt\lVert{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}\rVert_{\mathrm{F}}^{2}\leq
P_{\text{TX}}.$ (7b)

Note that the problem $(\text{P2})$ cannot be solved in general because the
power constraint (7b) is coupled with ${\mathbf{F}}_{\text{RF}}$ and the noise
${\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{n}}$ after analog combining is
not necessarily white. However, under the condition that
${\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N^{\text{RF}}_{\text{r}}}$
and
${\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N^{\text{RF}}_{\text{t}}}$
hold, the optimal solution of the problem (P2) is given by
$\\{{\mathbf{W}}_{\text{BB}}=\hat{{\mathbf{W}}}_{\text{BB}},{\mathbf{F}}_{\text{BB}}=\hat{{\mathbf{F}}}_{\text{BB}}\\}$,
where [32]

$\displaystyle\hat{{\mathbf{W}}}_{\text{BB}}=[{\mathbf{U}}_{\text{eff}}]_{:,1:N_{\text{s}}},\hskip
5.0pt\hat{{\mathbf{F}}}_{\text{BB}}=[{\mathbf{V}}_{\text{eff}}]_{:,1:N_{\text{s}}}{\mathbf{P}}^{1/2}_{\text{eff}}.$
(8)

In (8),
${\mathbf{U}}_{\text{eff}}\in\mathbb{C}^{N^{\text{RF}}_{\text{r}}\times
N^{\text{RF}}_{\text{r}}}$ and
${\mathbf{V}}_{\text{eff}}\in\mathbb{C}^{N^{\text{RF}}_{\text{t}}\times
N^{\text{RF}}_{\text{t}}}$ each denote the unitary matrices whose columns are
the left and right singular vectors of ${\mathbf{H}}_{\text{eff}}$, i.e., the
singular value decomposition (SVD) of ${\mathbf{H}}_{\text{eff}}$ is expressed
as
${\mathbf{H}}_{\text{eff}}={\mathbf{U}}_{\text{eff}}{\boldsymbol{\Sigma}}_{\text{eff}}{\mathbf{V}}^{\mathrm{H}}_{\text{eff}}$,
where
$|[{\boldsymbol{\Sigma}}_{\text{eff}}]_{m,m}|\geq|[{\boldsymbol{\Sigma}}_{\text{eff}}]_{n,n}|,\forall
m,n\in\\{1,\dots,\text{min}(N^{\text{RF}}_{\text{r}},N^{\text{RF}}_{\text{t}})\\}$
such that $m<n$. From now on, unless otherwise stated, we will express the SVD
of a matrix such that its singular values are in descending order of their
absolute values, just as in ${\boldsymbol{\Sigma}}_{\text{eff}}$. The
waterfilling power allocation matrix ${\mathbf{P}}_{\text{eff}}^{1/2}$ for
${\mathbf{H}}_{\text{eff}}$ is given by
${\mathbf{P}}_{\text{eff}}^{1/2}=\text{diag}\left(\big{[}\sqrt{P_{1}},...,\sqrt{P_{N_{\text{s}}}}\medspace\big{]}\right)$,
where
$P_{l}=\left(\frac{1}{\eta}-\frac{\sigma^{2}_{n}}{|[{\boldsymbol{\Sigma}}_{\text{eff}}]_{l,l}|^{2}}\right)^{+},\forall
l\in\\{1,\dots,N_{\text{s}}\\}$, and $\eta$ is chosen such that
$\sum_{l=1}^{N_{\text{s}}}P_{l}=P_{\text{TX}}$. The result in (8) indicates
that, when the additional constraint is imposed on (P1) so that each of
${\mathbf{W}}_{\text{RF}}$ and ${\mathbf{F}}_{\text{RF}}$ has orthonormal
columns, the baseband combiner $\hat{{\mathbf{W}}}_{\text{BB}}$ and precoder
$\hat{{\mathbf{F}}}_{\text{BB}}$ maximize the spectral efficiency $R$ in (2)
for given ${\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}}$, and
${\mathbf{F}}_{\text{RF}}$. In fact, $\hat{{\mathbf{W}}}_{\text{BB}}$ and
$\hat{{\mathbf{F}}}_{\text{BB}}$ achieve the maximum spectral
efficiency111From now on, we use $R_{\text{max}}({\mathbf{H}})$ to denote the
maximum spectral efficiency that can be achieved under the transmit power
constraint $P_{\text{TX}}$ and channel ${\mathbf{H}}$ of appropriate
dimension. $R_{\text{max}}({\mathbf{H}}_{\text{eff}})$ that can be attained
under ${\mathbf{H}}_{\text{eff}}$, i.e.,

$\displaystyle
R\Bigr{|}_{\begin{subarray}{c}{\mathbf{W}}_{\text{BB}}=\hat{{\mathbf{W}}}_{\text{BB}}\\\
{\mathbf{F}}_{\text{BB}}=\hat{{\mathbf{F}}}_{\text{BB}}\end{subarray}}$
$\displaystyle=\sum_{l=1}^{N_{\text{s}}}\log_{2}\left(1+\frac{P_{l}}{\sigma^{2}_{n}}|[{\boldsymbol{\Sigma}}_{\text{eff}}]_{l,l}|^{2}\right)=R_{\text{max}}({\mathbf{H}}_{\text{eff}}).$
(9)

We therefore aim to maximize $R_{\text{max}}({\mathbf{H}}_{\text{eff}})$ in
(9) by properly designing ${\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}}$, and
${\mathbf{F}}_{\text{RF}}$ according to the given channels
${\mathbf{H}}_{\text{TR}},{\mathbf{H}}_{\text{TI}},$ and
${\mathbf{H}}_{\text{IR}}$. That is, we focus on solving the effective channel
design problem, which is formulated as

$\displaystyle\text{(P3)}\hskip
57.0pt\max_{{\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}},{\mathbf{F}}_{\text{RF}}}\medspace
R_{\text{max}}({\mathbf{H}}_{\text{eff}})$ (10a) $\displaystyle\hskip
10.0pt\text{subject to }\hskip
21.0pt{\boldsymbol{\Phi}}=\text{diag}([e^{j\theta_{1}},\dots,e^{j\theta_{M}}]),$
(10b) $\displaystyle\hskip
82.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (10c) $\displaystyle\hskip
82.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n,$ (10d) $\displaystyle\hskip
82.0pt{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N_{\text{r}}^{\text{RF}}},{\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N_{\text{t}}^{\text{RF}}}.$
(10e)

Note that, given the transmit power constraint $P_{\text{TX}}$ and noise power
$\sigma^{2}_{n}$, the value of the objective function
$R_{\text{max}}({\mathbf{H}}_{\text{eff}})$ of the problem (P3) is solely
determined by
$\left\\{|[{\boldsymbol{\Sigma}}_{\text{eff}}]_{l,l}|^{2}\right\\}_{l=1}^{N_{\text{s}}}$,
which are necessarily the $N_{\text{s}}$ largest eigenvalues of
${\mathbf{H}}_{\text{eff}}{\mathbf{H}}_{\text{eff}}^{\mathrm{H}}$.

###  III-B IRS Reflection Matrix Design

Although the effective channel design problem (P3) developed in Section III-A
involves less variables than the original problem (P1) of interest, it is
still challenging to find the optimal solution of (P3) since the IRS
reflection matrix ${\boldsymbol{\Phi}}$ directly influences the total combined
channel ${\mathbf{H}}_{\text{tot}}$, which in turn determines the appropriate
analog combiner ${\mathbf{W}}_{\text{RF}}$ and precoder
${\mathbf{F}}_{\text{RF}}$. In order to address this difficulty, we propose in
this subsection the IRS reflection matrix design that smartly leverages the
structure of mmWave MIMO channels to significantly increase the spectral
efficiency that ${\mathbf{H}}_{\text{tot}}$ can support. After adjusting
${\boldsymbol{\Phi}}$ according to the proposed IRS design, we will construct
${\mathbf{W}}_{\text{RF}}$ and ${\mathbf{F}}_{\text{RF}}$ that successfully
translate the spectral efficiency gain in
${\mathbf{H}}_{\text{tot}}={\mathbf{H}}_{\text{TR}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}$
to that in
${\mathbf{H}}_{\text{eff}}={\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}_{\text{tot}}{\mathbf{F}}_{\text{RF}}$,
details of which are explained in Section III-C.

Let the SVD of ${\mathbf{H}}_{\text{tot}}$ be expressed as
${\mathbf{H}}_{\text{tot}}={\mathbf{U}}_{\text{tot}}{\boldsymbol{\Sigma}}_{\text{tot}}{\mathbf{V}}^{\mathrm{H}}_{\text{tot}}$.
Then, similar to $R_{\text{max}}({\mathbf{H}}_{\text{eff}})$ in (9), the
maximum achievable spectral efficiency
$R_{\text{max}}({\mathbf{H}}_{\text{tot}})$ under ${\mathbf{H}}_{\text{tot}}$
depends only on
$\left\\{|[{\boldsymbol{\Sigma}}_{\text{tot}}]_{l,l}|^{2}\right\\}_{l=1}^{N_{\text{s}}}$,
or equivalently the $N_{s}$ largest eigenvalues of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$, for given
$P_{\text{TX}}$ and $\sigma^{2}_{n}$. To mathematically characterize the
eigenvalues of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$, we first
define the receive array response matrix ${\mathbf{A}}^{i}_{\text{r}}$,
complex gain matrix ${\mathbf{G}}_{i}$, and transmit array response matrix
${\mathbf{A}}^{i}_{\text{t}}$ for the mmWave channel ${\mathbf{H}}_{i}$ in (3)
as

$\displaystyle{\mathbf{A}}^{i}_{\text{r}}$
$\displaystyle=\begin{bmatrix}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{i,0},\theta^{\text{r}}_{i,0})&\cdots&{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{i,N_{\text{path}}^{i}-1},\theta^{\text{r}}_{i,N_{\text{path}}^{i}-1})\end{bmatrix},$
$\displaystyle\hskip
33.0pt{\mathbf{G}}_{i}=\text{diag}\big{(}[\alpha_{i,0},\dots,\alpha_{i,N_{\text{path}}^{i}-1}]\big{)},$
$\displaystyle{\mathbf{A}}^{i}_{\text{t}}$
$\displaystyle=\begin{bmatrix}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{i,0},\theta^{\text{t}}_{i,0})&\cdots&{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{i,N_{\text{path}}^{i}-1},\theta^{\text{t}}_{i,N_{\text{path}}^{i}-1})\end{bmatrix}.$
(11)

The channel matrix ${\mathbf{H}}_{\text{i}}$ can then be decomposed as

$\displaystyle{\mathbf{H}}_{i}={\mathbf{A}}^{i}_{\text{r}}{\mathbf{G}}_{i}({\mathbf{A}}^{i}_{\text{t}})^{\mathrm{H}},$
(12)

where we assume, without loss of generality, that
$|[{\mathbf{G}}_{i}]_{m,m}|\geq|[{\mathbf{G}}_{i}]_{n,n}|,\forall
m,n\in\\{1,\dots,N_{\text{path}}^{i}\\}$ such that $m<n$. We now pay
particular attention to
${\mathbf{H}}_{\text{TR}}{\mathbf{H}}_{\text{TI}}^{\mathrm{H}}$, which can be
written as

$\displaystyle{\mathbf{H}}_{\text{TR}}{\mathbf{H}}_{\text{TI}}^{\mathrm{H}}={\mathbf{A}}_{\text{r}}^{\text{TR}}{\mathbf{G}}_{\text{TR}}({\mathbf{A}}_{\text{t}}^{\text{TR}})^{\mathrm{H}}{\mathbf{A}}_{\text{t}}^{\text{TI}}{\mathbf{G}}_{\text{TI}}^{\mathrm{H}}({\mathbf{A}}_{\text{r}}^{\text{TI}})^{\mathrm{H}}.$
(13)

Since the AoDs of different propagation paths can be considered as continuous
random variables that are independent from one another, it follows that the
event
$E=\big{\\{}\phi^{\text{t}}_{\text{TR},s}\neq\phi^{\text{t}}_{\text{TI},j},\theta^{\text{t}}_{\text{TR},s}\neq\theta^{\text{t}}_{\text{TI},j},\forall
s\in\\{0,\ldots,N^{\text{TR}}_{\text{path}}-1\\},\forall
j\in\\{0,\ldots,N^{\text{TI}}_{\text{path}}-1\\}\big{\\}}$ occurs with
probability one [33, 26]. Then, by the asymptotic orthogonality of UPA array
response vectors [34], we have

$\displaystyle({\mathbf{A}}_{\text{t}}^{\text{TR}})^{\mathrm{H}}{\mathbf{A}}_{\text{t}}^{\text{TI}}\rightarrow{\boldsymbol{0}}^{N^{\text{TR}}_{\text{path}}\times
N^{\text{TI}}_{\text{path}}},\medspace\text{as
}N_{\text{t}}\rightarrow\infty,$ (14)

which implies that each element of
${\mathbf{H}}_{\text{TR}}{\mathbf{H}}_{\text{TI}}^{\mathrm{H}}$ converges to
$0$ in the limit of large $N_{\text{t}}$. Using similar arguments, we can show
that

$\displaystyle{\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$
$\displaystyle={\mathbf{H}}_{\text{TR}}{\mathbf{H}}_{\text{TR}}^{\mathrm{H}}+{\mathbf{H}}_{\text{TR}}{\mathbf{H}}_{\text{TI}}^{\mathrm{H}}{\boldsymbol{\Phi}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}^{\mathrm{H}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}{\mathbf{H}}_{\text{TR}}^{\mathrm{H}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}{\mathbf{H}}_{\text{TI}}^{\mathrm{H}}{\boldsymbol{\Phi}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}^{\mathrm{H}}$
$\displaystyle\rightarrow\sum_{s=0}^{N_{\text{path}}^{\text{TR}}-1}|\alpha_{\text{TR},s}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s}){\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s})^{\mathrm{H}}+\sum_{j=0}^{N_{\text{path}}^{\text{TI}}-1}|\alpha_{\text{TI},j}|^{2}{\mathbf{q}}_{j}{\mathbf{q}}_{j}^{\mathrm{H}},$
(15)

as $N_{\text{t}}\rightarrow\infty$, where
${\mathbf{q}}_{j}={\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j}),\medspace
j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$.

We now present the following lemma to establish the connection between the
magnitude of ${\mathbf{q}}_{j}$ and IRS reflection matrix
${\boldsymbol{\Phi}}$.

###### Lemma 1:

For any $j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$,
$\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}$ satisfies

$\displaystyle\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}\leq\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}),$
(16)

where the equality holds if
${\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})$
is the eigenvector corresponding to the maximum eigenvalue
$\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})$
of ${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}$, i.e.,
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}){\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j}).$

Proof: See Appendix A.  $\square$

With the aid of Lemma 1, we now show in the following proposition that the IRS
reflection matrix ${\boldsymbol{\Phi}}$ that asymptotically maximizes
$\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}$ can be obtained in closed-form.

###### Proposition 1:

Let $j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$ and define the vector
${\mathbf{v}}\in\mathbb{C}^{M\times 1}$ that satisfies
${\boldsymbol{\Phi}}=\text{diag}({\mathbf{v}})$. Then, when
${\mathbf{v}}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$,
it holds that

$\displaystyle\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}\rightarrow\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}),\medspace\text{as
}N_{\text{r}},M\rightarrow\infty.$ (17)

Proof: Set $j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$ and
${\mathbf{v}}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$.
We can then write

$\displaystyle{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})$
$\displaystyle=\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})){\mathbf{v}}={\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0}).$
(18)

By a similar argument used to derive (14), we have for each
$m\in\\{0,\dots,N_{\text{path}}^{\text{IR}}-1\\}$ that

$\displaystyle{\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},m},\theta^{\text{t}}_{\text{IR},m})$
$\displaystyle={\mathbf{A}}_{\text{t}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}^{\mathrm{H}}({\mathbf{A}}_{\text{r}}^{\text{IR}})^{\mathrm{H}}{\mathbf{A}}_{\text{r}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},m},\theta^{\text{t}}_{\text{IR},m})$
$\displaystyle\rightarrow{\mathbf{A}}_{\text{t}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}^{\mathrm{H}}{\mathbf{G}}_{\text{IR}}[{\mathbf{I}}_{N_{\text{path}}^{\text{IR}}}]_{:,m+1}=|\alpha_{\text{IR},m}|^{2}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},m},\theta^{\text{t}}_{\text{IR},m}),$
(19)

as $N_{\text{r}},M\rightarrow\infty$. Since Lemma 1 states that
$\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})$
when
${\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})$
is the eigenvector of
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}$ associated
with
$\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})$,
we can prove Proposition 1 by showing that $|\alpha_{\text{IR},0}|^{2}$
asymptotically becomes the maximum eigenvalue of
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}$ in the limit
of large $N_{\text{r}}$ and $M$. We first note that, by the inequality
$\text{rank}({\mathbf{H}}^{\mathrm{H}}_{\text{IR}}{\mathbf{H}}_{\text{IR}})\leq\text{min}(M,N_{\text{path}}^{\text{IR}})$,
${\mathbf{H}}^{\mathrm{H}}_{\text{IR}}{\mathbf{H}}_{\text{IR}}$ can have at
most $N_{\text{path}}^{\text{IR}}$ nonzero eigenvalues, counting
multiplicities, for sufficiently large $M$ such that $M\geq
N_{\text{path}}^{\text{IR}}$. In fact, (19) implies that the number of nonzero
eigenvalues of ${\mathbf{H}}^{\mathrm{H}}_{\text{IR}}{\mathbf{H}}_{\text{IR}}$
approaches $N_{\text{path}}^{\text{IR}}$ as $N_{\text{r}},M\rightarrow\infty$.
Since $|\alpha_{\text{IR},0}|^{2}\geq|\alpha_{\text{IR},m}|^{2}$ for each $m$,
we have

$\displaystyle\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})\rightarrow|\alpha_{\text{IR},0}|^{2},\text{
as }N_{\text{r}},M\rightarrow\infty.$ (20)

This completes the proof of Proposition 1.  $\square$

To examine how setting ${\boldsymbol{\Phi}}$ as described in Proposition 1
modifies the value of
$\lVert{\mathbf{q}}_{p}\rVert^{2}_{2},p\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}\setminus\\{j\\},$
we first express ${\mathbf{q}}_{p}$ as

$\displaystyle{\mathbf{q}}_{p}$
$\displaystyle={\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},p},\theta^{\text{r}}_{\text{TI},p})={\mathbf{A}}_{\text{r}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},p},\theta^{\text{r}}_{\text{TI},p})){\mathbf{v}}.$
(21)

By direct computation, we can express the $(m+1)$-th element of
$({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},p},\theta^{\text{r}}_{\text{TI},p})){\mathbf{v}}$
as

$\displaystyle[({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},p},\theta^{\text{r}}_{\text{TI},p})){\mathbf{v}}]_{m+1}={\mathbf{a}}(f_{m,p},g_{m,p})^{\mathrm{H}}{\mathbf{a}}(f_{0,j},g_{0,j}),$
(22)

where
$m\in\\{0,\dots,N_{\text{path}}^{\text{IR}}-1\\},f_{m,p}=\sin(\phi^{\text{t}}_{\text{IR},m})\sin(\theta^{\text{t}}_{\text{IR},m})-\sin(\phi^{\text{r}}_{\text{TI},p})\sin(\theta^{\text{r}}_{\text{TI},p})$,
and
$g_{m,p}=\cos(\theta^{\text{t}}_{\text{IR},m})-\cos(\theta^{\text{r}}_{\text{TI},p})$.
The vector ${\mathbf{a}}(f,g)\in\mathbb{C}^{M\times 1}$ is expressed as

$\displaystyle{\mathbf{a}}(f,g)=\frac{1}{\sqrt{M}}\Big{[}1,\dots,e^{j\frac{2\pi
d}{\lambda}(m_{h}f+m_{v}g)},\dots,e^{j\frac{2\pi
d}{\lambda}((M^{h}-1)f+(M^{v}-1)g)}\Big{]}^{\mathrm{T}}.$ (23)

Here, $0\leq m_{h}<M^{h}$ and $0\leq m_{v}<M^{v}$ each denote the horizontal
and vertical indices for the IRS elements, where $M=M^{h}M^{v}$. Since the
vector ${\mathbf{a}}(f,g)$ has the same structure as an $M\times 1$ UPA array
response vector, it follows that each element of
$({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},p},\theta^{\text{r}}_{\text{TI},p})){\mathbf{v}}$
tends to zero as $M\rightarrow~{}\infty$. We can thus conclude from (21) that

$\displaystyle\lVert{\mathbf{q}}_{p}\rVert^{2}_{2}\rightarrow 0,\text{ as
}M\rightarrow\infty,$ (24)

for any $p\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}\setminus\\{j\\}$.

We now explain in the following proposition how the eigenvalues of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}_{\text{tot}}^{\mathrm{H}}$
asymptotically behave when ${\boldsymbol{\Phi}}$ is set as described in
Proposition 1.

###### Proposition 2:

Let $j^{\star}\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$ and
$s\in\\{0,\dots,N_{\text{path}}^{\text{TR}}-1\\}$. When
${\boldsymbol{\Phi}}=\text{diag}({\mathbf{v}})$,
${\mathbf{v}}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j^{\star}},\theta^{\text{r}}_{\text{TI},{j^{\star}}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$,
it holds that

$\displaystyle{\mathbf{H}}_{\text{tot}}{\mathbf{H}}_{\text{tot}}^{\mathrm{H}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})\rightarrow|\alpha_{\text{TI},j^{\star}}|^{2}|\alpha_{\text{IR},0}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}),$
(25)
$\displaystyle{\mathbf{H}}_{\text{tot}}{\mathbf{H}}_{\text{tot}}^{\mathrm{H}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s})\rightarrow|\alpha_{\text{TR},s}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s}),$
(26)

as $N_{\text{t}},N_{\text{r}},M\rightarrow\infty$.

Proof: Set
${\mathbf{v}}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j^{\star}},\theta^{\text{r}}_{\text{TI},{j^{\star}}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$,
$j^{\star}\in\\{0,\ldots,N^{\text{TI}}_{\text{path}}-1\\}$. According to (12),
(18), and the asymptotic orthogonality of UPA array response vectors, we can
express ${\mathbf{q}}_{j^{\star}}$ as

$\displaystyle{\mathbf{q}}_{j^{\star}}$
$\displaystyle={\mathbf{A}}_{\text{r}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})\rightarrow\alpha_{\text{IR},0}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}),\text{
as }M\rightarrow\infty.$ (27)

Combining (24) and (27) with (15), we have

$\displaystyle{\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}\rightarrow\sum_{s=0}^{N_{\text{path}}^{\text{TR}}-1}|\alpha_{\text{TR},s}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s}){\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s})^{\mathrm{H}}$
$\displaystyle\hskip
120.0pt+|\alpha_{\text{TI},j^{\star}}|^{2}|\alpha_{\text{IR},0}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}){\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})^{\mathrm{H}},$
(28)

as $N_{\text{t}},M\rightarrow\infty$. Also, it holds for each
$s\in\\{0,\ldots,N^{\text{TR}}_{\text{path}}-1\\}$ that

$\displaystyle{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s})^{\mathrm{H}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})\rightarrow
0,\text{ as }N_{\text{r}}\rightarrow\infty,$ (29)

since the event
$T_{s}=\big{\\{}\phi^{\text{r}}_{\text{TR},s}\neq\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{TR},s}\neq\theta^{\text{r}}_{\text{IR},0}\big{\\}}$
occurs with probability one. It then follows from (28) and (29) that

$\displaystyle{\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})\rightarrow|\alpha_{\text{TI},j^{\star}}|^{2}|\alpha_{\text{IR},0}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}),\text{
as }N_{\text{t}},N_{\text{r}},M\rightarrow\infty,$ (30)

which finishes the proof of (25). The proof of (26) can be done in a similar
manner.  $\square$

According to Proposition 2,
$|\alpha_{\text{TI},j^{\star}}|^{2}|\alpha_{\text{IR},0}|^{2}$ and
$\big{\\{}|\alpha_{\text{TR},s}|^{2}\big{\\}}_{s=0}^{N_{\text{path}}^{\text{TR}}-1}$
are the asymptotic eigenvalues of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$ when
${\mathbf{v}}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j^{\star}},\theta^{\text{r}}_{\text{TI},{j^{\star}}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$.
As $R_{\text{max}}({\mathbf{H}}_{\text{tot}})$ monotonically increases with
the eigenvalues of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$ for given
$P_{\text{TX}}$ and $\sigma_{n}^{2}$, we set the IRS reflection matrix as
${\boldsymbol{\Phi}}^{\star}=\text{diag}({\mathbf{v}}^{\star})$, where

$\displaystyle{\mathbf{v}}^{\star}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},0},\theta^{\text{r}}_{\text{TI},{0}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0}),$
(31)

so that $|\alpha_{\text{TI},0}|^{2}|\alpha_{\text{IR},0}|^{2}$ becomes the
eigenvalue of
${\mathbf{H}}_{\text{tot}}{\mathbf{H}}^{\mathrm{H}}_{\text{tot}}$ in the limit
of large $N_{\text{t}},N_{\text{r}}$ and $M$.

To investigate how the structure of mmWave MIMO channels is adjusted by the
proposed IRS reflection matrix design, we first express the reflected channel
${\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}^{\star}{\mathbf{H}}_{\text{TI}}$
as

$\displaystyle{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}^{\star}{\mathbf{H}}_{\text{TI}}$
$\displaystyle={\mathbf{A}}_{\text{r}}^{\text{IR}}{\mathbf{G}}_{\text{IR}}({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}{\boldsymbol{\Phi}}^{\star}{\mathbf{A}}_{\text{r}}^{\text{TI}}{\mathbf{G}}_{\text{TI}}({\mathbf{A}}_{\text{t}}^{\text{TI}})^{\mathrm{H}}.$
(32)

Using the expression in (23), we can write the element in the $(m+1)$-th row
and $(j+1)$-th column of
${\mathbf{C}}=({\mathbf{A}}_{\text{t}}^{\text{IR}})^{\mathrm{H}}{\boldsymbol{\Phi}}^{\star}{\mathbf{A}}_{\text{r}}^{\text{TI}}$
as

$\displaystyle[{\mathbf{C}}]_{m+1,j+1}=\begin{cases}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},m},\theta^{\text{t}}_{\text{IR},m})^{\mathrm{H}}{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})\hskip
8.0pt\text{ if }j=0,\\\
{\mathbf{a}}(f_{m,j},g_{m,j})^{\mathrm{H}}{\mathbf{a}}(f_{0,0},g_{0,0})\hskip
31.0pt\text{if }j\neq 0,\end{cases}$ (33)

for each $m\in\\{0,\dots,N_{\text{path}}^{\text{IR}}-1\\}$ and
$j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$. It is evident from (33)
that the asymptotic behavior of $[{\mathbf{C}}]_{m+1,j+1}$ can be
characterized as

$\displaystyle[{\mathbf{C}}]_{m+1,j+1}\rightarrow\delta_{m0}\delta_{j0},\text{
as }M\rightarrow\infty,$ (34)

where $\delta_{xy}$ is the Kronecker delta function that takes the value of 1
if and only if $x=y$. As a result, the adjusted total channel
${\mathbf{H}}_{\text{tot}}^{\star}={\mathbf{H}}_{\text{TR}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}^{\star}{\mathbf{H}}_{\text{TI}}$
satisfies

$\displaystyle{\mathbf{H}}_{\text{tot}}^{\star}\rightarrow\sum_{s=0}^{N_{\text{path}}^{\text{TR}}-1}\alpha_{\text{TR},s}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{TR},s},\theta^{\text{t}}_{\text{TR},s})^{\mathrm{H}}+\alpha_{\text{TI},0}\alpha_{\text{IR},0}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{TI},0},\theta^{\text{t}}_{\text{TI},0})^{\mathrm{H}},$
(35)

as $M\rightarrow\infty$. It can thus be concluded that, by asymptotically
maximizing the eigenvalue associated with the dominant transmit and receive
path pair at the IRS, the proposed IRS design successfully establishes a
strong communication link between the TX and RX. Also, since
$\alpha_{\text{TI},0}$ and $\alpha_{\text{IR},0}$ are independent, we have

$\displaystyle\mathbb{E}\left[|\alpha_{\text{TI},0}|^{2}|\alpha_{\text{IR},0}|^{2}\right]\geq
M^{2}c,$ (36)

where
$c=N_{\text{t}}N_{\text{r}}10^{-0.1(PL(d_{\text{TI}})+PL(d_{\text{IR}}))}/N_{\text{path}}^{\text{TI}}N_{\text{path}}^{\text{IR}}$.
The inequality in (36) shows that, for a fixed value of $c$, the average
magnitude of the asymptotic eigenvalue
$|\alpha_{\text{TI},0}|^{2}|\alpha_{\text{IR},0}|^{2}$ of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$
scales at least quadratically with $M$. This suggests that the proposed design
will offer significant increases in spectral efficiency for IRS-aided mmWave
MIMO systems, where the IRS passive elements are expected to be deployed in
large numbers thanks to their low costs and hardware simplicity [17, 19].

###  III-C Analog Beamformer Design

In this subsection, we propose the analog beamformer design that leverages the
structure of the favorably adjusted total channel
${\mathbf{H}}_{\text{tot}}^{\star}={\mathbf{H}}_{\text{TR}}+{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}^{\star}{\mathbf{H}}_{\text{TI}}$.
Specifically, we construct the analog precoder ${\mathbf{F}}_{\text{RF}}$ and
combiner ${\mathbf{W}}_{\text{RF}}$ so that the effective channel
${\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}_{\text{tot}}^{\star}{\mathbf{F}}_{\text{RF}}$
can asymptotically support the maximum spectral efficiency
$R_{\text{max}}({\mathbf{H}}^{\star}_{\text{tot}})$ achievable with fully-
digital beamforming.

With the IRS reflection matrix designed according to Section III-B, i.e.,
${\boldsymbol{\Phi}}={\boldsymbol{\Phi}}^{\star}$, the effective channel
design problem (P3) reduces to

$\displaystyle\text{(P4)}\hskip
45.0pt\max_{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}}\medspace
R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}})$
(37a) $\displaystyle\hskip 10.0pt\text{subject to }\hskip
21.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (37b) $\displaystyle\hskip
82.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n,$ (37c) $\displaystyle\hskip
82.0pt{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N_{\text{r}}^{\text{RF}}},{\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N_{\text{t}}^{\text{RF}}}.$
(37d)

Removing the constant modulus constraints (37b), (37c) of the problem (P4), we
can formulate the relaxed problem $(\text{P4}^{\prime})$ as

$\displaystyle(\text{P4}^{\prime})\hskip
45.0pt\max_{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}}\medspace
R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}})$
(38a) $\displaystyle\hskip 10.0pt\text{subject to }\hskip
21.0pt{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N_{\text{r}}^{\text{RF}}},{\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N_{\text{t}}^{\text{RF}}}.$
(38b)

With the SVD of ${\mathbf{H}}_{\text{tot}}^{\star}$ expressed as
${\mathbf{H}}^{\star}_{\text{tot}}={\mathbf{U}}^{\star}_{\text{tot}}{\boldsymbol{\Sigma}}^{\star}_{\text{tot}}({\mathbf{V}}^{\star}_{\text{tot}})^{\mathrm{H}}$,
it is possible to explicitly obtain the optimal solution of
$(\text{P4}^{\prime})$, as described in the following proposition.

###### Proposition 3:

Let $\\{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}\\}$ be a feasible
solution to the problem $(\text{P4}^{\prime})$. Then, it holds that

$\displaystyle
R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}})\leq
R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star}),$ (39)

where the equality holds if
${\mathbf{W}}_{\text{RF}}=[{\mathbf{U}}_{\text{tot}}^{\star}]_{:,1:N_{\text{r}}^{\text{RF}}}$
and
${\mathbf{F}}_{\text{RF}}=[{\mathbf{V}}_{\text{tot}}^{\star}]_{:,1:N_{\text{t}}^{\text{RF}}}$.
That is,
$\\{\hat{{\mathbf{W}}}_{\text{RF}}^{\star}=[{\mathbf{U}}_{\text{tot}}^{\star}]_{:,1:N_{\text{r}}^{\text{RF}}},\hat{{\mathbf{F}}}_{\text{RF}}^{\star}=[{\mathbf{V}}_{\text{tot}}^{\star}]_{:,1:N_{\text{t}}^{\text{RF}}}\\}$
is optimal for the problem $(\text{P4}^{\prime})$.

Proof: Suppose there exists a feasible solution
$\\{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}\\}$ of
$(\text{P4}^{\prime})$ such that
$R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}})>R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star})$.
Also, let
$\\{\hat{{\mathbf{W}}}^{\prime}_{\text{BB}},\hat{{\mathbf{F}}}^{\prime}_{\text{BB}}\\}$
denote the optimal solution of $(\text{P2})$, with ${\mathbf{H}}_{\text{eff}}$
set as
${\mathbf{H}}_{\text{eff}}={\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}}$.
Then, the spectral efficiency equal to
$R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}_{\text{RF}})$
can be attained under ${\mathbf{H}}_{\text{tot}}^{\star}$ by using the hybrid
combiner ${\mathbf{W}}_{\text{RF}}\hat{{\mathbf{W}}}^{\prime}_{\text{BB}}$ and
precoder ${\mathbf{F}}_{\text{RF}}\hat{{\mathbf{F}}}^{\prime}_{\text{BB}}$.
This contradicts the definition of
$R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star})$ and the proof of the
inequality in (39) is complete.

We now verify the optimality of
$\\{\hat{{\mathbf{W}}}_{\text{RF}}^{\star},\hat{{\mathbf{F}}}_{\text{RF}}^{\star}\\}$
for the problem $(\text{P4}^{\prime})$ by showing that
$R_{\text{max}}(\hat{{\mathbf{H}}}_{\text{eff}}^{\star})=R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star})$,
where
$\hat{{\mathbf{H}}}_{\text{eff}}^{\star}=(\hat{{\mathbf{W}}}_{\text{RF}}^{\star})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}\hat{{\mathbf{F}}}_{\text{RF}}^{\star}$.
Direct computation reveals that
$\hat{{\mathbf{H}}}_{\text{eff}}^{\star}(\hat{{\mathbf{H}}}_{\text{eff}}^{\star})^{\mathrm{H}}$
is the diagonal matrix that satisfies

$\displaystyle[\hat{{\mathbf{H}}}_{\text{eff}}^{\star}(\hat{{\mathbf{H}}}_{\text{eff}}^{\star})^{\mathrm{H}}]_{n,n}=\begin{cases}|[{\boldsymbol{\Sigma}}^{\star}_{\text{tot}}]_{n,n}|^{2}\hskip
15.0pt&\text{if }1\leq n\leq N_{\text{min}}^{\text{RF}},\\\
0&\text{otherwise,}\end{cases}$ (40)

where
$N_{\text{min}}^{\text{RF}}=\text{min}(N_{\text{r}}^{\text{RF}},N_{\text{t}}^{\text{RF}})$.
Since $N_{\text{s}}\leq N_{\text{min}}^{\text{RF}}$, the $N_{\text{s}}$
largest eigenvalues of
$\hat{{\mathbf{H}}}_{\text{eff}}^{\star}(\hat{{\mathbf{H}}}_{\text{eff}}^{\star})^{\mathrm{H}}$
coincide with those of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$,
which finishes the proof of Proposition 3.  $\square$

To develop the analog beamformer design that provides an asymptotically
optimal solution of (P4), we will now examine the asymptotic property of
$\hat{{\mathbf{W}}}_{\text{RF}}^{\star}$ and
$\hat{{\mathbf{F}}}_{\text{RF}}^{\star}$. Let us first focus on
$\hat{{\mathbf{W}}}_{\text{RF}}^{\star}=[{\mathbf{U}}_{\text{tot}}^{\star}]_{:,1:N_{\text{r}}^{\text{RF}}}$,
whose column vectors are necessarily the eigenvectors of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$.
The result in (28) indicates that the number of nonzero eigenvalues of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$,
counting multiplicities, approaches $N_{\text{path}}^{\text{TR}}+1$ as
$N_{\text{t}},N_{\text{r}},M\rightarrow\infty$. Since Proposition 2 states
that
$\\{{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR}})\\}_{s=0}^{N_{\text{path}}^{\text{TR}}-1}$
and
${\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})$
asymptotically become the $N_{\text{path}}^{\text{TR}}+1$ eigenvectors
associated with the nonzero eigenvalues of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$,
we can conclude that, as $N_{\text{t}},N_{\text{r}},M\rightarrow\infty$, each
of the eigenvectors that correspond to the nonzero eigenvalues of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$
and are in the columns of $\hat{{\mathbf{W}}}_{\text{RF}}^{\star}$ can be
expressed as a scalar multiple of one of the column vectors of
$\hat{{\mathbf{U}}}^{\star}_{\text{tot}}=\begin{bmatrix}{\mathbf{A}}^{\text{TR}}_{\text{r}}&{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})\\\
\end{bmatrix}\in\mathbb{C}^{N_{\text{r}}\times(N_{\text{path}}^{\text{TR}}+1)}$.
By similar argument, we can also show that each eigenvector that is associated
with a nonzero eigenvalue of
$({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}$
and is in one of the columns of
$\hat{{\mathbf{F}}}_{\text{RF}}^{\star}=[{\mathbf{V}}_{\text{tot}}^{\star}]_{:,1:N_{\text{t}}^{\text{RF}}}$
approaches a scalar multiple of one of the column vectors of
$\hat{{\mathbf{V}}}^{\star}_{\text{tot}}=\begin{bmatrix}{\mathbf{A}}^{\text{TR}}_{\text{t}}&{\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{TI},0},\theta^{\text{t}}_{\text{TI},0})\end{bmatrix}\in\mathbb{C}^{N_{\text{t}}\times(N_{\text{path}}^{\text{TR}}+1)}$
in the limit of large $N_{\text{t}},N_{\text{r}}$, and $M$.

Motivated by the above formulation, we now construct the analog precoder
${\mathbf{F}}_{\text{RF}}^{\star}$ and combiner
${\mathbf{W}}_{\text{RF}}^{\star}$ that are highly effective in IRS-aided
mmWave MIMO systems with a large number of antennas and IRS elements. First,
we construct the set
$\\{[{\mathbf{H}}_{\text{tot}}^{\star}{\mathbf{A}}_{\text{t}}]_{:,c_{u}}\\}_{u=1}^{N_{\text{t}}^{\text{RF}}}$
that consists of the $N_{\text{t}}^{\text{RF}}$ column vectors of
${\mathbf{H}}_{\text{tot}}^{\star}{\mathbf{A}}_{\text{t}}$ with the
$N_{\text{t}}^{\text{RF}}$ largest $\ell_{2}$-norm, where
$c_{u}\in\\{1,\dots,N_{\text{path}}^{\text{TR}}+N_{\text{path}}^{\text{TI}}\\}$.
We then set the analog precoder ${\mathbf{F}}_{\text{RF}}^{\star}$ as

$\displaystyle{\mathbf{F}}_{\text{RF}}^{\star}=\left[[{\mathbf{A}}_{\text{t}}]_{:,c_{1}},\dots,[{\mathbf{A}}_{\text{t}}]_{:,c_{N_{\text{t}}^{\text{RF}}}}\right].$
(41)

Similarly, we construct the analog combiner ${\mathbf{W}}_{\text{RF}}^{\star}$
as

$\displaystyle{\mathbf{W}}_{\text{RF}}^{\star}=\left[[{\mathbf{A}}_{\text{r}}]_{:,p_{1}},\dots,[{\mathbf{A}}_{\text{r}}]_{:,p_{N_{\text{r}}^{\text{RF}}}}\right],$
(42)

where
$\\{[({\mathbf{H}}_{\text{tot}}^{\star})^{\mathrm{H}}{\mathbf{A}}_{\text{r}}]_{:,p_{v}}\\}_{v=1}^{N_{\text{r}}^{\text{RF}}}$
is the set of $N_{\text{r}}^{\text{RF}}$ column vectors of
$({\mathbf{H}}_{\text{tot}}^{\star})^{\mathrm{H}}{\mathbf{A}}_{\text{r}}$ with
the $N_{\text{r}}^{\text{RF}}$ largest $\ell_{2}$-norm, and
$p_{v}\in\\{1,\dots,N_{\text{path}}^{\text{TR}}+N_{\text{path}}^{\text{IR}}\\}$.
According to Proposition 3 and the asymptotic property of
$\hat{{\mathbf{W}}}_{\text{RF}}^{\star}$ and
$\hat{{\mathbf{F}}}_{\text{RF}}^{\star}$ discussed in the preceding paragraph,
we see that the effective channel
${\mathbf{H}}^{\star}_{\text{eff}}=({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}^{\star}_{\text{RF}}$
satisfies

$\displaystyle R_{\text{max}}({\mathbf{H}}_{\text{eff}}^{\star})\rightarrow
R_{\text{max}}(\hat{{\mathbf{H}}}_{\text{eff}}^{\star})=R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star}),$
(43)

as $N_{\text{t}},N_{\text{r}},M\rightarrow\infty$. Therefore, even with
limited number of RF chains, the proposed analog beamformer design
successfully constructs the effective channel
${\mathbf{H}}^{\star}_{\text{eff}}$ whose maximum supportable spectral
efficiency $R_{\text{max}}({\mathbf{H}}_{\text{eff}}^{\star})$ is very close
to $R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star})$ in the systems of
interest, thereby harvesting the substantial spectral efficiency gain provided
by the IRS reflection matrix design in Section III-B.

###  III-D Baseband Beamformer Design

In the previous two subsections, we described how to design the IRS reflection
matrix and analog beamformer for the channel matrices
${\mathbf{H}}_{\text{TR}},{\mathbf{H}}_{\text{TI}}$ and
${\mathbf{H}}_{\text{IR}}$. In this subsection, we propose the design of the
baseband combiner and precoder that are nearly optimal for the effective
channel
${\mathbf{H}}^{\star}_{\text{eff}}=({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}{\mathbf{F}}^{\star}_{\text{RF}}$.
As explained in Section III-A, when
$({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{W}}^{\star}_{\text{RF}}={\mathbf{I}}_{N^{\text{RF}}_{\text{r}}}$
and
$({\mathbf{F}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{F}}^{\star}_{\text{RF}}={\mathbf{I}}_{N^{\text{RF}}_{\text{t}}}$
hold, the baseband combiner $\hat{{\mathbf{W}}}^{\star}_{\text{BB}}$ and
precoder $\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$ that maximize the spectral
efficiency in (6) for given
${\mathbf{H}}_{\text{eff}}={\mathbf{H}}_{\text{eff}}^{\star}$ are

$\displaystyle\hskip
20.0pt\hat{{\mathbf{W}}}^{\star}_{\text{BB}}=[{\mathbf{U}}^{\star}_{\text{eff}}]_{:,1:N_{\text{s}}},\hskip
5.0pt\hat{{\mathbf{F}}}^{\star}_{\text{BB}}=[{\mathbf{V}}^{\star}_{\text{eff}}]_{:,1:N_{\text{s}}}({\mathbf{P}}^{\star}_{\text{eff}})^{1/2},$
(44)

where the SVD of ${\mathbf{H}}_{\text{eff}}^{\star}$ is expressed as
${\mathbf{H}}_{\text{eff}}^{\star}={\mathbf{U}}_{\text{eff}}^{\star}{\boldsymbol{\Sigma}}_{\text{eff}}^{\star}({\mathbf{V}}_{\text{eff}}^{\star})^{\mathrm{H}}$.
The power allocation matrix $({\mathbf{P}}^{\star}_{\text{eff}})^{1/2}$ is
given by
$({\mathbf{P}}^{\star}_{\text{eff}})^{1/2}=\text{diag}\left(\big{[}\sqrt{P^{\star}_{1}},...,\sqrt{P^{\star}_{N_{\text{s}}}}\big{]}\right)$,
where
$P^{\star}_{l}=\left(\frac{1}{\eta^{\star}}-\frac{\sigma^{2}_{n}}{|[{\boldsymbol{\Sigma}}^{\star}_{\text{eff}}]_{l,l}|^{2}}\right)^{+},\forall
l\in\\{1,\dots,N_{\text{s}}\\}$, and $\eta^{\star}$ is chosen such that
$\sum_{l=1}^{N_{\text{s}}}P^{\star}_{l}=P_{\text{TX}}$. As both
${\mathbf{W}}^{\star}_{\text{RF}}$ and ${\mathbf{F}}^{\star}_{\text{RF}}$ have
UPA array response vectors in their respective columns,
$({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{W}}^{\star}_{\text{RF}}$
and
$({\mathbf{F}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{F}}^{\star}_{\text{RF}}$
each approach ${\mathbf{I}}_{N^{\text{RF}}_{\text{r}}}$ and
${\mathbf{I}}_{N^{\text{RF}}_{\text{t}}}$ as
$N_{\text{t}},N_{\text{r}}\rightarrow\infty$. Therefore, using
$\hat{{\mathbf{W}}}^{\star}_{\text{BB}}$ and
$\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$ as the baseband combiner and precoder
is nearly optimal in mmWave systems, where large antenna arrays are employed
at the TX and RX. Note that, while
${\mathbf{W}}^{\star}_{\text{RF}}\hat{{\mathbf{W}}}^{\star}_{\text{BB}}$ can
be directly used as the hybrid combiner, there is no guarantee that the hybrid
precoder
${\mathbf{F}}^{\star}_{\text{RF}}\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$
satisfies the transmit power constraint (5c) of the problem (P1), i.e.,
$\lVert{\mathbf{F}}^{\star}_{\text{RF}}\hat{{\mathbf{F}}}^{\star}_{\text{BB}}\rVert_{\mathrm{F}}^{2}$
might be greater than $P_{\text{TX}}$. With this difference in mind, we adopt
${\mathbf{W}}_{\text{BB}}^{\star}=\hat{{\mathbf{W}}}^{\star}_{\text{BB}}$ and
${\mathbf{F}}_{\text{BB}}^{\star}=\gamma^{\star}\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$
as the baseband combiner and precoder, where

$\displaystyle\gamma^{\star}=\frac{\sqrt{P_{\text{TX}}}}{\|{\mathbf{F}}^{\star}_{\text{RF}}\hat{{\mathbf{F}}}^{\star}_{\text{BB}}\|_{\mathrm{F}}}$
(45)

is the normalization factor that ensures the transmit power constraint is met
with equality, i.e.,
$\|{\mathbf{F}}^{\star}_{\text{RF}}{\mathbf{F}}^{\star}_{\text{BB}}\|^{2}_{\mathrm{F}}=P_{\text{TX}}$.
Since the asymptotic orthogonality of UPA array response vectors guarantees
that ${\mathbf{F}}^{\star}_{\text{BB}}$ converges to
$\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$ as $N_{\text{t}}\rightarrow\infty$,
we can conclude from (43) that the proposed hybrid precoder
${\mathbf{F}}^{\star}_{\text{RF}}{\mathbf{F}}^{\star}_{\text{BB}}$ and
combiner ${\mathbf{W}}^{\star}_{\text{RF}}{\mathbf{W}}^{\star}_{\text{BB}}$
achieve spectral efficiency that asymptotically approaches
$R_{\text{max}}({\mathbf{H}}_{\text{eff}}^{\star})$, or equivalently
$R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star})$, as
$N_{\text{t}},N_{\text{r}},M\rightarrow\infty$. Therefore, in typical IRS-
aided mmWave MIMO systems with a large number of antennas and IRS elements,
the proposed hybrid beamformer design can perform close to the optimal fully-
digital beamformer design, providing the systems with considerable benefits in
terms of cost and energy efficiency.

##  IV Extension of Joint Design to MIMO-OFDM Systems

As one of the key features of mmWave communications is the usage of large
bandwidth, it is important to investigate the design of IRS reflection matrix
and hybrid beamformer for frequency-selective mmWave channels. In this
section, we extend the proposed joint design of IRS reflection matrix and
hybrid beamformer in Section III to broadband MIMO-OFDM systems.

###  IV-A System Model and Problem Formulation

Consider an IRS-aided MIMO-OFDM system where the TX performs digital precoding
to the symbol vector ${\mathbf{s}}[k]\in\mathbb{C}^{N_{\text{s}}\times 1}$ in
the frequency domain by using the baseband precoder
${\mathbf{F}}_{\text{BB}}[k]\in\mathbb{C}^{N_{\text{t}}^{\text{RF}}\times
N_{\text{s}}}$ at each subcarrier $k\in\\{0,\dots,K-1\\}$. The precoded symbol
vector is then transformed into the time domain through the $K$-point inverse
fast Fourier transform (FFT) followed by cyclic prefix (CP) addition at each
of the $N_{\text{t}}^{\text{RF}}$ RF chains. Subsequently, the TX applies the
analog precoder ${\mathbf{F}}_{\text{RF}}$ to the transformed vector to
produce the final transmitted signal. We assume that ${\mathbf{s}}[k]$
satisfies
$\mathbb{E}[{\mathbf{s}}[k]{\mathbf{s}}[k]^{\mathrm{H}}]={\mathbf{I}}_{N_{\text{s}}}$,
and that the TX is subject to the per-subcarrier power constraint, i.e.,
$\lVert{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}[k]\rVert_{\mathrm{F}}^{2}\leq
P_{\text{TX}}[k]$. The received signal at the subcarrier $k$ can be expressed
as

$\displaystyle{\mathbf{y}}[k]$
$\displaystyle=({\mathbf{H}}_{\text{TR}}[k]+{\mathbf{H}}_{\text{IR}}[k]{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}[k]){\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}[k]{\mathbf{s}}[k]+{\mathbf{n}}[k]={\mathbf{H}}_{\text{tot}}[k]{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}[k]{\mathbf{s}}[k]+{\mathbf{n}}[k]$
(46)

where ${\mathbf{H}}_{\text{TR}}[k]\in\mathbb{C}^{N_{\text{r}}\times
N_{\text{t}}},{\mathbf{H}}_{\text{TI}}[k]\in\mathbb{C}^{M\times
N_{\text{t}}}$, and
${\mathbf{H}}_{\text{IR}}[k]\in\mathbb{C}^{N_{\text{r}}\times M}$ denote the
frequency-domain channel matrices at the subcarrier $k$ from the TX to RX,
from the TX to IRS, and from the IRS to RX, respectively. The total combined
channel matrix at the subcarrier $k$ from the TX to RX is denoted by
${\mathbf{H}}_{\text{tot}}[k]={\mathbf{H}}_{\text{TR}}[k]+{\mathbf{H}}_{\text{IR}}[k]{\boldsymbol{\Phi}}{\mathbf{H}}_{\text{TI}}[k]$,
and the AWGN vector ${\mathbf{n}}[k]$ has entries that are i.i.d with
${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2}_{n})$. The RX first applies the
analog combiner ${\mathbf{W}}_{\text{RF}}$ to the received signal in the time
domain, and then transforms it into the frequency domain by removing the CP
and performing the $K$-point FFT at each of its $N_{\text{r}}^{\text{RF}}$ RF
chains. Finally, the RX uses the baseband combiner
${\mathbf{W}}_{\text{BB}}[k]\in\mathbb{C}^{N_{\text{r}}^{\text{RF}}\times
N_{\text{s}}}$ to obtain the processed received signal
$\tilde{{\mathbf{y}}}[k]={\mathbf{W}}_{\text{BB}}[k]^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{y}}[k]$
at each subcarrier $k$. For each $i\in\\{\text{TR, TI, IR}\\}$, the channel
matrix ${\mathbf{H}}_{i}[k]$ at the subcarrier $k$ is expressed as [9]

$\displaystyle{\mathbf{H}}_{i}[k]=\sum_{q=0}^{N_{\text{path}}^{i}-1}\alpha_{i,q}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{i,q},\theta^{\text{r}}_{i,q}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{i,q},\theta^{\text{t}}_{i,q})^{\mathrm{H}}e^{-j2\pi
qk/K}.$ (47)

The achievable spectral efficiency over the subcarrier $k$ is given by

$\displaystyle
R[k]=\log_{2}\text{det}({\mathbf{I}}_{N_{\text{s}}}+{\mathbf{R}}_{\bar{{\mathbf{n}}}[k]}^{-1}{\mathbf{W}}_{\text{BB}}[k]^{\mathrm{H}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}[k]{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}[k]{\mathbf{F}}_{\text{BB}}[k]^{\mathrm{H}}{\mathbf{F}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}[k]^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}{\mathbf{W}}_{\text{BB}}[k]),$
(48)

where
${\mathbf{R}}_{\bar{{\mathbf{n}}}[k]}=\sigma^{2}_{n}{\mathbf{W}}_{\text{BB}}[k]^{\mathrm{H}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{W}}_{\text{RF}}{\mathbf{W}}_{\text{BB}}[k]$
is the covariance matrix of
$\bar{{\mathbf{n}}}[k]={\mathbf{W}}_{\text{BB}}[k]^{\mathrm{H}}{\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{n}}[k]$.

The problem of designing the IRS reflection matrix and hybrid beamformer that
maximize the spectral efficiency of MIMO-OFDM systems can be formulated as

$\displaystyle\text{(P5)}\hskip 13.0pt\max_{\begin{subarray}{c}\hskip
12.0pt\\{{\mathbf{W}}_{\text{BB}}[k],{\mathbf{F}}_{\text{BB}}[k]\\}_{k=0}^{K-1},\\\
{\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}},{\mathbf{F}}_{\text{RF}}\end{subarray}}\hskip
2.0pt\sum_{k=0}^{K-1}R[k]$ (49a) $\displaystyle\hskip 4.0pt\text{subject to
}\hskip
7.0pt{\boldsymbol{\Phi}}=\text{diag}([e^{j\theta_{1}},\dots,e^{j\theta_{M}}]),$
(49b) $\displaystyle\hskip
54.0pt\lVert{\mathbf{F}}_{\text{RF}}{\mathbf{F}}_{\text{BB}}[k]\rVert_{\mathrm{F}}^{2}\leq
P_{\text{TX}}[k],$ (49c) $\displaystyle\hskip
54.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (49d) $\displaystyle\hskip
54.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n.$ (49e)

Following the similar steps used to derive the problem (P3) in Section III-A,
we can formulate the effective channel design problem for MIMO-OFDM systems as

$\displaystyle\text{(P6)}\hskip
35.0pt\max_{{\mathbf{W}}_{\text{RF}},{\boldsymbol{\Phi}},{\mathbf{F}}_{\text{RF}}}\medspace\sum_{k=0}^{K-1}R_{\text{max}}({\mathbf{H}}_{\text{eff}}[k])$
(50a) $\displaystyle\hskip 4.0pt\text{subject to }\hskip
11.0pt{\boldsymbol{\Phi}}=\text{diag}([e^{j\theta_{1}},\dots,e^{j\theta_{M}}]),$
(50b) $\displaystyle\hskip
58.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (50c) $\displaystyle\hskip
58.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n,$ (50d) $\displaystyle\hskip
58.0pt{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N_{\text{r}}^{\text{RF}}},{\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N_{\text{t}}^{\text{RF}}},$
(50e)

where $R_{\text{max}}({\mathbf{H}}_{\text{eff}}[k])$ denotes the maximum
achievable spectral efficiency under the effective channel
${\mathbf{H}}_{\text{eff}}[k]={\mathbf{W}}^{\mathrm{H}}_{\text{RF}}{\mathbf{H}}_{\text{tot}}[k]{\mathbf{F}}_{\text{RF}}$
at the subcarrier $k$. Note that, due to the lack of baseband processing
capabilities at the IRS [27, 25], the design of ${\boldsymbol{\Phi}}$ must
take the channel matrices of all the subcarriers into account. Likewise,
${\mathbf{F}}_{\text{RF}}$ and ${\mathbf{W}}_{\text{RF}}$ are set to be
identical for all the subcarriers since both of them are applied after the
inverse FFT operation. These two characteristics of MIMO-OFDM systems make it
impossible to directly apply the joint design proposed in Section III to
frequency-selective mmWave channels.

###  IV-B IRS Reflection Matrix Design for MIMO-OFDM Systems

To tackle the effective channel design problem (P6) for MIMO-OFDM systems, we
adopt the strategy similar to the one described in Section III, where we first
designed the IRS reflection matrix ${\boldsymbol{\Phi}}$, and then constructed
the analog precoder ${\mathbf{F}}_{\text{RF}}$ and combiner
${\mathbf{W}}_{\text{RF}}$ accordingly. We first define the complex gain
matrix ${\mathbf{G}}_{i}[k]$ for the mmWave channel ${\mathbf{H}}_{i}[k]$ in
(47) as

$\displaystyle{\mathbf{G}}_{i}[k]=\text{diag}\left(\big{[}\alpha_{i,0},\dots,\alpha_{i,N_{\text{path}}^{i}-1}\cdot
e^{-j2\pi(N_{\text{path}}^{i}-1)k/K}\big{]}\right),$ (51)

where, without loss of generality, we assume that
$|[{\mathbf{G}}_{i}[k]]_{m,m}|\geq|[{\mathbf{G}}_{i}[k]]_{n,n}|,\forall
m,n\in\\{1,\dots,N_{\text{path}}^{i}\\}$ such that $m<n$. We can then express
the decomposition of ${\mathbf{H}}_{i}[k]$ as
${\mathbf{H}}_{i}[k]={\mathbf{A}}^{i}_{\text{r}}{\mathbf{G}}_{i}[k]({\mathbf{A}}^{i}_{\text{t}})^{\mathrm{H}}$,
which has the identical structure as that of ${\mathbf{H}}_{i}$ in (12).
Therefore, by following the similar steps used to derive (28), we can show
that
${\mathbf{H}}^{\star}_{\text{tot}}[k]={\mathbf{H}}_{\text{TR}}[k]+{\mathbf{H}}_{\text{IR}}[k]{\boldsymbol{\Phi}}^{\star}{\mathbf{H}}_{\text{TI}}[k]$
satisfies

$\displaystyle{\mathbf{H}}^{\star}_{\text{tot}}[k]({\mathbf{H}}^{\star}_{\text{tot}}[k])^{\mathrm{H}}\rightarrow\sum_{s=0}^{N_{\text{path}}^{\text{TR}}-1}|\alpha_{\text{TR},s}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s}){\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TR},s},\theta^{\text{r}}_{\text{TR},s})^{\mathrm{H}}$
$\displaystyle\hskip
150.0pt+|\alpha_{\text{TI},0}|^{2}|\alpha_{\text{IR},0}|^{2}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0}){\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{IR},0},\theta^{\text{r}}_{\text{IR},0})^{\mathrm{H}},$
(52)

as $N_{\text{t}},M\rightarrow\infty$, where
${\boldsymbol{\Phi}}^{\star}=\text{diag}({\mathbf{v}}^{\star})$ is the IRS
reflection matrix proposed in Section III-B, and ${\mathbf{v}}^{\star}$ is
given in (31). Since (28) and (52) indicate that
${\mathbf{H}}^{\star}_{\text{tot}}[k]({\mathbf{H}}^{\star}_{\text{tot}}[k])^{\mathrm{H}}$
and
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$
converge to the same matrix for each $k\in\\{0,\dots,K-1\\}$, the IRS
reflection matrix ${\boldsymbol{\Phi}}^{\star}$ has the same asymptotic effect
on the eigenvalues of
${\mathbf{H}}_{\text{tot}}^{\star}[k]({\mathbf{H}}_{\text{tot}}^{\star}[k])^{\mathrm{H}}$
as it does on those of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$.
As the inequality in (36) guarantees that the asymptotic eigenvalue
$|\alpha_{\text{TI},0}|^{2}|\alpha_{\text{IR},0}|^{2}$ of
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$
and
${\mathbf{H}}_{\text{tot}}^{\star}[k]({\mathbf{H}}_{\text{tot}}^{\star}[k])^{\mathrm{H}}$
increases at least quadratically with the number of IRS elements $M$, it can
be concluded that the proposed IRS reflection matrix design in Section III-B
will provide both narrowband and broadband MIMO systems with considerable
gains in spectral efficiency.

###  IV-C Analog Beamformer Design for MIMO-OFDM Systems

In the previous subsection, we showed that every frequency-domain channel
matrix ${\mathbf{H}}_{i}[k]$ at the subcarrier $k$ shares the same receive
array response matrix ${\mathbf{A}}_{\text{r}}^{i}$ and transmit array
response matrix ${\mathbf{A}}_{\text{t}}^{i}$. In this subsection, we exploit
this particular property of frequency-selective mmWave channels to extend the
proposed analog beamformer design in Section III-C to MIMO-OFDM systems. With
${\boldsymbol{\Phi}}={\boldsymbol{\Phi}}^{\star}$, the effective channel
design problem (P6) is simplified as

$\displaystyle\text{(P7)}\hskip
33.0pt\max_{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}}\medspace\sum_{k=0}^{K-1}R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}[k]{\mathbf{F}}_{\text{RF}})$
(53a) $\displaystyle\hskip 10.0pt\text{subject to }\hskip
10.0pt|[{\mathbf{W}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{r}}},\medspace\forall
m,n,$ (53b) $\displaystyle\hskip
71.0pt|[{\mathbf{F}}_{\text{RF}}]_{m,n}|=1/\sqrt{N_{\text{t}}},\medspace\forall
m,n,$ (53c) $\displaystyle\hskip
71.0pt{\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{W}}_{\text{RF}}={\mathbf{I}}_{N_{\text{r}}^{\text{RF}}},{\mathbf{F}}_{\text{RF}}^{\mathrm{H}}{\mathbf{F}}_{\text{RF}}={\mathbf{I}}_{N_{\text{t}}^{\text{RF}}}.$
(53d)

Now consider
${\mathbf{F}}^{\star}_{\text{RF},k^{*}}\in\mathbb{C}^{N_{\text{t}}\times
N_{\text{t}}^{\text{RF}}}$ and
${\mathbf{W}}^{\star}_{\text{RF},k^{*}}\in\mathbb{C}^{N_{\text{r}}\times
N_{\text{r}}^{\text{RF}}}$, which denote the analog precoder and combiner
obtained by applying the proposed analog beamformer design in Section III-C to
${\mathbf{H}}^{\star}_{\text{tot}}[k^{*}]$ for some
$k^{*}\in\\{0,\dots,K-1\\}$. Then, it holds for each $k\in\\{0,\dots,K-1\\}$
that
$R_{\text{max}}(({\mathbf{W}}^{\star}_{\text{RF},k^{*}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}[k]{\mathbf{F}}^{\star}_{\text{RF},k^{*}})$
asymptotically approaches the maximum achievable spectral efficiency
$R_{\text{max}}({\mathbf{H}}^{\star}_{\text{tot}}[k])$ under
${\mathbf{H}}^{\star}_{\text{tot}}[k]$. To see this, note that, as discussed
in Section IV-B,
${\mathbf{H}}^{\star}_{\text{tot}}[k]({\mathbf{H}}^{\star}_{\text{tot}}[k])^{\mathrm{H}}$
and
${\mathbf{H}}^{\star}_{\text{tot}}({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}$
converge to the same matrix as $N_{\text{t}},M\rightarrow\infty$. Likewise, it
can be shown that
$({\mathbf{H}}^{\star}_{\text{tot}}[k])^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}[k]$
and
$({\mathbf{H}}^{\star}_{\text{tot}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}$
converge to the same $N_{\text{t}}\times N_{\text{t}}$ matrix as
$N_{\text{r}},M\rightarrow\infty$. Furthermore, since Proposition 3 implies
that
$R_{\text{max}}({\mathbf{W}}_{\text{RF}}^{\mathrm{H}}{\mathbf{H}}_{\text{tot}}^{\star}[k]{\mathbf{F}}_{\text{RF}})\leq
R_{\text{max}}({\mathbf{H}}^{\star}_{\text{tot}}[k])$ for each feasible
solution $\\{{\mathbf{W}}_{\text{RF}},{\mathbf{F}}_{\text{RF}}\\}$ of the
problem (P7), we have that
$\\{{\mathbf{W}}_{\text{RF}}={\mathbf{W}}^{\star}_{\text{RF},k^{*}},{\mathbf{F}}_{\text{RF}}={\mathbf{F}}^{\star}_{\text{RF},k^{*}}\\}$
approaches the optimal solution of (P7) as
$N_{\text{t}},N_{\text{r}},M\rightarrow\infty$.

With these formulations in mind, we now explain the analog beamformer design
for frequency-selective mmWave channels. First, we use the analog beamformer
design proposed in Section III-C to create the block matrices
${\mathbf{F}}^{\text{block}}_{\text{RF}}\in\mathbb{C}^{N_{\text{t}}\times
KN_{\text{t}}^{\text{RF}}}$ and
${\mathbf{W}}^{\text{block}}_{\text{RF}}\in\mathbb{C}^{N_{\text{r}}\times
KN_{\text{r}}^{\text{RF}}}$, where

$\displaystyle{\mathbf{F}}^{\text{block}}_{\text{RF}}$
$\displaystyle=\begin{bmatrix}{\mathbf{F}}^{\star}_{\text{RF},0}&{\mathbf{F}}^{\star}_{\text{RF},1}&\dots&{\mathbf{F}}^{\star}_{\text{RF},K-1}\end{bmatrix},{\mathbf{W}}^{\text{block}}_{\text{RF}}$
$\displaystyle=\begin{bmatrix}{\mathbf{W}}^{\star}_{\text{RF},0}&{\mathbf{W}}^{\star}_{\text{RF},1}&\dots&{\mathbf{W}}^{\star}_{\text{RF},K-1}\end{bmatrix}.$
(54)

Then, the proposed analog precoder and combiner are given by

$\displaystyle{\mathbf{F}}_{\text{RF}}^{\star}=\left[[{\mathbf{A}}_{\text{t}}]_{:,e_{1}},\dots,[{\mathbf{A}}_{\text{t}}]_{:,e_{N_{\text{t}}^{\text{RF}}}}\right],$
(55)
$\displaystyle{\mathbf{W}}_{\text{RF}}^{\star}=\left[[{\mathbf{A}}_{\text{r}}]_{:,f_{1}},\dots,[{\mathbf{A}}_{\text{r}}]_{:,f_{N_{\text{r}}^{\text{RF}}}}\right],$
(56)

where
$\\{[{\mathbf{A}}_{\text{t}}]_{:,e_{u}}\\}_{u=1}^{N_{\text{t}}^{\text{RF}}}$
is the set of $N_{\text{t}}^{\text{RF}}$ vectors that appear most frequently
in the columns of ${\mathbf{F}}^{\text{block}}_{\text{RF}}$,
$e_{u}\in\\{1,\dots,N_{\text{path}}^{\text{TR}}+N_{\text{path}}^{\text{TI}}\\}$,
$\\{[{\mathbf{A}}_{\text{r}}]_{:,f_{v}}\\}_{v=1}^{N_{\text{r}}^{\text{RF}}}$
is the set of $N_{\text{r}}^{\text{RF}}$ vectors that appear most frequently
in the columns of ${\mathbf{W}}^{\text{block}}_{\text{RF}}$, and
$f_{v}\in\\{1,\dots,N_{\text{path}}^{\text{TR}}+N_{\text{path}}^{\text{IR}}\\}$.
Analogous to (43), we have

$\displaystyle\sum_{k=0}^{K-1}R_{\text{max}}({\mathbf{H}}_{\text{eff}}^{\star}[k])\rightarrow\sum_{k=0}^{K-1}R_{\text{max}}({\mathbf{H}}_{\text{tot}}^{\star}[k]),$
(57)

as $N_{\text{t}},N_{\text{r}},M\rightarrow\infty$, where
${\mathbf{H}}_{\text{eff}}^{\star}[k]=({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}[k]{\mathbf{F}}^{\star}_{\text{RF}}$.
The result in (57) demonstrates that the proposed analog precoder
${\mathbf{F}}^{\star}_{\text{RF}}$ and combiner
${\mathbf{W}}^{\star}_{\text{RF}}$ are asymptotically optimal for the problem
(P7), and thus are highly effective in practical mmWave MIMO-OFDM systems.

###  IV-D Baseband Beamformer Design for MIMO-OFDM Systems

In contrast to the IRS reflection matrix and analog precoder/combiner, the
baseband precoder and combiner can be designed differently for each
subcarrier. This flexibility greatly simplifies the generalization of the
proposed baseband beamformer design from narrowband to broadband MIMO systems.
That is, at each subcarrier $k\in\\{0,\dots,K-1\\}$, we can directly utilize
the baseband beamformer design in Section III-D to construct the baseband
combiner
${\mathbf{W}}^{\star}_{\text{BB}}[k]\in\mathbb{C}^{N_{\text{r}}^{\text{RF}}\times
N_{\text{s}}}$ and precoder
${\mathbf{F}}^{\star}_{\text{BB}}[k]\in\mathbb{C}^{N_{\text{t}}^{\text{RF}}\times
N_{\text{s}}}$ that asymptotically attain the maximum achievable spectral
efficiency $R_{\text{max}}({\mathbf{H}}_{\text{eff}}^{\star}[k])$ under
${\mathbf{H}}_{\text{eff}}^{\star}[k]=({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{H}}^{\star}_{\text{tot}}[k]{\mathbf{F}}^{\star}_{\text{RF}}$.
Denoting the SVD of ${\mathbf{H}}_{\text{eff}}^{\star}[k]$ as
${\mathbf{H}}_{\text{eff}}^{\star}[k]={\mathbf{U}}_{\text{eff}}^{\star}[k]{\boldsymbol{\Sigma}}_{\text{eff}}^{\star}[k]({\mathbf{V}}_{\text{eff}}^{\star}[k])^{\mathrm{H}}$,
we can express ${\mathbf{W}}^{\star}_{\text{BB}}[k]$ and
${\mathbf{F}}^{\star}_{\text{BB}}[k]$ as

$\displaystyle{\mathbf{W}}^{\star}_{\text{BB}}[k]$
$\displaystyle=[{\mathbf{U}}^{\star}_{\text{eff}}[k]]_{:,1:N_{\text{s}}},$
$\displaystyle{\mathbf{F}}^{\star}_{\text{BB}}[k]$
$\displaystyle=\gamma^{\star}[k][{\mathbf{V}}^{\star}_{\text{eff}}[k]]_{:,1:N_{\text{s}}}({\mathbf{P}}^{\star}_{\text{eff}}[k])^{1/2},$
(58)

where
$({\mathbf{P}}^{\star}_{\text{eff}}[k])^{1/2}=\text{diag}\left(\left[\sqrt{P^{\star}_{1}[k]},...,\sqrt{P^{\star}_{N_{\text{s}}}[k]}\medspace\right]\right)$,
$P^{\star}_{l}[k]=\left(1/\eta^{\star}[k]-\sigma^{2}_{n}/|[{\boldsymbol{\Sigma}}^{\star}_{\text{eff}}[k]]_{l,l}|^{2}\right)^{+},\forall
l\in\\{1,\dots,N_{\text{s}}\\}$, and $\eta^{\star}[k]$ is chosen such that
$\sum_{l=1}^{N_{\text{s}}}P^{\star}_{l}[k]=P_{\text{TX}}[k]$. Similar to
$\gamma^{\star}$ in (45), the normalization factor $\gamma^{\star}[k]$ is
defined as

$\displaystyle\gamma^{\star}[k]=\frac{\sqrt{P_{\text{TX}}[k]}}{\|{\mathbf{F}}^{\star}_{\text{RF}}[{\mathbf{V}}^{\star}_{\text{eff}}[k]]_{:,1:N_{\text{s}}}({\mathbf{P}}^{\star}_{\text{eff}}[k])^{1/2}\|_{\mathrm{F}}},$
(59)

so that the transmit power constraint is met with equality at each subcarrier,
i.e.,
$\|{\mathbf{F}}^{\star}_{\text{RF}}{\mathbf{F}}_{\text{BB}}^{\star}[k]\|_{\mathrm{F}}^{2}=P_{\text{TX}}[k],\medspace\forall
k\in\\{0,\dots,K-1\\}$. Since
$({\mathbf{F}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{F}}^{\star}_{\text{RF}}\rightarrow{\mathbf{I}}_{N^{\text{RF}}_{\text{t}}}$
and
$({\mathbf{W}}^{\star}_{\text{RF}})^{\mathrm{H}}{\mathbf{W}}^{\star}_{\text{RF}}\rightarrow{\mathbf{I}}_{N^{\text{RF}}_{\text{r}}}$
as $N_{\text{t}},N_{\text{r}}\rightarrow\infty$, it follows directly from the
discussions in Section III-D and (57) that the proposed hybrid beamformer
$\\{{\mathbf{W}}^{\star}_{\text{RF}}{\mathbf{W}}^{\star}_{\text{BB}}[k],{\mathbf{F}}^{\star}_{\text{RF}}{\mathbf{F}}_{\text{BB}}^{\star}[k]\\}_{k=0}^{K-1}$
achieves spectral efficiency that approaches arbitrarily close to
$\sum_{k=0}^{K-1}R_{\text{max}}({\mathbf{H}}^{\star}_{\text{tot}}[k])$ as
$N_{\text{t}},N_{\text{r}},M\rightarrow\infty$. Given that IRS-aided mmWave
MIMO-OFDM systems typically employ a large number of antennas and IRS
elements, it can be concluded that, even with a significantly reduced number
of RF chains, the proposed hybrid beamformer can attain performance close to
that of fully-digital beamformer, as will be demonstrated in Section VI.

##  V Complexity Analysis of Proposed Joint Designs

In this section, we analyze the computational complexities of the proposed
joint designs of IRS reflection matrix and hybrid beamformer for narrowband
and broadband mmWave MIMO systems. A summary of the proposed designs are
provided in Algorithm 1 and Algorithm 2. Since the systems of interest
typically employ a large number of antennas and IRS elements, we assume
throughout the analysis that $N_{\text{t}}$, $N_{\text{r}}$, and $M$ are much
greater than $N^{\text{RF}}_{\text{t}},N^{\text{RF}}_{\text{r}}$ and
$N_{\text{path}}^{i},i\in\\{\text{TR, TI, IR}\\}$. Under this assumption, the
complexities of the proposed IRS design and hybrid beamformer design for
narrowband MIMO systems are given by $\mathcal{O}(N_{\text{t}}N_{\text{r}}M)$
and
$\mathcal{O}(N_{\text{t}}N_{\text{r}}+N_{\text{t}}^{\text{RF}}N_{\text{r}}^{\text{RF}}N_{\text{min}}^{\text{RF}})$,
respectively. Therefore, the proposed joint design for narrowband systems has
the complexity of $\mathcal{O}(N_{\text{t}}N_{\text{r}}M)$. Similarly, the
complexity of the proposed joint design for MIMO-OFDM systems is calculated to
be $\mathcal{O}(KN_{\text{t}}N_{\text{r}}M)$.

Table I shows the comparison of the computational complexities of the proposed
joint designs, MO-based design [26], and geometric mean decomposition
(GMD)-based design [27]. By smartly leveraging the structures of frequency-
flat and frequency-selective mmWave channels, the proposed designs achieve the
lowest complexity for both narrowband and broadband MIMO-OFDM systems.

Algorithm 1 Design of IRS Reflection Matrix and Hybrid Beamformer for
Narrowband MIMO Systems

1:Compute the IRS reflection matrix
${\boldsymbol{\Phi}}^{\star}=\text{diag}({\mathbf{v}}^{\star})$, where
${\mathbf{v}}^{\star}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},0},\theta^{\text{r}}_{\text{TI},{0}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$.

2:Construct the analog precoder ${\mathbf{F}}_{\text{RF}}^{\star}$ and
combiner ${\mathbf{W}}_{\text{RF}}^{\star}$ according to (41) and (42).

3:Obtain the baseband combiner
${\mathbf{W}}_{\text{BB}}^{\star}=\hat{{\mathbf{W}}}^{\star}_{\text{BB}}$ and
precoder
${\mathbf{F}}_{\text{BB}}^{\star}=\gamma^{\star}\hat{{\mathbf{F}}}^{\star}_{\text{BB}}$
from (44) and (45).

Algorithm 2 Design of IRS Reflection Matrix and Hybrid Beamformer for
Broadband MIMO-OFDM Systems

1:Set the IRS reflection matrix as
${\boldsymbol{\Phi}}^{\star}=\text{diag}({\mathbf{v}}^{\star})$, where
${\mathbf{v}}^{\star}=M\text{diag}({\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},0},\theta^{\text{r}}_{\text{TI},{0}})^{\mathrm{H}}){\mathbf{a}}_{\text{t}}(\phi^{\text{t}}_{\text{IR},0},\theta^{\text{t}}_{\text{IR},0})$.

2:Construct the analog precoder ${\mathbf{F}}_{\text{RF}}^{\star}$ and
combiner ${\mathbf{W}}_{\text{RF}}^{\star}$ according to (55) and (56).

3:Obtain the baseband combiner ${\mathbf{W}}_{\text{BB}}^{\star}[k]$ and
precoder ${\mathbf{F}}_{\text{BB}}^{\star}[k]$ for each subcarrier
$k\in\\{0,\dots,K-1\\}$ from (58) and (59).

TABLE I: Computational Complexities of Different Designs

Systems | Design | Computational Complexity
---|---|---
Narrowband MIMO | Proposed | $\mathcal{O}(N_{\text{t}}N_{\text{r}}M)$
MO-based [26] | $\mathcal{O}(N_{\text{t}}N_{\text{r}}M+N_{\text{t}}N_{\text{r}}\text{min}(N_{\text{t}},N_{\text{r}}))$
GMD-based [27] | $\mathcal{O}(N_{\text{t}}N_{\text{r}}M+N_{\text{t}}N_{\text{r}}\text{min}(N_{\text{t}},N_{\text{r}}))$
MIMO-OFDM | Proposed | $\mathcal{O}(KN_{\text{t}}N_{\text{r}}M)$
MO-based [26] | $\mathcal{O}(KN_{\text{t}}N_{\text{r}}M+KN_{\text{t}}N_{\text{r}}\text{min}(N_{\text{t}},N_{\text{r}}))$
| GMD-based [27] | $\mathcal{O}(KN_{\text{t}}N_{\text{r}}M+KN_{\text{t}}N_{\text{r}}\text{min}(N_{\text{t}},N_{\text{r}}))$

##  VI Simulation Results

In this section, we present simulation results to demonstrate the
effectiveness of the proposed joint designs of IRS reflection matrix and
hybrid beamformer for narrowband and broadband MIMO systems. We assume that
the TX equipped with a UPA of $N_{\text{t}}=8\times 8=64$ antennas
communicates to the RX with a UPA of $N_{\text{r}}=4\times 4=16$ antennas. The
number of RF chains at the TX and RX is set to be
$N_{\text{t}}^{\text{RF}}=N_{\text{r}}^{\text{RF}}=4$. Unless otherwise
specified, we assume that the IRS is equipped with a UPA of $M=16\times
16=256$ passive elements and that ${\mathbf{H}}_{i}$ contains
$N_{\text{path}}^{i}=8$ propagation paths, $\forall i\in\\{\text{TR, TI,
IR}\\}$. The distance
$d_{\text{TR}}\in[d_{\text{TI}}+d_{\text{IR}}-10\medspace\text{m},d_{\text{TI}}+d_{\text{IR}}\medspace\text{m})$
between the TX and RX follows a uniform distribution over its range given
$d_{\text{TI}}$ and $d_{\text{IR}}$, each of which are uniformly distributed
over $[50\medspace\text{m},60\medspace\text{m}]$ and
$[10\medspace\text{m},20\medspace\text{m}]$. The distance-dependent path loss
$PL(d_{i})$ is modeled as

$\displaystyle
PL(d_{i})\medspace[\text{dB}]=\alpha+10\beta\log_{10}(d_{i})+\xi,$ (60)

where $\xi\sim{\mathcal{N}}(0,\sigma^{2})$. According to the experimental data
for 28 GHz channels in [31], the parameters in (60) are set to be
$\alpha=61.4,\beta=2,\sigma=5.8\medspace\text{dB}$ for a line-of-sight (LOS)
path of ${\mathbf{H}}_{i}$, and
$\alpha=72.0,\beta=2.92,\sigma=8.7\medspace\text{dB}$ for its non-line-of-
sight (NLOS) paths. To evaluate the effectiveness of different IRS reflection
matrix designs more accurately, we assume that each path of
${\mathbf{H}}_{\text{TR}}$ is a NLOS path that passes through tinted-glass
walls to experience an additional penetration loss of $40.1\medspace\text{dB}$
[35]. The element spacing, noise power, and number of data streams are each
set to be $d=\lambda/2,\sigma^{2}_{n}=-91\medspace\text{dBm}$, and
$N_{\text{s}}=4$. Lastly, all the simulation results are averaged over 10,000
channel realizations.

###  VI-A Narrowband MIMO Systems

In this subsection, we evaluate the performance of the proposed joint design
of IRS reflection matrix and hybrid beamformer for narrowband MIMO systems.
Fig. 2 shows the spectral efficiency achieved by the proposed design as a
function of the transmit power constraint $P_{\text{TX}}$. We also plotted in
Fig. 2 the performances of when the phase shift of each IRS element is
randomly selected from $[0\medspace\text{rad},2\pi\medspace\text{rad})$ and
when there is no IRS, each of which are labeled as “Random IRS” and “No IRS”.
The results show that, with hybrid beamforming, the proposed design
outperforms MO-based and GMD-based designs that require higher complexities.
In addition, the spectral efficiency of the proposed design with hybrid
beamforming is almost the same as that with fully-digital beamforming,
demonstrating the effectiveness of the hybrid beamformer design presented in
Section III. In contrast, there is a significant performance gap between GMD-
based design with hybrid beamforming and that with fully-digital beamforming.
This is primarily because the design aims to construct hybrid beamformers that
achieve low BERs instead of high achievable rates. It can also be observed
that there are negligible differences among the spectral efficiency achieved
by MO-based design, Random IRS, and No IRS. This indicates that, in order to
obtain performance gains from IRS, it is important to take into account the
presence of the direct channel and design the IRS reflection matrix carefully.

Figure 2: Spectral efficiency achieved by different designs as a function of
the transmit power constraint $P_{\text{TX}}$. Figure 3: Spectral efficiency
achieved by different designs as a function of the number of IRS elements $M$.

To investigate the impact that the number of IRS elements $M$ has on the
proposed design and other benchmarks, the spectral efficiency of different
designs is plotted as a function of $M$ in Fig. 3, where
$P_{\text{TX}}=40\medspace\text{dBm}$ and $M^{h}=M^{v}=\sqrt{M}$. The figure
shows that, as discussed in Section III-B, the spectral efficiency achieved by
the proposed design significantly increases with $M$. In contrast, the
performances of Random IRS and MO-based design do not improve monotonically
with $M$. This indicates that, regardless of how many IRS elements are
utilized, the judicious design of IRS reflection matrix is necessary to
achieve spectral efficiency gains from IRS. Furthermore, the proposed design
with hybrid beamforming performs just as well as that with fully-digital
beamforming at all values of $M$, while GMD-based design requires fully-
digital beamformers to perform comparably to the proposed design. This result
demonstrates that the proposed joint design is highly suitable for mmWave MIMO
systems, where fully-digital beamforming necessitates a large number of RF
chains and thus is prohibitively expensive.

Figure 4: Spectral efficiency achieved by different designs as a function of
the number of paths $N_{\text{path}}$.

Fig. 4 plots the spectral efficiency of different designs against the number
of paths $N_{\text{path}}=N_{\text{path}}^{i},i\in\\{\text{TR, TI, IR}\\}$,
when $P_{\text{TX}}=40\medspace\text{dBm}$. As shown in the figure, the
spectral efficiency of the proposed design and GMD-based design increases as
$N_{\text{path}}$ decreases, while that of Random IRS and MO-based design does
not change significantly according to $N_{\text{path}}$. This implies that the
proposed design is well suited to mmWave systems in which communications are
generally performed in environments with low scattering and small number of
paths.

Figure 5: Absolute value of an element of
$({\mathbf{A}}_{\text{t}}^{\text{TR}})^{\mathrm{H}}{\mathbf{A}}_{\text{t}}^{\text{TI}}$
as a function of the number of transmit antennas $N_{\text{t}}$. Figure 6:
Spectral efficiency achieved by different designs as a function of the angular
range parameter $\nu$.

Since the effectiveness of the proposed joint design depends on how
uncorrelated the array response vectors corresponding to different paths are
to each other, we evaluate in Fig. 5 the absolute value of an element of the
matrix
$({\mathbf{A}}_{\text{t}}^{\text{TR}})^{\mathrm{H}}{\mathbf{A}}_{\text{t}}^{\text{TI}}$
as a function of $N_{\text{t}}$. The figure shows that each element of the
matrix rapidly approaches $0$ as $N_{\text{t}}$ increases, demonstrating that
the result in (14) is highly relevant to mmWave systems where large antenna
arrays are typically deployed. In addition, Fig. 6 shows the spectral
efficiency of different designs when the azimuth and elevation AoAs/AoDs are
each uniformly distributed over
$[0\medspace\text{rad},2\nu\pi\medspace\text{rad})$ and
$[0\medspace\text{rad},\nu\pi\medspace\text{rad})$, where $\nu\in[0,1]$ and
the transmit power constraint is set as $P_{\text{TX}}=40\medspace\text{dBm}$.
The result shows that the spectral efficiency achieved by the proposed joint
design does not change significantly according to $\nu$. Such consistency can
be attributed to the fact that the array response vectors corresponding to
different paths are weakly correlated in the systems of interest, as implied
by Fig. 5. Also, when hybrid beamforming is used, the proposed design
outperforms all the other benchmarks at each value of $\nu$. This demonstrates
that the design can be effectively employed in practical IRS-aided mmWave
systems where a large number of antennas makes hybrid beamforming an
economical and efficient alternative to fully-digital beamforming.

In order to examine how the channel estimation error affects the performance
of the proposed design, we plot in Fig. 7 the spectral efficiency achieved by
different designs when the estimated channel $\hat{{\mathbf{H}}}_{i}$ is used
to construct the IRS reflection matrix and hybrid beamformer. For each
$i\in\\{\text{TR, TI, IR}\\}$, the estimated channel $\hat{{\mathbf{H}}}_{i}$
is given by

$\displaystyle\hat{{\mathbf{H}}}_{i}=\sum_{q=0}^{N_{\text{path}}^{i}-1}\hat{\alpha}_{i,q}{\mathbf{a}}_{\text{r}}(\hat{\phi}^{\text{r}}_{i,q},\hat{\theta}^{\text{r}}_{i,q}){\mathbf{a}}_{\text{t}}(\hat{\phi}^{\text{t}}_{i,q},\hat{\theta}^{\text{t}}_{i,q})^{\mathrm{H}},$
(61)

where the estimated complex gain and azimuth AoA of the $q$-th path are
respectively denoted by $\hat{\alpha}_{i,q}=(1+\delta)\alpha_{i,q}$ and
$\hat{\phi}^{\text{r}}_{i,q}=\phi^{\text{r}}_{i,q}+\delta\text{ [deg]}$. The
estimation noise $\delta$ is assumed to be uniformly distributed over
$[-\rho,\rho]$ [29], and other estimated AoAs/AoDs are defined in a similar
manner. The transmit power constraint is fixed at $P_{\text{TX}}=40\text{
dBm}$. The result in Fig. 7 shows that the proposed design with imperfect CSI
outperforms the benchmarks with perfect CSI at each value of $\rho$. In
addition, the presence of the channel estimation error does not lead to
significant degradation in the performance of the proposed design. It can thus
be concluded that the proposed design is highly robust against the channel
estimation error that inevitably occurs in practical communication systems.

Figure 7: Spectral efficiency achieved by different designs as a function of
the estimation error parameter $\rho$.

(a) $K=16$

(b) $K=64$

Figure 8: Spectral efficiency of different designs in MIMO-OFDM systems with
$K$ subcarriers as a function of the transmit power constraint
$P_{\text{TX}}$.

###  VI-B Broadband MIMO-OFDM Systems

In this subsection, we investigate the performance of the proposed joint
design of IRS reflection matrix and hybrid beamformer for broadband MIMO-OFDM
systems. We assume that equal power is allocated to each subcarrier, i.e.,
$P_{\text{TX}}[k]=P_{\text{TX}}/K,\medspace\forall k\in\\{0,\dots,K-1\\}$. In
Figs. 8 (a) and (b), the spectral efficiency that different designs achieve in
MIMO-OFDM systems with $K=16$ and $K=64$ subcarriers is plotted as a function
of $P_{\text{TX}}$. The figures show that the proposed design outperforms all
the other benchmarks at both values of $K$, regardless of whether fully-
digital or hybrid beamforming architectures are utilized. Also, the proposed
design with hybrid beamforming achieves spectral efficiency very close to that
of the design with fully-digital beamforming. In contrast, a significant
performance gap exists between GMD-based design with fully-digital and hybrid
beamforming, indicating that the proposed hybrid beamformer design is superior
in terms of maximizing spectral efficiency. As demonstrated by the results in
Fig. 8, the proposed joint design, which carefully exploits the angular
sparsity of frequency-selective mmWave channels, is capable of providing
substantial improvements in the spectral and energy efficiency of MIMO-OFDM
systems.

##  VII Conclusions

In this paper, we studied narrowband and broadband IRS-aided mmWave MIMO
systems with hybrid beamforming architectures. We first formulated the problem
of designing the IRS reflection matrix and analog beamformer for narrowband
MIMO systems into the effective channel design problem. By leveraging the
sparse-scattering structure and large dimension of mmWave channels, we
developed the joint design of IRS reflection matrix and hybrid beamformer for
narrowband MIMO systems. We generalized the proposed joint design for
narrowband MIMO systems to broadband MIMO-OFDM systems by carefully exploiting
the sparsity of frequency-selective mmWave channels in the angular domain.
Simulation results demonstrated that the proposed designs can provide the
systems of interest with significant spectral efficiency gains and outperform
the existing state-of-the-art designs while requiring lower computational
complexity. Interesting future research directions include the investigation
of the IRS reflection matrix and beamformer design that minimizes the inter-
user interference in IRS-aided multi-user systems with hybrid beamforming
architectures.

##  Appendix A Proof of Lemma 1

Let $j\in\\{0,\dots,N_{\text{path}}^{\text{TI}}-1\\}$. Define a unitary matrix
${\mathbf{U}}$ and a diagonal matrix ${\boldsymbol{\Lambda}}$ such that
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}={\mathbf{U}}{\boldsymbol{\Lambda}}{\mathbf{U}}^{\mathrm{H}}$,
where $[{\boldsymbol{\Lambda}}]_{m,m}$ is necessarily an eigenvalue of
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}$, $\forall
m\in\\{1,\dots,M\\}$. Since it holds for each $m$ that
$[{\boldsymbol{\Lambda}}]_{m,m}\leq\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})$,
$\lVert{\mathbf{q}}_{j}\rVert_{2}^{2}$ can be bounded as

$\displaystyle\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}$
$\displaystyle={\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})^{\mathrm{H}}{\boldsymbol{\Phi}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})=\sum_{m=1}^{M}[{\boldsymbol{\Lambda}}]_{m,m}|[{\mathbf{U}}^{\mathrm{H}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})]_{m}|^{2}$
$\displaystyle\leq\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})\lVert{\mathbf{U}}^{\mathrm{H}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})\rVert_{2}^{2}=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}),$
(62)

where the last equality follows from
$\lVert{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})\rVert_{2}^{2}=1$.
We now prove the equality condition of the inequality (16) in Lemma 1. If
${\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}){\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})$,
$\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}$ can be written as

$\displaystyle\lVert{\mathbf{q}}_{j}\rVert^{2}_{2}$
$\displaystyle=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}})\lVert{\boldsymbol{\Phi}}{\mathbf{a}}_{\text{r}}(\phi^{\text{r}}_{\text{TI},j},\theta^{\text{r}}_{\text{TI},j})\rVert_{2}^{2}=\lambda_{0}({\mathbf{H}}_{\text{IR}}^{\mathrm{H}}{\mathbf{H}}_{\text{IR}}).$
(63)

This completes the proof of Lemma 1.  $\square$

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