# Analog Beamforming Enabled Multicasting: Finite-Alphabet Inputs and
Statistical CSI

Yanjun Wu, Zhong Xie, Zhuochen Xie, Chongjun Ouyang, and Xuwen Liang
Yanjun Wu, Zhong Xie, and Xuewen Liang are with the Innovation Academy for
Microsatellites of Chinese Academy of Science, Shanghai 201304, China, also
with the Key Lab for Satellite Digitalization Technology, Chinese Academy of
Science, Shanghai 201210, China. Yanjun Wu is also with University of Chinese
Academy of Sciences, Beijing 100049, China. Zhong Xie is also with the
Shanghai Engineering Center for Microsatellites, Shanghai 201210, China.
Xuewen Liang is also with ShanghaiTech University, Shanghai 201210, China.
(email<EMAIL_ADDRESS><EMAIL_ADDRESS>liangxw@shanghaitech.edu.cn).Zhong Xie is with the National Space Science
Center, Chinese Academy of Sciences, Beijing 100190, China (email:
xiezhong@nssc.ac.cn).Chongjun Ouyang is with the School of Electrical and
Electronic Engineering, College of Engineering and Architecture, University
College Dublin, Dublin, D04 V1W8, Ireland (e-mail: chongjun.ouyang@ucd.ie).

###### Abstract

The average multicast rate (AMR) is analyzed in a multicast channel utilizing
analog beamforming with finite-alphabet inputs, considering statistical
channel state information (CSI). New expressions for the AMR are derived for
non-cooperative and cooperative multicasting scenarios. Asymptotic analyses
are conducted in the high signal-to-noise ratio regime to derive the array
gain and diversity order. It is proved that the analog beamformer influences
the AMR through its array gain, leading to the proposal of efficient
beamforming algorithms aimed at maximizing the array gain to enhance the AMR.

###### Index Terms:

Analog beamforming, finite-alphabet inputs, multicast, statistical channel
state information (CSI).

##  I Introduction

Wireless multicasting refers to transmitting common information, such as
regular system updates, financial data, and headline news, to several user
terminals (UTs) simultaneously [1]. Recently, multicasting has been regarded
as a promising technology for content distribution and delivery in the rising
trend of content-centric wireless networks [2].

Employing multiple antennas can enhance multicast transmission rates [1].
However, its fully digital implementation that entails a dedicated radio
frequency (RF) chain for each antenna, faces challenges such as high hardware
costs. Various approaches have been proposed to address this issue, and one
promising solution is analog beamforming, which utilizes phase shifters (PSs)
to connect an RF chain with multiple antennas. Analog beamforming has been
extensively studied in multicasting, with conventional works often relying on
instantaneous channel state information (CSI); see [3, 4] and revelent works
for more details. Yet, obtaining instantaneous CSI in practical systems is
challenging due to its rapid time variation, leading to significant signal
processing overhead and limiting the practical implementation of resulting
beamforming algorithms. In contrast, statistical CSI, such as spatial
correlation, tends to be more stable over time [5]. Therefore, leveraging
statistical CSI for analog beamforming design presents a promising approach in
practical multiple-antenna multicast systems, which has received great
research attention; see [8, 7, 6] and revelent works for more details.

Most existing research on analog beamforming-based multicasting using
statistical CSI focuses on Gaussian-distributed input signals. However,
practical transmission signals are often drawn from finite constellation
alphabets, such as quadrature amplitude modulation (QAM) [5]. These finite
constellations lead to reduced mutual information (MI), especially in high
signal-to-noise ratio (SNR) scenarios [9]. Despite its importance, the
analysis of average multicast rate (AMR) for finite input constellations and
the corresponding statistical CSI-based analog beamforming design has been
less explored.

To address the existing research gap, this letter analyzes the AMR and designs
analog beamforming strategies for multicast channels utilizing finite-alphabet
inputs and statistical CSI. Our contributions are as follows: i) We analyze
the AMR achieved by finite-alphabet inputs in both non-cooperative and
cooperative multicast transmissions; ii) We derive novel expressions for the
AMR in each scenario; iii) We provide high-SNR approximations for the AMR in
both scenarios, highlighting the influence of the analog beamformer on the AMR
through its array gain; iv) Leveraging insights from the asymptotic analyses,
we propose efficient analog beamforming algorithms aimed at maximizing array
gain to boost the AMR.

##  II System Model

###  II-A Transmission Model

In a multicast channel, an access point (AP) disseminates a common message to
$K$ single-antenna UTs. The multicast transmission is enabled by a cost-
efficient analog beamforming architecture using a single RF chain and $N$
phased array antennas. The transmitted signal can be expressed as follows:

${\mathbf{x}}=\sqrt{P}{\mathbf{f}}{{s}}.$ (1)

The terms appearing in (1) are defined as follows:

  * •

${s}\in{\mathbbmss{C}}$ is the encoded data symbol taken from an $M$-ary
constellation alphabet ${\mathcal{X}}=\\{{\mathsf{x}}_{m}\\}_{m=1}^{M}$ with
equal probabilities, which is subject to ${\mathbbmss{E}}\\{s\\}=0$ and
${\mathbbmss{E}}\\{\lvert s\rvert^{2}\\}=1$.

  * •

${\mathbf{f}}=\frac{1}{\sqrt{N}}[{\rm{e}}^{-{\rm{j}}\theta_{1}},\ldots,{\rm{e}}^{-{\rm{j}}\theta_{N}}]^{\mathsf{T}}\in{\mathbbmss{C}}^{N\times
1}$ is the analog beamforming vector, where $\theta_{n}\in(-\pi,\pi]$ denotes
the phase shift introduced by the PS connecting the RF chain with the $n$th
antenna for $n=1,\ldots,N$.

  * •

$P$ is the power budget.

The received signal at UT $k\in{\mathcal{K}}\triangleq\\{1,\ldots,K\\}$ reads

$y_{k}={\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{x}}+n_{k}=\sqrt{P}{\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{f}}{{s}}+n_{k},$
(2)

where ${\mathbf{h}}_{k}\in{\mathbbmss{C}}^{N\times 1}$ is the UT $k$-to-AP
channel vector, and $n_{k}\sim{\mathcal{CN}}(0,\sigma_{k}^{2})$ denotes the
additive white Gaussian noise (AWGN), and $\sigma_{k}^{2}$ is the noise power.
The received SNR at UT $k$ is given by
$\gamma_{k}=\frac{P}{\sigma_{k}^{2}}\lvert{\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{f}}\rvert^{2}$.
For brevity, we assume that $\sigma_{k}=\sigma^{2}$ ($\forall k$) and define
$\overline{\gamma}\triangleq\frac{P}{\sigma^{2}}$ as the average SNR.

###  II-B Channel Model

We consider a correlated Rayleigh fading model, which yields
${\mathbf{h}}_{k}\sim{\mathcal{CN}}({\mathbf{0}},{\mathbf{R}}_{k})$ ($\forall
k$) [10]. Here,
${\mathbf{R}}_{k}={\mathbbmss{E}}\\{{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\mathsf{H}}\\}\succeq{\mathbf{0}}$
represents the correlation matrix for UT $k$. Thus, we obtain
${\mathbf{h}}_{k}\doteq{\mathbf{R}}_{k}^{{1}/{2}}{\mathbf{g}}_{k}$, where
${\mathbf{g}}_{k}\sim{\mathcal{CN}}({\mathbf{0}},{\mathbf{I}})$ and $\doteq$
means equivalent in statistical distribution. Throughout this letter, we
consider a scenario in which each UT $k$ possesses full CSI about its own
effective channel ${\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{f}}$, while the AP
only has statistical CSI about the channels of all $K$ UTs, including the
distributions of $\\{{\mathbf{h}}_{k}\\}_{k=1}^{K}$ and the correlation
matrices $\\{{\mathbf{R}}_{k}\\}_{k=1}^{K}$.

The above set-ups indicate that $\mathbf{f}$ is independent with the
instantaneous realization of $\\{{\mathbf{h}}_{k}\\}_{k=1}^{K}$. Recalling
that ${\mathbf{h}}_{k}\doteq{\mathbf{R}}_{k}^{{1}/{2}}{\mathbf{g}}_{k}$ and
${\mathbf{g}}_{k}\sim{\mathcal{CN}}({\mathbf{0}},{\mathbf{I}})$, it follows
that

${\mathbf{f}}^{\mathsf{H}}{\mathbf{h}}_{k}\doteq{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}^{{1}/{2}}{\mathbf{g}}_{k}\sim{\mathcal{CN}}(0,{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}).$
(3)

The effective channel ${\mathbf{f}}^{\mathsf{H}}{\mathbf{h}}_{k}$ is thus
Gaussian distributed, and the effective channel gain
$\lvert{\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{f}}\rvert^{2}$ is exponentially
distributed. Therefore, the cumulative distribution function (CDF) and
probability density function (PDF) of $\gamma_{k}$ are given by
$F_{k}\left(x\right)=1-{\rm e}^{-\frac{1}{\overline{\gamma}_{k}}x}$ and
$f_{k}\left(x\right)=\frac{1}{\overline{\gamma}_{k}}{\rm
e}^{-\frac{1}{\overline{\gamma}_{k}}x}$, respectively, for $x>0$, where
$\overline{\gamma}_{k}=\overline{\gamma}{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}$
is the average SNR at UT $k$.

In the subsequent sections, two scenarios will be discussed: i) the non-
cooperative scenario where the UTs independently receive the common message,
and ii) the cooperative scenario where the UTs collectively receive the common
message.

##  III Non-Cooperative Scenario

In this section, we explore the non-cooperative scenario.

###  III-A Performance Evaluation

####  III-A1 Multicast Rate

For clarity, we use ${I}_{M}^{\mathcal{X}}\left(\gamma\right)$ to denote the
MI between the input symbol ${\mathsf{X}}\in{\mathcal{X}}$ and the output
signal ${\mathsf{Y}}$ over a Gaussian channel
${\mathsf{Y}}=\sqrt{\gamma}{\mathsf{X}}+{\mathsf{N}}$, where $\gamma$ is the
SNR, ${\mathsf{X}}$ is taken from the equiprobable constellation
$\mathcal{X}$, and ${\mathsf{N}}\sim{\mathcal{CN}}(0,1)$ is the noise.
Particularly, we have [5]

$\begin{split}{I}_{M}^{\mathcal{X}}\left(\gamma\right)&=\log_{2}{{M}}-\frac{1}{M\pi}\sum\nolimits_{m=1}^{M}\int_{\mathbbmss{C}}{{\rm
e}^{-\lvert u-\sqrt{\gamma}{\mathsf{x}}_{m}\rvert^{2}}}\\\
&\times{\log_{2}{\Big{(}\sum\nolimits_{m^{\prime}=1}^{M}{\rm e}^{\lvert
u-\sqrt{\gamma}{\mathsf{x}}_{m}\rvert^{2}-\lvert
u-\sqrt{\gamma}{\mathsf{x}}_{m^{\prime}}\rvert^{2}}\Big{)}}}{\rm
d}u.\end{split}$ (4)

Based on the signal model (2), the MI between the multicast symbol $s$ and the
received signal of UT $k$, $y_{k}$, is given by
${{I}}_{M}^{\mathcal{X}}(\gamma_{k})$, which describes the maximal error-free
transmission rate of UT $k$. In a non-cooperative multicast channel, the
maximum transmission rate of the common message is determined by the MI of the
UT’s channel with the poorest quality [1], which can be written as follows:

$\displaystyle\min\nolimits_{k\in\\{1,\ldots,K\\}}{{I}}_{M}^{\mathcal{X}}(\gamma_{k})\overset{\diamond}{=}{{I}}_{M}^{\mathcal{X}}(\min\nolimits_{k\in\\{1,\ldots,K\\}}\gamma_{k}),$
(5)

where the equality in $\diamond$ is due to the fact of
${I}_{M}^{\mathcal{X}}\left(\cdot\right)$ being monotonically increasing.

####  III-A2 Channel Statistics and AMR

Define
$\overline{\gamma}_{\rm{non}}\triangleq(\sum_{k=1}^{K}\frac{1}{\overline{\gamma}_{k}})^{-1}$.
Then the PDF and CDF of
$\gamma_{\rm{non}}\triangleq\min\nolimits_{k\in\\{1,\ldots,K\\}}\gamma_{k}$
are given by

$\displaystyle f_{\gamma_{\rm{non}}}(x)$ $\displaystyle=\frac{{\rm
d}(1-\prod\nolimits_{k=1}^{K}(1-F_{k}(x)))}{{\rm
d}x}={\overline{\gamma}_{\rm{non}}^{-1}}{\rm
e}^{-\frac{x}{\overline{\gamma}_{\rm{non}}}},$ (6) $\displaystyle
F_{\gamma_{\rm{non}}}(x)$
$\displaystyle=\int_{0}^{x}f_{\gamma_{\rm{non}}}(x){\rm{d}}x=1-{\rm
e}^{-\frac{x}{\overline{\gamma}_{\rm{non}}}},$ (7)

respectively. As outlined in Section II-B, the AP only has access to the
statistical CSI of all UTs for analog beamforming design. Consequently, we
define the performance evaluation metric as the AMR which is expressed as
follows:

$\displaystyle\overline{\mathcal{I}}_{M}^{\mathcal{X}}={{\mathbbmss{E}}_{{\gamma}_{\rm{non}}}\\{{{I}}_{M}^{\mathcal{X}}({\gamma}_{\rm{non}})\\}}=\int_{0}^{\infty}{{I}}_{M}^{\mathcal{X}}(x)f_{\gamma_{\rm{non}}}(x){\rm
d}x$ (8)
$\displaystyle=\int_{0}^{\infty}{{I}}_{M}^{\mathcal{X}}(\overline{\gamma}_{\rm{non}}x){\rm
e}^{-x}{\rm
d}x\overset{(a)}{\approx}\sum\nolimits_{t=1}^{T}w_{t}{{I}}_{M}^{\mathcal{X}}({\overline{\gamma}_{\rm{non}}}v_{t}),$
(9)

where the step “$(a)$” is based on the Gauss–Laguerre quadrature rule [11];
$\left\\{w_{t}\right\\}$ and $\left\\{v_{t}\right\\}$ denote the weight and
abscissas factors of the Gauss–Laguerre integration; $T$ is a complexity-
accuracy tradeoff parameter. Numerical simulation suggest that setting $T=50$
can generally achieve $10^{-14}$ accuracy, and this approximation can be used
for fast performance evaluation.

####  III-A3 Asymptotic Analysis

In the sequel, we set the transmit power $P$ or the average SNR
$\overline{\gamma}$ to infinity to unveil more system design insights. For
clarity, we rewrite (8) as follows:

$\displaystyle\overline{\mathcal{I}}_{M}^{\mathcal{X}}=$
$\displaystyle\left.{I}_{M}^{\mathcal{X}}(x)F_{\gamma_{\rm{non}}}({x})\right|_{0}^{\infty}-\int_{0}^{\infty}F_{\gamma_{\rm{non}}}({x}){\rm
d}{I}_{M}^{\mathcal{X}}(x)$ (10) $\displaystyle=$
$\displaystyle\log_{2}{M}-\int_{0}^{\infty}F_{\gamma_{\rm{non}}}({x}){\rm
d}{I}_{M}^{\mathcal{X}}(x),$ (11)

where the derivation from (10) to (11) is due to the facts that
$F_{\gamma_{\rm{non}}}(0)=0$, $F_{\gamma_{\rm{non}}}(\infty)=1$,
${I}_{M}^{\mathcal{X}}(0)=0$, and ${I}_{M}^{\mathcal{X}}(\infty)=\log_{2}{M}$
[5]. As $\overline{\gamma}\rightarrow\infty$, we have

$\displaystyle\lim\nolimits_{\overline{\gamma}\rightarrow\infty}\overline{\gamma}_{\rm{non}}^{-1}=\lim\nolimits_{\overline{\gamma}\rightarrow\infty}\sum\nolimits_{k=1}^{K}\frac{1}{\overline{\gamma}{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}=0,$
(12)

which together with the fact that $\lim_{x\rightarrow 0}{\rm e}^{-x}\simeq
1-x$, yields

$\displaystyle\lim\nolimits_{\overline{\gamma}\rightarrow\infty}F_{\gamma_{\rm{non}}}(x)\simeq\frac{x}{\overline{\gamma}_{\rm{non}}}=\sum\nolimits_{k=1}^{K}\frac{x}{\overline{\gamma}{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}.$
(13)

Inserting (13) into (11) gives

$\displaystyle\lim_{\overline{\gamma}\rightarrow\infty}\overline{\mathcal{I}}_{M}^{\mathcal{X}}\simeq\log_{2}{M}-\frac{1}{\overline{\gamma}}\sum\nolimits_{k=1}^{K}\frac{{\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}\left(x\right);2]}{{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}},$
(14)

where ${\mathsf{MMSE}}_{M}^{\mathcal{X}}\left(x\right)=\frac{{\rm d}}{{\rm
d}x}{{I}}_{M}^{\mathcal{X}}\left(x\right)$ denotes the minimum mean-square
error (MMSE) function [5, 12, 13], and
${\mathcal{M}}\left[f\left(x\right);t\right]=\int_{0}^{\infty}x^{t-1}f\left(x\right){\rm
d}x$ denotes the Mellin transform of $f\left(x\right)$ [14]. Before further
discussions, we introduce two famous lemmas taken from [14, 13], which will be
utilized to characterize the high-SNR behavior of the AMR.

######  Lemma 1.

If $f\left(x\right)$ is ${\mathcal{O}}\left(x^{m}\right)$ as $x\rightarrow
0^{+}$ and ${\mathcal{O}}\left(x^{n}\right)$ as $x\rightarrow+\infty$, then
$\left|{\mathcal{M}}\left[f\left(x\right);t\right]\right|<\infty$ when
$-m<t<-n$.

######  Lemma 2.

Given a finite discrete constellation alphabet
$\mathcal{X}=\left\\{{\mathsf{x}}_{k}\right\\}_{k=1}^{K}$, it has
${\rm{\mathsf{MMSE}}}_{M}^{\mathcal{X}}(x)\leq{\mathcal{O}}(x^{-\frac{1}{2}}{\rm
e}^{-\frac{x}{8}d_{{\mathcal{X}},{\min}}^{2}})$ where
$d_{{\mathcal{X}},{\min}}\triangleq\min_{m\neq
m^{\prime}}\left|{\mathsf{x}}_{m}-{\mathsf{x}}_{m^{\prime}}\right|$.

With these preliminaries, we characterize the asymptotic behavior of the AMR
in the high-SNR regime as follows.

######  Theorem 1.

In the high-SNR regime, the AMR satisfies

$\displaystyle\lim\nolimits_{\overline{\gamma}\rightarrow\infty}\overline{\mathcal{I}}_{M}^{\mathcal{X}}\simeq\log_{2}{M}-({d_{M,{\rm{non}}}^{\mathcal{X}}\overline{\gamma}})^{-1},$
(15)

where
$d_{M,{\rm{non}}}^{\mathcal{X}}=\frac{1}{{\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}\left(x\right);2]\sum_{k=1}^{K}\frac{1}{{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}}\in\left(0,\infty\right)$
and its value can be calculated by method of numerical integration.

###### Proof:

According to Lemma 2,
$\lim_{x\rightarrow\infty}{\rm{\mathsf{MMSE}}}_{M}^{\mathcal{X}}(x)=o({\rm
e}^{-\frac{x}{8}d_{{\mathcal{X}},{\min}}^{2}})$. Moreover, $\lim_{x\rightarrow
0^{+}}{{\mathsf{MMSE}}}_{M}^{\mathcal{X}}(x)=1$ [12]. In summary,
${\rm{\mathsf{MMSE}}}_{M}^{\mathcal{X}}(x)$ is ${\mathcal{O}}(x^{0})$ as
$x\rightarrow 0^{+}$ and ${\mathcal{O}}(x^{-\infty})$ as
$x\rightarrow+\infty$, which together with Lemma 1, yields
$\lvert{\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}(x);2]\rvert<\infty$.
Since ${\mathsf{MMSE}}_{M}^{\mathcal{X}}\left(x\right)>0$ [12], we have
$d_{M,{\rm{non}}}^{\mathcal{X}}\in\left(0,\infty\right)$ which can be
calculated numerically. The final results follow immediately. ∎

######  Remark 1.

Theorem 1 suggests that the AMR achieved by finite-alphabet input signals
converges to $\log_{2}{M}$ as the SNR increases and its rate of convergence is
determined by $d_{M,{\rm{non}}}^{\mathcal{X}}$.

Remark 1 exposes the performance gap between Gaussian and finite inputs. Based
on [1], the AMR achieved by Gaussian inputs approaches infinity in the high-
SNR regime. However, owing to the characteristics of the input signals, the
AMR achieved by finite inputs converges to some constant as the average SNR
increases.

######  Remark 2.

Motivated by [15], many performance criteria in fading channels can be
characterized by
${\mathsf{A}}\pm(G_{\mathsf{c}}\cdot{\mathsf{SNR}})^{-G_{\mathsf{d}}}$ in the
high-SNR region, where $\mathsf{SNR}$ denotes the average SNR. Here,
$\mathsf{A}$ denotes the performance limitation, $G_{\mathsf{c}}$ is termed
the array gain, and $G_{\mathsf{d}}$ is referred to as the diversity order.
These three parameters collectively describe the high-SNR behavior of various
performance metrics. On this basis, we observe from the results in (15) that
the diversity order and array gain of the AMR are given by $1$ and
$d_{M,{\rm{non}}}^{\mathcal{X}}$, respectively.

######  Remark 3.

The terms appearing in the array gain $d_{M,{\rm{non}}}^{\mathcal{X}}$,
$\sum_{k=1}^{K}\frac{1}{{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}$
is determined by the analog beamformer and correlation matrix, whereas
${\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}(x);2]$ is determined by the
adopted constellation alphabet.

Remark 3 offers valuable guidelines for designing the phase shifts in
${\bm{\phi}}=[{\rm{e}}^{-{\rm{j}}\theta_{1}},\ldots,{\rm{e}}^{-{\rm{j}}\theta_{N}}]^{\mathsf{T}}$
to enhance the AMR. Specifically, considering that ${\bm{\phi}}$ influences
the asymptotic AMR through the array gain, it proves efficient to designate
the array gain as the objective function when optimizing ${\bm{\phi}}$, aiming
to achieve an asymptotically optimal AMR.

###  III-B Analog Beamforming Design

The following section will propose efficient methods to design the phase
shifts in order to optimize the AMR. Based on (9), the corresponding problem
is formulated as follows:

$\displaystyle{\mathcal{P}}_{1}:~{}\max_{{\bm{\phi}}}$
$\displaystyle~{}{\mathsf{AMR}}_{\rm{non}}({\bm{\phi}})\triangleq\int_{0}^{\infty}{{I}}_{M}^{\mathcal{X}}(\overline{\gamma}_{\rm{non}}x){\rm
e}^{-x}{\rm d}x$ (16a) $\displaystyle{\rm{s.t.}}$
$\displaystyle~{}\lvert\phi_{n}\rvert=1,~{}n=1,\ldots,N,$ (16b)

where $\phi_{n}={\rm{e}}^{-{\rm{j}}\theta_{n}}$ for $n=1,\ldots,N$. Note that

$\displaystyle\frac{{\rm d}{{\mathsf{AMR}}_{\rm{non}}({\bm{\phi}})}}{{\rm
d}\overline{\gamma}_{\rm{non}}}=\overline{\gamma}_{\rm{non}}\int_{0}^{\infty}{\mathsf{MMSE}}_{M}^{\mathcal{X}}(\overline{\gamma}_{\rm{non}}x){\rm
e}^{-x}{\rm d}x,$ (17)

which together with the fact of ${\mathsf{MMSE}}_{M}^{\mathcal{X}}(x)>0$,
suggests that ${{\mathsf{AMR}}_{\rm{non}}({\bm{\phi}})}$ increases with
$\overline{\gamma}_{\rm{non}}$. As a result, problem ${\mathcal{P}}_{1}$ can
be equivalently simplified as follows:

$\displaystyle{\mathcal{P}}_{2}:\max_{\bm{\phi}}$
$\displaystyle~{}{\overline{\gamma}_{\rm{non}}}=\Big{(}\sum\nolimits_{k=1}^{K}\frac{1/\overline{\gamma}}{{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}\Big{)}^{-1}\triangleq
f_{1}({\bm{\phi}})$ (18a) $\displaystyle{\rm{s.t.}}$
$\displaystyle~{}\lvert\phi_{n}\rvert=1,~{}n=1,\ldots,N.$ (18b)

The results in (18a) suggest that maximizing the AMR is equivalent to
maximizing the array gain $d_{M,{\rm{non}}}^{\mathcal{X}}$ or minimizing
$\sum\nolimits_{k=1}^{K}\frac{1}{{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}$.
Taking this and the result in Remark 3 together, it can be concluded that
maximizing the array gain can maximize the AMR achieved by finite-alphabet
input signals both explicitly and asymptotically.

As shown in (18b), the modulus of each element in $\bm{\phi}$ is fixed to 1,
which, thus, defines a Riemannian manifold (RM)
${\mathcal{D}}\triangleq\\{{\mathbf{x}}=[x_{1},\cdots,x_{N}]^{\mathsf{T}}:\lvert
x_{n}\rvert=1,n=1,\ldots,N\\}$ [16]. Hence, ${\mathcal{P}}_{2}$ can be solved
by exploiting conjugate gradient descending (CGD) on a manifold. To use this
method, it is necessary to calculate the complex gradient of
$f_{1}({\bm{\phi}})$ w.r.t. $\bm{\phi}$ in the Euclidean space, which is given
by

$\displaystyle{\nabla}_{\bm{\phi}}f_{1}=\Big{(}\sum\nolimits_{k=1}^{K}\frac{N/\overline{\gamma}}{{\bm{\phi}}^{\mathsf{H}}{\mathbf{R}}_{k}{\bm{\phi}}}\Big{)}^{-2}\sum\nolimits_{k=1}^{K}\frac{N/\overline{\gamma}{\mathbf{R}}_{k}{\bm{\phi}}}{({\bm{\phi}}^{\mathsf{H}}{\mathbf{R}}_{k}{\bm{\phi}})^{2}}.$

The gradient of $f_{1}({\bm{\phi}})$ on the manifold $\mathcal{D}$ is given by
[16]

$\displaystyle{{\mathsf{grad}}}_{\bm{\phi}}f_{1}={\nabla}_{\bm{\phi}}f_{1}-\Re\left\\{{\nabla}_{\bm{\phi}}f_{1}\odot{\bm{\phi}}^{*}\right\\}\odot{\bm{\phi}}.$
(19)

The CGD on an RM (RM-CGD) consists of multiple iterations, each of which
contains two steps [16, Chapter 4]. The search direction is selected in the
first step. Denote the search starting-point on $\mathcal{D}$ at the $i$th
iteration as $\bm{\phi}_{i}$. Then, the search direction ${\bm{\eta}}_{i}$ at
this point lies in $\bm{\phi}_{i}$’s tangent space
${T}_{\bm{\phi}_{i}}{\mathcal{D}}$ [16]. By definition,
${T}_{\bm{\phi}_{i}}{\mathcal{D}}$ includes all the vectors that tangentially
pass through ${\bm{\phi}_{i}}$. By [16, eq. (8.26)], we have

$\displaystyle{\bm{\eta}}_{i}={{\mathsf{grad}}}_{{\bm{\phi}}_{i}}f_{1}+\zeta_{i}{\mathcal{T}}_{{\bm{\phi}}_{i}}({\bm{\eta}}_{i-1}),$
(20)

where
${\mathcal{T}}_{{\bm{\phi}}_{i}}({\bm{\eta}}_{i-1})={\bm{\eta}}_{i-1}-\Re\left\\{{\bm{\eta}}_{i-1}\odot{\bm{\phi}}_{i}^{*}\right\\}\odot{{\bm{\phi}}_{i}}$
calculates the projection of ${\bm{\eta}}_{i}$ on
${T}_{{\bm{\phi}}_{i}}{\mathcal{D}}$, and $\zeta_{i}$ denotes the Polak-
Ribière parameter [16, pp. 181]. In the second step, the search step size and
the next search starting-point are determined. The search starting-point for
the $(i+1)$th iteration is given by
${\bm{\phi}}_{i+1}={\mathsf{unt}}({\bm{\phi}}_{i}+\alpha_{i}{\bm{\eta}}_{i})$,
where $\alpha_{i}$ is the Armijo step size [16, pp. 62], and
${\mathsf{unt}}([x_{1},\ldots,x_{N}]^{\mathsf{T}})=[x_{1}/|x_{1}|,\ldots,x_{N}/|x_{N}|]^{\mathsf{T}}$.

The whole procedure for solving ${\mathcal{P}}_{2}$ is summarized in Algorithm
1. As proved in [16], Algorithm 1 is guaranteed to converge to a critical
point of problem $\mathcal{P}_{2}$, where the Riemannian gradient of the
objective function is zero, i.e.,
${{\mathsf{grad}}}_{\bm{\phi}}f_{1}={\mathbf{0}}$ [16]. Besides, the
complexity of Algorithm 1 scales with ${\mathcal{O}}(N^{1.5})$, which is of a
polynomial order.

Algorithm 1 Manifold Optimization-Based Algorithm

1: Initialize ${\bm{\phi}}_{0}$,
${\bm{\eta}}_{0}={{\mathsf{grad}}}_{{\bm{\phi}}_{0}}{f_{1}}$, and $i=0$;

2: repeat

3: Find the next search point
${\bm{\phi}}_{i+1}={\mathsf{unt}}({\bm{\phi}}_{i}+\alpha_{i}{\bm{\eta}}_{i})$;

4: Compute the Riemannian gradient
${{\mathsf{grad}}}_{{\bm{\phi}}_{i+1}}{f_{1}}$;

5: Choose the conjugate direction ${\bm{\eta}}_{i+1}$ by (20);

6: Update $i=i+1$;

7: until Convergence.

##  IV Cooperative Scenario

###  IV-A Performance Evaluation

Having characterized the non-cooperative AMR, our attention now shifts to the
cooperative case. The cooperative system is defined to be the same as the
multicast channel, but with all UTs coordinating to perform joint detection of
the public message. Specifically, we assume that the UTs share their effective
CSI and exchange information received from the AP. In scenarios where UTs can
cooperate, the multicast channel simplifies to a single-user $K\times N$
multiple-antenna system, and the corresponding transmission rate serves as the
upper bound for the non-cooperative AMR.

####  IV-A1 Channel Statistics

When the $K$ UTs work cooperatively, the maximal ratio combining (MRC)
technology can be applied to improve the multicast rate. In this case, the
instantaneous multicast rate is given by
${{I}}_{M}^{\mathcal{X}}(\gamma_{{\rm{MRC}}})$, where
$\gamma_{{\rm{MRC}}}=\sum_{k=1}^{K}\gamma_{k}$. As derived in Section II-B,
the PDF of $\gamma_{k}$ is given by
$f_{k}\left(x\right)=\frac{1}{\overline{\gamma}_{k}}{\rm
e}^{-\frac{1}{\overline{\gamma}_{k}}x}$ for $x>0$. Therefore, the PDF of
$\gamma_{{\rm{MRC}}}$ can be directly derived from [17, eq. (2.9)], which can
be written as follows:

$\displaystyle
f_{\gamma_{{\rm{MRC}}}}(x)=\frac{{\overline{\gamma}}_{\min}^{K}}{\prod_{k=1}^{K}{\overline{\gamma}}_{k}}\sum_{l=0}^{\infty}\frac{\psi_{l}x^{K+l-1}}{{\overline{\gamma}}_{\min}^{K+l}\Gamma(K+l)}{\rm
e}^{-\frac{x}{{\overline{\gamma}}_{\min}}},$ (21)

where $\Gamma\left(z\right)=\int_{0}^{\infty}{\rm e}^{-t}t^{z-1}{\rm d}t$ is
the Gamma function [11, eq. (8.310.1)],
${\overline{\gamma}}_{\min}=\min_{k\in\\{1,\ldots,K\\}}{\overline{\gamma}}_{k}$,
${\psi_{0}}=1$, and $\psi_{l}$ ($l\geq 1$) can be calculated recursively by

$\displaystyle{\psi_{l}}=\sum\nolimits_{i=1}^{l}{\left[{\sum\nolimits_{k=1}^{K}{{{\left({1-{{{{\overline{\gamma}}_{\min}}}}/{{{{\overline{\gamma}}_{k}}}}}\right)}^{i}}}}\right]}\frac{\psi_{l-i}}{{l}}.$
(22)

Following similar steps in obtaining (9), the cooperate AMR, i.e.,
$\overline{\mathcal{I}}_{M}^{\mathcal{X}}={\mathbbmss{E}}\\{{{I}}_{M}^{\mathcal{X}}(\gamma_{{\rm{MRC}}})\\}$,
can be approximated as follows:

$\displaystyle\overline{\mathcal{I}}_{M}^{\mathcal{X}}\approx\frac{{\overline{\gamma}}_{\min}^{K}}{\prod_{k=1}^{K}{\overline{\gamma}}_{k}}\sum_{l=0}^{\infty}\sum_{t=1}^{T}\frac{\psi_{l}w_{t}{{I}}_{M}^{\mathcal{X}}({\overline{\gamma}}_{\min}v_{t})}{v_{t}^{1-K-l}\Gamma(K+l)}.$
(23)

####  IV-A2 Asymptotic Analysis

Let us now investigate the asymptotic AMR in the high-SNR regime. Similar to
the former derivations for the non-cooperative AMR, we reexpress the AMR as
$\overline{\mathcal{I}}_{M}^{\mathcal{X}}=\log_{2}{M}-\int_{0}^{\infty}F_{\gamma_{\rm{MRC}}}({x})\mathsf{MMSE}_{M}^{\mathcal{X}}(x){\rm
d}x$ in order to facilitate the derivation, where
$F_{\gamma_{\rm{MRC}}}(\cdot)$ denotes the CDF of $\gamma_{\rm{MRC}}$. By
definition, we have
$F_{\gamma_{\rm{MRC}}}({x})=\int_{0}^{x}f_{\gamma_{\rm{MRC}}}({x}){\rm{d}}x$,
which yields

$\displaystyle
F_{\gamma_{\rm{MRC}}}({x})=\frac{{\overline{\gamma}}_{\min}^{K}}{\prod_{k=1}^{K}{\overline{\gamma}}_{k}}\sum\nolimits_{l=0}^{\infty}\frac{\psi_{l}\Upsilon(l+K,{x}/{{\overline{\gamma}}_{\min}})}{\Gamma(K+l)},$
(24)

where $\Upsilon\left(s,x\right)=\int_{0}^{x}t^{s-1}{\rm e}^{-t}{\rm d}t$ is
the lower incomplete Gamma function [11, eq. (8.350.1)]. As
$\overline{\gamma}\rightarrow\infty$, we have

$\displaystyle\lim_{\overline{\gamma}\rightarrow\infty}\frac{1}{{{\overline{\gamma}}_{\min}}}=\lim_{\overline{\gamma}\rightarrow\infty}\frac{1}{\min_{k\in\\{1,\ldots,K\\}}\overline{\gamma}{\mathbf{f}}^{\mathsf{H}}{\mathbf{R}}_{k}{\mathbf{f}}}=0.$
(25)

which together with the fact of $\lim_{x\rightarrow
0}\Upsilon\left(s,x\right)\simeq\frac{x^{s}}{s}$ [11, eq. (8.354.1)], yields

$\displaystyle\lim\nolimits_{\overline{\gamma}\rightarrow\infty}F_{\gamma_{\rm{MRC}}}({x})$
$\displaystyle\simeq\frac{{\overline{\gamma}}_{\min}^{K}}{\prod\nolimits_{k=1}^{K}{\overline{\gamma}}_{k}}\frac{\psi_{0}}{{\Gamma\left(K\right)}K}\left(\frac{x}{\overline{\gamma}_{\min}}\right)^{K}$
(26)
$\displaystyle={x^{K}}{{\overline{\gamma}}^{-K}\left(K!\right)^{-1}{\prod\nolimits_{k=1}^{K}{\overline{\gamma}}_{k}^{-1}}}.$
(27)

As a result, the asymptotic AMR satisfies

$\displaystyle\lim\nolimits_{\overline{\gamma}\rightarrow\infty}\overline{\mathcal{I}}_{M}^{\mathcal{X}}\simeq\log_{2}{M}-({d_{M,{\rm{coop}}}^{\mathcal{X}}}{\overline{\gamma}})^{-K},$
(28)

where
$d_{M,{\text{coop}}}^{\mathcal{X}}=({{K!{\prod_{k=1}^{K}{\overline{\gamma}}_{k}}}/{{\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}(t);K+1]}})^{1/K}\in\left(0,\infty\right)$
can be calculated numerically.

######  Remark 4.

Eq. (28) shows that the diversity order and array gain of the cooperative AMR
are $K$ and $d_{M,{\rm{coop}}}^{\mathcal{X}}$, respectively.

######  Remark 5.

Eq. (28) indicates that analog beamforming affects the asymptotic cooperative
AMR through the array gain $d_{M,{\rm{coop}}}^{\mathcal{X}}$. For the terms
within $d_{M,{\rm{coop}}}^{\mathcal{X}}$,
${\prod_{k=1}^{K}{\overline{\gamma}}_{k}}$ is determined by the analog
beamformer, whereas
${{\mathcal{M}}[{\mathsf{MMSE}}_{M}^{\mathcal{X}}(t);K+1]}$ is influenced by
the nature of the input signals.

Comparing (28) to (14), it is evident that the utilization of cooperation can
enhance the diversity order by a factor of $K$. Actually, owning to the
beamforming gain, employing cooperation can also improve the array gain.
However, it should be noted that user cooperation necessitates the receivers
to exchange their instantaneous CSI and received signals for MRC. This leads
to increased feedback overhead compared to the non-cooperative scenario.

###  IV-B Analog Beamforming Design

Now, let us delve into the design of analog beamforming. Mathematically, the
optimization problem for enhancing cooperative AMR can be formulated as
follows:

$\displaystyle{\mathcal{P}}_{3}:\max\nolimits_{\bm{\phi}}$
$\displaystyle~{}{\mathsf{AMR}}_{\rm{coop}}({\bm{\phi}})={\mathbbmss{E}}\\{{{I}}_{M}^{\mathcal{X}}(\gamma_{{\rm{MRC}}})\\}$
(29a) $\displaystyle{\rm{s.t.}}$
$\displaystyle~{}\lvert\phi_{n}\rvert=1,~{}n=1,\ldots,N.$ (29b)

Intuitively, ${\mathcal{P}}_{3}$ can be tackled using the method of RM-CGD.
However, the terms within
${\mathbbmss{E}}\\{{{I}}_{M}^{\mathcal{X}}(\gamma_{{\rm{MRC}}})\\}$, such as
$\overline{\gamma}_{k}$, $\overline{\gamma}_{\min}$, and $\psi_{l}$, are all
functions of $\bm{\phi}$. Additionally, $\psi_{l}$ lacks closed-form
expressions and can only be computed recursively using (22). Consequently,
computing $\nabla_{\bm{\phi}}{\mathsf{AMR}}_{\rm{coop}}$ is challenging,
rendering manifold optimization techniques impractical. Due to the intractable
form of ${\mathsf{AMR}}_{\rm{coop}}(\cdot)$ and the constant-modulus
constraints, traditional optimization strategies struggle with
${\mathcal{P}}_{3}$. As a compromise, we turn to heuristic algorithms, known
for their effectiveness in addressing NP-hard problems. Specifically, we adopt
the genetic algorithm (GA) to find a satisfactory solution to
${\mathcal{P}}_{3}$. Given the widespread use of GA in wireless
communications, we omit the algorithm details here and provide only the pseudo
code in Algorithm 2 for brevity.

Algorithm 2 Genetic Algorithm

1: Initialize $i=0$ and a population of $N_{\rm{pop}}$ individuals
${\mathcal{W}}_{i}=\\{{\bm{\theta}}_{\ell}=[\theta_{\ell,1},\ldots,\theta_{\ell,N}]^{\mathsf{T}}|\theta_{\ell,n}\in\left(-\pi,\pi\right],\ell=1,\ldots,N_{\rm{pop}},n=1,\ldots,N\\}$;

2: repeat

3: Select the optimal individual ${\bm{\theta}}_{i}^{\star}$ that maximizes
the AMR from ${\mathcal{W}}_{i}$;

4: Sample the parents for the $i$th generation from ${\mathcal{W}}_{i}$;

5: Generate the $i$th generation by the crossover$/$mutation oprtaions;

6: Denote the new generation as ${\mathcal{W}}_{i+1}$ and set $i=i+1$;

7: until Convergence.

In Algorithm 2, the AMR achieved by every individual in each generation
requires comparison. However, the values of ${{I}}_{M}^{\mathcal{X}}(\cdot)$
in (29a) or (23) can only be computed using Monte-Carlo simulation or
numerical integration methods [5, 12, 13]. To alleviate the computational
burden, optimizing the array gain $d_{M,{\rm{coop}}}^{\mathcal{X}}$ is a
pragmatic compromise. This is because the analog beamformer affects the
asymptotic AMR through the array gain, which takes a much simpler form than
the AMR itself, as indicated by Remark 5. The corresponding optimization
problem is formulated as follows:

$\displaystyle\max\nolimits_{\bm{\phi}}~{}{f}_{2}(\bm{\phi})\triangleq\sum\nolimits_{k=1}^{K}\log({\bm{\phi}}^{\mathsf{H}}{\mathbf{R}}_{k}{\bm{\phi}}),~{}{\rm{s.t.}}~{}\eqref{P3_Objective1}.$
(30)

This problem can be solved using the RM-CGD method, akin to Algorithm 1. For
brevity, we omit the details here. Notably, the complex gradient of
${f}_{2}(\bm{\phi})$ w.r.t. the vector $\bm{\phi}$ in the Euclidean space can
be expressed as follows:

$\displaystyle{\nabla}_{\bm{\phi}}{f}_{2}=\sum\nolimits_{k=1}^{K}{\mathbf{R}}_{k}{\bm{\phi}}({\bm{\phi}}^{\mathsf{H}}{\mathbf{R}}_{k}{\bm{\phi}})^{-1}.$
(31)

It is crucial to note that ${f}_{2}(\bm{\phi})$ is independent of
${{I}}_{M}^{\mathcal{X}}\left(\cdot\right)$ and, therefore, exhibits a simpler
form than ${\mathsf{AMR}}_{\rm{coop}}({\bm{\phi}})$. Consequently, the RM-CGD
method involves lower complexity than the GA-based method.

##  V Numerical Results

In this section, we present numerical results to validate the accuracy of the
derived analytical results and assess the efficacy of the proposed algorithms.
The long-term statistics of the channels, i.e.,
$\\{{\mathbf{R}}_{k}\\}_{k=1}^{K}$, are generated following the methodologies
outlined in [10]. Unless explicitly stated otherwise, the simulation
parameters are set as follows: $K=4$, $N=5$, $T=50$, and $N_{\rm{pop}}=50$.
All optimization variables are initialized randomly.

(a) $\overline{\gamma}=-20$ dB.

(b) 4-QAM, $N=5$, $\overline{\gamma}=-40$ dB.

Figure 1: The non-cooperative (Figure 1(a)) and cooperative (Figure 1(b)) AMR
versus the number of iterations.

In Figure 1, we examine the convergence performance of the proposed algorithms
for both non-cooperative and cooperative transmissions. Regarding non-
cooperative multicast transmission, the convergence behavior of the proposed
RM-CGD algorithm is depicted in Figure 1(a) for varying modulation orders and
antenna numbers. It is evident that the proposed algorithm achieves
convergence in less than 10 iterations. Moreover, an increase in modulation
order or antenna number enhances the non-cooperative AMR. Shifting our focus
to cooperative transmission, Figure 1(b) illustrates the convergence behavior
of the proposed GA. It is observed that, with an increase in the number of
iterations or population generations, the AMR rises until convergence is
reached. Notably, augmenting $N_{\rm{pop}}$ results in fewer iterations
required for convergence.

(a) Explicit AMR (4-QAM).

(b) Asymptotic AMR (RM-CGD).

Figure 2: The AMR versus the SNR. The analytical results are calculated by (9)
and (23). The asymptotic results are calculated by (14) and (28).

In Figure 2, the AMR is plotted against the average SNR $\overline{\gamma}$.
As depicted in Figure 2(a), the proposed algorithms achieve a higher AMR
compared to analog beamforming with random phase shifts. Furthermore, the
cooperation among UTs significantly enhances the AMR in contrast to the non-
cooperative scenario. However, this performance gain comes at the cost of
increased feedback overhead. Notably, it is observed that the proposed RG-CGD
method achieves nearly the same cooperative AMR as the GA method. As
previously analyzed, the former incurs lower complexity than the latter,
making the RG-CGD method a more favorable choice for practical
implementations.

In Figure 2(a), the AMR of finite-alphabet inputs converges to $\log_{2}{M}$
in the large limit of $\overline{\gamma}$. According to (15) and (28), the
rate of AMR converging to $\log_{2}{M}$ equals the rate of
${\mathcal{I}}_{\rm{con}}=\log_{2}{M}-\overline{\mathcal{I}}_{M}^{\mathcal{X}}$
converging to zero. To show the convergence rate, we plot
${\mathcal{I}}_{\rm{con}}$ versus $\overline{\gamma}$ in Figure 2(b). The
results demonstrate a close match between the derived asymptotic expressions,
i.e., $({d_{M,{\rm{non}}}^{\mathcal{X}}\overline{\gamma}})^{-1}$ and
$\left({d_{M,{\rm{coop}}}^{\mathcal{X}}}{\overline{\gamma}}\right)^{-K}$, and
the numerical results in the high-SNR regime. This alignment affirms the
accuracy of the diversity order derived in the previous section. Additionally,
the results in Figure 2(b) confirm that the utilization of cooperation
enhances the diversity order.

##  VI Conclusion

We analyzed the AMR in a multicast channel employing analog beamforming with
finite-alphabet inputs, considering statistical CSI at the AP. Theoretical
analyses revealed the impact of the analog beamformer on AMR through its array
gain. On this basis, we proposed efficient analog beamforming algorithms that
maximize array gain to enhance the AMR.

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