# Entropic measure of directional emissions in microcavity lasers

Kyu-Won Park  Research Institute of Mathematics, Seoul National University,
Seoul 08826, Korea  Chang-Hyun Ju  Department of Electronic Engineering,
Yeungnam University, Gyeongsan 38541, Korea  Kabgyun Jeong<EMAIL_ADDRESS>Research Institute of Mathematics, Seoul National University, Seoul 08826,
Korea  School of Computational Sciences, Korea Institute for Advanced Study,
Seoul 02455, Korea

###### Abstract

We propose a new notion of the directional emission in microcavity lasers.
First, Shannon entropy of the far-field profiles in the polar coordinate can
quantify the degree of unidirectionality of the emission, while previous
notions about the unidirectionality can not efficiently measure in the robust
range against a variation of the deformation parameter. Second, a divergence
angle of the directional emission is defined phenomenologically in terms of
full width at half maximum, and it is only easily applicable to a simple peak
structure. However, Shannon entropy of semi-marginal probability of the far-
field profiles in the cartesian coordinate can present equivalent results, and
moreover it is applicable to even the cases with a complicated peak structure
of the emission.

Microcavity lasers have recently garnered considerable attention owing to
their applicability as optimal candidate models for studying wave chaos
cao2015dielectric ; liu2018chaotic  and non-Hermitian quantum systems
cao2015dielectric ; takata2022imaginary ; park2020maximal  as well as their
optoelectronic applications and photonics vahala2003optical  such as speckle-
free full-field imaging redding2015optical , broadband coupling
jiang2017optical . In addition, recent studies reveal many interesting physics
in phase space related to the photon transport chen2019combining ;
qian2021combining . In particular, the key to such applications is that
microcavity lasers have high quality ($Q$) factors and directional emission
simultaneously wiersig2008combining ; yan2009directional . High-$Q$ modes
guarantee a low threshold lasing; in addition, they can be adopted for bio-
molecule detections armani2007label ; vollmer2008whispering  and nano
particles zhu2010chip ; chen2017exceptional . However they are limited by
isotopic emissions and low output power. Hence, directional emission is also
required to support a high output power, and also facilitate easy coupling to
a waveguide for optoelectronic circuits. Therefore, to date, several
microcavity lasers have been studied to achieve these properties
simultaneously wang2009deformed ; yang2016mode ; song2011highly ;
peter2018control ; wang2018high ; yang2021design .

In these studies, the discussions about a high $Q$-factor have been addressed
quantitatively and systemically, because the $Q$-factor can be well defined by
$Q=\frac{f_{r}}{\Delta f}$ where $f_{r}$ is a resonance frequency and $\Delta
f$ is a resonance width, or equivalently, $Q=\frac{f_{r}}{2|f_{i}|}$ in the
context of a complex eigenfrequency $f_{c}=f_{r}+jf_{i}$. However, regarding
directional emission, we consider that its definition is relatively subtle and
it remains not well established up to date. In this paper, thus, we introduce
new measures of the unidirectional emission by exploiting the notion of
Shannon entropy, and this suggestion holds that an entropic measure of the
unidirectional emission is more accurate and efficient than former
alternatives related to the emission window ryu2011designing ;
ryu2019optimization . It is possible to validate these results by
demonstrating that, former alternatives fail to detect the degree of the
directional emission, our new methods can.

To further engage this discussion, we consider a prevalent and basic limaçon-
shaped microcavity laser as a candidate for the directional emission with a
high $Q$-factor wiersig2008combining ; yan2009directional ; shinohara2009ray ;
song2009chaotic . In this reason, it has been extensively studied to date. The
geometrical boundary of the limaçon-shaped microcavity is defined as follows:

$\displaystyle R(\theta)=R_{0}(1+\chi\cos\theta),$ (1)

where $\theta$ is the angle in the polar coordinate, $\chi$ is the deformation
parameter, and $R_{0}(=1)$ is the radius of circles at $\chi=0$. The limaçon-
shaped microcavity laser with an effective index of refraction $n=3.3$ (for
InGaAsP semiconductor microcavity) are treated in ray simulation. Some of the
representative far-field profiles (FFPs) in the limaçon-shaped cavity is shown
in Fig. 1. The FFPs are obtained from transmitted rays by using Fresnel
equations for transverse magnetic (TM) modes. The figures (a), (b), and (c) in
Fig. 1 are plotted in the cartesian coordinate within the range of $|x|\leq 5$
and $|y|\leq 5$ at each deformation $\chi=0.43$, $\chi=0.454$, and
$\chi=0.478$, respectively. The subfigures (d), (e), and (f) in Fig. 1 are
FFPs plotted in the polar coordinate, corresponding to FFPs in the cartesian
coordinate, respectively. We call the angles in the range of
$|\theta|\leq\frac{\pi}{4}$ as ‘emission window’, which was first introduced
by Refs. ryu2011designing ; ryu2019optimization  for the definition of
unidirectionality.

Figure 1: Far-field profiles in a limaçon-shaped cavity. The figures (a), (b),
and (c) are the far-field profiles in the cartesian coordinate within
$x\in[-5,5]$ and $y\in[-5,5]$ at each deformation $\chi=0.43$, $\chi=0.454$,
and $\chi=0.478$, respectively. The figures (d), (e), and (f) are also far-
field profiles plotted in the polar coordinate corresponding to the far-field
profiles in the cartesian coordinate. The angles in the range of
$|\theta|\leq\frac{\pi}{4}$ is the emission window. The inset in (c) shows the
boundary shape of cavity at each $\chi=0.43$ (black lower line), 0.454 (red
center line), and 0.478 (blue upper line), respectively.

It was reported that the deformation parameter $\chi=0.43$ is the optimal
value for unidirectional emission wiersig2008combining . Moreover, they have
concluded that the results are robust against any variation of the deformation
parameter in the range of $0.43\leq\chi\leq 0.49$ wiersig2008combining .
Consistent with their results, the overall profiles of all figures in Fig. 1
appear similar to each other. However, if we observe the morphologies of FFPs
for Fig. 1(d), (e), and (f) simultaneously, we can suggest that FFPs at
$\chi=0.454$ exhibit a lager unidirectionality than the others.

To validate this suggestion, we address three types of measure for the
unidirectionality. The blue squares ($U_{W}$) and red circles ($U_{C}$) in
Fig. 2(a) are measures of the unidirectionality associated with the emission
windows. More precisely, the unidirectionality $U_{C}$ marked by red circles
is defined as follow shim2013adiabatic ; song2010directional :

$\displaystyle U_{C}=\frac{\int^{2\pi}_{0}I(\theta)\cos\theta
d\theta}{\int^{2\pi}_{0}I(\theta)d\theta}.$ (2)

Here, $I(\theta)$ denotes the intensity of angular distribution of FFPs
jiang2016whispering . The $\cos\theta$ as a window function determines the
extent to which the emission directionality deviates from unidirectionality.
Actually, the positive and negative $U_{C}$ represent tendencies toward a
forward and backward emission, respectively, and $U_{C}=0$ corresponds to the
bidirectional or isotropic emission of the microcavity laser. In our case, the
values of $U_{C}$ almost increase linearly from $U_{C}\simeq 0.3$ to
$U_{C}\simeq 0.35$ in the range of $0.43\leq\chi\leq 0.478$. This result
implies that the unidirectionality increases as the deformation parameter
$\chi$ increases, unlike observing the morphologies of FFPs for Fig. 1(d),
(e), and (f). Note that we here discretize $2\pi$ into $3600$ pieces for
numerical calculation, i.e., $d\theta\sim\Delta\theta=0.1^{\circ}$.

The other measure of the unidirectionality $U_{W}$ marked by blue squares is
also defined as follow ryu2011designing ; ryu2019optimization :

$\displaystyle
U_{W}=\frac{\int^{\frac{\pi}{4}}_{\frac{-\pi}{4}}I(\theta)d\theta}{\int^{2\pi}_{0}I(\theta)d\theta},$
(3)

where the integral range of numerator runs from $-\frac{\pi}{4}$ to
$\frac{\pi}{4}$. The angles in this range of $|\theta|\leq\frac{\pi}{4}$ is
so-called the emission window as mentioned before.

Figure 2: Three types of measure of unidirectionality in the limaçon-shaped
cavity. The blue squares ($U_{W}$) and red circles ($U_{C}$) in (a) are
measures of unidirectionality associated with emission window. They almost
increase linearly as a function of the deformation. The orange circles in (b)
are normalized Shannon entropies and they have a local minimum at
$\chi=0.454$.

Hence, the meaning of this definition is clearly ratio between accumulated
intensities of FFPs within emission window and the total intensity of FFPs.
The values of $U_{W}$ also increase almost linearly from $U_{W}\simeq 0.56$ to
$U_{W}\simeq 0.59$ in the range of $0.43\leq\chi\leq 0.478$. Consequently,
these values of $U_{W}$ are also inconsistent with our observation. We can
conjecture that this discrepancy is attributed to the fact that $U_{W}$
increases proportionally to the integral value of the emission window,
regardless of the detailed structure of the emission window. Furthermore, it
should be noticed that $U_{C}$ exhibits a similar trend with $U_{W}$.

Next, we introduce entropic unidirectionality. Accordingly, we first have to
obtain a Shannon entropy associated with FFPs. Shannon entropy is a relevant
measure of the average amount of information for a random variable with a
given probability distribution function shannon1948mathematical . It was
firstly developed and utilized in communication theory shannon1948mathematical
. However, recently it has been also exploited in various areas such as bio-
system fuhrman2000application , economics eagle2010network , atomic physics
gonzalez2003shannon , and microcavity laser park2018shannon .

The discrete Shannon entropy for the intensity of FFPs is formally defined as:

$\displaystyle S_{N}=-\frac{1}{\log K}\sum^{K}_{i=1}\rho_{i}\log\rho_{i},$ (4)

where $\rho_{i}$ represents the probability distribution obtained under the
normalization condition $\sum^{K}_{i=1}I(\theta_{i})=1$. That is, the random
variable $X$ is an angular coordinate ($\Theta$) with the probability
distribution $\\{\rho_{i}\\}=\\{P(\Theta=\theta_{i})\\}$. In addition, the
$\frac{1}{\log K}$ is a normalization factor such that the value of Shannon
entropy is restricted in the range of $0\leq S_{N}\leq 1$, with $K=3600$ as
mentioned above. The orange circles in Fig. 2 (b) represent normalized Shannon
entropy ($S_{N}$) calculated by this definition. The value of $S_{N}$ has a
local maximum ($0.898$) at $\chi=0.43$ and local minimum ($0.888$) at
$\chi=0.454$, respectively.

Our recent works have confirmed that Shannon entropy can be beneficial in
measuring the delocalization of the given probability distributions
park2021indicators . Hence, the minimum (or maximum) value of the Shannon
entropy indicates the maximum localization (or delocalization) of the
intensity of FFPs in the polar coordinate under the normalization condition
$\sum^{K}_{i=1}I(\theta_{i})=1$. Consequently, the local minimum value of
$S_{N}$ at $\chi=0.454$ directly verifies the maximal undirectionality.
Moreover, the maximal entropy, $\log K$, corresponds to the isotropic
distribution that is not a bidirectional emission in the polar coordinate.
This result validates the conjecture that by exploiting the Shannon entropy,
our measure of unidirectionality is coincident with our observations in Fig.
1.

Figure 3: FFPs in an oval-shaped microcavity laser. The subfigures (a), (b),
and (c) are FFPs in the cartesian coordinate restricted in the range of
$x\in[-5,5]$ and $y\in[-5,5]$ at each deformation $\varepsilon=0.043$,
$\varepsilon=0.05$, and $\varepsilon=0.058$, respectively. The subfigures (d),
(e), and (f) are also FFPs plotted in the polar coordinate corresponding to
FFPs in the cartesian coordinate. The inset in (c) shows the boundary shape of
cavity at each $\varepsilon=0.043$ (black upper line), 0.05 (red center line),
and 0.058 (blue lower line), respectively.

For more generality of our argument and to understand the discrepancy between
Fig. 2(a) and Fig. 2(b), let us consider an oval-shaped microcavity laser
lee2018extremely . The FFPs are also obtained from transmitted rays by using
Fresnel equations for transverse electric (TE) modes with an effective index
of refraction $n=3.3$. The geometrical boundary condition of an oval shaped-
cavity, which is the deformed from an ellipse, is defined as follow:

$\displaystyle\frac{x^{2}}{a^{2}}+(1+\varepsilon x)\frac{y^{2}}{b^{2}}=1.$ (5)

For convenience, we substitute the deformation parameter $\chi$ to
$\varepsilon$. It was reported that optimized condition for a directional
emission is $a=1.0$, $b=1.03$, and $\varepsilon=0.05$ where $a$ and $b$ are
major and minor axis of an ellipse, and $\varepsilon$ is the deformation
parameter, respectively lee2018extremely . According to their results, we
conduct ray simulation in the range of $0.043\leq\varepsilon\leq 0.058$ at the
fixed value of $a=1.0$ and $b=1.03$. We plot some of the representative FFPs,
i.e., Fig. 3(a), (b), and (c) in the cartesian coordinate restricted in
$x\in[-5,5]$ and $y\in[-5,5]$ at each deformation $\varepsilon=0.043$,
$\varepsilon=0.05$, and $\varepsilon=0.058$. The corresponding FFPs in the
polar coordinate are displayed in Fig. 3(d), (e), and (f).

Figure 4: Three types of measure of the unidirectionality in the oval-shaped
microcavity. The blue squares and red circles in (a) are measures of the
unidirectionality associated with the emission window. Both of them have local
maximal values at $\varepsilon=0.05$. The orange circles in (b) are normalized
Shannon entropies. The values of orange circles have a local minimum value at
$\varepsilon=0.05$.

On the contrast to Fig. 1, we can easily notice that the overall FFPs
depending on the deformation parameter vary manifestly, and it can be
naturally expected that the local maximum of the unidirectionality is obtained
at $\varepsilon=0.05$ by observing Fig. 3(b) and (e), and comparing other
subfigures in Fig. 3. In this case, the local minimum value of $S_{N}$, and
local maximum values of $U_{W}$ and $U_{C}$ are attained at $\varepsilon=0.05$
simultaneously, i.e., entropic measure of the unidirectionality $S_{N}$ agrees
well with the the former measure of the unidirectionality ($U_{C},U_{W}$)
related to the emission window. Consequently, this fact implies that $U_{C}$
and $U_{W}$ only can detect the unidirectionality when the overall FFPs vary
significantly with a manifest variation of the emission window. However, our
new measure for the unidirectionality by employing the Shannon entropy can
efficiently capture the unidirectionality in any case. Notice that the
relative difference between the local maximum and the local minimum of $S_{N}$
in Fig. 4 is much larger than that of Fig. 2. This fact coincides with our
initial intuition.

Another aspect for definition of the directional emission along with the
unidirectionality is a so-called ‘divergence angle’. This concept presents an
analogy for the full width at half maximum (FWHM) yan2009directional . In our
case, the divergence angle estimates the width of the peak emission (i.e.,
lobe around $\theta=0$) yan2009directional ; wang2009deformed  of the emission
intensity in the polar coordinate, and this can indicate the degree of the
spread of FFPs onto $y$-axis in the cartesian coordinate. Under these
assumptions, we fist introduce the concept of semi-marginal probability
distribution. When the joint probability distribution function of random
variables $X$ and $Y$ is given by $\rho(x,y)$, the marginal probability
distribution of $Y$ is $\rho(y)=\int\rho(x,y)dx$ where integral is carried out
over all points in the range of $(X,Y)$ for which $Y=y$. We can interpret the
intensity of FFPs, $I(x,y)$, in the cartesian coordinate as a joint
probability distribution function under the normalization condition $\int\int
I(x,y)dxdy=1$, i.e., the random variables $X$ and $Y$ are components of the
cartesian coordinate with the joint probability distribution function
$\rho(x,y)={P(X=x,Y=y)}$. The integral $\int\rho(x,y)dx$ is performed over the
interval $x\in[0,5]$ to solely handle the forward emission. This is why we
call it as semi-marginal probability distribution. Note that we have
discretized ($x$,$y$)-coordinate in the range $x\in[-5,5]$ and $y\in[-5,5]$
into $1000\times 1000$ grid for the numerical calculation. Then, the discrete
Shannon entropy from the $\rho(y_{j})$ is defined by

$\displaystyle S_{y}=-\frac{1}{\log
K}\sum^{K}_{j=1}\rho_{y_{j}}\log\rho_{y_{j}},$ (6)

where the semi-marginal distribution
$\rho_{y_{j}}=\sum^{K}_{i=1}\rho(x_{i},y_{j})$ and $K=1000$.

The blue squares in Fig. 5(a) represent the divergence angle ($D_{A}$) and the
red spheres in Fig. 5(a) represent the Shannon entropy ($S_{y}$) of the semi-
marginal probabilities in the oval-shaped microcavity laser at each
$\varepsilon=0.043$, $\varepsilon=0.05$, and $\varepsilon=0.058$. Note that
these two plots show similar trends and this results support our previous
assumption. Three insets in Fig. 5 represent semi-marginal probability
densities ($\rho_{y}$). All peaks of $\rho_{y}$ are located at
$\varepsilon\simeq\pm 1$, and careful examination reveals that the third one
is more spread out than the others.

Figure 5: (a) The blue squares are markers of the divergence angle ($D_{A}$)
and the red circles are those of Shannon entropy of the semi-marginal
probability densities ($S_{y}$) in the oval-shaped microcavity laser at each
$\varepsilon=0.043$, $\varepsilon=0.05$, and $\varepsilon=0.058$. These two
plots show similar trends. Three insets are semi-marginal probabilities
related to $S_{y}$. (b) The orange squares are markers of Shannon entropy of
semi-marginal probabilities ($S_{y}$) in the limaçon-shaped microcavity laser
at $\chi=0.43$, $\chi=0.454$, and $\chi=0.478$. Three insets are also semi-
marginal probabilities related to $S_{y}$.

In the case of complex peak structures, as illustrated in Fig. 1(d), (e), and
(f), we can barely define $D_{A}$ (i.e., FWHM). However, we can quantitatively
and systematically measure the spread of the emission peak by exploiting
$S_{y}$, and the obtained results are presented in Fig. 5(b). The orange
squares in Fig. 5(b) indicate Shannon entropy ($S_{y}$) of semi-marginal
probability ($\rho_{y}$) in the limaçon shaped-cavity at $\chi=0.43$,
$\chi=0.454$, and $\chi=0.478$. The minimum value of $S_{y}$ in Fig. 5(b) is
substantially larger than that of $S_{y}$ in Fig. 5(a), which can be confirmed
by comparison between Fig. 1(e) and Fig. 3(e).

Consequently, we can say that the oval-shaped micro-cavity laser has larger
unidirectionality and smaller $S_{y}$ (i.e., $D_{A}$) than the limaçon-shaped
microcavity laser at least in our examples. In contrast, the limaçon-shaped
microcavity laser has more robust range against any variation of the
deformation parameter than the oval-shaped microcavity laser.

We have present new measures for a directional emission in microcavity lasers
by exploiting the Shannon entropy. There are primarily two aspects of the
directional emission—the unidirectionality and the divergence angle. Shannon
entropy obtained from the normalized intensity of angular distributions of
FFPs can measure the unidirectionality even if former notions can not
effectively detect the directional emission when the emission is robust
against a variation of the deformation parameter. However, it should be
noticed that our method can be wrong in some extreme cases. For example,
extreme bidirectional emission can yields a very low value of the Shannon
entropy. In order to exclude such cases, each FFPs must have a single broad
but dominant emission. To guarantee this condition, an overlap between FFPs
and Gaussian normal distribution must be larger than a critical value as the
role of auxiliary measure. In the proposed examples, the overlaps are always
larger than 0.6 with a mean value $\langle\theta\rangle=0$ (corresponding to
the emission peak) and standard deviation $\sigma=45^{\circ}$ (corresponding
to the emission window). Shannon entropy obtained from the semi-marginal
probability densities of FFPs in the cartesian coordinates can provide
equivalent results to the divergence angle; moreover, it can be applied to
even in the case of complicated peak structures.

Our new measure is entirely defined by normalized FFPs (as probability
distributions), regardless of the origin of emissions. Thus, our measure is
applicable to the FFPs obtained from the wave simulations and, furthermore, it
can be applicable to any shape of microcavity lasers and, various antenna
structures. We hope that our results can help to design and to modulate a
micocavity lasers for a better directional emission.

## Acknowledgments

This work was supported by the National Research Foundation of Korea, a grant
funded by the Ministry of Education (Grant Nos. NRF-2021R1I1A1A01052331 & NRF-
2021R1I1A1A01042199) and the Ministry of Science and ICT (Grant No.
NRF-2020M3E4A1077861).

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#  Energy-Efficient Throughput Maximization in
mmWave MU-Massive-MIMO-OFDM:
Genetic Algorithm based Resource Allocation ††thanks: This work was supported
in part by Huawei Technologies Canada and in part by the Natural Sciences and
Engineering Research Council of Canada.

Asil Koc, Farhan Bishe, Tho Le-Ngoc  Department of Electrical and Computer
Engineering, McGill University, Montreal, QC, Canada
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>tho.le-
<EMAIL_ADDRESS>
###### Abstract

This paper develops a new genetic algorithm based resource allocation (GA-RA)
technique for energy-efficient throughout maximization in multi-user massive
multiple-input multiple-output (MU-mMIMO) systems using orthogonal frequency
division multiplexing (OFDM) based transmission. We employ a hybrid precoding
(HP) architecture with three stages: (i) radio frequency (RF) beamformer, (ii)
baseband (BB) precoder, (iii) resource allocation (RA) block. First, a single
RF beamformer block is built for all subcarriers via the slow time-varying
angle-of-departure (AoD) information. For enhancing the energy efficiency, the
RF beamformer aims to reduce the hardware cost/complexity and total power
consumption via a low number of RF chains. Afterwards, the reduced-size
effective channel state information (CSI) is utilized in the design of a
distinct BB precoder and RA block for each subcarrier. The BB precoder is
developed via regularized zero-forcing technique. Finally, the RA block is
built via the proposed GA-RA technique for throughput maximization by
allocating the power and subcarrier resources. The illustrative results show
that the throughput performance in the MU-mMIMO-OFDM systems is greatly
enhanced via the proposed GA-RA technique compared to both equal RA (EQ-RA)
and particle swarm optimization based RA (PSO-RA). Moreover, the performance
gain ratio increases with the increasing number of subcarriers, particularly
for low transmission powers.

###### Index Terms:

Massive MIMO, hybrid precoding, OFDM, genetic algorithm, resource allocation,
multi-user, beamforming.

##  I Introduction

Millimeter wave (mmWave) communication has attracted considerable attention
for the fifth-generation (5G) and beyond wireless communication systems by
addressing bandwidth limitation issue [1]. Bandwidth demands have arisen in
accordance with the intense growth in user demands in wireless networks and
emerging advanced technologies requiring high-throughput data communications.
Hence, the mmWave communications is an enabling technology by providing a
considerable large bandwidth (i.e., 30 GHz - 300 GHz). Moreover, transmitting
information on such high frequencies favorizes implementing large-dimensional
antenna arrays in relatively smaller dimensions. Nevertheless, making a shift
to mmWave communications leads to some challenging issues, notably higher
atmospheric absorption causing severe path loss due to its short
wavelength[1]. To alleviate the path loss effect in mmWave communication, it
is required to form high-gain directional beams, which can be done by
implementing large antenna arrays, introducing the massive multiple-input
multiple-output (mMIMO) technology[2].

Hybrid precoding (HP) is a promising technique for the multi-user mMIMO (MU-
mMIMO) systems [3]. The two-stage HP architecture interconnects the radio
frequency (RF) and baseband (BB) stages via a limited number of power-hungry
RF chains [4, 5, 6, 7, 8, 9]. Thus, HP remarkably enhances the energy-
efficiency compared to the single-stage fully digital precoding (FDP). On the
other hand, the wideband signal transmission is considered to achieve
extremely high data rates, especially in the mmWave communications. To address
the frequency selectivity issue, the orthogonal frequency division
multiplexing (OFDM) technique is widely considered for the wideband signal
transmission [10]. In [10, 11, 12], OFDM-based HP scheme is investigated for
mMIMO systems, where one RF beamformer is designed for all subcarriers, while
the number of BB precoders should be the same as the number of subcarriers.

The recent advances in artificial intelligence (AI) enable the development of
advanced communication systems [13]. In the field of AI, the genetic algorithm
(GA) is a evolutionary population-based optimization technique to solve the
complex real-world problems under given constraints [14, 15].

In this paper, we propose a novel genetic algorithm based resource allocation
(GA-RA) technique for energy-efficient throughput maximization in the MU-
mMIMO-OFDM system. Also, the angular-based hybrid precoding (AB-HP) technique
proposed in [5] is adopted for the OFDM-based transmission. Thus, the system
model is divided into three stages: (i) RF beamformer, (ii) BB precoder, (iii)
resource allocation (RA). First, the RF beamformer is designed via the slow
time-varying angle-of-departure (AoD) information. Next, the BB precoder is
built for each subcarrier using the reduced-size effective channel state
information (CSI) for the corresponding subcarrier. Finally, the proposed GA-
RA technique is applied for the throughput maximization by efficiently
allocating the power and subcarrier resources among the users. The numerical
results present that the proposed GA-RA considerably improves the throughput
(i.e., sum-rate)111We use the terms of throughput and sum-rate
interchangeably. performance compared to the equal RA (EQ-RA) scheme. Also,
when the number of subcarriers increases, the performance gain improves,
especially for the low transmit powers.

The rest of this paper is organized as follows. The system model is introduced
in Section II. The AB-HP scheme is expressed in Section III. Then, Section IV
provides the RA problem formulation for the throughput maximization. The
proposed GA-RA technique is introduced in Section V. After presenting the
illustrative results in Section VI, the paper is concluded in Section VII.

##  II System Model

We consider an OFDM-based downlink transmission via $C$ subcarriers for the
MU-mMIMO systems as shown in Fig. 1. Here, $K$ single-antenna users located in
$G$ groups are served by a BS having a uniform rectangular array (URA) with
$M=M_{x}\times M_{y}$ antennas222 Based on the URA structure, $M_{x}$ and
$M_{y}$ denote the number of antennas along $x$-axis and $y$-axis,
respectively. There are two main reasons for utilizing the URA structure: (i)
it packs a larger number of antennas in a two-dimensional (2D) grid under the
physical-limited space requirements in practical applications, (ii) it enables
three-dimensional (3D) beamforming by employing both azimuth and elevation
domains [2, 5, 7, 12, 16, 6]. . The BS utilizes $N_{RF}$ RF chains for
interconnecting the BB-stage and RF-stage. According to the multi-carrier
transmission, the data signal ${\bf{d}}\left[i\right]$ for each subcarrier
$i=1,\cdots,C$ is respectively passed through a power allocation block
${\bf{P}}\left[i\right]=\textrm{diag}\left(\sqrt{p_{1,i}},\cdots,\sqrt{p_{K,i}}\right)\in\mathbb{R}^{K\times
K}$ and a digital BB precoder ${\bf B}\left[i\right]=\left[{\bf
b}_{1,i},\cdots,{\bf b}_{K,i}\right]\in\mathbb{C}^{N_{RF}\times K}$, where
$p_{k,i}$ is the allocated power for the $k^{th}$ user at subcarrier $i$.
After applying the inverse fast Fourier transforms (IFFT) and adding the
cyclic prefixes (CP), the analog RF beamformer ${\bf F}\in\mathbb{C}^{M\times
N_{RF}}$ developed via low-cost phase-shifters is employed, which is identical
for all subcarriers333Since the analog RF beamformer is designed via the AoD
information, it is reasonable to assume that each subcarrier experiences a
similar AoD support (i.e., mean azimuth/elevation AoD and their spread). [10,
11, 12]. Then, the transmitted data signal at subcarrier $i$ is written as:

${\bf s}\left[i\right]={\bf FB}\left[i\right]{\bf P}\left[i\right]{\bf
d}\left[i\right]\in\mathbb{C}^{M}.$ (1)

Considering the URA structure [5] and the 3D geometry-based mmWave channel
model [17], the channel vector for the $k^{th}$ user at subcarrier
$i=1,\cdots,C$ is defined as:

${\bf{h}}_{k}^{T}\left[i\right]=\sum_{l=1}^{L}\tau^{-\eta}_{k_{l},i}z_{k_{l},i}{\bm{\phi}}^{T}\big{(}{{\gamma_{x,k_{l},i}},{\gamma_{y,k_{l},i}}}\big{)}e^{-j\frac{2\pi
li}{C}}={\bf{z}}^{T}_{k}\left[i\right]{{\bf{\Phi}}}_{k}\left[i\right],$ (2)

where $L$ is the number of paths, $\tau_{k_{l},i}$ and
$z_{k_{l},i}\sim\mathcal{CN}\big{(}0,\frac{1}{L}\big{)}$ are respectively the
distance and complex path gain of $l^{th}$ path at subcarrier $i$, $\eta$ is
the path loss exponent,
${\bm{\phi}}\big{(}\cdot,\cdot\big{)}\in\mathbb{C}^{M}$ is the phase response
vector,
$\gamma_{x,k_{l},i}=\sin\left(\theta_{k_{l},i}\right)\cos\left(\psi_{k_{l},i}\right)$
and
$\gamma_{y,k_{l},i}=\sin\left(\theta_{k_{l},i}\right)\sin\left(\psi_{k_{l},i}\right)$
are the coefficients reflecting the elevation AoD (EAoD) and azimuth AoD
(AAoD) for the corresponding path. Here,
$\theta_{k_{l},i}\in\big{[}\theta_{k}-\delta_{k}^{\theta},\theta_{k}+\delta_{k}^{\theta}\big{]}$
is the EAoD with mean $\theta_{k}$ and spread $\delta_{k}^{\theta}$,
$\psi_{k_{l},i}\in\big{[}\psi_{k}-\delta_{k}^{\psi},\psi_{k}+\delta_{k}^{\psi}\big{]}$
is the AAoD with mean $\psi_{k}$ and spread $\delta_{k}^{\psi}$.

Figure 1: Hybrid precoding and genetic algorithm based resource allocation
(GA-RA) in MU-mMIMO-OFDM systems.

Then, the phase response vector is modeled as [5]:

$\displaystyle{\bm{\phi}}\left({{\gamma_{x},\gamma_{y}}}\right)$
$\displaystyle=\big{[}{1,{e^{-j2\pi d{{\gamma_{x}}}}},\cdots,{e^{-j2\pi
d\left({{M_{x}}-1}\right){{\gamma_{x}}}}}}\big{]}^{T}$ (3)
$\displaystyle\otimes\big{[}{1,{e^{-j2\pi d{{\gamma_{y}}}}},\cdots,{e^{-j2\pi
d\left({{M_{y}}-1}\right){{\gamma_{y}}}}}}\big{]}^{T}\in\mathbb{C}^{M},$

where $d$ is the antenna element spacing normalized by the wavelength. As
indicated in (2), the instantaneous channel vector is represented via the fast
time-varying path gain vector ${\bf
z}_{k}=\left[\tau^{-\eta}_{k_{1}}z_{k_{1}},\cdots,\tau^{-\eta}_{k_{L}}z_{k_{L}}\right]^{T}\in\mathbb{C}^{L}$
and slow time-varying phase response matrix
${{\bf{\Phi}}}_{k}\in\mathbb{C}^{L\times M}$ as a function of AoD information.

By using (1) and (2), the received signal at the $k^{th}$ user at subcarrier
$i$ is given by:

$\displaystyle{r_{k,i}}$
$\displaystyle={\bf{h}}_{k}^{T}\left[i\right]{\bf{s}}\left[i\right]+{w_{k,i}}={\bf{h}}_{k}^{T}\left[i\right]{\bf{FB}}\left[i\right]{\bf
P}\left[i\right]{\bf d}\left[i\right]+{w_{k,i}}$ (4)
$\displaystyle=\underbrace{\sqrt{{p_{{k,i}}}}{\bf{h}}_{{k}}^{T}\left[i\right]{\bf{F}}{{\bf{b}}_{{k,i}}}{d_{{k,i}}}}_{\textrm{Desired
Signal}}+\underbrace{\sum_{t\neq
k}^{{K}}{\sqrt{{p_{{t,i}}}}{\bf{h}}_{{k}}^{T}\left[i\right]{\bf{F}}{{\bf{b}}_{{t,i}}}{d_{{t,i}}}}}_{\textrm{Inter-
User Interference}}+\underbrace{w_{k,i}}_{\textrm{Noise}},$

where $w_{k,i}\sim\mathcal{CN}\big{(}0,\sigma^{2}\big{)}$ is the complex
circularly symmetric Gaussian noise. Then, we derive the instantaneous sum-
rate (i.e., throughput) expression at subcarrier $i$ as follows:

$R_{\textrm{sum},i}\left({\bf F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right)=\sum_{k=1}^{K}\log_{2}\left(1+\frac{{{p_{{k,i}}}}|{\bf{h}}_{{k}}^{T}\left[i\right]{\bf{F}}{{\bf{b}}_{{k,i}}}|^{2}}{\sum\limits_{t\neq
k}^{K}{{p_{{t,i}}}}|{\bf{h}}_{{k}}^{T}\left[i\right]{\bf{F}}{{\bf{b}}_{{t,i}}}|^{2}+\sigma^{2}}\right).$
(5)

The overall sum-rate across all $C$ subcarriers is calculated as
$R_{\textrm{sum}}=\sum_{i=1}^{C}\mathbb{E}\left\\{R_{\textrm{sum},i}\left({\bf
F},{\bf B}\left[i\right],{\bf P}\left[i\right]\right)\right\\}$ in the unit of
[bps/Hz]. Similarly, the average sum-rate per subcarrier is obtained as
$\frac{R_{\textrm{sum}}}{C}$ in the unit of [bps/Hz/subcarrier]. Hence, we
formulate the throughput maximization problem as follows:

$\displaystyle\max_{\left\\{{\bf F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right\\}}$ $\displaystyle\hskip
17.22217ptR_{\textrm{sum},i}\left({\bf F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right)$ (6) s.t.
$\displaystyle\mathbb{E}\left\\{\left\|{\bf{s}}\left[i\right]\right\|^{2}_{2}\right\\}=\sum_{k=1}^{K}{p_{k,i}}\left\|{\bf{F}}{{\bf{b}}_{k,i}}\right\|_{2}^{2}\leq\frac{P_{T}}{C},\hskip
4.30554ptp_{k,i}\geq 0,\forall k,$
$\displaystyle|\left[{\bf{F}}\right]_{m,n}|=\frac{1}{\sqrt{M}},\forall m,n,$

where the constraints indicate the transmit power constraint per subcarrier
[10] (i.e., $\frac{P_{T}}{C}$) and the constant modulus (CM) constraint due to
the utilization of phase-shifters at the RF-stage. However, it is a non-convex
optimization due to the allocated powers entangled with each other (please see
(5)) and the CM constraint at the RF beamformer. Thus, we first design the RF
beamformer and the BB precoder via AB-HP in Section III, then we propose the
GA-RA technique in Section V for optimizing the allocated powers across all
subcarriers.

##  III Hybrid Precoding

In this section, we develop the angular-based hybrid precoding (AB-HP)
architecture for MU-mMIMO-OFDM systems to reduce the number of RF chains,
suppress the inter-user interference and lower the CSI overhead size. After
designing the RF beamformer and BB precoder in this section, the proposed GA-
RA technique is expressed in Section V.

###  III-A RF Beamformer

The slow-time varying AoD information444In [16], it is shown that the AoD
parameters (i.e., mean and spread) can be efficiently obtained via an offline
deep learning and geospatial data-based estimation technique instead of the
traditional online channel sounding. is utilized at the RF beamformer for
reducing the large CSI overhead size in MU-mMIMO-OFDM systems. Considering the
$K$ users located in $G$ groups as shown in Fig. 1, the channel matrix for
group $g$ at subcarrier $i$ is given by:

${\bf H}_{g}\left[i\right]=\left[{\bf h}_{1}\left[i\right],\cdots,{\bf
h}_{K_{g}}\left[i\right]\right]^{T}={\bf
Z}_{g}\left[i\right]{\bf\Phi}_{g}\left[i\right]\in\mathbb{C}^{K_{g}\times M},$
(7)

where ${\bf Z}_{g}\left[i\right]=\big{[}{\bf z}_{1}\left[i\right],\cdots,{\bf
z}_{K_{g}}\left[i\right]\big{]}^{T}\in\mathbb{C}^{K_{g}\times L}$,
${\bf\Phi}_{g}\left[i\right]\in\mathbb{C}^{L\times M}$, $K_{g}$ is the number
of users in group $g$ with $K=\sum_{g=1}^{G}K_{g}$. Then, the full-size
channel matrix at subcarrier $i$ is defined as ${\bf
H}\left[i\right]=\left[{\bf H}_{1}^{T}\left[i\right],\cdots,{\bf
H}_{G}^{T}\left[i\right]\right]^{T}\in\mathbb{C}^{K\times M}$. Here, we assume
that the users in the same groups experience a similar AoDs as in [5, 7, 6,
8]. Therefore, $G$ blocks are designed for the RF beamformer as follows:

${\bf F}=\left[{\bf F}_{1},\cdots,{\bf F}_{G}\right]\in\mathbb{C}^{M\times
N_{RF}},$ (8)

where ${\bf F}_{g}\in\mathbb{C}^{M\times N_{RF,g}}$ is the RF beamformer for
group $g$ with $N_{RF}=\sum\nolimits_{g=1}^{G}N_{RF,g}$. By using (7) and (8),
the effective channel matrix seen from the BB-stage is obtained as:

$\bm{\mathcal{H}}\left[i\right]={\bf H}\left[i\right]{\bf
F}=\left[{\begin{array}[]{*{20}{c}}{{{\bf{H}}_{1}}\left[i\right]{{\bf{F}}_{1}}}&{{{\bf{H}}_{1}}\left[i\right]{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{1}}\left[i\right]{{\bf{F}}_{G}}}\\\
{{{\bf{H}}_{2}}\left[i\right]{{\bf{F}}_{1}}}&{{{\bf{H}}_{2}}\left[i\right]{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{2}}\left[i\right]{{\bf{F}}_{G}}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{{{\bf{H}}_{G}}\left[i\right]{{\bf{F}}_{1}}}&{{{\bf{H}}_{G}}\left[i\right]{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{G}}\left[i\right]{{\bf{F}}_{G}}}\\\
\end{array}}\right]\in\mathbb{C}^{K\times N_{RF}},$ (9)

where the diagonal block matrix ${\bf H}_{g}\left[i\right]{\bf F}_{g}={\bf
Z}_{g}\left[i\right]{\bf\Phi}_{g}\left[i\right]{\bf
F}_{g}\in\mathbb{C}^{K_{g}\times N_{RF,g}}$ is the effective channel matrix
and the off-diagonal block matrix ${\bf H}_{t}\left[i\right]{\bf F}_{g}={\bf
Z}_{t}\left[i\right]{\bf\Phi}_{t}\left[i\right]{\bf
F}_{g}\in\mathbb{C}^{K_{t}\times N_{RF,g}}$ is the effective interference
channel matrix, $\forall t\neq g$. Therefore, the RF beamformer design targets
for achieving the following two objectives: (i) maximizing the beamforming
gain in the desired direction (i.e., $\textrm{Span}\left({\bf
F}_{g}\right)\subset\textrm{Span}\left({\bf\Phi}_{g}\left[i\right]\right)$),
(ii) successfully suppress the interference among user groups (i.e.,
$\textrm{Span}\left({\bf F}_{g}\right)\subset\cup_{t\neq
g}\textrm{Null}\left({\bf\Phi}_{t}\left[i\right]\right)$). As proven in [5],
both objectives are accomplished by building the RF beamformer ${\bf F}_{g}$
via the steering vector ${\bf
e}\left(\gamma_{x},\gamma_{y}\right)=\frac{1}{\sqrt{M}}{\bm{\phi}}^{*}\left({{\gamma_{x},\gamma_{y}}}\right)\in\mathbb{C}^{M}$
with $\left(\gamma_{x},\gamma_{y}\right)$ angle-pairs covering the AoD support
of desired user group and excluding the AoD supports of the other user groups
(please see (3) for ${\bm{\phi}}\left({{\gamma_{x},\gamma_{y}}}\right)$). For
covering the complete 3D elevation and azimuth angular space with minimum
number of angle-pairs, $M$ orthogonal quantized angle-pairs are defined as
${{\lambda^{x}_{u}}=-1+\frac{2u-1}{{{M_{x}}}}}$ for $u=1,\cdots,{M_{x}}$ and
${{\lambda^{y}_{n}}=-1+\frac{2n-1}{{{M_{y}}}}}$ for $n=1,\cdots,{M_{y}}$.
Considering that $N_{RF,g}$ quantized angle-pairs covers the AoD support of
user group $g$ [5, eq. (13)], we build the RF beamformer for group $g$ as
follows:

${\bf F}_{g}=\big{[}{\bf
e}\big{(}\lambda_{u_{1}}^{x},\lambda_{n_{1}}^{y}\big{)},\cdots,{\bf
e}\big{(}\lambda_{u_{N_{RF,g}}}^{x},\lambda_{n_{N_{RF,g}}}^{y}\big{)}\big{]}\in\mathbb{C}^{M\times
N_{RF,g}}.$ (10)

Finally, the complete RF precoder ${\bf F}$ satisfying the CM constraint given
in (6) is derived by substituting (10) into (8). It is worthwhile to mention
that the RF beamformer is a unitary matrix (i.e., ${\bf F}^{H}{\bf F}={\bf
I}_{N_{RF}}$).

###  III-B BB Precoder

As seen in Fig. 1, we develop $C$ distinct BB precoders for each subcarriers.
Here, the main objective is to further suppress the residual inter-user
interference. By utilizing the reduced-size effective channel matrix
$\bm{\mathcal{H}}\left[i\right]$, the regularized zero-forcing (RZF) technique
is applied for each subcarrier. Thus, the BB precoder at subcarrier $i$ is
defined as:

${\bf
B}\left[i\right]=\big{[}\left(\bm{\mathcal{H}}\left[i\right]\right)^{H}\bm{\mathcal{H}}\left[i\right]+K\alpha{\bf
I}_{K}\big{]}^{-1}\left(\bm{\mathcal{H}}\left[i\right]\right)^{H}\in\mathbb{C}^{N_{RF}\times
K},$ (11)

where $\alpha=\frac{\sigma^{2}}{P_{T}}$ is the regularization parameter [5].

##  IV Problem Formulation

After deriving the closed-form solutions for RF beamformer ${\bf F}$ and BB
precoder ${\bf B}\left[i\right]$ for all subcarriers $i=1,\cdots,C$, the
throughput maximization problem given in (6) turns into a resource allocation
(RA) problem (i.e., allocating power and subcarrier resources among the
users). Thus, for a given ${\bf F}$ and ${\bf B}\left[i\right]$, we formulate
the RA optimization problem as follows:

$\displaystyle\max_{{\bf P}\left[i\right]}{}$ $\displaystyle
R_{\textrm{sum},i}\left({\bf F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right)$ (12) s.t.
$\displaystyle\mathbb{E}\left\\{\left\|{\bf{s}}\left[i\right]\right\|^{2}_{2}\right\\}=\sum_{k=1}^{K}{{p_{k,i}}{\bf{b}}_{k,i}^{H}{{\bf{b}}_{k,i}}}\leq\frac{P_{T}}{C},~{}~{}p_{k,i}\geq
0,\forall k,$

where $R_{\textrm{sum},i}\left({\bf F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right)$ is the sum-rate at subcarrier $i$ defined in (5).
However, it is still a non-convex optimization problem due to the allocated
power $p_{k,i}$ interchangeably located in the numerator and denominator in
(5). Thus, the traditional optimization algorithms may not be utilized to
solve the RA problem.

##  V Genetic Algorithm based Resource Allocation

Genetic algorithm (GA) is one of the most well-known nature-inspired
evolutionary optimization algorithms [15]. It can address the shortcomings of
traditional optimization algorithms, which are not able to find the optimal
solution for non-convex problems. According to the GA, each solution of the
problem is considered a chromosome, where the genes on a chromosome are
defined as the problem variables. The chromosomes/genes evolve through
generations via three main steps: (i) selection, (ii) crossover (iii) mutation
[15, 18].

We here propose a new genetic algorithm based resource allocation (GA-RA)
technique for MU-mMIMO-OFDM systems. As the RA optimization problem defined in
(12) aims to find the optimal allocated powers for $K$ users at subcarrier
$i$, it lies on $K$-dimensional search space. Thus, each chromosome contains
$K$ genes. Since we deal with the continuous values, the continuous GA is
adopted instead of the binary GA. Consider the $n^{th}$ chromosome
representing the allocated powers of the $q^{th}$ generation at subcarrier $i$
as follows:

${\bf
P}_{n}^{(q)}[i]=\textrm{diag}\left(\sqrt{p_{1,i,n}^{(q)}},\cdots,\sqrt{p_{K,i,n}^{(q)}}\right)\in\mathbb{R}^{K\times
K},\vspace{0.5ex}$ (13)

where $p_{k,i,n}^{\left(q\right)}$ is the $k^{th}$ user power at subcarrier
$i$. For satisfying the total transmitted power constraint for each subcarrier
given in (12), each chromosome is normalized as:

$\hat{\bf{P}}_{n}^{(q)}[i]=\frac{{\bf{P}}_{n}^{(q)}[i]}{\varepsilon_{n}^{(q)}[i]}=\textrm{diag}\left(\sqrt{\hat{p}_{1,i,n}^{(q)}},\cdots,\sqrt{\hat{p}_{K,i,n}^{(q)}}\right)\in\mathbb{R}^{K\times
K},\vspace{0.5ex}$ (14)

where
$\varepsilon_{n}^{(q)}[i]=\sqrt{\frac{P_{T}/C}{\sum_{k=1}^{K}{{\hat{p}_{k,i,n}^{(q)}}{\bf{b}}_{k,i}^{H}{{\bf{b}}_{k,i}}}}}$
and $\hat{p}_{k,i,n}^{(q)}\in[0,1]$. The fitness function corresponding to the
$n^{th}$ chromosome at subcarrier $i$ is calculated via
$R_{\textrm{sum},i}\big{(}{\bf F},{\bf B}\left[i\right],{\bf
P}_{n}^{\left(q\right)}\left[i\right]\big{)}$ defined in (5). In the proposed
GA-RA, the population size is defined as $N_{p}$, which is kept as the same
through $Q$ generations. The first generation is initialized randomly, where
each allocated power is uniformly distributed as
$\hat{p}_{k,i,n}^{\left(1\right)}\sim U(0,\kappa)$ with
$\kappa\in\left(0,1\right]$. Afterwards, the top $\rho_{\textrm{mating}}$
percent of the population is selected to form the mating pool, and the rest
are discarded to free up space for new offspring. In order to generate new
chromosomes, we need to choose parents from the mating pool. As new offspring
inherits the traits from their parents, if only the best ones are selected the
algorithm may be trapped in a local optimum point. To prevent this issue, it
is needed to choose parents on a random basis while preserving our desire to
choose highly qualified parents[15]. Thus, we employ the tournament selection
method [19]. For selecting each parent, $\rho_{\textrm{sub}}$ percent of the
mating pool is randomly chosen as a subset. Then, based on the fitness value,
the best chromosome of the subset is selected as a parent. Each pair of
parents produces a new pair of offspring through the crossover. In the
proposed approach, we apply a mixture of two crossover methods: (i) linear
crossover[20], (ii) uniform crossover[18]. According to the a given pair of
parents as $\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i]$ and
$\hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i]$ with the randomly chosen $n_{1}$
and $n_{2}$ indices via the tournament selection method, the linear crossover
finds $\hat{\bf{P}}_{o_{1}}^{\left(q\right)}[i]$ and
$\hat{\bf{P}}_{o_{2}}^{\left(q\right)}[i]$ as the new offspring as follows:

$\displaystyle\hat{\bf{P}}_{o_{1}}^{{\left(q\right)}}[i,k]=\begin{cases}\frac{3\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k]-\hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i,k]}{2},&0\leq\frac{3\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k]-\hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i,k]}{2}\leq
1\\\ \hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k],&\text{otherwise}\end{cases}$
(15)
$\displaystyle\hat{\bf{P}}_{o_{2}}^{{\left(q\right)}}[i,k]=\begin{cases}\frac{3\hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i,k]-\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k]}{2},&0\leq\frac{3\hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i,k]-\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k]}{2}\leq
1\\\ \hat{\bf{P}}_{n_{2}}^{\left(q\right)}[i,k],&\text{otherwise}\end{cases}$

where we abuse the notation $\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i,k]$ to
represent the $k^{th}$ diagonal entry of
$\hat{\bf{P}}_{n_{1}}^{\left(q\right)}[i]$. It is important to note that the
linear crossover operation might generate infeasible solutions violating the
condition of $\hat{p}_{n_{K},i}^{(q)}\in[0,1]$. To prevent this from
happening, infeasible genes are replaced by the original parent’s genes.
Subsequently, the uniform crossover is applied by randomly swapping the genes
of $\hat{\bf{P}}_{o_{1}}^{\left(q\right)}[i]$ with the corresponding genes of
$\hat{\bf{P}}_{o_{2}}^{\left(q\right)}[i]$. The crossover process is applied
until reaching the initial population size of $N_{p}$. Afterwards, the top
$\rho_{\textrm{elite}}$ percent of them are selected as elite members,
implying that they do not undergo mutation to keep the good solutions. The
rest of the population are mutated by adding a Gaussian noise distributed as
$\mathcal{N}(0,10^{-4})$ to the randomly selected genes based on the mutation
rate of $\rho_{\textrm{mut}}$. All mutated genes are bounded in the interval
of $[0,1]$. Here, we advocate the exploitative behavior of the GA-RA technique
by introducing crossover, whereas the mutation encourages the exploratory
behavior of the GA-RA technique for preventing it from converging to a local
optimum solution. The whole procedure is repeated through $Q$ generations.
Finally, the best solution of the last generation is reported as the power
allocation block at subcarrier $i$ as follows:

${\bf{P}}[i]=\varepsilon_{best}^{(Q)}[i]\hat{\bm{P}}_{best}^{(Q)}[i]$ (16)

The summary of the proposed GA-RA technique is expressed in detail in
Algorithm 1.

Algorithm 1 Proposed GA-RA technique

0: ${\bf F}$, ${\bf B}[i]$, $P_{T}$, $N_{p}$, $Q$, $\kappa$,
$\rho_{\textrm{mating}}$,$\rho_{\textrm{sub}}$,$\rho_{\textrm{elite}}$,
$\rho_{\textrm{mut}}$

0: ${\bf P}[1]$, ${\bf P}[2]$,$\cdots$, ${\bf P}[C]$

1: for $i=1:C$ do

2: for $n=1:N_{p}$ do

3: Initialize $\hat{\bf{P}}_{n}^{\left(1\right)}[i]$ with diagonal entries
$\hat{p}_{k,i,n}^{\left(1\right)}\sim U(0,\kappa)$.

4: end for

5: for $q=1:Q$ do

6: Keep top $\rho_{\textrm{mating}}$ percent of population in $q^{th}$
generation as mating pool and discard the rest.

7: for $p=1:\frac{(1-\rho_{\textrm{mating}})}{2}N_{p}$ do

8: for $j=1:2$ do

9: For tournament selection, create a subset by randomly choosing
$\rho_{\textrm{sub}}$ percent of mating pool.

10: Select the best subset member as parent
$\hat{\bf{P}}_{n_{j}}^{\left(q\right)}[i]$.

11: end for

12: Apply linear crossover via (15) to calculate new offspring
$\hat{\bf{P}}_{o_{1}}^{\left(q\right)}[i]$ and
$\hat{\bf{P}}_{o_{2}}^{\left(q\right)}[i]$.

13: Update $\hat{\bf{P}}_{o_{1}}^{\left(q\right)}[i]$ and
$\hat{\bf{P}}_{o_{2}}^{\left(q\right)}[i]$ via uniform crossover by randomly
swapping their corresponding genes.

14: end for

15: Include all offspring to the population.

16: Select top $\rho_{\textrm{elite}}$ percent of population as elite members,
mutate the rest based on mutation rate of $\rho_{\textrm{mut}}$.

17: Carry elite/mutated population to $(q+1)^{th}$ generation.

18: end for

19: Find ${\bf P}[i]$ via (16).

20: end for

21: return ${\bf P}[1]$, ${\bf P}[2]$,$\cdots$, ${\bf P}[C]$

##  VI Illustrative Results

This section illustrates Monte-Carlo simulation results to evaluate the sum-
rate and energy-efficiency performance of the proposed GA-RA technique in MU-
mMIMO-OFDM systems. Table I presents the simulation parameters considering the
3D microcell scenario [21]. We have considered a BS equipped with a square URA
with $M=256$ antenna elements. For the proposed GA-RA technique, we assume the
population size as $N_{p}=100$ and the number of generations as $Q=10$, unless
otherwise stated. Also, other parameters are chosen as $\kappa=\frac{1}{20}$,
$\rho_{\textrm{mating}}=40\%$,$\rho_{\textrm{sub}}=10\%$,
$\rho_{\textrm{elite}}=80\%$, $\rho_{\textrm{mut}}=60\%$.

The energy-efficiency of the MU-mMIMO-OFDM systems is evaluated by taking the
ratio of the overall sum-rate $R_{\textrm{sum}}$ and the total power
consumption $P_{\textrm{total}}$ as [5, 6, 9]:

$\beta=\frac{R_{\textrm{sum}}}{P_{\textrm{total}}}=\frac{\sum_{i=1}^{C}\mathbb{E}\left\\{R_{\textrm{sum},i}\left({\bf
F},{\bf B}\left[i\right],{\bf
P}\left[i\right]\right)\right\\}}{P_{T}+N_{RF}P_{RF}+N_{PS}P_{PS}}~{}\textrm{[bps/Hz/W]},$
(17)

where $P_{T}$ is the transmission power, $P_{RF}$ ($P_{PS}$) is the power
consumption per each RF chain (phase-shifter), $N_{RF}$ ($N_{PS}$) is the
number of RF chains (phase-shifters). We here assume that $P_{RF}=250$ mW and
$P_{PS}=1$ mW as in [9]. It is important to highlight that the FDP
architecture has $N_{RF}=M=256$ RF chains and does not require any phase-
shifters (i.e., $N_{PS}=0$). On the other hand, the proposed AB-HP
architecture requires $N_{RF}=36$ RF chains and $N_{PS}=N_{RF}\times M=9216$
phase-shifters to support ${M=256}$ antennas in the considered simulation
setup. Thus, the number of RF chains reduces from $256$ to $36$ (i.e., $87\%$
reduction in hardware cost/complexity and CSI overhead size).

TABLE I: Simulation parameters. # of antennas [21] | $M=16\times 16=256$
---|---
Cell radius [21] | 100m
BS height [21] —. User height [21] | 10m —. 1.5m-2.5m
User-BS horizontal distance | 10m-90m
# of groups | $G=3$
# of users in each group | $K_{g}=\frac{K}{G}$
Mean EAoD —. Mean AAoD |  $\theta_{g}=60^{\circ}$ —. $\psi_{g}=21^{\circ}+120^{\circ}\left(g-1\right)$
EAoD spread —. AAoD spread |  $\delta_{g}^{\theta}=15^{\circ}$ —. $\delta_{g}^{\psi}=11^{\circ}$
Path loss exponent [7] | $\eta=3.76$
Noise PSD [7] |  $-174$ dBm/Hz
Channel bandwidth [7] |  $10$ kHz
# of paths | $L=10$
Antenna spacing (in wavelength) | ${d=0.5}$
# of network realizations | $1000$

The sum-rate and energy-efficiency are respectively plotted versus the number
of subcarriers in Fig. 2(a) and Fig. 2(b), where $P_{T}=40$ dBm and $K=15$
users. Here, the proposed GA-RA technique is compared with equal RA (EQ-RA)
and particle swarm optimization based RA (PSO-RA)555Even though PSO algorithm
is employed for power allocation optimization over single subcarrier
transmission in [7], it nevertheless serves as benchmark. [7]. Both GA-RA and
PSO-RA techniques consider $Q=10$ generations (e.g., iterations). Also, the
power values are equally distributed among the users for all subcarriers in
EQ-RA (i.e., $p_{1,1}=\cdots=p_{k,i}=\cdots=p_{K,C}$, $\forall k,i$). In the
EQ-RA scheme, the performance comparison is investigated between FDP and HP
schemes. The numerical results in Fig. 2(a) reveal that HP with EQ-RA closely
achieves the sum-rate provided by its FDP counterpart. Additionally, Fig. 2(b)
illustrates that HP with EQ-RA considerably improves the energy-efficiency
compared to the FDP scheme by means of reduced hardware cost/complexity. For
instance, when there are $C=16$ subcarriers, the energy-efficiency increases
from $32$ bps/Hz/W to $80$ bps/Hz/W. In comparison to EQ-RA, the proposed GA-
RA technique enhances the sum-rate and energy-efficiency by $28.6\%$,
$33.5\%$, $43.1\%$ for ${C=4,16,64}$ subcarriers, respectively. Hence, the
performance gap between GA-RA and EQ-RA keeps increasing for larger number of
subcarriers. On the other hand, the numerical results demonstrate that the
proposed GA-RA technique outperforms PSO-RA by further enhancing both sum-rate
and energy-efficiency.

(a) Sum-Rate

(b) Energy-Efficiency

Figure 2: Performance comparison among HP and FDP schemes ($P_{T}=40$ dBm and
$K=15$ users).

The average sum-rate per subcarrier is plotted versus the total transmit power
for $C=8$ and $C=64$ subcarriers in Fig. 3, where $K=3,9,15$ downlink users
are served. It is observed that the proposed GA-RA technique significantly
enhances the system capacity. For example, when the transmit power is
$P_{T}=40$ dBm, applying the proposed GA-RA technique with $Q=10$ generations
enhances the sum-rate performance for $C=8$ subcarriers by $19.66\%$,
$29.4\%$, $30.5\%$ and for $C=64$ subcarriers by $25.41\%$, $38\%$, $43.06\%$
compared with EQ-RA technique for $K=3,9,15$ users, respectively. In addition,
the performance gap between GA-RA and EQ-RA increases for the larger number of
users. On the other hand, when there are less number of users (e.g., $K=3$
users), the sum-rate performance is saturated in earlier generations.

(a) $C=8$ subcarriers

(b) $C=64$ subcarriers

Figure 3: Sum-rate per subcarrier versus the transmitted power.

Fig. 4(a) compares the average sum-rate per subcarrier performance of GA-RA
and EQ-RA techniques versus the number of subcarriers, where the transmit
power is chosen as $P_{T}=20,40,60\textrm{ dBm}$. Here, we observe that the
sum-rate improvement provided by the GA-RA technique is almost constant for
different subcarrier scenarios. Specifically, the performance of GA-RA is
better than EQ-RA by approximately $48.5$ bps/Hz/subcarrier at $P_{T}=40,60$
dBm. However, at $P_{T}=20$ dBm, the performance improvement is $47.6$
bps/Hz/subcarrier for $C=1$ subcarrier and it decays $38.2$ bps/Hz/subcarrier
for $C=64$ subcarriers. Moreover, it is shown that by increasing the number of
subcarriers, the sum-rate per subcarrier performance degrades since less power
is allocated to each subcarrier as discussed in expressed in (12).

By defining the sum-rate gain ratio as $\frac{R_{sum,GA}^{(Q)}}{R_{sum,EQ}}$,
Fig. 4(b) depicts that the proposed GA-RA technique enhances further the sum-
rate gain ratio by increasing the number of subcarriers. The importance of
this fact appears is more apparent in the case of low transmitted power.

##  VII Conclusions

In this work, a novel genetic algorithm based resource allocation (GA-RA)
technique has been developed for energy-efficient throughput maximization in
the MU-mMIMO-OFDM systems. Furthermore, the angular-based hybrid precoding
(AB-HP) scheme has been developed for the OFDM-based downlink transmission to
reduce the number of RF chains and the CSI overhead size. The AB-HP scheme
first develops a single RF beamformer block for all subcarriers via AoD
information, then builds a distinct BB precoder for each subcarrier via the
reduced-size effective CSI. Next, the GA-RA technique allocates power and
subcarrier resources among the users. The promising numerical results
demonstrate that the GA-RA technique greatly enhances the sum-rate (i.e.,
throughput) performance of MU-mMIMO-OFDM systems compared to the conventional
equal RA (EQ-RA). Also, the performance gain ratio with GA-RA increases for
the larger number of subcarriers, especially in case of low transmission
power.

(a)

(b)

Figure 4:  (a) Sum-rate per subcarrier and (b) sum-rate gain ratio versus
number of subcarriers ($K=15$ users).

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Wenzhe Liu1, Wei Xiao1, Meng Wang1, Shan Yang2, Yupeng Shi1, Yuyong Kang1, Dan
Su2, Shidong Shang1, Dong Yu2

# A High Fidelity and Low Complexity Neural Audio Coding

###### Abstract

Audio coding is an essential module in the real-time communication system.
Neural audio codecs can compress audio samples with a low bitrate due to the
strong modeling and generative capabilities of deep neural networks. To
address the poor high-frequency expression and high computational cost and
storage consumption, we proposed an integrated framework that utilizes a
neural network to model wide-band components and adopts traditional signal
processing to compress high-band components according to psychological hearing
knowledge. Inspired by auditory perception theory, a perception-based loss
function is designed to improve harmonic modeling. Besides, generative
adversarial network (GAN) compression is proposed for the first time for
neural audio codecs. Our method is superior to prior advanced neural codecs
across subjective and objective metrics and allows real-time inference on
desktop and mobile.

Index Terms: neural audio/speech codec, perception-based loss, GAN
compression, hybrid codec

##  1 Introduction

In recent years, video conferencing has become a new normal to help us connect
with others anytime and anywhere. Audio coding is one of the most important
technologies in real-time communication, which aims to compress the audio
signal with the minimum number of bits while retaining its quality.

Traditional coders can be grouped into two categories, waveform coders and
parametric coders [1]. They remove the redundancy in the audio signal by using
the knowledge of psycho-acoustics and speech signal characteristics. Hybrid
coders such as Opus [2] and AMR [3] combine waveform coders and parametric
coders by designing a careful pipeline.

Although the traditional coder such as Opus is successfully applied in WebRTC,
Zoom and YouTube, the speech quality will degrade at very low bitrates. In
order to improve the communication experiment and reduce bandwidth costs,
neural audio codecs are proposed with the development of deep learning. Deep
learning-based audio coders can be classified into two main streams, hybrid
neural audio codecs [4, 5] and end-to-end audio coding [6, 7, 8]. The former
leverages generative models as the decoder to generate the speech waveform
based on acoustic parameters which are hand-crafted or obtained from
conventional parametric coders. For example, LPCNet [4] is proposed to improve
the speech quality from low-bitrate Opus bitstream thanks to the high-quality
speech synthesis capability of generative models. Lyra [5] feeds quantized log
mel spectra to WaveGRU and outperforms Opus at low rates. Another follows the
``encoder-quantizer-decoder'' architecture, where the encoder compresses the
speech signal and extracts the latent representation, and the decoder recovers
the signal from the quantized features. Compared to hybrid coders, end-to-end
coders assign the transformation from acoustic features to latent
representations to the encoding side, which reduce the computational cost of
the device, and demonstrates better performance. The most popular architecture
is VQ-VAE [9]. SoundStream [6]/lyra-v2111https://github.com/google/lyra and
Encodec [7] take the time domain samples as input and achieve excellent audio
quality with end-to-end adversarial training. Besides, TF-Net [8] has also
made good performance by compressing and modeling in the frequency domain.
However, vector quantizer (VQ) raises the storage costs of applications while
improving encoding efficiency.

Among all these approaches, there are two problems: first, the reconstruction
speech quality and bandwidth (usually from 8 kHz to 12 kHz) are not
satisfactory, and quantized noise and/or spectral artifacts are produced due
to limitations of generative models' capabilities. Another problem is that
existing neural audio coding frameworks that require superior speech
representation usually have high computational and memory costs, and do not
meet the real-time requirement. To address the above problems, we design a
novel audio coding framework that combines deep neural networks and signal
processing technologies referring to classical coders. It is well known that
high-frequency bands need only a limited number of bits to reconstruct with
bandwidth extension (BWE). Therefore, the codec for wideband which consumes
more bits is replaced by the neural network. Aiming to overcome the limited
spectrum modeling capability of generative models, a perception-based
magnitude loss function and postfilter are proposed to constant critical-band
energies and suppress the quantization noise. In order to boost the inference
speed, lightweight operators are utilized. Meanwhile, GAN compression
especially for audio coding is employed for the first time to improve the
performance of the lightweight generator.

In summary, this paper makes the following contributions:

  * •

We propose a novel super wideband (SWB) audio codec framework that combines
the generative model and classical codecs to achieve superior audio quality
but low complexity.

  * •

We introduce GAN compression especially for audio coding for the first time,
and a series of low-resource operations (such as pooling, repeat, grouped
convolution, and the scaler quantizer) replace original modules to reduce
compute consumption and storage footprint.

  * •

We design a perception-based loss function and postfilter to further improve
audio quality.

The rest of the paper is organized as follows. In Section 2, we illustrate the
overall diagram. We present the experimental setup in Section 3. The results
and analysis are given in Section 4, and some conclusions are drawn in Section
5.

Figure 1: Overall diagram of the proposed framework.

##  2 Proposed system

###  2.1 Overview

The diagram of the proposed codec Penguins is shown in Figure 1. Different
from current neural codecs [6, 7, 8] that rely completely on deep learning for
audio compression, our proposed system consists of two main parts, namely the
wideband neural coding (WB-NC) module, and the super wideband bandwidth
extension (SWB-BWE) module. It is known that the neural vocoder often causes
spectral artifacts, especially at high frequencies due to its weak high-
frequency representation capability. On the other hand, conventional coders
allocate only a few bitrates to high-frequency bands. Therefore, WB-NC
leverages GAN [10] with the advantage of fast inference speed and lightweight
networks to model broadband components. SWB-BWE employs a bitrate of 2kbps for
the super wideband components encoding and the wideband improvement. The
decomposition and synthesis of the wideband signal and the high-band signal
are achieved by two-channel quadrature mirror filters (QMF) [11].

The SWB-BWE module refers to [12, 13] to transmit the spectrum information in
the modified discrete cosine transform (MDCT) domain. The WB-NC module is
composed of a sequence of three components, as illustrated in Figure 1. First,
an encoder is utilized to extract the raw audio waveform to a compressed
acoustic representation. After that, a quantizer discretizes the compressed
feature with the target bitrate. Finally, a decoder reconstructs the signal
from the discrete embeddings. This model is trained in an end-to-end fashion
together with discriminators. To alleviate the lack of harmonic expression
capability of GANs, a postfilter [14] is applied during inference. Each module
will be described in detail in the following.

###  2.2 Generator

The encoder, the quantizer, the decoder, and the post-filter comprise the
generator $G_{S}$ of the WB-NC module. The encoder components consist of a 1D
convolutional layer, a pre-processing block, followed by $N$ downsampling
blocks. Finally, a 1D convolution with $D$ channels and a kernel size of 3 is
used to generate the compressed acoustic representation. The first causal
convolutional layer takes 320 samples (20 ms) as the input and transforms them
into learnable embeddings. The channel of the first convolution is set to 16.
The pre-processing block and downsampling blocks are used to continuously
compress the time dimension until it is 1, where the pre-processing block is
composed of a causal convolution with 16 channels and a downsampling layer
with a downsample rate of 2, and the downsampling block includes three grouped
residual units with dilation rate $d=\\{1,3,9\\}$ and a downsampling layer
with downsampling rate $r=\\{2,4,4,5\\}$. The kernel sizes of all
convolutional layers are 3, and the number of channels is doubled in the
downsampling blocks. Different from previous codecs, average pooling replaces
the stride convolution as the downsampling layer due to its extremely low
computational cost. Therefore, one acoustic embedding is mapped from $2\times
2\times 4\times 4\times 5=320$ audio samples.

Between the encoder and the decoder, the quantizer is inserted to turn the
output of the encoder into discrete values. Due to large codebooks brought by
the residual vector quantization (RVQ) of the neural codecs increasing the
storage occupation of RTC software, the scalar quantizer (SQ) is used as an
alternative solution to quantize each of the embeddings after $tanh(\cdot)$
individually without increasing storage requirement.

Following the quantizer, the decoder reconstructs the input speech signal
utilizing discrete features, which is similar to the encoder but is the mirror
version of the encoder. The decoder component uses upsampling layers instead
of downsampling layers. Considering the calculation burden, we replace the
transpose convolution commonly used in previous work with the repeat
operation, and the upsampling rates are in the reverse order of the
downsampling rates.

###  2.3 Discriminator

Our adversarial training framework relies on multi-resolution STFT-based (MR-
STFT) patch discriminators $D_{S}$, which capture spectral structures of
different frequency resolutions. We also investigate other discriminators
commonly found in the vocoder such as multi-scale discriminator (MSD) [15] and
multi-period discriminator (MPD) [16], which have no performance benefits and
long training time. 6 different scales are used with FFT points of
$K=\\{60,120,240,480,960,1920\\}$. Each discriminator takes the magnitude
spectrum and its logarithmic spectrum is concatenated as input and is composed
of 7 2D convolution layers with a kernel size of $(3,3)$ and a stride of
$(1,1)$ or $(2,2)$.

###  2.4 Training loss functions

The training loss is a combination of a reconstruction loss term, an
adversarial loss term, and the feature match loss term. where the
reconstruction loss consists of a multi-resolution full-band and subband
short-time Fourier transform (STFT) loss and a proposed perception-based
magnitude (PM) loss.

For the multi-resolution STFT loss, we minimize the spectral convergence loss
[17] and the L1 distance in the logarithmic magnitude spectral domain with
different FFT analysis parameters, which can be written as:

$\mathcal{L}_{s}(X)=\sum_{r}{\left(||log(X_{r})-log(\widehat{X}_{r})||_{1}+\frac{||X_{r}-\widehat{X}_{r}||_{F}}{||\widehat{X}_{r}||_{F}}\right)},$
(1)

where $X_{r}$ and $\widehat{X}_{r}$ are the spectrum of the clean speech and
the predict waveform with FFT-point of $2^{r}$.

For full-band and subband cases, their corresponding loss functions are
defined as $\mathcal{L}_{s}(S)$ and $\mathcal{L}_{s}(S^{sub})$, where $S$ and
$S^{sub}$ represent the magnitude spectrum of the complete signal $s$ and
subband signals $s^{sub}$ decomposed by pseudo-quadrature mirror filters
(PQMF), respectively. It is known that always constant energy in the critical
bands is important for the codec, and perceptible quantization noise is often
due to the noise energy in spectral valleys being in excess of the auditory
masking. The PM loss function is proposed to solve these problems. For the
former, $\mathcal{L}_{s}(E)$ focuses on the energy in the designed auditory-
perceptual scale, where $E$ is the equivalent rectangular bandwidth (ERB)
spectra. For the latter, $\mathcal{L}_{val}=\sum_{r=9}^{11}{M\cdot
S_{r}^{p}\cdot||(S_{r}-\widehat{S}_{r})||_{1}}$ pays more attention to
spectral valleys, where $S_{r}^{p}$ is a weight implemented by $p$-law
compression of the spectrum, which can enlarge small values and reduce
spectral peaks when $p\textless 0$. $M$ is a mask of voice activate detection
(VAD) to avoid scaling up the values in silence frames. The loss function can
be expressed as:

$\mathcal{L}_{pm}=\mathcal{L}_{spec}(E)+\mathcal{L}_{valley}.$ (2)

The adversarial loss follows LS-GAN [18], which forces the generator to fool
the discriminator when training the generator. On the other hand, when the
discriminator is trained, the LS-GAN helps the discriminator recognize the
clean samples as 1, and the samples coded by WB-NC as 0. The loss functions
for the generator $G_{S}$ and the discriminator $D_{S}$ are expressed as:

$\displaystyle\mathcal{L}_{adv}=\mathbb{E}\left[(1-D_{S}(\hat{s}))^{2}\right],$
(3)
$\displaystyle\mathcal{L}_{D_{S}}=\mathbb{E}\left[(D_{S}(s)-1)^{2}+(D_{S}(\hat{s}))^{2}\right],$
(4)

Additionally, the feature match loss [15] is calculated to minimize the L1
distance between feature maps of the discriminator for real and generated
audio, which has been demonstrated to be effective in previous work.

$\mathcal{L}_{f}=\mathbb{E}\left[\frac{1}{L}\sum_{l=0}^{L-1}{|D_{S}^{l}(s)-D_{S}^{l}(\hat{s})|}\right],$
(5)

which $L$ denotes the number of the discriminator's layer.

The overall generator loss is a weighted sum of the above loss terms:

$\mathcal{L}_{G_{S}}=\mathcal{L}_{s}(S)+\mathcal{L}_{s}(S^{sub})+\lambda_{pe}\cdot\mathcal{L}_{pm}+\lambda_{adv}\cdot\mathcal{L}_{adv}+\lambda_{f}\cdot\mathcal{L}_{f}.$
(6)

###  2.5 GAN compression

As can be seen from the previous section, the computational complexity and
number of parameters of our proposed model are far less than those of previous
work. A series of low-computational-cost alternative operations are applied,
which inevitably degrade the performance of the speech reconstruction. In
addition to the improvement of speech coding quality using well-designed loss
functions mentioned in section 2.4, we also employ model compression methods
namely GAN compression [19, 20] to improve the performance of the proposed
model. The adversarial training of GAN is unstable since its generator and
discriminator are updated alternately as opponents compete. Consequently, GAN
requires a specific compression method, in difference to CNN compression. To
the best of our knowledge, this is the first proposed GAN-knowledge
distillation (KD) algorithm specific to audio codecs. We pre-train a high-
computational complexity model (teacher model $\\{G_{T},D_{T}\\}$) as the
teacher, which transfers the information to the proposed model (student model
$\\{G_{S},D_{S}\\}$) with GAN-KD. In the KD stage, the parameters of the
teacher model's discriminator are used to initialize the discriminator of the
lightweight model. We calculate the distance between the intermediate outputs
of $D_{T}$ and that of $D_{S}$, and the features of these two models'
generators are also calculated in the distillation loss. The KD loss can be
expressed as:

$\mathcal{L}_{KD}=\lambda_{G}\sum_{i}{||O_{i}^{G_{T}}-O_{i}^{G_{S}}||_{1}}+\lambda_{D}\sum_{i}{||O_{i}^{D_{T}}-O_{i}^{D_{S}}||_{1}}$
(7)

where $O_{i}^{\Theta}$ is the output of $i$-th layer of the model
$\Theta\in\\{G_{S};D_{S};G_{T},D_{T}\\}$. Additionally, we design a training
schedule for GAN-KD for stable and effective training. The learning rate for
the generator is updated with a cosine decay from 2e-4, and that for the
discriminator is always 1e-5. The weight decay is scheduled with an increasing
cosine schedule leading to a larger regularization for the later steps.

##  3 Experiments

###  3.1 Dataset

In the experiments, we use the LibriTTS (English) [21], DNS Challenge (English
and Mandarin) [22], and private datasets (English and Mandarin) as the speech
corpus. The noise clips from the DNS challenge are utilized as the noise set.
MIR-1k [23] and FMA [24] datasets are used as the music set. All clips are
resampled at 16 kHz, and the packet loss case is simulated for packet loss
concealment (PLC).

The evaluation metrics include objective and subjective evaluations. For the
objective evaluation, the test set is from ITU-T P.501 Annex
C222https://handle.itu.int/11.1002/1000/14271, which is a multilingual dataset
prepared for ITU-T P.800 conformant applications and perceptual-based
objective speech quality prediction. Please note that we only trained the
model with English and Mandarin utterances, but evaluated under 8 languages.
ViSQOL [25] is chosen as the objective evaluation metric because it has been
used in previous literature [6, 7], although they can only assess wideband
speech. Besides, the more standard and commonly used POLQA [26] has also been
added to evaluate speech quality. Moreover, eSTOI [27] is used as the metric
of intelligibility. Subjective performance is evaluated by an ITU-T P.808
recommended crowdsourced subjective listening test, which is divided into WB
and SWB sections. 24 native listeners in Mandarin as subjects for tests, with
12 Mandarin utterances.

###  3.2 Training configurations

The batch size is set to 16 with 8 A100 GPUs . The FFT points $2^{r}$ of the
MR-STFT are from 512 to 2048. The weights of loss terms
$\\{\lambda_{pe},\lambda_{adv},\lambda_{f},\lambda_{G},\lambda_{D}\\}$ are
$\\{2,1,20,30,10\\}$.

###  3.3 Baselines

Opus is one of the most popular classical codecs which ranges from 6 kbps
narrowband monophonic audio to 510 kbps full-band stereophonic audio. We use
Opus as the traditional codec baseline. Lyra-v2 is based on a famous end-to-
end audio codec namely SoundStream, which supports bitrates at 3.2 kbps and 6
kbps for 32 kHz audio samples. Like Lyra-v2, Encodec is also a state-of-the-
art neural audio coder which compresses 24 kHz audio samples to 1.5-24 kbps.
These models are chosen as the deep learning-based baselines and implemented
officially.

##  4 Results and analysis

Table 1: Objective results of different audio codecs. † denotes that Encodec
operates audio sampels in 24 kHz sampling rate.

Method | Bitrate | Bandwidth | POLQA | ViSQOL | eSTOI(%)
---|---|---|---|---|---
Opus | 6 kbps | WB | 1.82 | 2.03 | 58.45
Opus | 8 kbps | WB | 2.96 | 3.46 | 76.03
Opus | 10 kbps | SWB | 3.26 | 3.71 | 77.93
Opus | 16 kbps | WB | 4.26 | 4.30 | 85.30
Lyra-v2 | 9.2 kbps | WB | 3.49 | 3.92 | 90.81
Encodec | 12 kbps |  SWB† | 3.72 | 4.16 | 94.21
WB-NC | 6 kbps | WB | 3.66 | 3.88 | 91.59
+PM loss | 6 kbps | WB | 3.86 | 4.19 | 93.00
+GAN-KD | 6 kbps | WB | 4.06 | 4.22 | 94.63
+SWB-BWE | 8 kbps | SWB | 4.16 | 4.20 | 94.37

Table 2: Real time factor (RTF) for neural audio codecs.

Model | Bitrate | RTF
---|---|---
Enc. | Dec.
Lyra-v2 | 9.2 kbps | 0.015 | 0.034
Encodec | 12 kbps | 0.103 | 0.094
Proposed | 6 kbps | 0.032 | 0.028

###  4.1 Ablation study

Table 1 shows the performance gains obtained by the PM loss, the GAN
compression, and the SWB-BWE. One can find that, when the PM loss is applied,
it yields 0.20, 0.31 and 1.41% improvements in POLQA, ViSQOL and eSTOI,
respectively. This is because the PM loss emphasizes band energy consistency
and spectrum valley noise minimization. 0.20 and 0.10 POLQA gains are achieved
by using GAN-KD and SWB-BWE, showing the effect of these strategies.

###  4.2 Comparison with other codecs

The result comparison in terms of POLQA, ViSQOL and eSTOI among different
audio codecs is shown in Table 1. From the results, Several observations can
be made. Firstly, neural audio codecs can reduce the bitrates while achieving
better performance compared with classical signal processing codecs. Secondly,
The proposed system significantly outperforms baselines at a lower bit rate.
From Lyra-v2 to our approach, around 0.67, 0.30 and 3.82% improvements are
achieved in terms of POLQA, ViSQOL and eSTOI, indicating the superiority of
the proposed framework. Compared with Encodec, our method achieves similar
ViSQOL and eSTOI, but 0.44 POLQA gain.

Figure 2: Subjective evaluation results. The red and blue dotted lines denote
the scores of the reference in the WB case and SWB case, respectively.

###  4.3 Subjective evaluation

The subjective evaluation results of different codecs are reported in Figure 2
, where *-WB and *-SWB represent wideband and super wideband subjective
listening testings. We find that the proposed system outperforms advanced
neural audio coders, which is consistent with objective evaluation. Note that
the performance of the proposed framework is close to Opus at 20 kbps which is
the recommended bit rate for wideband speech communication, demonstrating the
excellent performance of our approach.

###  4.4 Inference speed

In Table 2, we present the real-time factors (RTF) of neural audio codecs on a
single thread of a MacBook Pro 2019 (2.6 GHZ i7). Compared with Encodec, the
proposed model is better in terms of RTF. Meanwhile, the inference speed of
the proposed model is comparable to that of Lyra-v2. We intended the decoder
to have a lower computational complexity since it is common to send only one
bitstream but possibly decode multiple bitstreams during conferencing.
Besides, the inference speed of the proposed method optimized by
NCNN333https://github.com/Tencent/ncnn on iPhone XS is evaluated, The RTFs of
the encoder and the decoder are 0.017 and 0.012, showing the proposed
framework is also sufficient to run in real-time on mobile.

##  5 Conclusions

In this paper, we propose a hybrid codec that combines the traditional codec
and the neural codec to implement efficient compression and high-fidelity
reconstruction. Low computational cost operators, perception-based loss, and
GAN compression are proposed to improve the generation capability of the GAN.
Experimental results show that our framework outperforms prior arts across the
objective and subjective evaluation.

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