# Outage Performance Analysis of Full-Correlated Rayleigh MIMO Channels

Huan Zhang1, Guanghua Yang2, Zheng Shi${}^{\textrm{{\char 0\relax}}}$2,
Shaodan Ma1, and Hong Wang3
1Department of Electrical and Computer Engineering, University of Macau, China
2School of Intelligent Systems Science and Engineering  Jinan University
Zhuhai  China
3School of Communication and Information Engineering  Nanjing University of
Posts and Telecommunications  Najing  China

###### Abstract

The outage performance of multiple-input multiple-output (MIMO) technique has
received intensive attention to meet the stringent requirement of reliable
communications for 5G applications, e.g., mission-critical machine-type
communication (cMTC). To account for spatial correlation effects at both
transmit and receive sides, the full-correlated Rayleigh MIMO fading channels
are modeled according to Kronecker correlation structure in this paper. The
outage probability is expressed as a weighted sum of the generalized Fox’s H
functions. The simple analytical result empowers asymptotic outage analysis at
high signal-to-noise ratio (SNR), which not only reveal helpful insights into
understanding the behavior of fading effects, but also offer useful design
guideline for MIMO configurations. Particularly, the negative impact of the
spatial correlation on the outage probability is revealed by using the concept
of majorization, and the asymptotic outage probability is proved to be a
monotonically increasing and convex function of the transmission rate. In the
end, the analytical results are validated through extensive numerical
experiments.

###### Index Terms:

Asymptotic analysis, Mellin transform, MIMO, Outage probability, Spatial
correlation.

##  I Introduction

††The corresponding author is Zheng Shi. This work was supported in part by
the National Key Research and Development Program of China under Grant
2017YFE0120600, in part by National Natural Science Foundation of China under
Grants 61801192 and 61801246, in part by Guangdong Basic and Applied Basic
Research Foundation under Grant 2019A1515012136, in part by the Science and
Technology Planning Project of Guangdong Province under Grants 2018B010114002
and 2019B010137006, in part by the Science and Technology Development Fund,
Macau SAR (File no. 0036/2019/A1 and File no. SKL-IOTSC2018-2020), in part by
Open Research Foundation of National Mobile Communications Research Laboratory
of Southeast University under Grant 2018D09, and in part by the Research
Committee of University of Macau under Grant MYRG2018-00156-FST.

5G systems are anticipated to not only support extraordinarily high data rate
and capacity, but also provide ultra-reliability, scalability and low latency
for emerging communication paradigms, e.g., mission-critical machine-type
communications (cMTC) [1]. The multiple-input multiple-output (MIMO) is an
indispensable enabler that fulfills these requirements for 5G. Specifically,
the MIMO technique capitalizes on the spatial dimension to explore its
potentials of boosting the spectral efficiency and reliability [2]. Most of
the existing works concentrate on studying the information-theoretical
capacity of MIMO systems for the purpose of the spectral efficiency
enhancement[3, 4].

Aside from the spectral efficiency, the reliability of communications has
become of ever-increasing importance in the Internet-of-Things (IoT)
applications (e.g., automated transportation, industrial control and
augmented/virtual reality), ultra-reliable low-latency communications (URLLC)
and tactile Internet [5]. Since the outage probability is frequently used to
characterize the reception reliability, the outage probability of MIMO systems
also has attracted considerable attention in the literature [2, 6, 7, 8].
However, the prior works in [2, 6, 7, 8] did not take into account the
correlation between antenna elements, which exists in realistic propagation
environments because of mutual antenna coupling and close spacing between
adjacent elements [9]. The spatial correlation would remarkably impair the
reliability of MIMO systems. In [10], the character expansion method was
initially introduced to give a closed-form expression for the moment-
generating function (MGF) of the capacity under full-correlated (correlation
at both the transmitter and receiver) Rayleigh MIMO channels if the numbers of
transmit and receive antennas are identical. The same method was further
extended to derive the MGF of the capacity for the case with arbitrary numbers
of transmit and receive antennas in [11]. Unfortunately, the outage
probability for full-correlated Rayleigh MIMO systems was obtained in the
literature by relying upon either approximations or numerical inversions,
which impede the extraction of more helpful insights about the system
parameters. Finally, the analytical results are verified by numerical
analysis.

To address the above issues, The Mellin transform is applied in this paper to
derive exact and tractable representations for the outage probabilities. Upon
the exact expressions, the asymptotic analysis of the outage probability in
the high signal-to-noise ratio (SNR) regime is derived. The expression
demonstrates that full diversity can be achieved regardless of the presence of
spatial correlation. The qualitative relationship between the spatial
correlation and the outage probability is established by virtue of the concept
of majorization in [9]. Moreover, the transmission rate affects the outage
performance via the term of modulation and coding gain, and the asymptotic
outage probability is found to be an increasing and convex function of the
transmission rate.
_Notations:_ We shall use the following notations throughout the paper. Bold
uppercase and lowercase letters are used to denote matrices and vectors,
respectively. ${\bf A}^{\mathrm{H}}$, ${\bf A}^{-1}$ and ${\bf A}^{1/2}$
denote the conjugate transpose, matrix inverse and Hermitian square root of
matrix ${\bf A}$, respectively. $\mathrm{vec}$, $\rm{tr}$, $\mathrm{det}$ and
$\mathrm{diag}$ are the operators of vectorization, trace, determinant and
diagonalization, respectively. $\Delta\left({\bf A}\right)$ refers to the
Vandermonde determinant of the eigenvalues of matrix ${\bf A}$.
$\mathbf{0}_{n}$ and $\mathbf{I}_{n}$ stand for $1\times n$ all-zero vector
and $n\times n$ identity matrix, respectively. $\mathbb{C}^{m\times n}$
denotes the sets of $m\times n$-dimensional complex matrices. The symbol ${\rm
i}=\sqrt{-1}$ is the imaginary unit. $o(\cdot)$ denotes little-O notation.
$(\cdot)_{n}$ represents Pochhammer symbol. $|S|$ refers to the cardinality of
set $S$. Any other notations will be defined in the place where they occur.

##  II System Model

By considering a point-to-point MIMO system with $N_{t}$ transmit and
${N_{r}}$ receive antennas, the received signal vector
$\mathbf{y}\in\mathbb{C}^{{N_{r}}\times 1}$ is written as

$\mathbf{y}=\sqrt{\frac{P}{{N_{t}}}}\mathbf{H}\mathbf{x}+\mathbf{n},$ (1)

where $\mathbf{H}\in\mathbb{C}^{{N_{r}}\times{N_{t}}}$ is the matrix of the
channel coefficients, $\mathbf{x}\in\mathbb{C}^{{N_{t}}\times 1}$ denotes the
vector of transmitted signals, $\mathbf{n}\in\mathbb{C}^{{N_{r}}\times 1}$
represents the complex-valued additive white Gaussian noise vector with zero
mean and covariance matrix $\sigma^{2}\mathbf{I}_{N_{r}}$, and $P$ is the
total average transmitted power. Moreover, in order to account for the effect
of the antenna correlation, the channel matrix $\mathbf{H}$ is modeled herein
according to the Kronecker correlation structure as[12]

$\mathbf{H}={\mathbf{R}_{r}}^{{1}/{2}}{\bf H}_{w}{\mathbf{R}_{t}}^{{1}/{2}},$
(2)

where $\mathbf{H}_{w}\in\mathbb{C}^{{N_{r}}\times{N_{t}}}$ is a random matrix
whose entries are independent and identically distributed (i.i.d.), complex
circularly symmetric Gaussian random variables, i.e.,
$\mathrm{vec}\left(\mathbf{H}_{w}\right)\sim\mathcal{CN}\left(\mathbf{0}_{{N_{t}}{N_{r}}},\mathbf{I}_{N_{t}}\otimes\mathbf{I}_{N_{r}}\right)$,
$\mathbf{R}_{t}$ and $\mathbf{R}_{r}$ are respectively termed as the transmit
and receive correlation matrices, and both of them are positive semi-definite
Hermitian matrices. For the sake of simplicity, we assume that the correlation
matrices follow the constraints as ${\rm
tr}{\left(\mathbf{R}_{t}\right)}=N_{t}$ and ${\rm
tr}{\left(\mathbf{R}_{r}\right)}=N_{r}$. From the perspective of information
theory for MIMO system in [11], the outage probability can be expressed as

$\displaystyle p_{out}$
$\displaystyle=\Pr\left({{{\log}_{2}}\det\left({{{\bf{I}}_{N_{r}}}+{\rho}{\bf{H}}{{\bf{H}}^{\rm{H}}}}\right)<R}\right)$
(3)
$\displaystyle=\Pr\left(G\triangleq{\prod\limits_{i=1}^{{N_{r}}}{\left({1+{\rho}{\lambda_{i}}}\right)}}<{2^{R}}\right)=F_{G}(2^{R}),$

where $\rho={P}/{(\sigma^{2}N_{t})}$ stands for the average transmit SNR per
antenna, $\lambda_{1},\cdots,\lambda_{N_{r}}$ denotes the unordered
eigenvalues of ${{\bf{H}}}{\bf{H}}^{\rm{H}}$111It is worth noting that
${{\bf{H}}}{\bf{H}}^{\rm{H}}$ is a singular matrix and has at least
$(N_{r}-N_{t})$ zero eigenvalues if ${N_{t}}<{N_{r}}$., $F_{G}(x)$ denotes the
cumulative distribution function (CDF) of $G$, with the property [13, Exercise
7.25, p167], the outage probabilities for $N_{t}\geq N_{r}$ and $N_{t}<N_{r}$
can be derived in the same fashion. Hence, we assume $N_{t}\geq N_{r}$ in the
sequel unless otherwise specified. From (3), it boils down to determining the
distribution of the product of multiple shifted eigenvalues
$\boldsymbol{\lambda}=(\lambda_{1},\cdots,\lambda_{N_{r}})$.

##  III Analysis of Outage Probability

The outage probability is the fundamental performance metric to characterize
the error performance of decodings. However, the occurrence of the correlation
among eigenvalues will yield the involvement of a multi-fold integral in
deriving the expression of $F_{G}(2^{R})$. Nonetheless, the product form of
$G$ motivates us to apply Mellin transform to obtain the distribution of $G$
[14]. Specifically, the Mellin transform of the probability density function
(PDF) of $G$, $\left\\{{\mathcal{M}f_{G}}\right\\}\left(s\right)$, is given by

$\displaystyle\varphi(s)=\int\nolimits_{0}^{\infty}{\cdots\int\nolimits_{0}^{\infty}{\prod\limits_{i=1}^{N_{r}}{{{\left({1+{\rho}{\lambda_{i}}}\right)}^{s-1}}f\left({\boldsymbol{\lambda}}\right)d\lambda_{1}\cdots
d{\lambda_{N_{r}}}}}}.$ (4)

By utilizing the inverse Mellin transform together with its associated
property of integration [15, eq.(8.3.15)], the CDF of $G$ can be obtained as

$\displaystyle{F_{G}}\left(x\right)$
$\displaystyle=\left\\{{\mathcal{M}}^{-1}{\left[-\frac{1}{s}\varphi\left({s+1}\right)\right]}\right\\}\left(x\right)$
(5) $\displaystyle=\frac{{1}}{{2\pi{\rm{i}}}}\int\nolimits_{c-{\rm
i}\infty}^{c+{\rm
i}\infty}{\frac{{{x^{-s}}}}{-s}\varphi\left({s+1}\right)ds},$

where $c\in(-\infty,0)$, because the Mellin transform of
${F_{G}}\left(x\right)$ exists for any complex number $s$ in the fundamental
strip $-\infty<\Re{(s)}<0$ by noticing ${F_{G}}\left(x\right)=0$ for $x<1$ and
$\lim\nolimits_{x\to\infty}{F_{G}}\left(x\right)=1$ [16, p400].

###  III-A Exact Outage Probability

By favor of the character expansions, the joint distribution of the $N_{r}$
unordered strictly positive eigenvalues of $\mathbf{H}\mathbf{H}^{\mathrm{H}}$
is obtained by Ghaderipoor _et al._ in [17] as

$f\left({\boldsymbol{\lambda}}\right)=\sum\nolimits_{{{\bf{k}}_{N_{r}}}}{\frac{{{{\left({-1}\right)}^{\frac{{{N_{r}}\left({{N_{r}}-1}\right)}}{2}}}\mathcal{A}}}{{{N_{r}}!\Delta\left({\bf{K}}\right)}}\Delta\left({\boldsymbol{\lambda}}\right)\det\left({{{\left\\{{{\lambda_{i}}^{{k_{j}}+{N_{t}}-{N_{r}}}}\right\\}}_{{i,j}}}}\right)},$
(6)

where
$\mathcal{A}=\frac{{{\prod\nolimits_{i=1}^{N_{r}}{{a_{i}}^{N_{t}}}\prod\nolimits_{j=1}^{N_{t}}{{b_{j}}^{N_{r}}}}}}{{{\Delta\left({\bf{A}}\right)\Delta\left({\bf{B}}\right)}}{\prod\nolimits_{j=1}^{N_{r}}{\left({{k_{j}}+{N_{t}}-{N_{r}}}\right)!}}}\det{(\\{{{\left({-{a_{i}}}\right)}^{{k_{j}}}}\\}_{i,j})}\times\det({{{\\{{{b_{i}}^{{k_{j}}+N-M}}\\}}_{{i,1\leq
j\leq M}}},{{\\{{{b_{i}}^{N-j}}\\}}_{{i,M+1\leq j\leq N}}}})$, ${\bf
a}=(a_{1},\cdots,a_{N_{r}})$ and ${\bf b}=(b_{1},\cdots,b_{N_{t}})$ represent
the eigenvalue vectors of ${{\bf R}_{r}}^{-1}$ and ${{\bf R}_{t}}^{-1}$,
respectively, ${\bf k}_{N_{r}}=(k_{1},\cdots,k_{N_{r}})$ stands for all
irreducible representation of the general linear group
$\mathrm{GL}({N_{r}},\mathbb{C})$ and $k_{1}\geq\cdots\geq k_{N_{r}}$ are
integers, ${\bf A}$, $\bf B$ and $\bf K$ are the diagonalizations of vectors
$\bf a$, $\bf b$ and ${\bf k}_{N_{r}}$, respectively.

By substituting (6) into (4), the Mellin transform of $f_{G}(x)$ is expressed
as

$\displaystyle\varphi\left(s\right)$
$\displaystyle=\frac{{{{\left({-1}\right)}^{{N_{r}}\left({{N_{t}}-{N_{r}}}\right)}}{{{{\rho}}}^{-\frac{1}{2}{{{N_{r}}\left({{N_{r}}+1}\right)}}}}\prod\nolimits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}\prod\nolimits_{j=1}^{N_{t}}{{b_{j}}^{N_{r}}}}}{{\Delta\left({\bf{A}}\right)\Delta\left({\bf{B}}\right)\prod\nolimits_{i=1}^{N_{r}}{{{\left({s+i-2}\right)}^{i-1}}}}}$
(7)
$\displaystyle\times\det\left({\begin{array}[]{*{20}{c}}{{{\left\\{{\Psi\left({1,s+{N_{r}};\frac{{{a_{i}}{b_{j}}}}{\rho}}\right)}\right\\}}_{1\leq
i\leq{N_{r}},j}}}\\\ {{{\left\\{{{b_{j}}^{{N_{t}}-i}}\right\\}}_{{N_{r}}+1\leq
i\leq{N_{t}},j}}}\end{array}}\right).$

where $\Psi\left(\cdot,\cdot;\cdot\right)$ denotes confluent hypergeometric
function [18, eq. (9.210)].

###### Proof.

Please see Appendix A ∎

By substituting (7) into (5), the CDF of $G$ can then be obtained as shown in
the following theorem.

######  Theorem 1.

The CDF of $G$ is given by

$\displaystyle{F_{G}}\left(x\right)$
$\displaystyle=\Lambda{{\rho}^{\frac{1}{2}{{{N_{r}}\left({{N_{r}}-1}\right)}}}}\sum\limits_{{\boldsymbol{\sigma}}\in{S_{N_{t}}}}{\mathop{\rm
sgn}}\left({\boldsymbol{\sigma}}\right)\times$ (8)
$\displaystyle{\prod\limits_{i={N_{r}}+1}^{N_{t}}{{b_{{\sigma_{i}}}}^{{N_{t}}+{N_{r}}-i}}}\underbrace{Y_{{N_{r}},2{N_{r}}}^{{N_{r}},{N_{r}}}\left[{\left.{\begin{array}[]{*{20}{c}}\mathbf{C}\\\
\mathbf{D}\end{array}}\right|x{\prod\limits_{i=1}^{N_{r}}{\frac{{{a_{i}}{b_{{\sigma_{i}}}}}}{\rho}}}}\right]}_{\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)},$

where
$\Lambda=\frac{{{{\left({-1}\right)}^{{N_{r}}\left({{N_{t}}-{N_{r}}}\right)+\frac{1}{2}{{{N_{r}}\left({{N_{r}}-1}\right)}}}}}}{{\Delta\left({\bf{A}}\right)\Delta\left({\bf{B}}\right)}}$,
${\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)}$ is the generalized Fox’s H
function which is defined by using the integral of Mellin-Branes type and
provided by an efficient MATHEMATICA implementation as [19],
$\mathbf{C}=[{\left({1,1,0,1}\right),{{\left({{N_{r}},1,0,1}\right)}_{j=1,\cdots,{N_{r}}-1}}}]$,
and
$\mathbf{D}=[{{{({{N_{r}},1,\frac{{{a_{{i}}}{b_{\sigma_{i}}}}}{\rho},1})}_{i=1,\cdots,{N_{r}}}},\left({0,1,0,1}\right),{{\left({j,1,0,1}\right)}_{j=1,\cdots,{N_{r}}-1}}}]$.

###### Proof.

The proof is given in Appendix B. ∎

Substituting (8) into (3), the outage probability under fully correlated
Rayleigh MIMO channels can be obtained.

The generalized Fox’s H function involved in (8) is too complex to extract
insightful results. In order to obtain tractable results and gain more
insights, we have to recourse to the asymptotic analysis of the outage
probability at high SNR.

###  III-B Asymptotic Outage Probability

In order to derive the asymptotic expression at high SNR for the outage
probability in (8), the following lemma associated with the asymptotic
expression of ${\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)}$ is developed first.

######  Lemma 1.

As $\rho\to\infty$, ${\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)}$ is asymptotic
to

$\displaystyle{\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)}$
$\displaystyle={{{{\rho}}}^{-{N_{r}}^{2}}}\prod\limits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}{{b_{\sigma_{i}}}^{N_{r}}}\frac{1}{{2\pi{\rm{i}}}}\int\nolimits_{-c-{\rm
i}\infty}^{-c+{\rm
i}\infty}\frac{{\Gamma\left(s\right)}}{{\Gamma\left({1+s}\right)}}$ (9)
$\displaystyle\times\prod\limits_{i=1}^{N_{r}}{\frac{{\Gamma\left({s-{N_{r}}}\right)}}{{\Gamma\left({s-i+1}\right)}}}{\sum\limits_{{n_{{i}}}=0}^{\infty}{\frac{{{{\left({\frac{{{a_{{i}}}{b_{\sigma_{i}}}}}{\rho}}\right)}^{{n_{{i}}}}}}}{{{{\left({1+{N_{r}}-s}\right)}_{{n_{{i}}}}}}}}{x^{s}}ds}$
$\displaystyle+o\left({{\rho^{-{N_{t}}{N_{r}}-\frac{1}{2}N_{r}(N_{r}+1)}}}\right).$

###### Proof.

Please see Appendix C. ∎

By using Lemma 1, the asymptotic expression of the outage probability is given
by the following theorem.

######  Theorem 2.

At high transmit SNR, the outage probability asymptotically equals

$\displaystyle
p_{out}=\frac{{{{{{\rho}}}^{-{N_{r}}{N_{t}}}}}}{{\det\left({{{{\bf{R}}_{r}}}}\right)^{N_{t}}\det\left({{{{\bf{R}}_{t}}}}\right)^{N_{r}}}}g(R)+o\left({{\rho^{-{N_{t}}{N_{r}}}}}\right),$
(10)

where
$g(R)=G_{{N_{r}}+1,{N_{r}}+1}^{0,{N_{r}}+1}\left({\left.{\begin{array}[]{*{20}{c}}{1,{N_{t}}+1,\cdots,{N_{t}}+{N_{r}}}\\\
{0,1,\cdots,{N_{r}}}\end{array}}\right|2^{R}}\right)$ denotes the Meijer
G-function[18].

###### Proof.

Please see Appendix D. ∎

It is worth mentioning that the asymptotic analysis of the outage probability
for $N_{t}<N_{r}$ can be carried out in an analogous way owing to the property
[13, Exercise 7.25, p167]. Similar to (8) and (10), the exact and asymptotic
outage expressions for $N_{t}<N_{r}$ can be obtained by directly interchanging
${\bf R}_{t}$ and $N_{t}$ with ${\bf R}_{r}$ and $N_{r}$, respectively.

##  IV Discussions of Asymptotic Results

The asymptotic outage probabilities commonly exhibit the same basic
mathematical structure as [20, eq.(3.158)]

$p_{out}=\mathcal{S}({\bf R}_{t},{\bf
R}_{r}){\left({{\mathcal{C}(R)}}\rho\right)^{-d}}+o\left({{\rho^{-d}}}\right),$
(11)

where $\mathcal{S}({\bf R}_{t},{\bf R}_{r})$ quantifies the impact of spatial
correlation at transmit and receive sides, $\mathcal{C}(R)$ is the modulation
and coding gain, and $d$ stands for the diversity order. By identifying (10)
with (11), we get

$\mathcal{S}({\bf R}_{t},{\bf
R}_{r})=\frac{1}{{\det\left({{{{\bf{R}}_{r}}}}\right)^{N_{t}}\det\left({{{{\bf{R}}_{t}}}}\right)^{N_{r}}}},$
(12) $\mathcal{C}(R)={\left({g_{\bf{0}}}(R)\right)}^{-\frac{1}{N_{t}N_{r}}},$
(13)

and $d=N_{t}N_{r}$, respectively. To comprehensively understand the asymptotic
behavior of the outage probability, these impact factors are discussed
individually.

###  IV-A Diversity Order

The terminology of the diversity order can be used to measure the degree of
freedom of communication systems, which is defined as the ratio of the outage
probability to the transmit SNR on a log-log scale as

$d=\mathop{\lim}\limits_{\rho\to\infty}\frac{{\log{p_{out}}}}{{\log\rho}}.$
(14)

Hence, the diversity order indicates the decaying speed of the outage
probability with respect to the transmit SNR. As shown in (10), full diversity
can be achieved by MIMO systems regardless of the presence of the spatial
correlation, i.e., $d=N_{t}N_{r}$.

###  IV-B Modulation and Coding Gain

The modulation and coding gain $\mathcal{C}(R)$ quantifies the amount of the
SNR reduction required to reach the same outage probability when employing a
certain modulation and coding scheme (MCS). Accordingly, the increase of
$\mathcal{C}(R)$ is in favor of the improvement of the outage performance.
From (13), in order to disclose the behaviour of $\mathcal{C}(R)$, it suffices
to investigate the property of the function ${g_{\bf{0}}}(R)$. Towards this
end, we can arrive at following Theorem.

######  Theorem 3.

${g_{\bf{0}}}(R)$ is a monotonically increasing and convex function of the
transmission rate $R$.

###### Proof.

Similar to [14, Lemma 4], the detailed proof is omitted here to save space. ∎

Evidently from Theorem 3, the transmission rate is an increasing and convex
function of the asymptotic outage probability. Without dispute, the
monotonicity and convexity of ${g_{\bf{0}}}(R)$ can greatly facilitate the
optimal rate selection of MIMO systems if the asymptotic results are used.

###  IV-C Spatial Correlation

Although the effect of the spatial correlation can be quantified by
$\mathcal{S}({\bf R}_{t},{\bf R}_{r})$, it is also imperative to draw a
qualitative conclusion about the outage behaviour of the spatial correlation.
To characterize the spatial correlation, the majorization theory is usually
adopted as a powerful mathematical tool to establish a tractable framework [9,
21]. The majorization-based correlation model is defined as follows.

######  Definition 1.

For two $N\times N$ semidefinite positive matrices $\mathbf{R}_{1}$ and
$\mathbf{R}_{2}$, ${\bf r}_{1}=(r_{1,1},\cdots,r_{1,{N}})$ and ${\bf
r}_{2}=(r_{2,1},\cdots,r_{2,{N}})$ are defined as the vectors of the
eigenvalues of $\mathbf{R}_{1}$ and $\mathbf{R}_{2}$, respectively, where the
eigenvalues are arranged in descending order as
${r_{i,1}}\geq\cdots\geq{r_{i,{N}}},i\in\left\\{{1,2}\right\\}$. We denote
$\mathbf{R}_{1}\preceq\mathbf{R}_{2}$ and say the matrix $\mathbf{R}_{1}$ is
majorized by the matrix $\mathbf{R}_{2}$ if

$\sum\limits_{j=1}^{k}{{r_{1,j}}}\leq\sum\limits_{j=1}^{k}{{r_{2,j}}},\,\left({k\leq{N}-1}\right)\quad{\rm
and}\quad\sum\limits_{j=1}^{N}{{r_{1,j}}}=\sum\limits_{j=1}^{N}{{r_{2,j}}}.$

(15)

We also say ${\mathbf{R}}_{1}$ is more correlated than $\mathbf{R}_{2}$.

It is easily found by definition that ${\rm
diag}(1,1,\cdots,1)\preceq\mathbf{R}_{i}\preceq{\rm diag}({N},0,\cdots,0)$ if
${\rm tr}(\mathbf{R}_{i})=N$, where ${\rm diag}({N_{t}},0,\cdots,0)$ and ${\rm
diag}(1,1,\cdots,1)$ correspond to completely correlated and independent
cases, respectively. Notice that
$\det(\mathbf{R}_{i})=\prod\nolimits_{j=1}^{N}r_{i,j}$, the property of the
majorization in [22, F.1.a] proves that the determinant of the correlation
matrix is a Schur-concave function, where
$\det(\mathbf{R}_{1})\geq\det(\mathbf{R}_{2})$ if
$\mathbf{R}_{1}\preceq\mathbf{R}_{2}$, the interested reader is referred to
[22] for further details regarding the schur monotonicity. By recalling
$\mathcal{S}({\bf R}_{t},{\bf R}_{r})$ is the composition of the determinants
and using the fact associated with the composition involving Schur-concave
functions [22], we arrive at

$\mathcal{S}({\bf R}_{t_{1}},{\bf R}_{r_{1}})\leq\mathcal{S}({\bf
R}_{t_{2}},{\bf R}_{r_{2}}),$ (16)

whenever ${\bf R}_{t_{1}}\preceq{\bf R}_{t_{2}}$ and ${\bf
R}_{r_{1}}\preceq{\bf R}_{r_{2}}$. As a consequence, it is concluded that the
presence of the spatial correlation adversely impacts the outage performance.

##  V Numerical Results

In this section, numerical results are presented for verifications and
discussions. For notational convenience, we define $\mathbf{t}$ and
$\mathbf{r}$ as the row vectors of the eigenvalues of the transmit and receive
correlation matrices, i.e., $\mathbf{R}_{t}$ and $\mathbf{R}_{r}$.

###  V-A Verifications

Figs. 1 depicts the outage probability versus the transmit SNR. The labels
‘Sim.’, ‘Exa.’ and ‘Asy.’ in this figure indicates the simulated, exact and
asymptotic outage probabilities, respectively. As observed in fig. 1, the
exact and simulation results are in perfect agreement, which confirms the
correctness of the exact analysis. Besides, it can be seen that the asymptotic
results coincide well with the exact and simulation ones at high SNR, which
validates the asymptotic results as well.

Figure 1: Outage probability versus the transmit SNR $\rho$ with ${N_{t}}=3$,
${N_{r}}=2$, ${\mathbf{t}}=(2.7,0.2,0.1)$, ${\mathbf{r}}=(1.9,0.1)$, and
$R=$2bps/Hz.

###  V-B Coding and Modulation Gain

Fig. 2 illustrates the impacts of the transmission rate $R$ on the coding and
modulation gain ${\mathcal{C}(R)}$ for different numbers of transmit and
receive antennas. It is shown in Fig. 2 that ${\mathcal{C}(R)}$ increases with
the number of antennas, which further justifies the benefit of using MIMO.
Additionally, it can be observed from Fig. 2 that the increase of the
transmission rate impairs the coding and modulation gain ${\mathcal{C}(R)}$,
which consequently leads to the deterioration of the outage performance. This
is consistent with the asymptotic analysis in Section IV-B. Aside from
degrading the outage performance, the increase of the transmission rate causes
the enhancement of the system throughput. The two opposite effects force us to
properly select the transmission rate in practice. Fortunately, the optimal
rate selection can be eased by using the asymptotic outage probability thanks
to its increasing monotonicity and convexity with respect to the transmission
rate.

Figure 2: Coding and modulation gain ${\mathcal{C}(R)}$ versus the
transmission rate $R$.

###  V-C Impact of Spatial Correlation

In Fig. 3, the outage probability is plotted against the transmit SNR under
three different transmit correlation matrices, i.e., ${\bf{R}}_{t_{1}}$,
${\bf{R}}_{t_{2}}$ and ${\bf{R}}_{t_{3}}$. For notational simplicity, the
vectors of the eigenvalues are set as ${\mathbf{t}}_{1}=(1,1,1)$,
${\mathbf{t}}_{2}=(2.3,0.5,0.2)$ and ${\mathbf{t}}_{3}=(2.7,0.2,0.1)$. It is
clear from the figure that the spatial correlation does not affect the
diversity order. Moreover, according to the concept of majorization, the
relationship of the transmit correlation matrices follows as
${{\bf{R}}_{t_{3}}}\succeq{{\bf{R}}_{t_{2}}}\succeq{{\bf{R}}_{t_{1}}}$, and
$\mathbf{R}_{t_{3}}$ are the most correlated correlation matrix among them. It
is readily observed in Fig. 3 that the outage probability curve associated
with $\mathbf{R}_{t_{3}}$ displays the worst performance. The numerical result
corroborates the validity of the analytical results in Section IV-C.

Figure 3: Outage probability versus the transmit SNR $\rho$ with
${N_{t}}={N_{r}}=3$ and ${\mathbf{r}}=(2.7,0.2,0.1)$.

##  VI Conclusions

This paper has derived novel representation for the outage probabilitiy of the
MIMO system by invoking Mellin transform. The compact and simple expression
not only has enabled the accurate evaluation of the outage probability, but
also has facilitated the asymptotic analysis under high SNR to gain a profound
understanding of fading effects and MIMO configurations, which has never been
performed in the literature. On the one hand, the asymptotic results have
revealed meaningful insights into the effects of the spatial correlation, the
number of antennas, and transmission rate. For instance, the spatial
correlation degenerates the outage performance, while full diversity can be
achieved even at the presence of the spatial correlation. On the other hand,
the asymptotic result have paved the way for the simplification of practical
system designs. For example, the increasing monotonicity and convexity of the
asymptotic outage probability will facilitate the proper selection of target
transmission rate.

## Appendix A

By substituting (6) into (4), using the identity
$\Delta\left({\boldsymbol{\lambda}}\right)=\det\left({{{\left\\{{{\lambda_{i}}^{j-1}}\right\\}}_{{i,j}}}}\right)$
and the Leibniz formula for the determinant expansion [23], we can obtain that

$\begin{aligned}
\varphi\left(s\right)&=\sum\nolimits_{{{\bf{k}}_{N_{r}}}}\frac{{{{\left({-1}\right)}^{\frac{{{N_{r}}\left({{N_{r}}-1}\right)}}{2}}}{\cal
A}}}{{{N_{r}}!\Delta\left({\bf{K}}\right)}}\sum\limits_{{{\boldsymbol{\sigma}}_{1}},{{\boldsymbol{\sigma}}_{2}}\in{S_{N_{r}}}}{\mathop{\rm
sgn}}\left({{{\boldsymbol{\sigma}}_{1}}}\right){\mathop{\rm
sgn}}\left({{{\boldsymbol{\sigma}}_{2}}}\right)\\\
&\times\prod\limits_{i=1}^{N_{r}}{\int\nolimits_{0}^{\infty}{{{\left({1+\rho\lambda}\right)}^{s+N_{r}-{\sigma_{1,i}}-1}}{\lambda^{{k_{{\sigma_{2,i}}}}+{\sigma_{1,i}}+\tau}}d\lambda}}.\end{aligned}$

(17)

where $S_{N_{r}}$ denotes the set of permutations of
$\left\\{1,2,\cdots,{N_{r}}\right\\}$,
$\boldsymbol{\sigma}_{l}\triangleq(\sigma_{l,1},\cdots,\sigma_{l,{N_{r}}})$
for $l\in\left(1,2\right)$ and ${\rm sgn}(\boldsymbol{\sigma}_{l})$ denotes
the signature of the permutation $\boldsymbol{\sigma}_{l}$, i.e., ${\rm
sgn}(\boldsymbol{\sigma}_{l})$ is $1$ whenever the minimum number of
transpositions necessary to reorder $\boldsymbol{\sigma}_{l}$ as
$(1,2,\cdots,{N_{r}})$ is even, and $-1$ otherwise [11].

######  Lemma 2.

If $\eta({\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}})$ is a function
of $\boldsymbol{\sigma}_{1}$ and ${\boldsymbol{\sigma}_{2}}$ irrespective of
the ordering of the elements in the permutations of the set of two-tuples
$\\{(\sigma_{1,l},\sigma_{2,l}):l\in[1,N_{r}]\\}$, the summation of
${{\mathop{\rm sgn}}\left({{\boldsymbol{\sigma}_{1}}}\right){\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}_{2}}}\right)}\eta({\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}})$
over all permutations of $\boldsymbol{\sigma}_{1}$ and
$\boldsymbol{\sigma}_{2}$ degenerates to

$\displaystyle\sum\nolimits_{{{\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}}\in{S_{N_{r}}}}}{{\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}_{1}}}\right){\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}_{2}}}\right)}\eta({\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}})$
(18)
$\displaystyle=\sum\nolimits_{{{\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}}\in{S_{N_{r}}}}}{{{\rm{sgn}}\left({{{\boldsymbol{\sigma}}}}\right){\mathop{\rm
sgn}}\left({\bar{\boldsymbol{\sigma}}}\right)\eta({\boldsymbol{\sigma}},\bar{\boldsymbol{\sigma}})}}$
$\displaystyle=N_{r}!\sum\nolimits_{{{\boldsymbol{\sigma}}\in{S_{N_{r}}}}}{{\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}}}\right)}\eta(\bar{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}),$

where ${\boldsymbol{\sigma}}=(\sigma_{1},\cdots,\sigma_{N_{r}})$ and
$\bar{\boldsymbol{\sigma}}=(1,\cdots,N_{r})$.

###### Proof.

Similar to the proofs of Leibniz formulae [11, eq.(2)] and [17, eqs.(64-65)],
denote by $S_{N_{r}}\times S_{N_{r}}$ the Cartesian product of $S_{N_{r}}$.
Hence, $(\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2})\in S_{N_{r}}\times
S_{N_{r}}$. We further establish the one-to-one mapping
$\vartheta(\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2})$ as a vector of
ordered pairs $(\sigma_{1,l},\sigma_{2,l})$ for $l\in[1,N_{r}]$, i.e.,
$\vartheta(\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2})\triangleq\left((\sigma_{1,l},\sigma_{2,l}):l\in[1,N_{r}]\right)$.
We thus reach the relation ${\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}_{1}}}\right){\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}_{2}}}\right)={\mathop{\rm
sgn}}\left({{\boldsymbol{\sigma}}}\right){\mathop{\rm
sgn}}\left({\bar{\boldsymbol{\sigma}}}\right)$ after a certain number of
transpositions to achieve
$\vartheta(\boldsymbol{\sigma},\bar{\boldsymbol{\sigma}})$ starting from
$\vartheta(\boldsymbol{\sigma}_{1},\boldsymbol{\sigma}_{2})$, where
${\mathop{\rm sgn}}\left({\bar{\boldsymbol{\sigma}}}\right)=1$. Secondary,
according to the definition of
$\eta({\boldsymbol{\sigma}_{1}},{\boldsymbol{\sigma}_{2}})$, the one-to-one
mapping and the cardinality of $S_{N_{r}}$, i.e., $|S_{N_{r}}|=N_{r}!$. Using
the same manner the second step holds. ∎

Basing on Lemma 2 and the Leibniz formula for the determinant expansion [23],
(17) can be simplified as

$\displaystyle\varphi\left(s\right)=\sum\nolimits_{{{\bf{k}}_{N_{r}}}}{\frac{{{{\left({-1}\right)}^{\frac{{{N_{r}}\left({{N_{r}}-1}\right)}}{2}}}{\cal
A}}}{{\Delta\left({\bf{K}}\right)}}}{\xi\left(s\right)},$ (19)

where
${\xi\left(s\right)}=\det[{\\{{\int_{0}^{\infty}{{{\left({1+\rho\lambda}\right)}^{s+{N_{r}}-i-1}}{\lambda^{{k_{j}}+i+\tau}}d\lambda}}\\}}_{i,j}]$.
By the change of variable $x=1/{{(1+\rho\lambda)}}$, $\xi\left(s\right)$ can
be expressed in terms of Beta function [18, eq.(8.380.1)] as

$\xi\left(s\right)={\rho^{-{k_{j}}-i-\tau-1}}{\rm{B}}\left({-s-{N_{r}}-{k_{j}}-\tau,{k_{j}}+i+\tau+1}\right).$
(20)

By using the relationship ${\rm
B}(\alpha,\beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$ [18,
eq.(8.384.1)] and the generalized Cauchy-Binet formula [17, Lemma 4], (19) can
be simplified as

$\begin{aligned}
\varphi\left(s\right)&=\frac{{{{\left({-1}\right)}^{{N_{r}}\left({{N_{t}}-{N_{r}}}\right)}}{\rho^{-\frac{1}{2}{N_{r}}\left({{N_{r}}+1}\right)}}\prod\nolimits_{i=1}^{{N_{r}}}{{a_{i}}^{{N_{r}}}}\prod\nolimits_{j=1}^{{N_{t}}}{{b_{j}}^{{N_{r}}}}}}{{\Delta\left({\bf{A}}\right)\Delta\left({\bf{B}}\right)\prod\nolimits_{i=1}^{{N_{r}}}{{{\left({s+i-2}\right)}^{i-1}}}}}\\\
&\times\det\left(\begin{array}[]{l}{\left\\{{\sum\limits_{k=0}^{\infty}{\frac{{\Gamma\left({-s-{N_{r}}-k+1}\right)}}{{\Gamma\left({-s-{N_{r}}+2}\right)}}{{\left({-\frac{{{a_{i}}{b_{j}}}}{\rho}}\right)}^{k}}}}\right\\}_{1\leq
i\leq{N_{r}},j}}\\\ {\left\\{{{b_{j}}^{{N_{t}}-i}}\right\\}_{{N_{r}}+1\leq
i\leq{N_{t}},j}}\end{array}\right)\end{aligned}.$

(21)

By using eqs. (8.380.1), (8.384.1), and (8.384.3) in [18], the infinite series
in (21) can be rewritten as

$\begin{aligned}
&\sum\limits_{k=0}^{\infty}{\frac{{\Gamma\left({-s-{N_{r}}-k+1}\right)}}{{\Gamma\left({-s-{N_{r}}+2}\right)}}{{\left({-\frac{{{a_{i}}{b_{j}}}}{\rho}}\right)}^{k}}}\\\
&=\int_{0}^{\infty}{{{\left({1+y}\right)}^{s+{N_{r}}-2}}{e^{-\frac{{{a_{i}}{b_{j}}}}{\rho}y}}dy}=\Psi\left({1,s+{N_{r}};\frac{{{a_{i}}{b_{j}}}}{\rho}}\right)\end{aligned},$

(22)

where the last equality holds by using (9.211.4) in [18],
$\Psi\left(\cdot,\cdot;\cdot\right)$ denotes confluent hypergeometric
function. By substituting (22) into (21), we finally arrive at (7).

## Appendix B

By applying the determinant expansion to (7), we get

$\begin{aligned}
&{F_{G}}\left(x\right)=\frac{1}{{2\pi{\rm{i}}}}\frac{{\Lambda\prod\nolimits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}\prod\nolimits_{j=1}^{N_{t}}{{b_{j}}^{N_{r}}}}}{{\rho^{\frac{1}{2}{{{N_{r}}\left({{N_{r}}+1}\right)}}}}}\sum\limits_{{\boldsymbol{\sigma}}\in{S_{N_{t}}}}{{\mathop{\rm
sgn}}\left({\boldsymbol{\sigma}}\right)}\\\
&\times{\prod\limits_{i={N_{r}}+1}^{N_{t}}{{b_{{\sigma_{i}}}}^{{N_{t}}-i}}}\int\nolimits_{c-{\rm
i}\infty}^{c+{\rm
i}\infty}\frac{{\Gamma\left({-s}\right)}}{{\Gamma\left({1-s}\right)}}\prod\limits_{j=1}^{{N_{r}}-1}\frac{{\Gamma\left({-s-{N_{r}}+1}\right)}}{{\Gamma\left({-s-j+1}\right)}}\\\
&\times\prod\limits_{i=1}^{N_{r}}\Psi\left({1,s+1+{N_{r}};\frac{{{a_{{i}}}{b_{\sigma_{i}}}}}{\rho}}\right){x^{-s}}ds\end{aligned}.$

(23)

By using the definition of
$\Xi\left({a,\alpha,A,\varphi}\right)={A^{\varphi+a+\alpha
s-1}}\Psi\left({\varphi,\varphi+a+\alpha s;A}\right)$ in [14] and identifying
the integration in (23) with the generalized Fox’s H function [19], (23) can
finally be represented as (8).

## Appendix C

Applying property 2 in [14] to (8) gives rise to

$\begin{aligned}
&{\mathcal{Y}_{{{\boldsymbol{\sigma}}}}(x)}={{{{\rho}}}^{-{N_{r}}^{2}}}\prod\limits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}{{b_{\sigma_{i}}}^{N_{r}}}\frac{1}{{2\pi{\rm{i}}}}\int\nolimits_{-c-{\rm
i}\infty}^{-c+{\rm
i}\infty}\frac{{\Gamma\left(s\right)}}{{\Gamma\left({1+s}\right)}}\\\
&\times\prod\limits_{j=1}^{{N_{r}}-1}{\frac{{\Gamma\left({s-{N_{r}}+1}\right)}}{{\Gamma\left({s-j+1}\right)}}}\prod\limits_{i=1}^{N_{r}}{\Psi\left({1,1+{N_{r}}-s;{\frac{{{a_{i}}{b_{{\sigma_{i}}}}}}{\rho}}}\right)}{x^{s}}ds\end{aligned}.$

(24)

We set $c<-{N_{t}}{N_{r}}+\frac{1}{2}{{N_{r}}({N_{r}}-1)}$. By using [18,
eq.(9.210.2)] and ignoring the higher order term
$o\left({{\rho^{-{N_{t}}{N_{r}}-\frac{1}{2}N_{r}(N_{r}+1)}}}\right)$ in (24),
together with the series expansion as
$\Psi\left({\alpha,\beta;x}\right)=\sum\nolimits_{n=0}^{\infty}{{{{\left(\alpha\right)}_{n}}}}{{{x^{n}}}}/{{{{\left(\beta\right)}_{n}}}}/{{n!}}$
[18, eq.(9.210.1)], where $(\cdot)_{n}$ is Pochhammer symbol, we finally
arrive at (9).

## Appendix D

Substituting (9) into (8) and swapping the orders of integration and summation
produces, after some basic algebraic manipulations, (8) can be further derived
as

$\begin{aligned}
p_{out}=&\frac{\Lambda\prod\nolimits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}\prod\nolimits_{j=1}^{N_{t}}{{b_{j}}^{N_{r}}}}{{\rho}^{\frac{1}{2}{{N_{r}}\left({{N_{r}}+1}\right)}}}\zeta(N_{r})\xi(N_{t})+o\left({\rho^{-{N_{t}}{N_{r}}}}\right)\end{aligned}.$
(25)

where
$\xi(N_{t})=\sum\limits_{{\boldsymbol{\sigma}}\in{S_{N_{t}}}}{{\mathop{\rm
sgn}}\left({\boldsymbol{\sigma}}\right)}\prod\limits_{i=1}^{N_{r}}{{{\left({-\frac{{{a_{{i}}}{b_{\sigma_{i}}}}}{\rho}}\right)}^{{n_{i}}}}}\prod\limits_{i={N_{r}}+1}^{N_{t}}{{b_{\sigma_{i}}}^{{N_{t}}-{i}}}$
and
$\zeta(N_{r})=\sum\limits_{{n_{1}},\cdots,{n_{N_{r}}}=0}^{\infty}{{\frac{1}{{2\pi{\rm{i}}}}\int\nolimits_{-c-{\rm
i}\infty}^{-c+{\rm
i}\infty}{\frac{{\Gamma\left(s\right)}}{{\Gamma\left({1+s}\right)}}\prod\limits_{i=1}^{N_{r}}{\frac{{\Gamma\left({s-{N_{r}}-{n_{i}}}\right)}}{{\Gamma\left({s-i+1}\right)}}}{2^{Rs}}ds}}}$.
By using the determinant expansion in $\xi(N_{t})$ and expressing
$\zeta(N_{r})$ in terms of Meijer G-function, leads to

$\begin{aligned}
\xi(N_{t})&={\prod\limits_{i=1}^{N_{r}}{{{\left({-\frac{{{a_{i}}}}{\rho}}\right)}^{{n_{i}}}}}}\det\left({\begin{array}[]{*{20}{c}}{{{\left\\{{{b_{j}}^{{n_{i}}}}\right\\}}_{1\leq
i\leq{N_{r}},j}}}\\\ {{{\left\\{{{b_{j}}^{{N_{t}}-i}}\right\\}}_{{N_{r}}+1\leq
i\leq{N_{t}},j}}}\end{array}}\right)\\\
&=\prod\limits_{i=1}^{{N_{r}}}{{{\left({-\frac{{{a_{i}}}}{\rho}}\right)}^{{n_{i}}}}}{\rm{sgn}}\left({\bf{n}}\right){(-1)^{{N_{r}}({N_{t}}-{N_{r}})}}\Delta\left({\bf{B}}\right)\end{aligned},$
(26)
$\leavevmode\resizebox{469.75499pt}{}{$\zeta(N_{r})=\sum\limits_{{\bf{n}}\in{\mathbb{N}^{N_{r}}}}{G_{{N_{r}}+1,{N_{r}}+1}^{0,{N_{r}}+1}\left({\left.{\begin{array}[]{*{20}{c}}{1,1+{N_{r}}+{n_{\rm{1}}},\cdots,1+{N_{r}}+{n_{N_{r}}}}\\\
{0,1,\cdots,{N_{r}}}\end{array}}\right|2^{R}}\right)}$},$ (27)

where ${\bf n}=(n_{1},\cdots,n_{N_{r}})$. Notice that any term with
$n_{i}=0,\cdots,{N_{t}}-{N_{r}}-1$ is equal to zero thanks to the basic
property of determinant, the dominant terms with $\bf n$ belonging to the set
of the permutations of
${\Omega_{N_{r}}}=\left\\{{{N_{t}}-{N_{r}},\cdots,{N_{t}}-1}\right\\}$ can
produce non-zero determinants. Substituting (27) and (26) into (25), and
ignoring the terms with both zero value of the determinant and the order of
$\rho$ larger than $N_{t}N_{r}$, $p_{out}$ can be asymptotically expanded as

$\begin{aligned}
p_{out}=&\frac{{\left({-1}\right)}^{\frac{1}{2}{{{N_{r}}\left({{N_{r}}-1}\right)}}}\prod\nolimits_{i=1}^{N_{r}}{{a_{i}}^{N_{r}}}\prod\nolimits_{j=1}^{N_{t}}{{b_{j}}^{N_{r}}}}{{{{{\rho}}}^{{N_{t}}{N_{r}}}}{\Delta\left({\bf{A}}\right)}}\\\
&\times
G_{{N_{r}}+1,{N_{r}}+1}^{0,{N_{r}}+1}\left({\left.{\begin{array}[]{*{20}{c}}{1,{N_{t}}+1,\cdots,{N_{t}}+{N_{r}}}\\\
{0,1,\cdots,{N_{r}}}\end{array}}\right|2^{R}}\right)\\\
&\times\sum\nolimits_{{\bf{n}}\in{\Omega_{N_{r}}}}{{\mathop{\rm
sgn}}\left({{\bf
n}}\right)\prod\limits_{i=1}^{N_{r}}{{a_{i}}^{{n_{i}}}}}+o\left({{\rho^{-{N_{t}}{N_{r}}}}}\right),\end{aligned}$

(28)

where the Meijer-G function can be extracted from the summation as a common
factor due to the fact that its value is independent of the order of the
elements of $\bf n$. Since the following equality holds

$\sum\nolimits_{{\bf{n}}\in{\Omega_{N_{r}}}}{{\mathop{\rm sgn}}\left({{\bf
n}}\right)\prod\limits_{i=1}^{N_{r}}{{a_{i}}^{{n_{i}}}}}={\left({-1}\right)}^{\frac{1}{2}{{{N_{r}}\left({{N_{r}}-1}\right)}}}\prod\limits_{i=1}^{N_{r}}{{a_{i}}^{{N_{t}}-{N_{r}}}}\Delta\left({\bf{A}}\right),$
(29)

the asymptotic $p_{out}$ can be finally derived as (10).

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# Rewritable Photonic Integrated Circuits Using Dielectric-assisted Phase-
change Material Waveguides

Forrest Miller  Department of Electrical and Computer Engineering, University
of Washington, 185 E Stevens Way NE, Seattle WA  Draper Scholar, The Charles
Stark Draper Laboratory, 555 Technology Square, Cambridge MA  Rui Chen
Department of Electrical and Computer Engineering, University of Washington,
185 E Stevens Way NE, Seattle WA  Johannes E. Fröch  Department of Electrical
and Computer Engineering, University of Washington, 185 E Stevens Way NE,
Seattle WA  Department of Physics, University of Washington, 3910 15th Ave.
NE, Seattle WA  Hannah Rarick  Department of Physics, University of
Washington, 3910 15th Ave. NE, Seattle WA  Sarah Geiger  The Charles Stark
Draper Laboratory, 555 Technology Square, Cambridge MA  Arka Majumdar
Department of Electrical and Computer Engineering, University of Washington,
185 E Stevens Way NE, Seattle WA

## Abstract

Photonic integrated circuits (PICs) can drastically expand the capabilities of
quantum and classical optical information science and engineering. PICs are
commonly fabricated using selective material etching, a subtractive process.
Thus, the chip’s functionality cannot be substantially altered once
fabricated. Here, we propose to exploit wide-bandgap non-volatile phase-change
materials (PCMs) to create rewritable PICs. A PCM-based PIC can be written
using a nano-second pulsed laser without removing any material, akin to
rewritable compact disks. The whole circuit can then be erased by heating, and
a new circuit can be rewritten. We designed a dielectric-assisted PCM
waveguide consisting of a thick dielectric layer on top of a thin layer of
wide-bandgap PCMs $Sb_{2}S_{3}$ and $Sb_{2}Se_{3}$. The low-loss PCMs and our
designed waveguides lead to negligible optical loss. Furthermore, we analyzed
the spatio-temporal laser pulse shape to write the PICs. Our proposed platform
will enable low-cost manufacturing and have a far-reaching impact on the rapid
prototyping of PICs, validation of new designs, and photonic education.

##  1 Introduction

Photonic Integrated Circuits (PICs) are becoming essential for various
applications, including optical communication [1], sensing [2], and quantum
information science [3]. While PICs can significantly expand and enhance the
performance of these systems, the fabrication methodology for the PICs is
complex and expensive: they require high-resolution lithography and etch
processes, which must take place in a sophisticated nanofabrication facility
[4]. Additionally, these processes are inherently subtractive, _i.e._ , once
fabricated, the wafer cannot be used to fabricate other structures. A low-cost
method to fabricate PICs and the ability to rewrite the PIC in the same wafer
can help with rapid prototyping.

Chalcogenide-based non-volatile phase change materials (PCMs) provide a
promising route to create such rewritable PICs [5, 6, 7]. These PCMs exhibit
large changes in their refractive index ($\Delta n>0.5$) when they undergo
structural phase transition between the amorphous (aPCM) and crystalline
(cPCM) states [8]. Crystallization can be actuated by holding the PCM above
its glass transition temperature ($T_{g}$) but below the melting temperature
($T_{mp}$) until a crystal lattice forms. Amorphization is achieved by melting
and rapidly quenching the PCM. Notably, this micro-structural phase transition
is non-volatile, _i.e._ , no external power is required to maintain the state
after the material phase is changed [9]. These PCMs have been cycled thousands
of times without degradation [10] and can potentially be switched for more
than $10^{12}$ times [11]. Consequently, PCMs are widely used in rewritable
compact disks (CDs) to store information. A writing laser fires pulses to heat
segments of the PCMs which, through amorphization or crystallization, write or
erase the stored information. The information is then read out using a probing
CW laser. However, rewritable CDs are fundamentally different from rewritable
PICs. In a CD, the probing light is reflected off the surface, but in PICs, it
propagates along the surface. Therefore, to build PICs, PCMs must provide
enough refractive index contrast to guide light in-plane while avoiding
significant optical loss.

While researchers have already experimentally demonstrated laser-written
rewritable meta-optics [12, 13] in PCM, an expensive femto-second laser was
used, and the light did not propagate in a waveguide over a long path. Another
work demonstrated only one-way writing of PICs in PCM [14], which lacks the
rewritable functionality. Some PIC structures have been written in
$Ge_{2}Sb_{2}Te_{5}$ (GST) using nano-second lasers [15], but the waveguide
loss was not considered.

Here, we present a design of a rewritable PIC based on PCMs. As the probing
light is guided in the high-index crystalline PCM, we must ensure near-zero
absorptive loss in the PCM. At 1.55 $\mu$m, GST is too lossy with an
extinction coefficient of cGST $\kappa_{cGST}\approx 1$[16]. However, wide-
bandgap PCMs, such as $Sb_{2}S_{3}$ and $Sb_{2}Se_{3}$, are suitable thanks to
their low absorption [10]. These wide-bandgap PCMs also exhibit large enough
index contrast between their amorphous and crystalline states ($\Delta
n_{SbS}=0.6$ and $\Delta n_{SbSe}$ = 0.77 at 1.55$\mu$m) [10] to confine an
optical mode. The loss can be further reduced via a dielectric-assisted PCM
structure (Fig. 1b). The propagation loss is estimated at 0.0100 dB/$\mu$m
(0.0086 dB/$\mu$m) using $Sb_{2}S_{3}$ ($Sb_{2}Se_{3}$). We envision that the
probing light will be coupled in and out of the chip using pre-fabricated
grating couplers, akin to input/output pins in an electronic integrated
circuit (Fig. 1a). Finally, we simulate switching dynamics to select the
spatio-temporal pulse shape of the writing laser that best achieves a complete
and reversible phase transition. Specifically, we show that a nano-second
pulsed laser can actuate the phase transition with a spatially Gaussian and
temporally rectangular shape. Our proposed rewritable PIC platform could
democratize PIC prototyping thanks to its low cost and reusability. This
frugal innovation can help with educating students and rapid prototyping of
PICs to validate designs.

Figure 1: a) Schematic of the proposed rewritable PIC platform. A nano-second
pulsed laser switches the PCM from the crystalline (light grey) to the
amorphous (dark grey) phase. The chip can then be uniformly heated to reset
the PCM to the crystalline phase, erasing the written PICs. Subsequently, a
different PIC can be written in the same region. b) The proposed dielectric-
assisted PCM waveguide geometry for low-loss waveguiding. A simulated guided
mode is overlayed, depicting a well-confined mode in the dielectric silicon
nitride layer for a low propagation loss. Air clads the upper face of the
$Si_{3}N_{4}$ layer.

##  2 Low-Loss PCM Waveguides

While wide-bandgap PCMs have negligible loss in the amorphous state, a non-
negligible loss is still present in the crystalline state. We investigated
prior works on PCM-integrated ring resonators and estimate the refractive
index at $\lambda=1.55\mu m$ in the crystalline state to be
$n_{cSbS}=3.33+0.016i$ [17], and $n_{cSbSe}=4.23+0.0043i$ [18, 19]. We
obtained these values by simulating the reported experimental structures and
adjusting the extinction coefficient of the PCMs until our simulated loss
matched the reported experimental results. The loss in the crystalline state
is critical since the probing light is confined in crystalline PCM, which has
a higher refractive index than the amorphous phase ($n_{aSbS}=2.76+0i$,
$n_{aSbSe}=3.22+0i$) but also higher loss. If the light is perfectly confined
in a $cSb_{2}S_{3}$ ($cSb_{2}Se_{3}$) core, then the unit propagation loss at
$\lambda$ becomes
$-20\log_{10}[\exp(-\frac{2\pi}{\lambda}\cdot\kappa_{cSbS(cSbSe)})]$,
resulting in a unit loss for $cSb_{2}S_{3}$ ($cSb_{2}Se_{3}$) $\approx
0.56(0.15)dB/\mu m$. While the loss is overestimated due to the perfect
confinement assumption, the result suggests the necessity of a careful
waveguide design to achieve a low propagation loss.

One straightforward way to reduce the optical loss is to reduce the PCM
thickness, which decreases the interaction between the optical mode and the
cPCM. Additionally, thick PCMs are generally harder to fully switch with 70nm
the thickest to date [20]. Moreover, thinner PCM layers enable higher
switching endurance [21, 5]. However, an extremely thin PCM layer does not
provide enough index contrast to guide a mode. We mitigate this trade-off by
exploiting a dielectric-assisted PCM waveguide architecture [22], where a
thick dielectric layer of $Si_{3}N_{4}$ [23] is deposited on a thin PCM layer
(Fig. 1a). This layer may experience small changes in refractive index during
writing, but we anticipate such changes will be negligible [24]. The optical
mode is mainly confined in the dielectric layer due to the geometry of the PCM
layer, mitigating the absorptive loss of cPCMs. Such a waveguide will be
written from the chip’s “erased” state, where the PCM layer is uniformly
crystalline. A PCM waveguide is created by selectively switching the PCM to
the amorphous state. We assume a layer of PCM with a thickness $T_{PCM}$ will
be deposited on 2 $\mu$m of thermal oxide ($SiO_{2}$) film on a silicon wafer.
A conformal capping material is needed to prevent material reflow and
oxidation during switching [25]. Therefore, we encapsulate the PCM with 20nm
of atomic layer deposited $Al_{2}O_{3}$ and then deposit a $Si_{3}N_{4}$ layer
with thickness $T_{clad}$ to create the material stack (Fig. 1b).

We optimize the waveguide geometry for low loss, using the Finite Element
Eigenmode (FEM) solver in Ansys Lumerical (Supplement S1). We start by
sweeping the $Sb_{2}S_{3}$ thickness $T_{PCM}$, and the $cSb_{2}S_{3}$ core
width $W_{core}$ (Fig. 2a). Unsurprisingly, a thinner $T_{PCM}$ yields a lower
loss. However, below a thickness of 12 nm, the PCM cannot confine a mode.
Since the PCM must guide the mode, we find a compromise between loss and mode
confinement at a layer thickness of 15 nm and a core width of 2.0 $\mu$m. The
thickness of the $Si_{3}N_{4}$ layer is designed around 0.4 $\mu$m as shown in
Fig. 2b. Our optimized structure exhibits a loss of 0.0100 dB/$\mu$m.

The loss in the $Sb_{2}Se_{3}$ waveguide exhibits much weaker dependence on
$W_{core}$ (Fig. 2c). This is due to the higher real refractive index of
$Sb_{2}Se_{3}$, which gives tighter mode confinement. We choose 1.5$\mu$m as
the core width to improve integration density. Similar to the $Sb_{2}S_{3}$
design, the loss increases with increasing $T_{PCM}$. Here we choose 20 nm as
the layer thickness. This $T_{PCM}$ is thin enough to switch and provide low
loss while reliably guiding a mode. The $Si_{3}N_{4}$ thickness was set to 400
nm to minimize loss while still confining the mode (Fig. 2d). This geometry
demonstrates a loss of 0.0086 $dB/\mu m$ at 1.55 $\mu m$.

We note that while a PCM thickness of 15nm is sufficient for a straight
waveguide, for a bent waveguide, we need slightly thicker PCM and/ or thinner
$Si_{3}N_{4}$. Based on our simulation, we found a $T_{PCM}$ of 25nm PCM and a
100nm $T_{clad}$ will be guide light across a 75$\mu$m (25 $\mu$m) bend using
$Sb_{2}S_{3}$ ($Sb_{2}Se_{3}$). These new parameters increase the loss by a
factor of 8 (4), but support bending, which is critical for any integrated
photonic device. With this architecture, we believe we can optically write
directional couplers, Y-splitters and ring/ disk resonators [26]. Components
like mode-converters or waveguide crossings rely on small features, and our
rewritable PIC may not be suitable to write such small features.

Figure 2: a) Waveguide loss (dB/$\mu$m) for a joint parameter sweep of the
$Sb_{2}S_{3}$ thickness and waveguide core ($cSb_{2}S_{3}$) width. b) The loss
and effective mode area as a function of the $Si_{3}N_{4}$ thickness in the
$Sb_{2}S_{3}$ design. c) Waveguide loss (dB/$\mu$m) for a joint parameter
sweep of the $Sb_{2}Se_{3}$ thickness and waveguide core ($cSb_{2}Se_{3}$)
width. d) The loss and effective mode area as a function of the $Si_{3}N_{4}$
thickness in the $Sb_{2}Se_{3}$ design.

##  3 Fixed In/out ports using grating couplers

The PICs presented here must interface with free-space optics. We propose to
accomplish this using fixed grating couplers on the chip, forming optical
input/output ports, between which designs can be written, erased, and re-
written (Fig. 1a). Edge coupling would also work with this design, but we
envision grating couplers enabling easier interfacing. These grating couplers
are formed by etching the $Si_{3}N_{4}$ layer since the optical modes for both
designs are primarily confined in the $Si_{3}N_{4}$ layer (Fig. 1b). While
this etching step is exactly what we intend to eliminate, we note that after
this one etch, thousands of PIC designs can be written and tested on this
platform without further etching. It is possible to write grating couplers
into the PCM, but such couplers suffer from low coupling efficiencies and must
be re-written after each anneal. We optimize these gratings for maximum
coupling efficiency using single mode fibers at $25^{o}$ of incidence using
Lumerical’s Finite Difference Time Domain (FDTD) simulation (Supplement S2).
In the $Sb_{2}S_{3}$ design, a grating pitch of 1.01 $\mu$m, a duty cycle of
0.8, and an etch depth of 180 nm resulted in a coupling efficiency of 21%. In
the $Sb_{2}Se_{3}$ design, a grating pitch of 0.97 $\mu$m, a duty cycle of
0.56, and an etch depth of 240 nm also resulted in a coupling efficiency of
21%. The gratings are connected to the waveguides via a tapered section with
end width of 4 $\mu$m for both PCMs, which yields a mode overlap of more than
95% with the guided modes.

Figure 3: A grating coupler and the two-dimensional cross-sectional view along
the dashed line. P, the optimal pitch, is 1.01 (0.97) $\mu$m for the
$Sb_{2}S_{3}$ ($Sb_{2}Se_{3}$) design. The optimal duty cycle, $\frac{a}{b}$,
is 0.8 (0.56) for the $Sb_{2}S_{3}$ ($Sb_{2}Se_{3}$) design. The etch depth
was 180 (240) nm for $Sb_{2}S_{3}$ ($Sb_{2}Se_{3}$). In both cases, a taper
end width, w, of 4 $\mu$m provides more than 95% mode overlap between the
coupler and the waveguide.

##  4 Spatio-temporal pulse shaping

The PICs can be arbitrarily patterned using a pulsed laser and a three-axis
translation stage. We simulated four pulse schemes with different spatio-
temporal pulse shapes (4a-d) in COMSOL Multiphysics and select the pulse that
best amorphizes the PCM. We expect the ideal shape to be a rectangular spatial
shape to provide a smooth boundary region and an increasing temporal shape to
produce uniformly switching in the depth direction. However, we found that a
Gaussian spatial shape and rectangular temporal shape can switch thin
$Sb_{2}S_{3}$, offering a simple experimental realization. 450 nm wavelength
pulses were delivered directly to the 15 nm film of $cSb_{2}S_{3}$
($n=4.12+1.80i$) in Fig. 1b, so reflection off the alumina and $Si_{3}N_{4}$
layers is not considered. The pulses had either a Gaussian or uniform spatial
distribution and either a rectangular or exponentially decaying temporal
distribution. All pulses last approximately 14 ns and the power was adjusted
to achieve a maximum temperature of approximately 900 ${}^{o}C$.

A successful amorphization pulse must satisfy four conditions. First, enough
thermal energy must be applied to melt the PCM completely. This requires the
delivered energy to heat PCM to its melting temperature and further overcome
the latent heat, given by the multiplication of enthalpy of fusion ($H_{f}$)
and mass of the PCM. Second, the cooling rate must exceed $\sim$ 1 $K/ns$ [8].
This ensures the PCM is frozen in the amorphous state [27] and does not
undergo unintentional recrystallization. Third, the absorbed thermal energy
must not heat the PCM above its boiling point $T_{b}$, which irreversibly
ablates the material. We obtain $Sb_{2}S_{3}$ parameters from the literature
for the simulations: the melting point is $T_{mp}=$ 547 ${}^{o}C$ [10], the
enthalpy of fusion is $H_{f}=$ 47.9 kJ/mol, the boiling point is $T_{b}=$ 1149
${}^{o}C$, the density is $4.56$ $g/cm^{3}$ [28], the specific heat is $575.4$
$J/(kg*K)$, and the thermal conductivity is $0.42$ $W/(m*K)$[17]. Material
parameters for $Si_{3}N_{4}$, $Al_{2}O_{3}$, and $SiO_{2}$ are found from
COMSOL’s materials library (Supplement S3). Lastly, the boundary region
between switched and non-switched regions defines the waveguide edge. This
boundary contains material heated to its melting point but not enough to
exceed its enthalpy of fusion. This implies a partial amorphization, where a
portion of the crystalline structure remains. This is undesirable as a more
abrupt index change at the boundary could lead to better mode confinement.

We examine different pulse conditions against these criteria. As shown in Fig.
4a-d, all pulse conditions exceed the melting point and achieve a faster
cooling rate than $1K/ns$. Our hypothesis with temporal modulation was that
the exponential ramp would more uniformly heat the PCM layer along its
thickness in the vertical direction. This statement is supported by the more
linear average temperature curve in 4c,d. However, since the $Sb_{2}S_{3}$
layer is very thin, there was little variation in temperature from the top to
the bottom of the PCM film. Therefore, we can consider pulses with rectangular
temporal modulation. In a thicker PCM layer, we anticipate that temporal
modulation of the beam power could enable more uniform heating in the depth
direction.

The performance difference between the spatial beam shapes is apparent in the
feature size and the boundary region width. A Gaussian mode is better if the
desired circuit requires fine features. This is a direct result of the
narrower intensity distribution of a Gaussian compared to a uniform beam. With
a Gaussian beam, the width of the switched PCM could be smaller than the
laser’s spot size if the laser power is tuned such that the beam’s full-width-
half-maximum is lower than the amorphization threshold. A uniform beam lacks
this ability to achieve a small feature size. However, its advantage is a
narrower boundary region in the PCM. The boundary region for the Gaussian
beams traverses a distance of 105 pm (Fig. 4e). In comparison, the boundary is
only 27 pm (Fig. 4f) for a spatially uniform beam. This shorter boundary
resembles the step-index profiles used in our previous simulations and leads
to better mode confinement. Our thermal simulation verifies that the thin PCM
layer can be switched entirely with nano-second laser pulses and offers a
simple experimental realization with a natural laser beam with a Gaussian
spatial and rectangular temporal shape.

Figure 4: a-d) The transient thermal dynamics for different pulses: a)
temporally rectangular, spatially Gaussian, b) temporally rectangular,
spatially uniform, c) temporally exponentially decaying, spatially Gaussian,
d) temporally exponentially decaying, spatially uniform. e, f) Static
temperature profile along the spatio-temporal line cut at y = $T_{SbS/2}$ and
t = 14 ns for e) a spatially Gaussian beam and f) a spatially uniform beam.
The Gaussian beam switches a smaller area of $Sb_{2}S_{3}$ than the uniform
beam, but the uniform beam has a shorter boundary region (the distance between
$T_{mp}$ and full amorphization). Thus a Gaussian beam can write finer
features, but a uniform beam will likely have better mode confinement.

##  5 Conclusion

In conclusion, we have proposed and numerically verified a rewritable, cost-
efficient PIC platform using wide-bandgap PCMs and a cost-efficient nano-
second pulse laser. PICs are envisioned to be written (amorphized) by nano-
second laser pulses and erased (crystallized) by rapid thermal annealing or
even a simple hotplate. We have designed a dielectric-assisted PCM waveguide
configuration, allowing low propagation loss for both $Sb_{2}Se_{3}$ and
$Sb_{2}S_{3}$. Efficient grating couplers, working as optical I/O ports, were
optimized for both types of waveguides. Comprehensive thermal transfer dynamic
simulations were used to verify and select a desirable spatio-temporal pulse
condition to ensure complete amorphization with nano-second pulses. This etch-
free platform could accelerate PIC fabrication and testing, potentially
democratizing PIC fabrication.

##  6 Backmatter

### Funding

Research was sponsored by DARPA and the Army Research Office and was
accomplished under Grant Number W911NF-21-1-0368. F.M. is supported by a
Draper Scholarship.

### Acknowledgments

Author Contributions A.M. and F.M. conceived the project. F.M. performed
simulation. R.C., J.F., and H.R. assisted with simulation. A.M. and S.G.
supervised the project. F.M. wrote the manuscript with input from all the
authors.

### Disclosures

The authors declare no conflicts of interest.

### Data Availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon request.

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