# Quartic Perturbation-based Outage-constrained Robust Design in Two-hop One-
way
Relay Networks

Sissi Xiaoxiao Wu, Sherry Xue-Ying Ni, Jiaying Li, and Anthony Man-Cho So This
work is supported by the National Natural Science Foundation of China under
Grant 61701315; by Shenzhen Technology R$\&$D Fund JCYJ20170817101149906 and
JCYJ20190808120415286; by Shenzhen University Launch Fund 2018018. S. X. Wu
and J. Li are with the College of Electronics and Information Engineering,
Shenzhen University, Shenzhen, China. S. X.-Y. Ni is with NM OPTIM Limited. A.
M.-C. So is with the Department of Systems Engineering and Engineering
Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
S.A.R., China. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work, we study a classic robust design problem in two-hop one-way
relay system. We are particularly interested in the scenario where channel
uncertainty exists in both the transmitter-to-relay and relay-to-receiver
links. By considering the problem design that minimizes the average amplify-
and-forward power budget at the relay side while satisfying SNR outage
requirements, an outage-constrained robust design problem involving quartic
perturbations is formulated to guarantee the robustness during transmission.
This problem is in general difficult as it involves constraints on the tail
probability of a high-order polynomial. Herein, we resort to moment inequality
and Bernstein-type inequality to tackle this problem, which provide convex
restrictions, or safe approximations, of the original design. We also analyze
the relative tightness of the two safe approximations for a quadratic
perturbation-based outage constrained problem. Our analysis shows that the
Bernstein-type inequality approach is less conservative than the moment
inequality approach when the outage rate is within some prescribed regime. To
our best knowledge, this is the first provable tightness result for these two
safe approximations. Our numerical simulations verify the superiority of the
robust design and corroborate the tightness results.

##  I Introduction

In recent decades, multi-hop relay technology has been widely used in long-
distance wireless communication systems to expand the coverage of
communication. For example, when devices are far away from each other, such a
technology can improve the “quality-of-service” (QoS) between transmitters and
receivers [1]. Moreover, multi-hop relay technology can be applied to other
advanced communication systems, such as device-to-device (D2D) communication
for which the equipment can act as an instant relay node [2, 3, 4], the
millimeter wave communication which can overcome the serious attenuation
phenomenon at high frequency [5], and cognitive radio (CR) networks in which
it could improve the coverage of cognitive network and the channel capacity of
the system [6, 7]. In the multi-hop relay transmission, a classic task is to
design the amplify-and-forward (AF) weights that guide the relays to adjust
their antennas towards the receiver. This design task is conditioned on that
the relay system acquires the channel state information (CSI) from both the
transmitter and the receiver. However, in practice, due to estimation error,
quantization error, or limited feedback, the available CSIs at the relays are
usually imperfect. It is well known that CSI uncertainties might lead to
serious performance degradation during transmission. This motivates us to
study robust designs for the multi-hop relay system.

In this paper, we consider the robust design problem in the context of two-hop
one-way relay beamforming, which is generally more involved than its non-
robust counterpart [8]. This robust design problem is considered in either a
worst-case setting or a chance-constrained setting. In both settings, it is
usually assumed that CSI errors are only present in either the transmitter-to-
relay link or the relay-to-receiver link. Under this assumption, the so-called
S-lemma can be applied to the worst-case setting to turn a semi-infinite
program (SIP) [9, 10] into a tractable semidefinite program (SDP) [11, 12];
while in the chance-constrained setting, the so-called sphere-bounding or
Bernstein-type inequality [13, 14, 15, 16, 17, 18, 19] can be applied to find
safe approximations of the original chance constraints. So far, there are very
few works considering a reliable robust design for the case where both the
transmitter-to-relay and relay-to-receiver links have errors, as this case
involves quartically perturbed constraints and is generally difficult. Our
previous work [20] solved a quartically-constrained robust design problem in a
worst-case setting. Our goal in this paper is to fill the gap in the chance-
constrained setting. In this work, we target the problem design that minimizes
the average AF power at the relays while satisfying the receivers’ signal-to-
noise ratio (SNR) constraints with a small outage probability, given that both
the transmitter-to-relay and relay-to-receiver links are subject to Gaussian
errors. To tackle the resulting outage constraints with quartic polynomials of
complex Gaussian random variables [21], there are two possible approaches: 1)
Reformulate the high-order chance constraints into safe tractable
approximations and employ the positive semidefinite relaxation (SDR) technique
[22] to solve the approximated problem; 2) simply ignore the higher-order
perturbation terms and deal only with quadratic chance constraints. For the
former, we can resort to a suitable moment inequality to develop safe
approximations of the quartically perturbed outage constraints. For the
latter, we can apply both the moment inequality and Bernstein-type inequality
to develop safe approximations of the approximating quadratically perturbed
outage constraints. The resulting approximations can then be solved by the SDR
technique, and a sub-optimal AF beamforming vector can be extracted from the
optimal SDR solution using a Gaussian randomization procedure.

In the literature, the common robust design only gives rise to a quadratically
perturbed chance constraint. For example, [13] first proposed and summarized
three methods to solve chance constraints involving quadratic forms, namely
sphere bounding, Bernstein-type inequality, and decomposition-based large
deviation inequality (LDI). The works [23, 24] applied Taylor’s expansion to
approximate the chance constraint of interest by one with quadratic
perturbations and then tackled it using the LDI approximation. In [14, 15,
16], the authors used Bernstein-type inequality to convert a quadratically
perturbed chance constraint into a deterministic form in the robust design of
CR networks. The sphere bounding method was used to propose a safe tractable
approximation of the quadratic chance constraint in [17, 18]. Furthermore,
[19] used the S-procedure and Bernstein-type inequality to tackle quadratic
chance constraints and compared the performance of these two methods. Even for
works that originally aim at tackling robust designs with high-order
perturbations, the traditional way is to ignore the higher-order terms and
apply standard techniques from robust optimization to simplify the constraints
[25, 26, 27, 28]. There are also other works that tackle quartic constraint
but not from a probabilistic perspective. For example, [20] employed the SDR
technique and tools from polynomial optimization to construct safe
approximations of such constrains; [29] introduced several auxiliary variables
to convert the quartic ISL constraint and the PAPR constraint into several
quadratic constraints and proposed an alternating direction method of
multipliers-based solution to handle them; [30] showed that the quartic
constraint associated with a certain hybrid precoding problem is automatically
satisfied and hence can be removed; [31] introduced a slack variable to
replace the higher-order terms in the SINR QoS constraint, thus converting it
into a linear matrix inequality. Generally speaking, finding a good solution
to robust designs involving quartic perturbations is a very difficult problem,
especially from a probabilistic perspective. Recently, [21] proposed a safe
tractable approximation of quartically perturbed chance constraints using
moment inequalities for Gaussian polynomials and the SDR technique. However,
no theoretical analysis was provided, and the relative tightness of the
Bernstein-type inequality approach and the moment inequality approach remains
unknown.

Our contribution in this work is fourfold. First, we introduce the two-hop
relay robust design problem with a quartically perturbed chance constraint,
which is seldom studied in prior work. Second, we apply the fourth-order
moment inequality to obtain a safe approximation of the said chance
constraint. Third, we apply the second-order moment inequality and the
Bernstein-type inequality to provide a safe approximation of the quadratically
perturbed chance constraint obtained by ignoring higher-order perturbation
terms. In addition, we provide an analytical bound to prove the relative
tightness of different restrictions. Our numerical results show that the
proposed approximation approaches are more reliable than the non-robust
counterpart. Also, the results suggest that by dealing with all the
perturbation terms in the chance constraint instead of keeping only the lower-
order ones, the resulting design is more robust against perturbations that do
not match the prior distributional information. Lastly, our comparison between
the second-order moment inequality-based approach and the Bernstein-type
inequality-based approach corroborate our relative tightness result.

The rest of the paper is organized as follows. In Section II, we provide the
system model and formulate the robust design problem for the two-hop one-way
relay system. Section III introduces the moment inequality-based approach,
which can provide safe approximations for both the quartically perturbed
chance constraint and the quadratically perturbed chance constraint. In
Section IV, the Bernstein-type inequality-based approach is proposed to find a
safe approximation of the quadratically perturbed chance constraint. Moreover,
in Section V, we establish theoretically the relative tightness of the moment
inequality-based and Bernstein-type inequality-based approaches. Simulation
results are presented in Section VI and we conclude our work in Section VII.

##  II System Model and Problem Formulation

In this work, we consider a classic scenario setting for a relay network
consisting of one transmit-receiver pair and there are $L$ relays between them
to assist the transmission. We assume that no direct link is involved in this
setting and both the transmitter and receiver are equipped with a single
antenna. Then, the information is transmitted from the transmitter to receiver
through two types of links. One is the _transmitter-to-relay links_ , via
which the transmitter sends common information to the relays. In this context,
the receive model is given by

${\bm{r}}(t)={\bm{f}}s(t)+{\bm{n}}(t),$ (1)

where $s(t)$ is the common information with $\mathbb{E}[|s(t)|^{2}]=P_{t}$ and
$P_{t}$ is the transmit power at the transmitter; ${\bm{f}}\in\mathbb{C}^{L}$
is the channel from the transmitter to the relays;
${\bm{n}}(t)=[n_{1}(t),\ldots,n_{\ell}(t),\ldots,n_{L}(t)]^{T}$ and
$n_{\ell}(t)$ is the white noise at relay-$\ell$ with variance
$\sigma_{\ell}^{2}$. The other is the _relay-to-receiver links_ , via which
relays amplify and forward the received signal to the receiver. In this paper,
we target at the relay beamforming scheme, in which the AF process at the
relay side is given by

${\bm{x}}(t)={\rm Diag}({\bm{w}}){\bm{r}}(t),$ (2)

where ${\bm{w}}=[w_{1},\ldots,w_{\ell},\ldots,w_{L}]^{T}$ and $w_{\ell}$ is
the AF weight at relay $\ell$. Under this model, the received signal can be
expressed as

$\displaystyle y(t)=$ $\displaystyle{\bm{g}}^{H}{\bm{x}}(t)+{v}(t),$ (3)

where ${\bm{g}}\in\mathbb{C}^{L}$ is the channel from the relays to the
receiver; ${v}(t)$ is the white noise at the receiver with variance
$\sigma_{{v}}^{2}$. Then, the SNR at the receiver can be expressed as

${\rm
SNR}=\frac{{\bm{w}}^{H}P_{t}({\bm{f}}\odot{\bm{g}}^{*})({\bm{f}}\odot{\bm{g}}^{*})^{H}{\bm{w}}}{{\bm{w}}^{H}{\rm
Diag}([|g^{1}|^{2}\sigma_{1}^{2},|g^{2}|^{2}\sigma_{2}^{2},\ldots,|g^{L}|^{2}\sigma_{L}^{2}]){\bm{w}}+\sigma_{{v}}^{2}}.$

Figure 1: The two-hop one-way relay network.

Under (2) and (3) we may explicitly express the power at the relays as
${\rm\mathbb{E}}[{\bm{w}}^{H}{\bm{D}}{\bm{w}}]$ and ${\rm SNR}$ as

$\begin{array}[]{ll}&\quad\displaystyle{\rm
SNR}=\frac{{\bm{w}}^{H}{{\bm{A}}}{\bm{w}}}{{\bm{w}}^{H}{{\bm{C}}}{\bm{w}}+1},\end{array}$

where

$\displaystyle{{\bm{A}}}=$ $\displaystyle
P_{t}(({\bm{f}}{\bm{f}}^{H})\odot(({\bm{g}}^{*})({\bm{g}}^{*})^{H}))/\sigma_{{v}}^{2},$
(4) $\displaystyle{{\bm{C}}}=$
$\displaystyle{\bm{\Sigma}}\odot({\bm{g}}{\bm{g}}^{H})/\sigma_{{v}}^{2},\quad$
(5) $\displaystyle{\bm{D}}=$ $\displaystyle
P_{t}{\bm{I}}\odot({\bm{f}}{\bm{f}}^{H})+{\bm{\Sigma}},$ (6)

$\sigma_{{v}}^{2}$ is the noise power at the destination, ${\bm{\Sigma}}={\rm
Diag}([\sigma_{1}^{2},...,\sigma_{L}^{2}])$, and ${\bm{f}}$, ${\bm{g}}$
represent the actual transmiter-to-relays channel and relays-to-receiver
channel, respectively. In the general case where both links are imperfect, we
write

${\bm{f}}={\bar{\bm{f}}}+\Delta{\bm{f}},\quad\quad{\bm{g}}={\bar{\bm{g}}}+\Delta{\bm{g}},$

where ${\bar{\bm{f}}}$ and ${\bar{\bm{g}}}$ are the estimated CSI;
$\Delta{\bm{f}}$ and $\Delta{\bm{g}}$ are the corresponding stochastic CSI
errors that respectively follow the distributions
$\Delta{\bm{f}}\sim\mathcal{CN}(0,{{\bm{E}}}_{f})$, ${{\bm{E}}}_{f}\succ 0$
and $\Delta{\bm{g}}\sim\mathcal{CN}(0,{{\bm{E}}}_{g})$, ${{\bm{E}}}_{f}\succ
0$. Here we adopt a Gaussian channel error model; $i.e.$,
${{\bm{E}}}_{f}=\epsilon^{2}{\bm{I}}$, ${{\bm{E}}}_{g}=\eta^{2}{\bm{I}}$ with
$\epsilon,\eta>0$. Equivalently, we may write

$\Delta{\bm{f}}=\epsilon{\bm{x}},\quad\quad\Delta{\bm{g}}=\eta{\bm{y}},$

where ${\bm{x}}$ and ${\bm{y}}$ are standard complex Gaussian vectors and
$\epsilon,\eta$ are known scalars to bound the error magnitudes [8, 25].

We are motivated to design the AF weight vector ${\bm{w}}$ so that the average
transmit power is minimized while the receiver’s SNR outage constraint is
satisfied. Specifically, we consider the following problem:

$\begin{array}[]{ll}\displaystyle\min_{{\bm{w}}\in\mathbb{C}^{L}}&\quad\displaystyle{\rm\mathbb{E}}[{\bm{w}}^{H}{\bm{D}}{\bm{w}}]\\\
\text{subject to}&\quad\displaystyle{\rm Pr}\\{{\rm
SNR}\leq\gamma\\}\leq\rho,\\\ \end{array}$ (7)

where $\gamma$ is the target SNR threshold and $\rho$ is a prescribed outage
rate we want to guarantee during the transmission. To further tackle Problem
(7), a typical relaxation is

$\begin{array}[]{ll}\displaystyle\min_{{\bm{W}}\in\mathbb{C}^{L\times
L}}&\quad\displaystyle{\rm\mathbb{E}}[{\bm{D}}\cdot{\bm{W}}]\\\ \text{subject
to}&\quad\displaystyle{\rm Pr}\\{Q({\bm{W}},{\bm{x}},{\bm{y}})\geq
0\\}\leq\rho,\\\ &\quad{\bm{W}}\succeq 0,\end{array}$ (8)

where

$\displaystyle Q({\bm{W}},{\bm{x}},{\bm{y}})$ (9) $\displaystyle=$
$\displaystyle\sigma_{{v}}^{2}+{\bm{W}}\cdot\big{(}{\bm{\Sigma}}\odot({\bm{g}}{\bm{g}}^{H})-\frac{P_{t}}{\gamma}(({\bm{f}}{\bm{f}}^{H})\odot(({\bm{g}}^{*})({\bm{g}}^{*})^{H}))\big{)}.$

Problem (8) is still difficult as $Q({\bm{W}},{\bm{x}},{\bm{y}})$ in the
outage constraint involves high-order perturbation terms. Our main task in the
sequel is to discuss how to deal with this challenge.

##  III Moment Inequality-Based Approach

In this section, we review and further develop the moment inequality approach
in [21] to construct safe tractable approximations of chance constraints with
quartic perturbations.

###  III-A The Fourth-order Moment Inequality Method

Key to the development of safe tractable approximations of quartically
perturbed chance constraints is the following moment inequality for quartic
polynomials in complex Gaussian random variables.

######  Theorem 1.

Let $\xi_{1},\ldots,\xi_{m}$ be independent standard real Gaussian random
variables. Consider the function
$f:\mathbb{C}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$:

$\displaystyle f({\bm{x}},{\bm{\xi}})=-a_{0}({\bm{x}})+\sum_{1\leq i\leq
m}\xi_{i}a_{i}({\bm{x}})+\sum_{1\leq j_{1},j_{2}\leq
m}\xi_{j_{1}}\xi_{j_{2}}a_{j_{1}j_{2}}({\bm{x}})$ $\displaystyle+\sum_{1\leq
k_{1},k_{2},k_{3}\leq
m}\xi_{k_{1}}\xi_{k_{2}}\xi_{k_{3}}a_{k_{1}k_{2}k_{3}}({\bm{x}})$ (10)
$\displaystyle+\sum_{1\leq\ell_{1},\ell_{2},\ell_{3},\ell_{4}\leq
m}\xi_{\ell_{1}}\xi_{\ell_{2}}\xi_{\ell_{3}}\xi_{\ell_{4}}a_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}({\bm{x}}),$

where
$a_{0},a_{i},a_{j_{1}j_{2}},a_{k_{1}k_{2}k_{3}},a_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}$
are affine functions of $\bm{x}$. Note that we allow the decision vector
${\bm{x}}$ to take complex values. However, we assume that the value
$f({\bm{x}},{\bm{\xi}})$ is real for any ${\bm{x}}\in\mathbb{C}^{n}$ and
${\bm{\xi}}\in\mathbb{R}^{m}$. Consider the chance constraint

$\Pr(f({\bm{x}},{\bm{\xi}})\geq 0)\leq\rho,$ (11)

where $\rho>0$ is given. The following hold:

  * (a)

For each ${\bm{x}}\in{{\mathbb{C}}^{n}}$ and
${\bm{\xi}}\in{{\mathbb{R}}^{m}}$, let

$\bar{f}({\bm{x}},{\bm{\xi}})\triangleq
f({\bm{x}},{\bm{\xi}})+{{a}_{0}}({\bm{x}}).$

Then, the function $\bm{x}\mapsto\bar{f}(\bm{x},\bm{\xi})^{2}$ is quadratic in
$\bm{x}$ and can be written in the form

$\displaystyle\bar{f}({\bm{x}},{\bm{\xi}})^{2}={\bm{v}}^{H}({\bm{x}}){\bm{U}}({\bm{\xi}}){\bm{v}}({\bm{x}})$
(12)

for some ${\bm{U}}({\bm{\xi}})\succeq{\bm{0}}$, where ${\bm{U}}({\bm{\xi}})$
is a Hermitian positive semidefinite matrix whose rows and columns are labeled
by the set of indices

$\mathscr{S}=\\{0,\underbrace{i,\ldots}_{1\leq i\leq
m},\underbrace{j_{1}j_{2},\ldots}_{1\leq j_{1},j_{2}\leq
m},\underbrace{k_{1}k_{2}k_{3},\ldots}_{1\leq k_{1},k_{2},k_{3}\leq
m},\underbrace{\ell_{1}\ell_{2}\ell_{3}\ell_{4},\ldots}_{1\leq\ell_{1},\ell_{2},\ell_{3},\ell_{4}\leq
m}\\}$

arranged in lexicographic order and
${\bm{v}}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{|\mathscr{S}|}$ is an affine
function. In particular, ${\bm{v}}({\bm{x}})$ is a vector whose $s$-th
component is $a_{s}({\bm{x}})$, where $s\in\mathscr{S}$.

  * (b)

Let ${\bm{U}}\triangleq{\mathbb{E}}[{\bm{U}}({\bm{\xi}})]\succeq{\bm{0}}$ and

$c(\rho)\triangleq\left\\{\begin{array}[]{ll}{{({q}(\rho)-1)}^{2}}\exp(\frac{2{q}(\rho)}{{q}(\rho)-1})&\
{\rm if}\ {q}(\rho)>2;\\\ 1/{\sqrt{\rho}}&\ {\rm if}\
{q}(\rho)=2,\end{array}\right.$ (13)

where

${\mathop{q}}\,(\rho)\triangleq\left\\{\begin{array}[]{ll}\frac{-\ln\rho+\sqrt{{{(\ln\rho)}^{2}}-8\ln\rho}}{4}&\
{\rm if}\ \rho\in(0,\exp(-8)];\\\ 2&\ {\rm otherwise},\end{array}\right.$ (14)

Then, the second-order cone constraint

${{a}_{0}}({\bm{x}})\geq c(\rho)||{{\bm{U}}^{1/2}}{\bm{x}}||$ (15)

serves as a safe tractable approximation of the chance constraint in (11).

This theorem was first proposed in [21] for robust beamforming design in a
two-way relay network. Therein, a partial proof was sketched while important
details were omitted due to the page limit. In this work, we will give a
complete proof in Appendix A-A. Now, let us apply Theorem 1 to our problem. We
define

$\displaystyle a_{0}({\bm{W}})$ $\displaystyle=$
$\displaystyle-\sigma_{{v,d}}^{2}-{\bm{W}}\cdot\big{(}{\bm{\Sigma}}\odot(\bar{\bm{g}}\bar{\bm{g}}^{H})-\frac{P_{t}}{\gamma}((\bar{\bm{f}}\bar{\bm{f}}^{H})\odot(({\bar{\bm{g}}}^{*})({\bar{\bm{g}}}^{*})^{H}))\big{)}.$

Then, we have

$\displaystyle\bar{f}({\bm{W}},{\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle f({\bm{W}},{\bm{x}},{\bm{y}})+a_{0}({\bm{W}})$ $\displaystyle=$
$\displaystyle{\bm{W}}\cdot{\bm{M}}({\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle{\rm vec}({\bm{W}})^{H}{\rm vec}({\bm{M}}({\bm{x}},{\bm{y}}))$

and

${\rm\mathbb{E}}\big{[}\bar{f}({\bm{W}},{\bm{x}},{\bm{y}})^{2}\big{]}={\rm
vec}({\bm{W}})^{H}{\bm{U}}{\rm vec}({\bm{W}}),$

where

${\bm{U}}={\rm\mathbb{E}}\big{[}{\rm vec}({\bm{M}}({\bm{x}},{\bm{y}})){\rm
vec}({\bm{M}}({\bm{x}},{\bm{y}}))^{H}\big{]}.$

The explicit forms of ${\bm{M}}({\bm{x}},{\bm{y}})$ and ${\bm{U}}$ will be
given in the sequel. Armed with Theorem 1, the following second-order cone
constraint serves as a safe approximation of the chance constraint ${\rm
Pr}\\{Q(\bm{W},\bm{x},\bm{y})\geq 0\\}\leq\rho$ in (8):

$a_{0}({\bm{W}})\geq c_{1}(\rho)\|{\bm{U}}^{1/2}{\rm vec}({\bm{W}})\|.$

This yields the following safe approximation of Problem (8):

$\begin{array}[]{ll}\displaystyle\min_{{\bm{W}}\in\mathbb{C}^{L\times
L}}&\quad\displaystyle(P_{t}{\bm{I}}\odot(\bar{\bm{f}}\bar{\bm{f}}^{H}+\epsilon^{2}{\bm{I}})+{\bm{\Sigma}})\cdot{\bm{W}}\\\
\text{subject to}&\quad\displaystyle a_{0}({\bm{W}})\geq
c_{1}(\rho)\|{\bm{U}}^{1/2}{\rm vec}({\bm{W}})\|,\\\ &\quad{\bm{W}}\succeq
0.\end{array}$ (16)

Note that Problem (16) can be readily handled by applying the SDR technique
and off-the-shelf convex solvers.

TABLE I: Explicit Expressions in Fourth-order Moment Inequality-based Approach ${\bm{M}}({\bm{x}},{\bm{y}})_{(i,i)}$ |  $\displaystyle{\bm{M}}({\bm{x}},{\bm{y}})_{(i,i)}=\eta\sigma_{i}^{2}({y_{i}}{\bar{g}}_{i}^{*}+{\bar{g}}_{i}{y_{i}}^{*})+\eta^{2}\sigma_{i}^{2}{y_{i}}{y_{i}}^{*}-\frac{P_{t}}{\gamma}\big{[}\big{(}\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}({y_{i}}^{*}{\bar{g}}_{i}+{\bar{g}}_{i}^{*}{y_{i}})$ $\displaystyle+\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}({x_{i}}{\bar{f}}_{i}^{*}+{\bar{f}}_{i}{x_{i}}^{*})\big{)}+\big{(}\eta^{2}{\bar{f}}_{i}{\bar{f}}_{i}^{*}{y_{i}}{y_{i}}^{*}+\epsilon^{2}{\bar{g}}_{i}{\bar{g}}_{i}^{*}{x_{i}}{x_{i}}^{*}+\epsilon\eta({x_{i}}{\bar{f}}_{i}^{*}+{\bar{f}}_{i}{x_{i}}^{*})({y_{i}}^{*}{\bar{g}}_{i}+{\bar{g}}_{i}^{*}{y_{i}})\big{)}$ $\displaystyle+\big{(}\epsilon\eta^{2}({x_{i}}{\bar{f}}_{i}^{*}+{\bar{f}}_{i}{x_{i}}^{*}){y_{i}}{y_{i}}^{*}+\eta\epsilon^{2}{x_{i}}{x_{i}}^{*}({y_{i}}^{*}{\bar{g}}_{i}+{\bar{g}}_{i}^{*}{y_{i}})\big{)}+\epsilon^{2}\eta^{2}{x_{i}}{x_{i}}^{*}{y_{i}}{y_{i}}^{*}\big{]}$
---|---
${\bm{M}}({\bm{x}},{\bm{y}})_{(k,\ell)}$ |  $\displaystyle{\bm{M}}({\bm{x}},{\bm{y}})_{(k,\ell)}=-\frac{P_{t}}{\gamma}\big{[}\big{(}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}({y_{k}}^{*}{\bar{g}}_{\ell}+{\bar{g}}_{k}^{*}{y_{\ell}})+\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}({x_{k}}{\bar{f}}_{\ell}^{*}+{\bar{f}}_{k}{x_{\ell}}^{*})\big{)}$ $\displaystyle+\big{(}\eta^{2}{{\bar{f}}_{k}}{{\bar{f}}_{\ell}}^{*}{y_{k}}^{*}{y_{\ell}}+\epsilon^{2}{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{x_{k}}{x_{\ell}}^{*}+\epsilon\eta({x_{k}}{\bar{f}}_{\ell}^{*}+{\bar{f}}_{k}{x_{\ell}}^{*})({y_{k}}^{*}{\bar{g}}_{\ell}+{\bar{g}}_{k}^{*}{y_{\ell}})\big{)}$ $\displaystyle+\big{(}\epsilon\eta^{2}({x_{k}}{\bar{f}}_{\ell}^{*}+{\bar{f}}_{k}{x_{\ell}}^{*}){y_{k}}^{*}{y_{\ell}}+\eta\epsilon^{2}{x_{k}}{x_{\ell}}^{*}({y_{k}}^{*}{\bar{g}}_{\ell}+{\bar{g}}_{k}^{*}{y_{\ell}})\big{)}+\epsilon^{2}\eta^{2}{x_{k}}{x_{\ell}}^{*}{y_{k}}^{*}{y_{\ell}}\big{]},\quad 1\leq i\leq L,1\leq k\neq\ell\leq L$
${\bm{U}}_{(i,j),(k,\ell)}$ |  ${\bm{U}}_{(i,j),(k,\ell)}=\left\\{\begin{array}[]{rl}\Sigma_{m=1}^{9}C_{m}(i)C_{m}(i)^{*}&\ {\rm if}~{}1\leq i=j=k=\ell\leq L;\\\ C_{9}(i)C_{9}(k)^{*}&\ {\rm if}\ i=j,\ k=\ell,\ i\neq k;\\\ C_{1}(i)D_{1}(i,\ell)^{*}+C_{3}(i)D_{3}(i,\ell)^{*}+C_{5}(i)D_{5}(i,\ell)^{*}&\ {\rm if}\ i=j,\ k\neq\ell,\ i=k;\\\ C_{2}(i)D_{2}(k,i)^{*}+C_{4}(i)D_{4}(k,i)^{*}+C_{8}(i)D_{8}(k,i)^{*}&\ {\rm if}\ i=j,\ k\neq\ell,\ i=\ell;\\\ D_{1}(i,j)C_{1}(i)^{*}+D_{3}(i,j)C_{3}(i)^{*}+D_{5}(i,j)C_{5}(i)^{*}&\ {\rm if}\ i\neq j,\ k=\ell,\ i=k;\\\ D_{2}(i,j)C_{2}(j)^{*}+D_{4}(i,j)C_{4}(j)^{*}+D_{8}(i,j)C_{8}(j)^{*}&\ {\rm if}\ i\neq j,\ k=\ell,\ j=k;\\\ \Sigma_{n=1}^{15}D_{n}(i,j)D_{n}(i,j)^{*}&\ {\rm if}~{}i\neq j,\ k\neq\ell,\ i=k,\ j=\ell;\\\ D_{1}(k,j)D_{1}(k,\ell)^{*}+D_{3}(k,j)D_{3}(k,\ell)^{*}+D_{5}(k,j)D_{5}(k,\ell)^{*}&\ {\rm if}~{}i\neq j,\ k\neq\ell,\ i=k,\ j\neq\ell;\\\ D_{2}(i,\ell)D_{2}(k,\ell)^{*}+D_{4}(i,\ell)D_{4}(k,\ell)^{*}+D_{8}(i,\ell)D_{8}(k,\ell)^{*}&\ {\rm if}~{}i\neq j,\ k\neq\ell,\ i\neq k,\ j=\ell;\\\ 0&\ {\rm otherwise.}\end{array}\right.$

A straightforward but tedious calculation yields
${\bm{M}}({\bm{x}},{\bm{y}})_{(i,j)}$ in Table I. To derive an explicit
expression for ${\bm{U}}$, we treat $(i,j)$ as an index of any vectorized
matrix in the sequel. In other words, for a matrix ${\bm{R}}$, we denote

${\rm vec}({\bm{R}})_{(i,j)}={\bm{R}}_{i,j}.$

Recall that ${\bm{U}}={\rm\mathbb{E}}\big{[}{\rm
vec}({\bm{M}}({\bm{x}},{\bm{y}})){\rm
vec}({\bm{M}}({\bm{x}},{\bm{y}}))^{H}\big{]}.$ Then,

$\displaystyle{\bm{U}}_{(i,j),(k,\ell)}$ $\displaystyle=$
$\displaystyle{\rm\mathbb{E}}\big{[}{\rm
vec}({\bm{M}}({\bm{x}},{\bm{y}}))_{(i,j)}{\rm
vec}({\bm{M}}({\bm{x}},{\bm{y}}))^{*}_{(k,\ell)}\big{]}$ $\displaystyle=$
$\displaystyle{\rm\mathbb{E}}\big{[}{\bm{M}}({\bm{x}},{\bm{y}})_{i,j}{\bm{M}}({\bm{x}},{\bm{y}})^{*}_{k,\ell}\big{]}.$

The entries of $\bm{U}$ are given as ${\bm{U}}_{(i,j),(k,\ell)}$ in Table I
with

$\left\\{\begin{array}[]{rl}&C_{1}(i)=\eta\sigma_{i}^{2}{\bar{g}}_{i}-\frac{P_{t}}{\gamma}(\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}{\bar{g}}_{i}+\eta\epsilon^{2}{\bar{g}}_{i});\\\
&C_{2}(i)=\eta\sigma_{i}^{2}{\bar{g}}_{i}^{*}-\frac{P_{t}}{\gamma}(\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}{\bar{g}}_{i}^{*}+\eta\epsilon^{2}{\bar{g}}_{i}^{*});\\\
&C_{3}(i)=-\frac{P_{t}}{\gamma}(\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}{\bar{f}}_{i}^{*}+\epsilon\eta^{2}{\bar{f}}_{i}^{*});\\\
&C_{4}(i)=-\frac{P_{t}}{\gamma}(\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}{\bar{f}}_{i}+\epsilon\eta^{2}{\bar{f}}_{i});\\\
&C_{5}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}^{*}{\bar{g}}_{i};\\\
&C_{6}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}^{*}{\bar{g}}_{i}^{*};\\\
&C_{7}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}{\bar{g}}_{i};\\\
&C_{8}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}{\bar{g}}_{i}^{*};\\\
&C_{9}(i)=\eta^{2}\sigma_{i}^{2}-\frac{P_{t}}{\gamma}(\eta^{2}{\bar{f}}_{i}{\bar{f}}_{i}^{*}+\epsilon^{2}{\bar{g}}_{i}{\bar{g}}_{i}^{*}+\epsilon^{2}\eta^{2})\end{array}\right.$

and

$\left\\{\begin{array}[]{rl}&D_{1}(k,\ell)=-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}{\bar{g}}_{\ell};\\\
&D_{2}(k,\ell)=-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}{\bar{g}}_{k}^{*};\\\
&D_{3}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{\bar{f}}_{\ell}^{*};\\\
&D_{4}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{\bar{f}}_{k};\\\
&D_{5}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{\ell}^{*}{\bar{g}}_{\ell};\\\
&D_{6}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{\ell}^{*}{\bar{g}}_{k}^{*};\\\
&D_{7}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{k}{\bar{g}}_{\ell};\\\
&D_{8}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{k}{\bar{g}}_{k}^{*};\\\
&D_{9}(k,\ell)=-\frac{P_{t}}{\gamma}\eta^{2}{\bar{f}}_{k}{\bar{f}}_{\ell}^{*};\\\
&D_{10}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon^{2}{\bar{g}}_{k}^{*}{\bar{g}}_{\ell};\\\
&D_{11}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta^{2}{\bar{f}}_{\ell}^{*};\\\
&D_{12}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta^{2}{\bar{f}}_{k};\\\
&D_{13}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon^{2}\eta{\bar{g}}_{\ell};\\\
&D_{14}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon^{2}\eta{\bar{g}}_{k}^{*};\\\
&D_{15}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon^{2}\eta^{2}.\\\
\end{array}\right.$

###  III-B The Second-order Moment Inequality Method

The aforementioned fourth-order moment inequality method can provide a
feasible solution to Problem (8). However, as it involves all orders (from
first to fourth) of channel uncertainties in the outage constraint, the
ultimate explicit expression is rather complicated. In the robust design, we
observe that channel uncertainties are usually small and thus higher-order
uncertainty terms contribute very little to the SNR quantity. Hence, we are
motivated to ignore the third and fourth-order uncertainty terms so as to
significantly simplify our problem.

To proceed, we let $f_{quad}({\bm{W}},{\bm{x}},{\bm{y}})$ be the quadratic
approximation of $f({\bm{W}},{\bm{x}},{\bm{y}})$, obtained by dropping the
cubic and quartic terms in $f({\bm{W}},{\bm{x}},{\bm{y}})$. Then, we consider
the following second-order approximation of the original chance constraint:

$\Pr(f_{quad}({\bm{W}},{\bm{x}},{\bm{y}})\geq 0)\leq\rho.$ (17)

To tackle the above constraint, we define

$\displaystyle\bar{f}_{quad}({\bm{W}},{\bm{x}},{\bm{y}})$ $\displaystyle=$
$\displaystyle f_{quad}({\bm{W}},{\bm{x}},{\bm{y}})+a_{0}({\bm{W}})$
$\displaystyle=$
$\displaystyle{\bm{W}}\cdot{\bm{M}}_{quad}({\bm{x}},{\bm{y}})$
$\displaystyle=$ $\displaystyle{\rm vec}({\bm{W}})^{H}{\rm
vec}({\bm{M}}_{quad}({\bm{x}},{\bm{y}}))$

and
${\rm\mathbb{E}}\big{[}\bar{f}_{quad}({\bm{W}},{\bm{x}},{\bm{y}})^{2}\big{]}={\rm
vec}({\bm{W}})^{H}{\bm{U}}_{quad}{\rm vec}({\bm{W}}),$ where
${\bm{U}}_{quad}={\rm\mathbb{E}}\big{[}{\rm
vec}({\bm{M}}_{quad}({\bm{x}},{\bm{y}})){\rm
vec}({\bm{M}}_{quad}({\bm{x}},{\bm{y}}))^{H}\big{]}$. The explicit forms of
${\bm{M}}_{quad}({\bm{x}},{\bm{y}})$ and $\bm{U}_{quad}$ are given in the
sequel. By Theorem 1, the second-order cone constraint

$a_{0}({\bm{W}})\geq c_{2}(\rho)\|{\bm{U}_{quad}}^{1/2}{\rm vec}({\bm{W}})\|$

serves as a safe approximation of the chance constraint (17). This yields the
following safe approximation of Problem (8):

$\begin{array}[]{ll}\displaystyle\min_{{\bm{W}}\in\mathbb{C}^{L\times
L}}&\quad\displaystyle(P_{t}{\bm{I}}\odot(\bar{\bm{f}}\bar{\bm{f}}^{H}+\epsilon^{2}{\bm{I}})+{\bm{\Sigma}})\cdot{\bm{W}}\\\
\text{subject to}&\quad\displaystyle a_{0}({\bm{W}})\geq
c_{2}(\rho)\|{\bm{U}}_{quad}^{1/2}{\rm vec}({\bm{W}})\|,\\\
&\quad{\bm{W}}\succeq 0.\end{array}$ (18)

Note that Problem (18) can be readily handled by applying SDR technique and
off-the-shelf convex solvers.

The explicit expression for $\bm{M}_{quad}({\bm{x}},{\bm{y}})$ and
$\bm{U}_{quad}$ can be obtained in a similar way. We provide the explicit
expression of $\bm{M}_{quad}({\bm{x}},{\bm{y}})$ and $\bm{U}_{quad}$ in Table
II with

$\left\\{\begin{array}[]{rl}&C_{1}(i)=\eta\sigma_{i}^{2}{\bar{g}}_{i}-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}{\bar{g}}_{i};\\\
&C_{2}(i)=\eta\sigma_{i}^{2}{\bar{g}}_{i}^{*}-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}{\bar{g}}_{i}^{*};\\\
&C_{3}(i)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}{\bar{f}}_{i}^{*};\\\
&C_{4}(i)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}{\bar{f}}_{i};\\\
&C_{5}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}^{*}{\bar{g}}_{i};\\\
&C_{6}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}^{*}{\bar{g}}_{i}^{*};\\\
&C_{7}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}{\bar{g}}_{i};\\\
&C_{8}(i)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{i}{\bar{g}}_{i}^{*};\\\
&C_{9}(i)=\eta^{2}\sigma_{i}^{2}-\frac{P_{t}}{\gamma}(\eta^{2}{\bar{f}}_{i}{\bar{f}}_{i}^{*}+\epsilon^{2}{\bar{g}}_{i}{\bar{g}}_{i}^{*})\end{array}\right.$

and

$\left\\{\begin{array}[]{rl}&D_{1}(k,\ell)=-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}{\bar{g}}_{\ell};\\\
&D_{2}(k,\ell)=-\frac{P_{t}}{\gamma}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}{\bar{g}}_{k}^{*};\\\
&D_{3}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{\bar{f}}_{\ell}^{*};\\\
&D_{4}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{\bar{f}}_{k};\\\
&D_{5}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{\ell}^{*}{\bar{g}}_{\ell};\\\
&D_{6}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{\ell}^{*}{\bar{g}}_{k}^{*};\\\
&D_{7}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{k}{\bar{g}}_{\ell};\\\
&D_{8}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon\eta{\bar{f}}_{k}{\bar{g}}_{k}^{*};\\\
&D_{9}(k,\ell)=-\frac{P_{t}}{\gamma}\eta^{2}{\bar{f}}_{k}{\bar{f}}_{\ell}^{*};\\\
&D_{10}(k,\ell)=-\frac{P_{t}}{\gamma}\epsilon^{2}{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}.\\\
\end{array}\right.$

Apparently, the above second-order moment inequality-based approach involves
less perturbation terms compared to its fourth-order counterpart. In fact,
compared to Problem (16), it is easier to find a feasible solution to Problem
(18).

TABLE II: Explicit Expressions in Second-order Moment Inequality-based Approach ${\bm{M}}({\bm{x}},{\bm{y}})_{(i,i)}$ |  $\displaystyle{\bm{M}}({\bm{x}},{\bm{y}})_{(i,i)}=\eta\sigma_{i}^{2}({y_{i}}{\bar{g}}_{i}^{*}+{\bar{g}}_{i}{y_{i}}^{*})+\eta^{2}\sigma_{i}^{2}{y_{i}}{y_{i}}^{*}-\frac{P_{t}}{\gamma}\big{[}\eta{\bar{f}}_{i}{\bar{f}}_{i}^{*}({y_{i}}^{*}{\bar{g}}_{i}+{\bar{g}}_{i}^{*}{y_{i}})+\epsilon{\bar{g}}_{i}{\bar{g}}_{i}^{*}({x_{i}}{\bar{f}}_{i}^{*}+{\bar{f}}_{i}{x_{i}}^{*})$ $\displaystyle+\eta^{2}{\bar{f}}_{i}{\bar{f}}_{i}^{*}{y_{i}}{y_{i}}^{*}+\epsilon^{2}{\bar{g}}_{i}{\bar{g}}_{i}^{*}{x_{i}}{x_{i}}^{*}+\epsilon\eta({x_{i}}{\bar{f}}_{i}^{*}+{\bar{f}}_{i}{x_{i}}^{*})({y_{i}}^{*}{\bar{g}}_{i}+{\bar{g}}_{i}^{*}{y_{i}})\big{]}$
---|---
${\bm{M}}({\bm{x}},{\bm{y}})_{(k,\ell)}$ |  $\displaystyle{\bm{M}}({\bm{x}},{\bm{y}})_{(k,\ell)}=-\frac{P_{t}}{\gamma}\big{[}\eta{\bar{f}}_{k}{\bar{f}}_{\ell}^{*}({y_{k}}^{*}{\bar{g}}_{\ell}+{\bar{g}}_{k}^{*}{y_{\ell}})+\epsilon{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}({x_{k}}{\bar{f}}_{\ell}^{*}+{\bar{f}}_{k}{x_{\ell}}^{*})$ $\displaystyle+\eta^{2}{{\bar{f}}_{k}}{{\bar{f}}_{\ell}}^{*}{y_{k}}^{*}{y_{\ell}}+\epsilon^{2}{\bar{g}}_{k}^{*}{\bar{g}}_{\ell}{x_{k}}{x_{\ell}}^{*}+\epsilon\eta({x_{k}}{\bar{f}}_{\ell}^{*}+{\bar{f}}_{k}{x_{\ell}}^{*})({y_{k}}^{*}{\bar{g}}_{\ell}+{\bar{g}}_{k}^{*}{y_{\ell}})\big{]},\quad 1\leq i\leq L,1\leq k\neq\ell\leq L$
${\bm{U}}^{quad}_{(i,j),(k,\ell)}$ |  ${\bm{U}}^{quad}_{(i,j),(k,\ell)}=\left\\{\begin{array}[]{rl}\Sigma_{m=1}^{9}C_{m}(i)C_{m}(i)^{*}&\ {\rm if}~{}1\leq i=j=k=\ell\leq L;\\\ C_{9}(i)C_{9}(k)^{*}&\ {\rm if}\ i=j,\ k=\ell,\ i\neq k;\\\ C_{1}(i)D_{1}(i,\ell)^{*}+C_{3}(i)D_{3}(i,\ell)^{*}+C_{5}(i)D_{5}(i,\ell)^{*}&\ {\rm if}\ i=j,\ k\neq\ell,\ i=k;\\\ C_{2}(i)D_{2}(k,i)^{*}+C_{4}(i)D_{4}(k,i)^{*}+C_{8}(i)D_{8}(k,i)^{*}&\ {\rm if}\ i=j,\ k\neq\ell,\ i=\ell;\\\ D_{1}(i,j)C_{1}(i)^{*}+D_{3}(i,j)C_{3}(i)^{*}+D_{5}(i,j)C_{5}(i)^{*}&\ {\rm if}\ i\neq j,\ k=\ell,\ i=k;\\\ D_{2}(i,j)C_{2}(j)^{*}+D_{4}(i,j)C_{4}(j)^{*}+D_{8}(i,j)C_{8}(j)^{*}&\ {\rm if}\ i\neq j,\ k=\ell,\ j=k;\\\ \Sigma_{n=1}^{10}D_{n}(i,j)D_{n}(i,j)^{*}&\ {\rm if}\ i\neq j,\ k\neq\ell,\ i=k,\ j=\ell;\\\ D_{1}(k,j)D_{1}(k,\ell)^{*}+D_{3}(k,j)D_{3}(k,\ell)^{*}+D_{5}(k,j)D_{5}(k,\ell)^{*}&\ {\rm if}\ i\neq j,\ k\neq\ell,\ i=k,\ j\neq\ell;\\\ D_{2}(i,\ell)D_{2}(k,\ell)^{*}+D_{4}(i,\ell)D_{4}(k,\ell)^{*}+D_{8}(i,\ell)D_{8}(k,\ell)^{*}&\ {\rm if}\ i\neq j,\ k\neq\ell,\ i\neq k,\ j=\ell;\\\ 0&\ {\rm otherwise.}\end{array}\right.$

##  IV The Bernstein-type Inequality Approach

Although the fourth-order moment inequality-based approach is much more
restricted and the feasibility rate can be lower, it can provide a more robust
AF design, compared to the second-order counterpart. Hence, in practice, we
can first try to find an AF weight via Problem (16). If it cannot provide a
feasible solution, we then drop the higher-order perturbations and resort to a
quadratically perturbed chance constraint problem, e.g., Problem (18), to find
a relatively good AF weight. In this section, we introduce another typical way
of tackling the quadratically perturbed chance constraint, i.e., to use the
so-called Bernstein-type inequality [13].

To be specific, we define ${\bm{\xi}}=\sqrt{2}\left(\begin{array}[]{c}{\rm
Re}({\bm{x}})\\\ {\rm Im}({\bm{x}})\\\ {\rm Re}({\bm{y}})\\\ {\rm
Im}({\bm{y}})\end{array}\right)\sim\mathcal{N}(0,{\bm{I}}_{4L})$, with ${\rm
Re}(\cdot)$ and ${\rm Im}(\cdot)$ denoting the real part and imaginary part of
a complex number, respectively. Then, the quadratic approximation can be
rewritten as

$\displaystyle{f}_{quad}({\bm{\xi}})=s_{0}({\bm{\xi}})+s_{1}({\bm{\xi}})+s_{2}({\bm{\xi}}),$
(19)

where $s_{0}({\bm{\xi}}),s_{1}({\bm{\xi}}),s_{2}({\bm{\xi}})$ represent the
constant term, linear term, and quadratic term in ${f}_{quad}({\bm{\xi}})$,
respectively. More precisely, we have

$\displaystyle s_{0}({{\bm{\xi}}})$ $\displaystyle=$
$\displaystyle-\sigma_{{v}}^{2}-{\bm{W}}\cdot\big{(}{\bm{\Sigma}}\odot(\bar{\bm{g}}\bar{\bm{g}}^{H})\big{)}+\frac{P_{t}}{\gamma}{\bm{W}}\cdot\big{(}{\bar{\bm{f}}}{\bar{\bm{f}}}^{H}\odot({\bar{\bm{g}}}^{*})({\bar{\bm{g}}}^{*})^{H}\big{)},$
$\displaystyle s_{1}({{\bm{\xi}}})$ $\displaystyle=$
$\displaystyle{\bm{W}}\cdot\big{(}\frac{P_{t}\epsilon}{\gamma}({\bm{x}}\bar{\bm{f}}^{H}+\bar{\bm{f}}{\bm{x}}^{H})\odot{\bar{\bm{g}}}^{*}({\bar{\bm{g}}}^{*})^{H}-\eta{\bm{\Sigma}}\odot(\bar{\bm{g}}{\bm{y}^{H}}+{\bm{y}}\bar{\bm{g}}^{H})\big{)}$
$\displaystyle+\frac{P_{t}\eta}{\gamma}{\bm{W}}\cdot\big{(}(\bar{\bm{f}}\bar{\bm{f}}^{H})\odot(({\bm{y}}^{*})(\bar{\bm{g}}^{*})^{H}+(\bar{\bm{g}}^{*})({\bm{y}}^{*})^{H}),$
$\displaystyle s_{2}({{\bm{\xi}}})$ $\displaystyle=$
$\displaystyle\frac{P_{t}}{\gamma}{\bm{W}}\cdot\big{(}\epsilon^{2}({\bm{x}}{\bm{x}}^{H})\odot({\bar{\bm{g}}}^{*})({\bar{\bm{g}}}^{*})^{H}+\eta^{2}(\bar{\bm{f}}\bar{\bm{f}}^{H})\odot(({\bm{y}}^{*})({\bm{y}}^{*})^{H})\big{)}$
$\displaystyle+\frac{P_{t}}{\gamma}{\bm{W}}\cdot\big{(}\eta\epsilon({\bm{x}}\bar{\bm{f}}^{H}+\bar{\bm{f}}{\bm{x}}^{H})\odot(({\bm{y}}^{*})(\bar{\bm{g}}^{*})^{H}+(\bar{\bm{g}}^{*})({\bm{y}}^{*})^{H})\big{)}$
$\displaystyle-\eta^{2}{\bm{W}}\cdot\big{(}{\bm{\Sigma}}\odot({\bm{y}}{\bm{y}}^{H})\big{)}.$

Define ${\bm{F}}=\bar{\bm{f}}\bar{\bm{f}}^{H}$,
${\bm{G}}=(\bar{\bm{g}}^{*})(\bar{\bm{g}}^{*})^{H}$, one can easily verify
that

$s_{1}({{\bm{\xi}}})={\bm{v}}_{{\bm{\xi}}}^{T}{\bm{\xi}},$

where

${\bm{v}}_{{\bm{\xi}}}={\sqrt{2}}\left(\begin{array}[]{c}\epsilon\frac{Pt}{\gamma}{\rm
Re}\big{(}{\bm{W}}\odot{\bm{G}}^{*})\bar{\bm{f}}\big{)}\\\
\epsilon\frac{Pt}{\gamma}{\rm
Im}\big{(}{\bm{W}}\odot{\bm{G}}^{*})\bar{\bm{f}}\big{)}\\\ -\eta{\rm
Re}({\bm{W}}\odot{\bm{\Sigma}}){\rm
Re}(\bar{\bm{g}})+\eta\frac{Pt}{\gamma}{\rm
Re}\big{(}{\bm{W}}^{*}\odot{\bm{F}})\bar{\bm{g}}\big{)}\\\ -\eta{\rm
Re}({\bm{W}}\odot{\bm{\Sigma}}){\rm
Im}(\bar{\bm{g}})+\eta\frac{Pt}{\gamma}{\rm
Im}\big{(}{\bm{W}}^{*}\odot{\bm{F}})\bar{\bm{g}}\big{)}\\\
\end{array}\right).$

In addition, consider

$\displaystyle{\bm{R}}=\frac{1}{2}\left(\begin{array}[]{cc}\epsilon^{2}\frac{P_{t}}{\gamma}{\bm{K}_{1}}&\epsilon\eta\frac{P_{t}}{\gamma}{\bm{K}_{2}}\\\
\epsilon\eta\frac{P_{t}}{\gamma}{\bm{K}_{3}}&-\eta^{2}{\bm{K}_{4}^{(1)}}+\eta^{2}\frac{P_{t}}{\gamma}{\bm{K}_{4}^{(2)}}\\\
\end{array}\right),$

where

$\displaystyle{\bm{K}_{1}}=\left(\begin{array}[]{cc}{\rm
Re}({\bm{W}}^{*}\odot{\bm{G}})&{\rm Im}({\bm{W}}^{*}\odot{\bm{G}})\\\ -{\rm
Im}({\bm{W}}^{*}\odot{\bm{G}})&{\rm Re}({\bm{W}}^{*}\odot{\bm{G}})\\\
\end{array}\right),$
$\displaystyle{\bm{K}_{4}^{(1)}}=\left(\begin{array}[]{cc}{\rm
Re}({\bm{W}}\odot{\bm{\Sigma}})&{\bm{0}}_{L\times L}\\\ {\bm{0}}_{L\times
L}&{\rm Re}({\bm{W}}\odot{\bm{\Sigma}})\\\ \end{array}\right),$
$\displaystyle{\bm{K}_{4}^{(2)}}=\left(\begin{array}[]{cc}{\rm
Re}({\bm{W}}\odot{\bm{F}}^{*})&{\rm Im}({\bm{W}}\odot{\bm{F}}^{*})\\\ -{\rm
Im}({\bm{W}}\odot{\bm{F}}^{*})&{\rm Re}({\bm{W}}\odot{\bm{F}}^{*})\\\
\end{array}\right),$

and ${\bm{K}_{2}},{\bm{K}_{3}}$ are given at the top of the next page.

$\displaystyle{\bm{K}_{2}}=\left(\begin{array}[]{cc}{\rm Diag}\big{(}{\rm
Re}({\bm{W}}(\bar{\bm{f}}\odot\bar{\bm{g}}^{*}))\big{)}+{\rm
Re}\big{(}{\bm{W}}\odot(\bar{\bm{g}}\bar{\bm{f}}^{T})\big{)}&{\rm
Diag}\big{(}{\rm
Im}({\bm{W}}^{*}(\bar{\bm{f}}^{*}\odot\bar{\bm{g}}))\big{)}+{\rm
Im}\big{(}{\bm{W}}\odot(\bar{\bm{g}}\bar{\bm{f}}^{T})\big{)}\\\ {\rm
Diag}\big{(}{\rm Im}({\bm{W}}(\bar{\bm{f}}\odot\bar{\bm{g}}^{*}))\big{)}+{\rm
Im}\big{(}{\bm{W}}\odot(\bar{\bm{g}}\bar{\bm{f}}^{T})\big{)}&{\rm
Diag}\big{(}{\rm
Re}({\bm{W}}^{*}(\bar{\bm{f}}^{*}\odot\bar{\bm{g}}))\big{)}-{\rm
Re}\big{(}{\bm{W}}\odot(\bar{\bm{g}}\bar{\bm{f}}^{T})\big{)}\\\
\end{array}\right)$ $\displaystyle{\bm{K}_{3}}=\left(\begin{array}[]{cc}{\rm
Diag}\big{(}{\rm Re}({\bm{W}}(\bar{\bm{f}}\odot\bar{\bm{g}}^{*}))\big{)}+{\rm
Re}\big{(}{\bm{W}}^{*}\odot(\bar{\bm{f}}\bar{\bm{g}}^{T})\big{)}&{\rm
Diag}\big{(}{\rm Im}({\bm{W}}(\bar{\bm{f}}\odot\bar{\bm{g}}^{*}))\big{)}+{\rm
Im}\big{(}{\bm{W}}^{*}\odot(\bar{\bm{f}}\bar{\bm{g}}^{T})\big{)}\\\ {\rm
Diag}\big{(}{\rm
Im}({\bm{W}}^{*}(\bar{\bm{f}}^{*}\odot\bar{\bm{g}}))\big{)}+{\rm
Im}\big{(}{\bm{W}}^{*}\odot(\bar{\bm{f}}\bar{\bm{g}}^{T})\big{)}&{\rm
Diag}\big{(}{\rm
Re}({\bm{W}}^{*}(\bar{\bm{f}}^{*}\odot\bar{\bm{g}}))\big{)}-{\rm
Re}\big{(}{\bm{W}}^{*}\odot(\bar{\bm{f}}\bar{\bm{g}}^{T})\big{)}\\\
\end{array}\right)$

We symmetrize ${\bm{R}}$ by setting
$\bar{{\bm{R}}}=\frac{1}{2}({{\bm{R}}}+{{\bm{R}}}^{T})$, so that
$s_{2}({{\bm{\xi}}})={\bm{\xi}}^{T}\bar{{\bm{R}}}{\bm{\xi}}$. Furthermore,
denote $s_{{\bm{\xi}}}=s_{0}({\bm{\xi}})$, so that we can rewrite the
quadratic approximation as

${f}_{quad}({\bm{\xi}})=s_{{\bm{\xi}}}+{\bm{v}}_{{\bm{\xi}}}^{T}{\bm{\xi}}+{\bm{\xi}}^{T}\bar{\bm{R}}{\bm{\xi}}.$

Now, we apply the Bernstein-type inequality to the chance constraint

${\rm
Pr}\\{s_{{\bm{\xi}}}+{\bm{v}}_{{\bm{\xi}}}^{T}{\bm{\xi}}+{\bm{\xi}}^{T}\bar{{\bm{R}}}{\bm{\xi}}\geq
0\\}\geq 1-\rho$

and obtain the following safe approximation of our problem:

$\begin{array}[]{ll}\displaystyle\min_{{\bm{W}},\lambda,\delta}&\displaystyle(P_{t}{\bm{I}}\odot(\bar{\bm{f}}\bar{\bm{f}}^{H}+\epsilon^{2}{\bm{I}})+{\bm{\Sigma}})\cdot{\bm{W}}\\\
\text{subject to}&\displaystyle{\rm Tr}(\bar{{\bm{R}}})-2\sqrt{{\rm
ln}(1/\rho)}\cdot\delta+2{\rm ln}(\rho)\cdot\lambda+s_{\bm{\xi}}\geq 0,\\\
&\sqrt{\|\bar{{\bm{R}}}\|^{2}_{F}+\frac{1}{2}\|{\bm{v}}_{\bm{\xi}}\|^{2}}\leq\delta,\\\
&\lambda{\bm{I}}_{4L}+\bar{{\bm{R}}}\succeq{\bm{0}},\\\ &\lambda\geq 0,\\\
&{\bm{W}}\succeq 0.\end{array}$ (20)

Again, the problem can be readily solved by the SDR technique and convex
solvers.

Remark 1: For both the moment inequality-based and Bernstein-type inequality-
based AF design, we target at solving Problem (8) to find a good AF weight
matrix ${\bm{W}}$, rather than an AF weight vector ${\bm{w}}$, since we have
relaxed the rank constraint in (7) for the sake of computational tractability.
Hence, after we have solved (8), we still need to apply the Gaussian
randomization algorithm [22]; more specifically, see Algorithm 1 in [32] on
how to find an approximate rank-one AF weight vector ${\bm{w}}$.

Remark 2: It is worth noting that solving Problem (16) yields a safe
approximate solution to Problem (8), while solving Problem (18) or Problem
(20) yields only approximate but not necessarily safe solution to Problem (8).
This is because we drop the higher-order perturbation terms from the SNR
expression.

##  V A Relative Tightness Analysis

When we tackle the second-order approximation of the chance constraint in
(11)—i.e., $\Pr(f_{quad}({\bm{W}},{\bm{x}},{\bm{y}})\geq 0)\leq\rho$—we resort
to two types of restriction approaches. The first is the second-order moment
inequality approach in (18) and the other is the Bernstein-type inequality
approach in (20). In this section, we provide a theoretical relative tightness
result for the two approaches. To proceed, we consider a general Gaussian
quadratic polynomial

$\displaystyle Q({\bm{\xi}})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\xi_{i}a_{i}+\sum_{i,j=1}^{m}\xi_{i}\xi_{j}a_{ij}={\bm{\xi}}^{T}{\bm{A}}{\bm{\xi}}+{\bm{a}}^{T}{\bm{\xi}},$

where ${\bm{\xi}}\sim\mathcal{N}(\mathbf{0},\bm{I})$. Then, our tightness
result can be summarized in the following theorem:

######  Theorem 2.

For the chance constraint

$\Pr(Q({\bm{\xi}})\geq t)\leq\rho,$ (21)

consider its two safe approximations:

  * •

Moment inequality-based safe approximation:

$t\geq c(\rho)\sqrt{{\mathbb{E}}\left[Q({\bm{\xi}})^{2}\right]}.$

  * •

Bernstein-type inequality-based safe approximation:

$\displaystyle\displaystyle-{\rm Tr}({{\bm{A}}})-2\sqrt{{\rm
ln}(1/\rho)}\cdot\delta+2{\rm ln}(\rho)\cdot\lambda+t\geq 0,$
$\displaystyle\sqrt{\|{{\bm{A}}}\|^{2}_{F}+\frac{1}{2}\|{\bm{a}}\|^{2}}\leq\delta,$
$\displaystyle\lambda{\bm{I}}_{4L}-{{\bm{A}}}\succeq{\bm{0}},$
$\displaystyle\lambda\geq 0.$

Then, for any $\rho\in(\exp(-8),0.00045)$, the moment inequality-based safe
approximation is always more conservative than the Bernstein-type inequality-
based safe approximation.

###### Proof.

We assume first that $\rho>\exp(-8)$ in the chance constraint (21). Using the
fact that ${\mathbb{E}}\left[\xi_{i}\xi_{j}\right]=\delta_{ij}$,
${\mathbb{E}}\left[\xi_{i}\xi_{j}\xi_{k}\right]=0$, and

$\displaystyle{\mathbb{E}}\left[\xi_{i}\xi_{j}\xi_{k}\xi_{l}\right]$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{c@{\quad}l}3&\mbox{if
}i=j=k=l,\\\ 1&\mbox{if there are two distinct indices},\\\
0&\mbox{otherwise},\end{array}\right.$

we can compute

$\displaystyle{\mathbb{E}}\left[Q({\bm{\xi}})^{2}\right]$ $\displaystyle=$
$\displaystyle{\bm{v}}^{T}{\mathbb{E}}\left[{\bm{\eta}}{\bm{\eta}}^{T}\right]{\bm{v}}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}a_{i}^{2}+3\sum_{i=1}^{m}a_{ii}^{2}+\sum_{1\leq
i\not=j\leq m}\left(a_{ii}a_{jj}+a_{ij}^{2}\right)$ $\displaystyle=$
$\displaystyle\|{\bm{a}}\|_{2}^{2}+\|{\bm{A}}\|_{F}^{2}+(\mbox{tr}({\bm{A}}))^{2}+\sum_{i=1}^{m}a_{ii}^{2}.$

Then, following the moment inequality-based approach, we have a safe tractable
approximation of (21):

$\displaystyle t$
$\displaystyle\geq\sqrt{\frac{{\mathbb{E}}\left[Q({\bm{\xi}})^{2}\right]}{\rho}}$
(23)
$\displaystyle=\left(\frac{\|{\bm{a}}\|_{2}^{2}+\|{\bm{A}}\|_{F}^{2}+(\mbox{tr}({\bm{A}}))^{2}+\sum_{i=1}^{m}a_{ii}^{2}}{\rho}\right)^{1/2}.$

On the other hand, the Bernstein-type inequality-based approach yields the
following safe tractable approximation of (21):

$t\geq\mbox{tr}({\bm{A}})+2\sqrt{\ln\frac{1}{\epsilon}}\sqrt{\|{\bm{A}}\|_{F}^{2}+\frac{1}{2}\|{\bm{a}}\|_{2}^{2}}+\left(2\ln\frac{1}{\epsilon}\right)s^{+}({\bm{A}}),$
(24)

where $s^{+}({\bm{A}})=\max\\{\lambda_{\max}({\bm{A}}),0\\}$. We claim that
for sufficiently small $\rho>\exp(-8)$, every feasible solution to (23) is
feasible for (24), which means that (23) is more conservative than (24).
Indeed, if (23) is satisfied, then

$\displaystyle t$
$\displaystyle\geq\frac{1}{\sqrt{\rho}}\cdot\sqrt{\|{\bm{a}}\|_{2}^{2}+\|{\bm{A}}\|_{F}^{2}+(\mbox{tr}({\bm{A}}))^{2}}$
$\displaystyle\geq\frac{1}{\sqrt{3\rho}}\left(\|{\bm{a}}\|_{2}+\|A\|_{F}+|\mbox{tr}({\bm{A}})|\right)$
(25)
$\displaystyle\geq\frac{1}{\sqrt{3\rho}}\left(\left[\sqrt{\frac{1}{16}\|{\bm{A}}\|_{F}^{2}+\|{\bm{a}}\|_{2}^{2}}+\frac{3}{4}\|{\bm{A}}\|_{F}\right]+|\mbox{tr}({\bm{A}})|\right)$
(26)
$\displaystyle\geq\mbox{tr}({\bm{A}})+\frac{1}{4\sqrt{3\rho}}\sqrt{\|{\bm{A}}\|_{F}^{2}+\frac{1}{2}\|{\bm{a}}\|_{2}^{2}}+\frac{\sqrt{3}}{4\sqrt{\rho}}s^{+}({\bm{A}}),$
(27)

where (25) and (26) follow from the fact that for any
${\bm{x}}\in\mathbb{R}^{n}$,

$\|{\bm{x}}\|_{1}\geq\|{\bm{x}}\|_{2}\geq\frac{1}{\sqrt{n}}\|{\bm{x}}\|_{1},$

and (27) holds as long as $\rho<1/3$ (note that
$s^{+}({\bm{A}})\leq\|{\bm{A}}\|_{F}$). Now, the proof of the claim will be
complete if

$\frac{1}{4\sqrt{3\rho}}\geq
2\sqrt{\ln\frac{1}{\rho}}\quad\mbox{and}\quad\frac{\sqrt{3}}{4\sqrt{\rho}}\geq
2\ln\frac{1}{\rho}.$

It can be verified that the above inequalities hold if $\rho<0.00045$. Hence,
(23) is more conservative than (24) when $\rho\in(\exp(-8),0.00045)$ (note
that $\exp(-8)\approx 0.0003)$. This completes the proof. ∎

It is worth remarking that a general tightness result for the moment
inequality-based and Bernstein-type inequality-based safe approximations is
still difficult to obtain, while Theorem 2 provides a provable result in a
small region of $\rho$. In the next section, we will provide numerical
validations via simulation results.

Figure 2: Minimum Power required versus the SNR threshold. For
$\epsilon^{2}=\eta^{2}=0.002$; SNR outage percentage $\rho=0.1$.

Figure 3: Minimum Power required versus the SNR threshold. For
$\epsilon^{2}=\eta^{2}=0.06$; SNR outage percentage $\rho=0.1$.

Figure 4: Histogram of the SNR satisfaction probability. For
$\epsilon^{2}=\eta^{2}=0.002$; SNR outage percentage $\rho=0.1$; SNR threshold
$\gamma=18$dB.

Figure 5: Histogram of the SNR satisfaction probability. For
$\epsilon^{2}=\eta^{2}=0.06$; SNR outage percentage $\rho=0.1$; SNR threshold
$\gamma=12$dB.

##  VI Numerical Simulations

In this section, we provide numerical simulations to compare four different AF
weight designs for our target problem, namely, the non-robust (NR) AF weight,
the fourth-order moment inequality-based (M4) AF weight, the second-order
moment inequality-based (M2) AF weight, and the Bernstein-type inequality-
based (B2) AF weight. The setup of the experiments is as follows. The number
of relays is set to be $L=4$; channels are generated by
$\bar{\bm{f}}\sim\mathcal{CN}(0,{\bm{I}})$,
$\bar{\bm{g}}\sim\mathcal{CN}(0,{\bm{I}})$ independently; channel errors are
generated by $\Delta{\bm{f}}\sim\mathcal{CN}(0,\epsilon^{2}{\bm{I}})$,
$\Delta{\bm{g}}\sim\mathcal{CN}(0,\eta^{2}{\bm{I}})$ independently, where the
variances are specified in the sequel; the noise power at each relay is set to
be $\sigma_{\ell}^{2}=0.25,\forall\ell$; the noise power at the receiver is
$\sigma_{v}^{2}=0.25$; the outage probability is denoted by $\rho$. In this
paper, we take $\rho=0.1$ to illustrate the performance of each scheme. Also,
if the $kth$ largest eigenvalue of the SDR solution is $10^{4}$ times larger
than the $(k+1)st$ largest eigenvalue, then the SDR solution is considered to
be of rank-$k$.

###  VI-A Comparison among the M4, M2, and B2 Approaches

In Fig. 5, we present the averaged minimum power budget needed to satisfy the
outage constraint as the SNR threshold varies from 3dB to $18$dB when
$\rho=0.1$, $\epsilon^{2}=\eta^{2}=0.002$. Specifically, we compare the
results for the non-robust and robust cases under different design approaches.
In Fig. 5, we may consider that the NR design discards all perturbation terms,
and thus the required power budget serves as a lower bound for all the robust
designs. We find that the M4 and M2 approaches require similar minimum power
budgets, while the B2 and NR approaches require slightly less power budget.
This is because the channel errors here are rather small and all the robust
designs can easily handle the uncertainties in this case. This implies that
the higher-order perturbation terms do not have much effect on the SNRs if the
error is small. Despite the similar power budgets, we can still see the
differences of the tightness among the four design approaches. In Fig. 5, we
test the outage cases of each approach by generating $1000$ complex Gaussian
channel perturbations with specific variance and evaluating the SNR
satisfaction probability for each channel realization (herein we calculate the
SNR by (9)). For each approach, we generate $100$ channel realizations and
then pick up those realizations that are feasible for all methods to get the
corresponding histograms in Fig. 5. As shown in Fig. 5, the SNR satisfaction
percentage of NR is the lowest, basically near $50\%$, which implies that the
NR design is not reliable in general. On the contrary, all robust designs are
very conservative and almost always satisfy the SNR constraint. It reveals
that both the moment inequality-based and Bernstein-type inequality-based
schemes are robust against channel errors. However, it is important to note
from the histogram that compared to the M4 and M2 designs, the B2 design
sometimes violates the SNR constraint. This implies that the moment
inequality-based approaches are more conservative than the Bernstein-type
inequality-based approach. This is consistent with our relative tightness
analysis in Section V, although the outage rate $\rho$ is not within the
target region.

In Fig. 5 and Fig. 5, we increase the variances of the channel errors to be
$0.06$. Since in this case the channel uncertainty is significantly enlarged,
we can see an obvious power budget gap among different strategies. We find
that M4 requires the largest power to support the robust design since the
fourth-order moment inequality is the strictest. B2 requires a relatively
lower power budget than M2 and M4, which implies that moment inequality-based
approaches are more conservative than the Bernstein-type inequality-based
approach. This is confirmed by the SNR satisfaction histogram (herein we
calculate the SNR by (9)). The results in Fig. 5 again demonstrate the
robustness of the moment inequality-based and Bernstein-type inequality-based
approaches, as the SNR satisfaction percentages of B2, M2, and M4 all exceed
our target threshold $0.9$. Actually, in this setting, the M2 and M4 designs
always satisfy the SNR, while the B2 design sometimes violates the SNR
constraint. This also confirms that the B2 approach is less conservative than
the M2 and M4 approaches.

To further investigate conservatism, we increase the variance of channel
uncertainty and compare the feasibility percentage, rank-one percentage, and
randomization feasibility percentage in Tables III and IV. In Table III, we
have $\rho=0.1$ and $\epsilon^{2}=\eta^{2}=0.06$; in Table IV, we have have
$\rho=0.1$ and $\epsilon^{2}=\eta^{2}=0.08$. Herein, “Feasibility%” denotes
the feasibility of the design problems (16), (18) and (20); “Rank $k\%$”
denotes the percentage of rank-$k$ solutions of the original design problem;
“Feasibility of Rand. in Rank $k$ Cases $\%$” denotes the rate of feasibility
of the randomization algorithm for rank-$k$ solutions, where we set the number
of randomizations to be $1000$. Note that “NaN” in the table indicates that
the percentage of a certain rank is $0$, and thus there is no randomization of
cases for this rank. The data in both tables reveal the same fact, i.e., M2 is
less conservative than M4, while B2 is less conservative than M2 and M4. This
is consistent with our analytical results in Section V, although the outage
rate $\rho$ for these cases is not within the target region in Theorem 2.

###  VI-B Comparison between M4 and M2 Approaches under Mismatched Noise and
Perturbations

To demonstrate that the M4 design is more robust than the M2 one, we set up
three types of experiments. The first one follows the same setting as that in
Fig. 5, where we have $\sigma_{\ell}^{2}=\sigma_{v}^{2}=0.25$,
$\epsilon^{2}=\eta^{2}=0.002$, $\rho=0.1$, and $\gamma=18$dB. We generate
$1000$ and $10000$ channel realizations, respectively. For each channel
realization, we solve M4 and M2 respectively to obtain the AF weights. For the
1000 (resp. 10000) channel realizations generated, 861 (resp. 8725) of them
are feasible for both the original design problems (16) and (18), and their
optimal solutions are rank-one. Then, for each feasible channel realization,
we generate $10000$ perturbation realizations to calculate the actual SNR (we
calculate the SNR by (9)) satisfaction rate $\hat{\rho}$ under M4 and M2 AF
weights, where $\hat{\rho}$ is calculated as the ratio of the SNR-satisfying
cases and the number of perturbation realizations (10000). In Table V, we show
the number of channel realizations that fall in different intervals of
$\hat{\rho}$. Clearly, both M4 and M2 approaches provide a good outage rate,
which is much less than $0.1$. However, we can still see that M4 is more
restricted than M2, as the former has more channel realizations that exhibit
higher $\hat{\rho}$ under the given perturbation realizations.

In the second experiment, we consider the scenario that the actual noise
levels $\hat{\sigma}_{{\ell}}^{2}$ and $\hat{\sigma}_{{v}}^{2}$, or the actual
perturbations $\hat{\epsilon}$ and $\hat{\eta}$, do not match the prior
information in the original design problem (7). That is, when we solve (7), we
set $\sigma_{\ell}^{2}=\sigma_{v}^{2}=0.25$, $\epsilon^{2}=\eta^{2}=0.002$,
$\rho=0.1$, and $\gamma=18$dB, while in practice, the noise levels change to
$\hat{\sigma}_{\ell}^{2}$ and $\hat{\sigma}_{v}^{2}$; or the perturbation
levels change to $\hat{\epsilon}^{2}$ and $\hat{\eta}^{2}$. Under the
interference of the mismatch, SNR satisfaction rates and the corresponding
number of channel realizations are shown in Table VI. We find that under M4,
there are more channel realizations that exhibit higher $\hat{\rho}$ under the
given mismatched realizations, which means that M4 is more robust than M2.

To further corroborate the robustness of M4, in Table VII, we investigate the
critical point at which the mismatch of the noise or the perturbation causes
the outage. To proceed, we set $\rho=0.1,\gamma=18$dB and solve the M4 and M2
design problems. Among $1000$ channel realizations, there are in total $861$
feasible cases for both M4 and M2. From the table, we can clearly see that as
the mismatch increases, when
$\hat{\sigma}_{\ell}^{2}=\hat{\sigma}_{v}^{2}=0.265$ or
$\hat{\epsilon}^{2}=\hat{\eta}^{2}=0.0032$, M2 may cause outage in one out of
$861$ realizations while M4 does not cause outage in all $861$ realizations.
This implies that M4 is more robust than M2, as the former takes into account
the exact SNR expression by keeping all higher-order perturbations.

TABLE III: Feasibility rate, rank-$k$ rate and feasibility rate of Randomization (Rand.): $\sigma_{\ell}^{2}=0.25,\forall\ell$, $\sigma_{v}^{2}=0.25$, $\epsilon^{2}=\eta^{2}=0.06$, $\rho=0.1$. SNR in dB | $\gamma=3$ | $\gamma=6$ | $\gamma=9$ | $\gamma=12$ | $\gamma=15$
---|---|---|---|---|---
Method | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2
Feasibility% | $0.45$ | $0.50$ | $0.96$ | $0.42$ | $0.47$ | $0.92$ | $0.36$ | $0.40$ | $0.82$ | $0.28$ | $0.29$ | $0.78$ | $0.22$ | $0.24$ | $0.63$
Rank $1$% | $1.000$ | $1.000$ | $0.844$ | $1.000$ | $1.000$ | $0.891$ | $0.972$ | $1.000$ | $0.951$ | $1.000$ | $1.000$ | $0.961$ | $1.000$ | $1.000$ | $0.984$
Rank $2$% | $0$ | $0$ | $0.031$ | $0$ | $0$ | $0.022$ | $0.028$ | $0$ | $0.012$ | $0$ | $0$ | $0.026$ | $0$ | $0$ | $0.016$
Rank $3$% | $0$ | $0$ | $0.010$ | $0$ | $0$ | $0.065$ | $0$ | $0$ | $0.037$ | $0$ | $0$ | $0.013$ | $0$ | $0$ | $0$
Rank $4$% | $0$ | $0$ | $0.115$ | $0$ | $0$ | $0.022$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
| Feasibility of Rand.
---
in Rank $2$ Cases%
NaN | NaN | $0.0507$ | NaN | NaN | $0.0280$ | $1.0000$ | NaN | $0.0750$ | NaN | NaN | $0.0345$ | NaN | NaN | $0.004$
| Feasibility of Rand.
---
in Rank $3$ Cases%
NaN | NaN | $0.0020$ | NaN | NaN | $0.0005$ | NaN | NaN | $0$ | NaN | NaN | $0$ | NaN | NaN | NaN
| Feasibility of Rand.
---
in Rank $4$ Cases%
NaN | NaN | $9\times 10^{-5}$ | NaN | NaN | $0$ | NaN | NaN | NaN | NaN | NaN | NaN | NaN | NaN | NaN
TABLE IV: Feasibility rate, rank-$k$ rate and feasibility rate of Randomization (Rand.): $\sigma_{\ell}^{2}=0.25,\forall\ell$, $\sigma_{v}^{2}=0.25$, $\epsilon^{2}=\eta^{2}=0.08$, $\rho=0.1$. SNR in dB | $\gamma=3$ | $\gamma=6$ | $\gamma=9$ | $\gamma=12$ | $\gamma=15$
---|---|---|---|---|---
Method | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2 | M4 | M2 | B2
Feasibility% | $0.17$ | $0.19$ | $0.91$ | $0.16$ | $0.19$ | $0.81$ | $0.16$ | $0.16$ | $0.72$ | $0.10$ | $0.13$ | $0.70$ | $0.06$ | $0.06$ | $0.44$
Rank $1$% | $1.000$ | $1.000$ | $0.714$ | $1.000$ | $1.000$ | $0.840$ | $1.00$ | $1.00$ | $0.833$ | $1.000$ | $1.000$ | $0.871$ | $1.000$ | $1.000$ | $0.864$
Rank $2$% | $0$ | $0$ | $0.077$ | $0$ | $0$ | $0.074$ | $0$ | $0$ | $0.115$ | $0$ | $0$ | $0.086$ | $0$ | $0$ | $0.114$
Rank $3$% | $0$ | $0$ | $0.033$ | $0$ | $0$ | $0.025$ | $0$ | $0$ | $0.039$ | $0$ | $0$ | $0.029$ | $0$ | $0$ | $0$
Rank $4$% | $0$ | $0$ | $0.176$ | $0$ | $0$ | $0.061$ | $0$ | $0$ | $0.013$ | $0$ | $0$ | $0.014$ | $0$ | $0$ | $0.022$
| Feasibility of Rand.
---
in Rank $2$ Cases%
NaN | NaN | $0.0733$ | NaN | NaN | $0.0583$ | NaN | NaN | $0.0453$ | NaN | NaN | $0.0187$ | NaN | NaN | $0.004$
| Feasibility of Rand.
---
in Rank $3$ Cases%
NaN | NaN | $0.0097$ | NaN | NaN | $0.0020$ | NaN | NaN | $0.0033$ | NaN | NaN | $0.0010$ | NaN | NaN | NaN
| Feasibility of Rand.
---
in Rank $4$ Cases%
NaN | NaN | $0.0003$ | NaN | NaN | $0.0008$ | NaN | NaN | $0.0020$ | NaN | NaN | $0$ | NaN | NaN | $0.2320$
TABLE V: SNR Satisfaction rates for different channel realizations: $\sigma_{\ell}^{2}=0.25,\forall\ell$, $\sigma_{v}^{2}=0.25$, $\epsilon^{2}=\eta^{2}=0.002$, $\rho=0.1$; $\gamma=18$dB. Number of Channel Realizations | $1000$ | $10000$
---|---|---
SNR Satisfaction Rate ($\hat{\rho}$) Method | M4 | M2 | M4 | M2
$1.000\geq\hat{\rho}>0.999$ | $852$ | $850$ | $8616$ | $8597$
$0.999\geq\hat{\rho}>0.998$ | $7$ | $8$ | $86$ | $95$
$0.998\geq\hat{\rho}>0.997$ | $2$ | $2$ | $8$ | $7$
$0.997\geq\hat{\rho}>0.996$ | $0$ | $0$ | $3$ | $7$
$0.996\geq\hat{\rho}>0.995$ | $0$ | $1$ | $1$ | $3$
$0.995\geq\hat{\rho}>0.994$ | $0$ | $0$ | $2$ | $3$
$0.994\geq\hat{\rho}>0.993$ | $0$ | $0$ | $1$ | $3$
$0.993\geq\hat{\rho}>0.992$ | $0$ | $0$ | $1$ | $2$
$0.992\geq\hat{\rho}>0.991$ | $0$ | $0$ | $0$ | $1$
$0.991\geq\hat{\rho}>0.990$ | $0$ | $0$ | $1$ | $0$
$0.990\geq\hat{\rho}\geq 0$ | $0$ | $0$ | $6$ | $7$
Number of Feasible Channel Realizations | $861$ | $8725$
TABLE VI: SNR Satisfaction rates under noise and perturbation mismatches: $\rho=0.1$, $\gamma=18$dB, channel realization $=1000$. |  | Original
---
(No Mismatch)
Noise Mismatch | Perturbation Mismatch
Channel Error ($\hat{\eta}^{2}=\hat{\epsilon}^{2}$) | 0.002 | 0.0022 | 0.0024 | 0.0028
Noise ($\hat{\sigma}_{\ell}^{2}=\hat{\sigma}_{v}^{2}$) | 0.25 | 0.252 | 0.256 | 0.258 | 0.25
SNR Satisfaction Rate ($\hat{\rho}$) Method | M4 | M2 | M4 | M2 | M4 | M2 | M4 | M2 | M4 | M2 | M4 | M2 | M4 | M2
$1.000\geq\hat{\rho}>0.999$ | $852$ | $850$ | $703$ | $694$ | $65$ | $59$ | $31$ | $26$ | $306$ | $294$ | $1$ | $0$ | $1$ | $0$
$0.999\geq\hat{\rho}>0.998$ | $7$ | $8$ | $156$ | $165$ | $285$ | $283$ | $85$ | $80$ | $538$ | $550$ | $130$ | $122$ | $0$ | $1$
$0.998\geq\hat{\rho}>0.997$ | $2$ | $2$ | $1$ | $1$ | $343$ | $346$ | $163$ | $158$ | $12$ | $12$ | $508$ | $503$ | $0$ | $0$
$0.997\geq\hat{\rho}>0.996$ | $0$ | $0$ | $0$ | $0$ | $127$ | $131$ | $203$ | $206$ | $2$ | $1$ | $206$ | $215$ | $2$ | $1$
$0.996\geq\hat{\rho}>0.995$ | $0$ | $1$ | $1$ | $0$ | $33$ | $33$ | $182$ | $181$ | $1$ | $1$ | $11$ | $13$ | $7$ | $6$
$0.995\geq\hat{\rho}>0.994$ | $0$ | $0$ | $0$ | $1$ | $4$ | $5$ | $111$ | $119$ | $1$ | $1$ | $0$ | $2$ | $19$ | $14$
$0.994\geq\hat{\rho}>0.993$ | $0$ | $0$ | $0$ | $0$ | $2$ | 2 | $36$ | $37$ | $0$ | $1$ | $1$ | $0$ | $74$ | $69$
$0.993\geq\hat{\rho}>0.992$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $25$ | $27$ | $0$ | $0$ | $1$ | $1$ | $182$ | $183$
$0.992\geq\hat{\rho}>0.991$ | $0$ | $0$ | $0$ | $0$ | $1$ | $1$ | $15$ | $17$ | $0$ | $0$ | $0$ | $2$ | $293$ | $292$
$0.991\geq\hat{\rho}>0.990$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $4$ | $4$ | $0$ | $0$ | $0$ | $0$ | $195$ | $200$
$0.990\geq\hat{\rho}\geq 0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $6$ | $6$ | $1$ | $1$ | $3$ | $3$ | $88$ | $95$
Number of Feasible Channel Realizations | $861$
TABLE VII: Noise mismatch and perturbation mismatch cause the outage: $\rho=0.1$, $\gamma=18$dB, channel realization $=1000$. |  | Original
---
(No Mismatch)
Noise Mismatch | Perturbation Mismatch
Noise ($\hat{\sigma}_{\ell}^{2}=\hat{\sigma}_{v}^{2}$) | 0.25 | 0.265 | 0.25
Channel Error ($\hat{\eta}^{2}=\hat{\epsilon}^{2}$) | 0.002 | 0.0032
SNR Satisfaction Rate ($\hat{\rho}$) Method | M4 | M2 | M4 | M2 | M4 | M2
$1.000\geq\hat{\rho}\geq 0.900$ | $861$ | $861$ | $861$ | $860$ | $861$ | $860$
$0.900>\hat{\rho}\geq 0$ (Outage) | $0$ | $0$ | $0$ | $1$ | $0$ | $1$
Number of Feasible Channel Realizations | $861$

###  VI-C Relative Tightness Verification for M2 and B2 Approaches

Lastly, we set the outage probability $\rho=0.00044$ to verify the conclusion
in Theorem 2, i.e., “ _for any $\rho\in(\exp(-8),0.00045)$, the moment
inequality-based safe approximation is always more conservative than the
Bernstein-type inequality-based safe approximation_.” We also set
$\sigma_{\ell}^{2}=\sigma_{v}^{2}=0.25$ and $\epsilon^{2}=\eta^{2}=0.0005$.
The results are presented in Fig. 6, Fig. 7, and Table VIII. We find that M2
requires more power than B2, which implies that moment inequality-based
approaches are more conservative than the Bernstein-type inequality-based
approach. The SNR satisfaction percentages of both B2 and M2 exceed our target
threshold $1-0.00044$ (herein we calculate the SNR by (19)), as M2 always
provides $100\%$ SNR satisfaction while B2 provides a slightly lower SNR
satisfaction percentage. It is interesting to see that B2 requires a similar
level of power as the non-robust case, while its SNR satisfaction is
significantly better than the latter. This shows the necessity of the robust
design. Table VIII is even more interesting, as the B2 design seems always
feasible while the M2 design is rarely feasible. Note that the rank-one
feasibility is $100\%$ for both B2 and M2. We can clearly see and conclude
that the moment inequality-based approach is more conservative than the
Bernstein-type inequality-based approach from this experiment, which is
consistent with our Theorem 2.

Figure 6: Minimum Power required versus the SNR threshold. For
$\epsilon^{2}=\eta^{2}=0.0005$; SNR outage percentage $\rho=0.00044$.

Figure 7: Histogram of the SNR satisfaction probability. For $\epsilon^{2}=\eta^{2}=0.0005$; SNR outage percentage $\rho=0.00044$; SNR threshold $\gamma=12$dB. TABLE VIII: Feasibility rate and the rank-one rate respectively for $\sigma_{\ell}^{2}=0.25,\forall\ell$, $\sigma_{v}^{2}=0.25$, $\epsilon^{2}=\eta^{2}=0.0005$; $\rho=0.00044$. SNR in dB | $\gamma=3$ | $\gamma=6$ | $\gamma=9$ | $\gamma=12$ | $\gamma=15$
---|---|---|---|---|---
Method | M2 | B2 | M2 | B2 | M2 | B2 | M2 | B2 | M2 | B2
Feasibility% | $0.07$ | $1.00$ | $0.06$ | $1.00$ | $0.06$ | $1.00$ | $0.06$ | $1.00$ | $0.04$ | $1.00$
Rank One% | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$ | $1.00$

##  VII Conclusions

In this paper we studied the robust design problem for two-hop one-way relay
beamforming. Specifically, we considered the scenario where both the
transmitter-to-relay and relay-to-receiver links are subject to errors. This
scenario is difficult and seldom studied in the literature as it involves
chance constraints with quartic perturbations. We provided different
reformulations of the chance-constrained robust design problem and further
analyzed the relative tightness of different reformulations. Numerical results
further confirmed the superiority of the proposed robust design. The quartic
perturbation-based outage-constrained robust design is indeed more
conservative. Nevertheless, by taking into account the higher-order
perturbations, the resulting design is more robust against mismatch of prior
distributional information. The SNR satisfaction rate and rank feasibility
tables verified the tightness results. In the future, many transceiver pairs
could be considered as a non-trivial extension of this work.

##  Appendix A Appendix

###  A-A Proof of Theorem 1

Given $\bar{f}({\bm{x}},{\bm{\xi}})=f({\bm{x}},{\bm{\xi}})+a_{0}(\bm{x})$, by
assumption, for each ${\bm{x}}\in\mathbb{C}^{n}$ the function
${\bm{x}}\mapsto\bar{f}({\bm{x}},{\bm{\xi}})$ is affine in
${\bm{x}}\in\mathbb{C}^{n}$. This implies that
${\bm{x}}\mapsto\bar{f}^{2}({\bm{x}},{\bm{\xi}})$ is a non-negative
homogeneous quadratic polynomial in ${\bm{x}}\in\mathbb{C}^{n}$. This
establishes (a) in Theorem 1.

To prove Theorem 1(b), we need the following lemma.

######  Lemma 1.

(cf. [33, Theorem 5.10]) For all $q\geq 2$,

${\mathbb{E}}\left[|\bar{f}({\bm{x}},{\bm{\xi}})|^{q}\right]^{1/q}\leq(q-1)^{2}{\mathbb{E}}\left[|\bar{f}({\bm{x}},{\bm{\xi}})|^{2}\right]^{1/2}.$

To prove Theorem 1(b), since
$\bar{f}({\bm{x}},{\bm{\xi}})^{2}={\bm{v}}^{T}({\bm{x}}){\bm{U}}({\bm{\xi}}){\bm{v}}({\bm{x}}),$
it follows that

$\displaystyle{\mathbb{E}}\left[|\bar{f}({\bm{x}},{\bm{\xi}})|^{2}\right]\geq$
$\displaystyle{\mathbb{E}}\left[\bar{f}({\bm{x}},{\bm{\xi}})^{2}\right]$
$\displaystyle=$
$\displaystyle{\bm{v}}^{T}({\bm{x}}){\mathbb{E}}\left[{\bm{U}}({\bm{\xi}})\right]{\bm{v}}({\bm{x}})$
$\displaystyle=$
$\displaystyle{\bm{v}}^{T}({\bm{x}}){\bm{U}}{\bm{v}}({\bm{x}}),$

where ${\bm{U}}={\mathbb{E}}\left[{\bm{U}}({\bm{\xi}})\right]$ is a Hermitian
positive semidefinite matrix, which can be computed explicitly, as each entry
of ${\bm{U}}({\bm{\xi}})$ involves only the expectation of a certain product
of standard Gaussian random variables.

For ease of notation, we let $\bar{q}=q(\rho)$, where $q(\rho)$ is defined in
(14). By Lemma 1 and Markov’s inequality, for any $\bar{q}\geq 2$, we have

$\displaystyle\Pr(|\bar{f}({\bm{x}},{\bm{\xi}})|\geq t)$ $\displaystyle\leq$
$\displaystyle\frac{{\mathbb{E}}\left[|\bar{f}({\bm{x}},{\bm{\xi}})|^{\bar{q}}\right]}{t^{\bar{q}}}$
$\displaystyle\leq$
$\displaystyle\frac{({\bar{q}}-1)^{2{\bar{q}}}\cdot{\mathbb{E}}\left[|\bar{f}({\bm{x}},{\bm{\xi}})|^{2}\right]^{{\bar{q}}/2}}{t^{\bar{q}}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{({\bar{q}}-1)^{2}\cdot\|{\bm{U}}^{1/2}{\bm{v}}({\bm{x}})\|_{2}}{t}\right)^{\bar{q}}.$

Thus, whenever $t\geq c(\rho)\|{\bm{U}}^{1/2}{\bm{v}}({\bm{x}})\|_{2}$, where
$c(\rho)$ is defined in (13), we have $\Pr(|\bar{f}({\bm{x}},{\bm{\xi}})|\geq
t)\leq\rho$. Whenever the second-order cone constraint in (15) holds, we have

$\displaystyle\Pr(f({\bm{x}},{\bm{\xi}})\geq 0)=$
$\displaystyle\Pr(\bar{f}({\bm{x}},{\bm{\xi}})\geq a_{0}(\bm{x}))$
$\displaystyle\leq$ $\displaystyle\Pr(|\bar{f}({\bm{x}},{\bm{\xi}})|\geq
a_{0}(\bm{x}))$ $\displaystyle\leq$
$\displaystyle\Pr(|\bar{f}({\bm{x}},{\bm{\xi}})|\geq t)$ $\displaystyle\leq$
$\displaystyle\rho.$

This implies that the (complex) second-order cone constraint (15) is a safe
tractable approximation of (11), as desired.

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