# Fluctuations of the SNR at the output of the MVDR with Regularized Tyler
Estimators

Khalil Elkhalil, Abla Kammoun, , Tareq Y. Al-Naffouri, , and Mohamed-Slim
Alouini This work was funded by a CRG3 grant ORS#2221 from the Office of
Competitive Research (OCRF) at King Abdullah University of Science and
Technology (KAUST). The authors are with the Electrical Engineering Program,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia;
e-mails: {khalil.elkhalil, abla.kammoun, tareq.alnaffouri,
<EMAIL_ADDRESS>
###### Abstract

This paper analyzes the statistical properties of the signal-to-noise ratio
(SNR) at the output of the Capon’s minimum variance distortionless response
(MVDR) beamformers when operating over impulsive noises. Particularly, we
consider the supervised case in which the receiver employs the regularized
Tyler estimator in order to estimate the covariance matrix of the
interference-plus-noise process using $n$ observations of size $N\times 1$.
The choice for the regularized Tylor estimator (RTE) is motivated by its
resilience to the presence of outliers and its regularization parameter that
guarantees a good conditioning of the covariance estimate. Of particular
interest in this paper is the derivation of the second order statistics of the
SINR. To achieve this goal, we consider two different approaches. The first
one is based on considering the classical regime, referred to as the $n$-large
regime, in which $N$ is assumed to be fixed while $n$ grows to infinity. The
second approach is built upon recent results developped within the framework
of random matrix theory and assumes that $N$ and $n$ grow large together.
Numerical results are provided in order to compare between the accuracies of
each regime under different settings.

###### Index Terms:

MVDR beamforming, robust estimators, regularized Tyler estimator, central
limit theorem.

##  I Introduction

The minimum variance distortionless response (MVDR) beamformer or the Capon’s
MVDR beamformer is widely used in sensor array signal processing applications
such as the inspection of direction of arrival (DOA) and the estimation of the
power of a given signal of interest (SOI) [1, 2]. The design of the MVDR
beamforming requires the receiver to acquire an estimate of the unknown
interference and noise covariance matrix. Several covariance estimators
constructed from signal-free observations can be employed. The most popular
ones are those based on the sample covariance matrix (SCM). Their popularity
owe to their low-complexity and the existence of a good understanding of their
behaviour. However, SCM based estimators are well-known to exhibit poor
performances when observations contain outliers. This drawback becomes more
acute in many applications such as radar and sonar processing where the noise
is known to present an impulsive behaviour [3, 4, 5]. A promising alternative
to the use of these estimators is represented by the class of robust scatter
estimators. The latter can be traced back to the early works of Huber [6] and
Maronna [7] in the seventies. With the emergence of new tools allowing the
understanding of many robust covariance estimators, there is today a rekindled
interest in the analysis of these estimators. The focus of our work is on the
regularized tyler estimator (RTE) for which a new wave of important results
have been obtained [8, 9, 10]. Of particular interest in this work is the
behavior of the SNR at the output of the MVDR filter in impulsive noise
environments. This question has not been addressed before. To the best of our
knowledge, the existing works have thus far focused on the behavior of the SNR
of the MVDR filter in Gaussian noise environments. In this course, the early
results have provided an asymptotic characterization of the SNR in the
limiting regime defined by both the number of samples and their dimensions
growing large with the same pace [11]. Subsequently, a second order analysis
gaining insights into the fluctuations of the SNR has been carried out in
[12]. The objective of this paper is to extend the aforementioned results
concerning the behaviour of the SNR at the output of the MVDR when the noise
sample covariance matrix is estimated using the RTE. Under this setting, we
establish in this paper a central limit theorem (CLT) of the SNR under two
different regimes. The first regime corresponds to the classical one obtained
by fixing the dimension of the observations and tending their number to
infinity. The second regime, on the other hand, merely consists in assuming
that the number of samples and their dimensions grow large at the same pace.
Both regimes can be of practical interest. Intuitively, the former is suitable
for scenarios in which the number of observations is much greater than the
array size, while the second is expected to be more accurate when the number
of samples and their dimensions are of the same order of magnitude. Such
intuition will be confirmed by a set of numerical results, comparing the
performance of both regimes in terms of some meaningful metrics.

The remainder of this paper is organized as follows. Section II reviews the
MVDR beamforming and the RTE of the covariance matrix. In section III, the
main result deriving the CLT of the SNR at the MVDR beamformer is provided.
Prior to concluding, the last section presents a set of numerical results
allowing to compare between both regimes.

_Notations:_ Throughout this paper, we use the following notations :
$\left(.\right)^{t}$, $\left(.\right)^{*}$ and $\textnormal{tr}\left(.\right)$
respectively stand for the transpose, the hermitian and the trace of a matrix.
Also, $\otimes$ denotes the Kronecker product between two matrices while
$\Re(.)$ and $\Im(.)$ respectively denote the real and the imaginary parts of
a matrix.

##  II Robust MVDR Filtering with RTE

###  II-A Robust MVDR

We consider a uniform linear array (ULA) with $N$ sensors, receiving a narrow
band source signal. The received vector at time $t$ can be represented by

${\bf y}(t)={\bf s}_{0}s(t)+{\bf x}(t),$

where ${\bf s}_{0}$ and $s(t)$ refer respectively to the array steering vector
and the source signal at time $t$, whereas ${\bf x}(t)$ stands for the
additive noise vector at time $t$. We assume that the distribution of the
noise is heavy-tailed belonging to the family of compound-Gaussian
distributions, i.e, ${\bf x}(t)$ can be put in the following form:

${\bf x}(t)=\sqrt{\tau(t)}\boldsymbol{\Sigma}_{N}^{\frac{1}{2}}{\bf w}(t),$
(1)

where ${\bf w}(t)$ is a $N\times 1$ standard Gaussian vector, $\tau(t)$ is a
positive random scalar called texture. Usually, $\tau(t)$ is drawn from heavy-
tailed distribution in order to account for the impulsive character of the
noise. Matrix $\boldsymbol{\Sigma}_{N}$ is the noise covariance matrix and is
assumed to take the following form [13]

$\mathbf{\Sigma}_{N}=\sigma_{0}^{2}\mathbf{I}_{N}+\sum_{i=1}^{q}\sigma_{i}^{2}\mathbf{a}\left(\theta_{i}\right)\mathbf{a}\left(\theta_{i}\right)^{*},$
(2)

where $q$ is the number of interferers, $\left\\{\theta_{i}\right\\}$,
$i\in\left\\{1,\cdots,q\right\\}$ are their corresponding angles of arrival,
and ${\bf a}(\theta)$ is the $N\times 1$ array steering vector given by:

$\left[{\bf a}(\theta)\right]_{k}=\exp(\jmath 2\pi(k-1)\theta)$

The received vector is processed by a beamformer in order to enhance the
desired signal while reducing the impact of the noise:

${z}={\bf u}^{*}{\bf y}(t).$

We consider in this paper the MVDR beamformer which seeks the best filter
${\bf u}$ that minimizes the power of the resulting noise while ensuring the
distortionless response of the beamformer towards the direction of the desired
source. The corresponding optimization problem is thus given by [13]:

$\mathbf{u}_{0}=\operatorname*{argmin}_{\mathbf{u}\in\mathbb{C}^{N\times
1}:\mathbf{u}^{*}\mathbf{s}_{0}=1}\mathbf{u}^{*}\mathbf{\Sigma}_{N}\mathbf{u},$
(3)

Using the Lagrange method, it can be shown that $\mathbf{u}_{0}$ has the
following closed-form expression:

$\mathbf{u}_{0}=\lambda\mathbf{\Sigma}^{-1}_{N}\mathbf{s}_{0},$ (4)

where $\lambda$ is the Lagrange multiplier satisfying
$\lambda=\frac{1}{\mathbf{s}_{0}^{*}\mathbf{\Sigma}^{-1}_{N}\mathbf{s}_{0}}$.

As shown by (4), the design of the MVDR beamforming requires the knowledge of
the noise covariance matrix. In practice, this unknown covariance matrix is
replaced by an estimate that is built from signal-free observations. In order
to ensure the robustness of the beamformer towards the impulsive character of
the noise, robust covariance estimators should be used. In this paper we focus
on the use of the regularized Tyler estimator (RTE).

###  II-B MVDR Beamforming Based on the RTE

We assume that the receiver has previously acquired $n$ free source signal
observations ${\bf x}_{1},\cdots,{\bf x}_{n}$ drawn from the same distribution
of ${\bf x}(t)$ in (1), i.e,

$\mathbf{x}_{i}=\sqrt{\tau_{i}}\mathbf{\Sigma}_{N}^{\frac{1}{2}}\mathbf{w}_{i},\>i=1,\cdots,n.$
(5)

The regularized robust scatter estimator is defined as the unique solution
$\widehat{\mathbf{C}}_{N}(\rho)$ to the following fixed-point equation:

$\widehat{\mathbf{C}}_{N}(\rho)=\left(1-\rho\right)\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbf{x}_{i}\mathbf{x}_{i}^{*}}{\frac{1}{N}\mathbf{x}_{i}^{*}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{x}_{i}}+\rho\mathbf{I}_{N},$
(6)

where $\rho\in\left(\max\left(0,1-\frac{n}{N}\right),1\right]$ is the
regularization parameter111The existence and uniqueness of $\widehat{\bf
C}_{N}(\rho)$ is proved in [14].. Note that the robustness of the RTE can be
easily seen from (6) which reveals its invariance towards the scaling of ${\bf
x}_{i}$ thus allowing the cancelling-out of the impact of $\tau_{i}$. Using
the RTE for covariance matrix estimation, the optimal MVDR beamforming vector
becomes

$\hat{\mathbf{u}}_{0}=\frac{1}{\mathbf{s}_{0}^{*}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s}_{0}$
(7)

Therefore, the SNR at the output of the MVDR beamforming is given by:

$\widehat{\textnormal{SNR}}\left(\rho\right)=\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}.$
(8)

##  III Asymptotic Behaviour of the MVDR Beamforming SNR

In this paper, our aim is to study the first and second-order statistics of
the SINR in (8). For the sake of tractability, this study is carried out under
two asymptotic regimes. The first one corresponds to $N$ and $n$ growing to
infinity such that $c_{N}\triangleq\frac{N}{n}\to c$ and is referred to as the
large-$(N,n)$ regime, whereas the second one considers the case of fixed $N$
with $n$ growing to infinity and will be coined the Large$-n$ regime.

###  III-A Asymptotic Behavior in the Large-$(N,n)$ Regime

In this section, we study the fluctuations of the SINR in the
large$(N,n)$-regime. To this end, we will essentially rely on the second order
analysis of the SNR at the output of the MVDR established in [12] and the
recent results concerning the behaviour of quadratic forms associated with the
RTE [10]. Details of the derivation are provided in Appendix 1. Before stating
our first main result, we will introduce some notations (see [10] and [12]).
We define $\gamma_{N}\left(\rho\right)$ to be the solution to the following
equation:

$\displaystyle
1=\frac{1}{N}\text{tr}\left[\boldsymbol{\Sigma}_{N}\left(\rho\gamma_{N}\left(\rho\right)\mathbf{I}_{N}+\left(1-\rho\right)\boldsymbol{\Sigma}_{N}\right)^{-1}\right].$

We also denote by $\delta$ the solution to the following fixed-point equation

$\displaystyle\delta=\frac{1}{n}\text{tr}\left[\boldsymbol{\Sigma}_{N}\left(\frac{1}{1+\delta}\boldsymbol{\Sigma}_{N}+\alpha\left(\rho\right)\mathbf{I}_{N}\right)^{-1}\right],$

where

$\displaystyle\alpha\left(\rho\right)=\frac{\rho\gamma_{N}\left(\rho\right)\left(1-(1-\rho)c_{N}\right)}{1-\rho}.$
$\displaystyle\overline{\textnormal{SNR}\left(\rho\right)}=\overline{\text{SNR}\left(\hat{\mathbf{w}}_{0,\text{MVDR}}\right)},\quad\sigma_{N,n}^{2}=\sigma_{s,M}^{2}.$

$\overline{\text{SNR}\left(\hat{\mathbf{w}}_{0,\text{MVDR}}\right)}$ and
$\sigma_{s,M}^{2}$ are defined in [12, Theorem 1] by replacing $\delta_{M}$ by
$\delta$, $\tilde{\delta}_{M}$ by $\frac{1}{1+\delta}$ and $\alpha$ by
$\alpha\left(\rho\right)$.

######  Theorem 1.

Assume that $\boldsymbol{\Sigma}_{N}$ is given by (2) where $q$ is fixed. In
the Large $\left(N,n\right)$\- regime where $(N,n)\rightarrow\infty$ with
$\frac{N}{n}\rightarrow c$, the quantity
$\sigma_{N,n}^{-1}\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\overline{\textnormal{SNR}\left(\rho\right)}\right)$
behaves as a standard normal distribution or equivalently

$\sigma_{N,n}^{-1}\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\overline{\textnormal{SNR}\left(\rho\right)}\right)\xrightarrow[(N,n)\to+\infty]{d}\mathcal{N}\left(0,1\right).$
(9)

###### Proof.

See Appendix A for a detailed proof. ∎

###  III-B Asymptotic Behavior in the large-$n$ Regime

In this section, we study the fluctuations of the SNR at the output of the
MVDR (8) in the large-$n$ regime. Our result will mainly build on the CLT of
the RTE that has recently been derived in [8]. Keeping the same notations as
in [8], the following theorem from [8] establishes the CLT of the robust-
scatter estimator:

######  Lemma 1.

[8] In the large-$n$ regime,

$\displaystyle\sqrt{n}\left(\textnormal{vec}\left(\widehat{\mathbf{C}}_{N}(\rho)\right)-\textnormal{vec}\left(\mathbf{\Sigma}_{0}(\rho)\right)\right)$

behaves as a zero-mean Gaussian distributed vector with covariance matrix
$\mathbf{M}_{1}$ and pseudo-covariance matrix $\mathbf{M}_{2}$ defined in [8],
where $\mathbf{\Sigma}_{0}(\rho)$ is the solution to the following equation

$\mathbf{\Sigma}_{0}(\rho)=N\left(1-\rho\right)\mathbb{E}\left[\frac{\mathbf{x}\mathbf{x}^{*}}{\mathbf{x}^{*}\boldsymbol{\Sigma}_{0}^{-1}(\rho)\mathbf{x}}\right]+\rho\mathbf{I}_{N},$
(10)

where the expectation is taken over the distribution of the random vectors
$\mathbf{x}_{i}$. 222 A simple way to evaluate numerically
$\boldsymbol{\Sigma}_{0}$ has been provided in [8]. It is merely based on
noticing that the eigenvectors of $\boldsymbol{\Sigma}_{0}$ are the same as
$\boldsymbol{\Sigma}_{N}$ while its eigenvalues satisfy a fixed point equation
as shown in [8],

The following theorem can be used in order to derive CLT for any functional of
the RTE under the large$-n$ regime. In particular, we will show in this work
how this CLT can be transferred to that of the SNR at the output of the MVDR
beamforming. Note that under large$-n,N$ regime a similar result cannot be
derived in general as the dimensions of $\widehat{\bf C}_{N}(\rho)$ increase
with the number of samples. Before stating our second main theorem, we shall
introduce the following quantities:

$\displaystyle\mathbf{B}=\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho).$
$\displaystyle\textnormal{SNR}_{0}\left(\rho\right)=\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\mathbf{B}\mathbf{s_{0}}}.$
$\displaystyle\Xi=\frac{1}{2}\begin{bmatrix}\Re(\mathbf{M}_{1})+\Re(\mathbf{M}_{2})&-\Im(\mathbf{M}_{1})+\Im(\mathbf{M}_{2})\\\
\Im(\mathbf{M}_{1})+\Im(\mathbf{M}_{2})&\Re(\mathbf{M}_{1})-\Re(\mathbf{M}_{2})\end{bmatrix},$
$\displaystyle\mathbf{c}^{*}=\frac{\left(\mathbf{s}_{0}^{*}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s}_{0}^{*}\mathbf{B}\mathbf{s_{0}}\right)^{2}}\biggl{[}\mathbf{s}_{0}^{*}\mathbf{B}\left[\left(\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{t}\otimes\mathbf{I}_{N}\right]$
$\displaystyle+\mathbf{s}_{0}^{*}\mathbf{\Sigma}_{0}^{-1}(\rho)\left[\left(\mathbf{B}\mathbf{s_{0}}\right)^{t}\otimes\mathbf{I}_{N}\right]\Biggr{]}$
$\displaystyle-2\frac{\mathbf{s}_{0}^{*}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}}{\mathbf{s}_{0}^{*}\mathbf{B}\mathbf{s_{0}}}\mathbf{s}_{0}^{*}\mathbf{\Sigma}_{0}^{-1}(\rho)\left[\left(\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{t}\otimes\mathbf{I}_{N}\right].$
$\displaystyle\widetilde{\mathbf{c}}=\begin{bmatrix}\Re\left(c\right)\\\
\Im\left(c\right)\end{bmatrix}.$

######  Theorem 2.

In the large$-n$ regime
$\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\textnormal{SNR}_{0}\left(\rho\right)\right)$
behaves as a Normal distribution with zero-mean and variance
$\sigma_{n}^{2}=\widetilde{\mathbf{c}}^{t}\Xi\widetilde{\mathbf{c}}$ or
equivalently

$\sigma_{n}^{-1}\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\textnormal{SNR}_{0}\left(\rho\right)\right)\xrightarrow[n\to+\infty]{d}\mathcal{N}\left(0,1\right).$
(11)

###### Proof.

See Appendix B for a detailed proof. ∎

##  IV Numerical Results

In all our simulations, we consider a uniform linear array (ULA) with elements
located half a wavelength apart. The desired signal is received at an
exploration angle $\theta_{0}=0$ deg, and the interfering signals are received
from the angles $-35$ and $70$ degrees. Moreover, all signals were received at
a power $10$ dB above the background noise. In all simulations, we fix the
number of antennas to $N=4$. Moreover, we assume that the number of
observations $n$ can not exceed $100$ observations ($\frac{n}{N}<25$), which
constitutes the total budget of the system in terms of samples used to
estimate the noise-plus-interference covariance matrix. This assumption is
quite practical and has been considered in many papers in the literature [11,
15, 16]. To assess the accuracy of the derived CLTs in both regimes, we will
use two different metrics, namely the symmetrized divergence
Kolmogorov–-Smirnov (KS) statistic and the $f-$ divergence denoted as
$\mathcal{D}_{f}\left(P||Q\right)$. These two metrics are generally employed
to quantify the difference between two continuous probability distributions
with CDFs $P$ and $Q$ respectively. The KS statistic between $P$ and $Q$ is
given by

$D\triangleq\sup_{x}\left|P\left(x\right)-Q\left(x\right)\right|.$ (12)

while the $f-$divergence with respect to a convex function $f$ satisfying
$f\left(1\right)=0$, is defined as follows

$\mathcal{D}_{f}\left(P||Q\right)=\int
f\left(\frac{p\left(x\right)}{q\left(x\right)}\right)q\left(x\right)dx,$ (13)

where $p$ and $q$ are respectively the corresponding PDFs of $P$ and $Q$. In
Table I, we summarize some selected instances of the functions $f$ that we use
in the letter.333Note that the Kullback-Leibler divergence defined in Table I
is not a distance. We thus use instead its modified version called the
symmetrised divergence given by
$\mathcal{D}_{KL}\left(P||Q\right)+\mathcal{D}_{KL}\left(Q||P\right)$.

Divergence | Corresponding $f\left(t\right)$
---|---
Hellinger distance, $\mathcal{H}\left(P||Q\right)$ | $\left(\sqrt{t}-1\right)^{2}$
Total variation distance, $\mathcal{T}\left(P||Q\right)$ | $\frac{1}{2}\left|t-1\right|$
Kullback-Leibler divergence, $\mathcal{D}_{KL}\left(P||Q\right)$ | $t\log t$
Table I: Selected instances of the $f-$divergence

To have a unified notation, we denote by $\mathcal{D}\left(p,q\right)$, the
distance between $p$ and $q$, where the metric can be either the KS statistic
or the $f$-divergence. With these metrics at hand, we compare the empirical
cumulative distribution function (CDF) of the following quantities

$\displaystyle
Q_{N,n}\triangleq\sigma_{N,n}^{-1}\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\overline{\textnormal{SNR}\left(\rho\right)}\right).$
$\displaystyle
Q_{n}\triangleq\sigma_{n}^{-1}\sqrt{n}\left(\widehat{\textnormal{SNR}}\left(\rho\right)-\textnormal{SNR}_{0}\left(\rho\right)\right).$

with that of the standard normal distribution $\mathcal{N}\left(0,1\right)$.
Letting $\overline{f}_{N,n},\overline{F}_{N,n}$ and
$\overline{f}_{n},\overline{F}_{n}$ be respectively the empirical PDFs and
CDFs of $Q_{N,n}$ and $Q_{n}$, we define the following distance metrics

$\begin{split}&\mathcal{D}_{N,n}\triangleq\mathcal{D}\left(f_{0,1},\overline{f}_{N,n}\right).\\\
&\mathcal{D}_{n}\triangleq\mathcal{D}\left(f_{0,1},\overline{f}_{n}\right).\end{split}$
(14)

where $f_{0,1}$ and $F_{0,1}$ denote respectively the PDF and CDF of the
standard normal distribution
444$f_{0,1}\left(x\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)$
and $F_{0,1}\left(x\right)=1-Q\left(x\right)$, where $Q\left(.\right)$ is the
Q-function..

Figure 1: Empirical cumulative distribution functions (CDF) of $Q_{n}$ and
$Q_{N,n}$ compared with the CDF of the standard Normal density for different
values of $\left(N,n\right)$. $\rho=0.65$. Figure 2: $\mathcal{D}_{n}$ and
$\mathcal{D}_{N,n}$ as defined in (14) versus $\frac{n}{N}$. $\rho=0.1$.
Figure 3: $\mathcal{D}_{n}$ and $\mathcal{D}_{N,n}$ as defined in (14) versus
$\frac{n}{N}$. $\rho=0.5$. Figure 4: $\mathcal{D}_{n}$ and $\mathcal{D}_{N,n}$
as defined in (14) versus $\frac{n}{N}$. $\rho=0.9$.

To begin with, we report in Figure 1 the empirical CDFs of both regimes along
with the standard normal CDF. As seen, the accuracy the $n$-large regime is
lower than the $\left(N,n\right)$-large regime for small values of $n$. As the
number of observations increases, the $\left(N,n\right)-$large regime becomes
less accurate. This is to be compared with the $n-$large regime which provides
a good fit for $n=60$ and $n=100$.

To reinforce the observations made in Figure 1, we display the distance
between the empirical distributions $\overline{f}_{n}$ and
$\overline{f}_{N,n}$ with the standard normal density for different values of
$N$ and $n$. In order to investigate the impact of $\rho$ on the performance,
we display the different instances of the distance for small ($\rho=0.1$
Figure 2), mid ($\rho=0.5$, Figure 3) and high ($\rho=0.9$, Figure 4) values
of $\rho$.

  * •

$\rho=0.1$: We observe in this case that the $(N,n)$-Large regime provides
more accurate results. This might be related to the fact that the empirical
average
$\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbf{x}_{i}\mathbf{x}_{i}^{*}}{\frac{1}{N}\mathbf{x}_{i}^{*}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{x}_{i}}$
is the dominant term in the expression of $\widehat{\bf C}_{N}(\rho)$. This
averaging is approximated in the $n$-Large regime by
$N\left(1-\rho\right)\mathbb{E}\left[\frac{\mathbf{x}\mathbf{x}^{*}}{\mathbf{x}^{*}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{x}}\right]$,
which can be not well-estimated as $n$ is limited to $100$. On the other hand,
the $(n,N)-$ large regime is more accurate since it leverages the double
averaging over $n$ and $N$.

  * •

$\rho=0.9$: In this case, $\rho$ is high, thus, the estimated SNR behaves as a
deterministic quantity since
$\widehat{\mathbf{C}}_{N}(\rho)\simeq\mathbf{I}_{N}$ which is the case for
$\mathbf{\Sigma}_{0}(\rho)$ as well. This is clearly expected by the large$-n$
regime. However, the large$-(n,N)$ regime fails to predict in an accurate way
the performances. One possible explanation can be related to the fact that as
$\rho$ tends to $1$, quantity $\alpha(\rho)$ converges to infinity, causing
the fluctuations to be not properly predicted.

  * •

$\rho=0.5:$ In this case, the main observation is that the
$\left(N,n\right)-$large regime has a better fit to the standard normal
distribution for lower values of $n$, while for $\frac{n}{N}\geq 10$, the
$n-$large regime starts to exhibit a better fit.

As a conclusion, for mid values of $\rho$, it is better to work under the
$\left(N,n\right)-$large regime as long as the number of observations is low.
As we get more observations, the $n-$large regime yields a better performance.

##  V Conclusion

In this paper, we have analyzed the asymptotic behaviour of the Capon’s MVDR
beamformer when using the regularized Tyler estimator (RTE) for both the
$n-$large and the $\left(N,n\right)-$large regimes. Based on recent results on
the convergence of the RTE, we have analyzed the fluctuations of the SNR at
the output of the MVDR. Using well known divergence metrics, we have examined
the accuracy of both regimes and determined which regime is more accurate and
thus more convenient to use.

## Appendix A

## Proof of Theorem 1

The proof hinges on recent results concerning the asymptotic behaviour of the
RTE developed in [10] and the second order analysis of the SNR at the output
of the MVDR derived in [12]. As we shall see next, these results lead together
to the sought-for CLT.

####  V-1 First ingredient: Approximation of the RTE estimate
$\widehat{\mathbf{C}}_{N}(\rho)$

Studying the RTE in the large$-n,N$ regime is not an easy task, as the RTE
does not follow a standard random matrix model. To overcome these issues, the
work in [10] shows that as far as quadratic forms are concerned the asymptotic
behaviour of the $\widehat{\mathbf{C}}_{N}(\rho)$ is the same as another
random object which, contrary to $\widehat{\mathbf{C}}_{N}(\rho)$ can be
studied using standard RMT tools. More formally, we have the following
convergence results,

$N^{1-\epsilon}|\mathbf{u}^{*}\widehat{\mathbf{C}}^{k}_{N}(\rho)\mathbf{v}-\mathbf{u}^{*}\widehat{\mathbf{S}}^{k}_{N}(\rho)\mathbf{v}|\xrightarrow[]{a.s.}0,$
(15)

where $\mathbf{u}$ and $\mathbf{v}$ are unit norm vectors in $\mathbb{C}^{N}$
and $\widehat{\bf S}_{N}(\rho)$ is given by

$\displaystyle\widehat{\mathbf{S}}_{N}(\rho)=\frac{\rho}{\alpha\left(\rho\right)}\frac{1}{n}\sum_{i=1}^{n}\mathbf{z}_{i}\mathbf{z}_{i}^{*}+\rho\mathbf{I}_{N},\mathbf{z}_{i}=\mathbf{\Sigma}_{N}^{\frac{1}{2}}\mathbf{w}_{i},\>i=1,\cdots,n.$

As will be shown next, this convergence implies that the SNR has the same
fluctuations if $\widehat{\bf C}_{N}$ is replaced by
$\widehat{\mathbf{S}}_{N}(\rho)$.

As the SNR is scale invariant, the fluctuations would be the same when
$\widehat{\bf C}_{N}$ is replaced by $\widetilde{\bf S}_{N}(\rho)$ given by:

$\widetilde{\bf
S}_{N}(\rho)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{z}_{i}\mathbf{z}_{i}^{*}+\alpha\left(\rho\right)\mathbf{I}_{N}.$
(16)

Interestingly, the approximation matrix in (16) follows the same structure of
a sample correlation matrix with a diagonal loading factor
$\alpha\left(\rho\right)$ that was considered in [11]. The fluctuations of the
SNR will be thus obtained by simply leveraging the results of [11].

####  V-2 Second-order Analysis of the SNR of Diagonally Loaded MVDR Filters

As discussed above, to prove Theorem 1, it suffices to show that the SNR has
the same fluctuations when the RTE is replaced by $\widehat{\bf S}_{N}(\rho)$.
As per the Slutsky Lemma, this amounts to showing that:

$\displaystyle\sqrt{n}\left(\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}-\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}}\right)$
$\displaystyle\xrightarrow[\frac{N}{n}\to c]{a.s.}0.$

To this end, we first decompose the above term as:

$\displaystyle\sqrt{n}\left(\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}-\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}}\right)$
$\displaystyle=\sqrt{n}\left(\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}-\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}\right)$
$\displaystyle+\sqrt{n}\left(\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}-\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}}\right)$
$\displaystyle\triangleq\xi_{1}+\xi_{2}.$

The term $\xi_{1}$ can be rewritten as

$\begin{split}\xi_{1}&=\frac{\sqrt{n}\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}}.\\\
&\times\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}+\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)\end{split}$
(17)

Then, by the results of (22), we have the following convergence

$\displaystyle\sqrt{n}\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)\xrightarrow[\frac{N}{n}\to
c]{a.s.}0$

Moreover, since,

$\displaystyle\left\|\widehat{\mathbf{C}}_{N}^{-1}(\rho)-\widehat{\mathbf{S}}_{N}^{-1}(\rho)\right\|\xrightarrow[]{a.s.}0.$

then, any well-behaved functional of $\widehat{\mathbf{C}}_{N}^{-1}(\rho)$
converges $a.s.$ to the same functional of
$\widehat{\mathbf{S}}_{N}^{-1}(\rho)$. In particular, we do have:

$\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\xrightarrow[\frac{N}{n}\to
c]{a.s.}0.$

and

$\displaystyle\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\xrightarrow[]{a.s.}0.$

All this leads to

$\displaystyle\xi_{1}\xrightarrow[\frac{N}{n}\to c]{P}0.$

We now handle the term $\xi_{2}$. By a similar reasoning, $\xi_{2}$ can be
rewritten as follows

$\displaystyle\xi_{2}$ $\displaystyle=$
$\displaystyle\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)}$
$\displaystyle\times\sqrt{n}\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right).$

We now refer to the special structure of $\mathbf{\Sigma}_{N}$ and rewrite
$\xi_{2}$ as follows

$\displaystyle\xi_{2}$
$\displaystyle=\frac{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{s_{0}}\right)}$
$\displaystyle\times\sqrt{n}\Biggl{[}\sigma_{0}^{2}\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-2}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-2}(\rho)\mathbf{s_{0}}\right)$
$\displaystyle+\sum_{i=1}^{q}\sigma_{i}^{2}\left(|\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|^{2}-|\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|^{2}\right)\Biggr{]}.$

Noticing that

$\displaystyle|\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|^{2}-|\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|^{2}$
$\displaystyle=\left(|\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|-|\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|\right)$
$\displaystyle\times\left(|\mathbf{s^{*}_{0}}\widehat{\mathbf{S}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|+|\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}_{N}^{-1}(\rho)\mathbf{a}\left(\theta_{i}\right)|\right).$

and resorting to the same arguments used in the control of $\xi_{1}$, it
follows that

$\displaystyle\xi_{2}\xrightarrow[\frac{N}{n}\to c]{a.s.}0.$

This concludes the proof of Theorem 1.

## Appendix B

## Proof of Theorem 2

For ease of presentation, we omit the argument $\rho$ in the SNR expressions.
According to [8], the asymptotic limit of $\widehat{\textnormal{SNR}}$ would
be

$\displaystyle\textnormal{SNR}_{0}=\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}}.$

The objective here is to study the fluctuations of the SNR around
$\textnormal{SNR}_{0}$. To this end, we decompose
$\sqrt{n}\left(\widehat{\textnormal{SNR}}-\textnormal{SNR}_{0}\right)$ by
subtracting and adding
$\sqrt{n}\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}}$
resulting in expression (18) given on the top of the next page.

$\begin{split}&\sqrt{n}\left(\widehat{\textnormal{SNR}}-\textnormal{SNR}_{0}\right)=\\\
&\underbrace{\frac{\sqrt{n}}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}}\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}+\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)}_{Q_{1}}\\\
&+\underbrace{\frac{\sqrt{n}\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)}\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)}_{Q_{2}}\end{split}.$
(18)

We will now treat subsequently the terms $Q_{1}$ and $Q_{2}$ defined in (18).
First, note that using the resolvent identity:

$\widehat{\mathbf{C}}^{-1}_{N}(\rho)-\mathbf{\Sigma}_{0}^{-1}(\rho)=\widehat{\mathbf{C}}^{-1}_{N}(\rho)\left(\mathbf{\Sigma}_{0}(\rho)-\widehat{\mathbf{C}}_{N}(\rho)\right)\mathbf{\Sigma}_{0}^{-1}(\rho).$
(19)

along with the relation:

$\displaystyle\mathbf{x^{*}}\mathbf{A}\mathbf{y}$
$\displaystyle=\textnormal{tr}\left(\mathbf{x^{*}}\mathbf{A}\mathbf{y}\right)$
$\displaystyle=\textnormal{vec}^{*}(\mathbf{x})\textnormal{vec}\left(\mathbf{Ay}\right)$
$\displaystyle=\mathbf{x}^{*}\left(\mathbf{y^{t}}\otimes\mathbf{I}_{N}\right)\textnormal{vec}(\mathbf{A}).$

for ${\bf x}\in\mathbb{C}^{N\times 1},{\bf y}\in\mathbb{C}^{N\times 1}$ and
${\bf A}\in\mathbb{C}^{N\times N}$, yields

$\displaystyle\sqrt{n}\>\mathbf{s^{*}_{0}}\left(\widehat{\mathbf{C}}^{-1}_{N}(\rho)-\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\mathbf{s_{0}}$
$\displaystyle=\sqrt{n}\>\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\left(\mathbf{\Sigma}_{0}(\rho)-\widehat{\mathbf{C}}_{N}(\rho)\right)\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}$
$\displaystyle=\sqrt{n}\>\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\left[\left(\mathbf{s_{0}^{t}}\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\otimes\mathbf{I}_{N}\right]$
$\displaystyle\times\textnormal{vec}\left(\mathbf{\Sigma}_{0}(\rho)-\widehat{\mathbf{C}}_{N}(\rho)\right).$

Using the result of Lemma 1, we have

$\displaystyle\sqrt{n}\>\textnormal{vec}\left(\widehat{\mathbf{C}}_{N}(\rho)-\mathbf{\Sigma}_{0}(\rho)\right)\xrightarrow[n\to+\infty]{d}\mathbf{x}\sim\mathcal{GCN}\left(0,\mathbf{M}_{1},\mathbf{M}_{2}\right),$

where $\mathcal{GCN}\left(\mathbf{0},\mathbf{M}_{1},\mathbf{M}_{2}\right)$
denotes the Generalized Complex Normal distribution with zero-mean, covariance
matrix $\mathbf{M}_{1}$ and pseudo-covariance matrix $\mathbf{M}_{2}$.
Finally, using the following convergence relations

$\displaystyle\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\left[\left(\mathbf{s_{0}^{t}}\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\otimes\mathbf{I}_{N}\right]$
$\displaystyle\xrightarrow[n\to+\infty]{a.s}\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\left[\left(\mathbf{s_{0}^{t}}\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\otimes\mathbf{I}_{N}\right].$
$\displaystyle\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}+\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\xrightarrow[n\to+\infty]{a.s}2\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}.$
$\displaystyle\frac{1}{\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}}\xrightarrow[n\to+\infty]{a.s}\frac{1}{\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}}.$

it follows from the Slutsky’s theorem [17] that:

$\begin{split}Q_{1}\xrightarrow[n\to+\infty]{d}&\frac{-2\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}}{\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}}\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\\\
&\times\left[\left(\mathbf{s_{0}^{t}}\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\otimes\mathbf{I}_{N}\right]\mathbf{x}\end{split}.$
(20)

We now handle $Q_{2}$. To this end, we treat the term
$\sqrt{n}\>\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)$
as follows

$\displaystyle\sqrt{n}\>\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)$
$\displaystyle=\sqrt{n}\>\Biggl{(}\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}$
$\displaystyle+\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\Biggr{)}$
$\displaystyle=\sqrt{n}\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\left(\mathbf{\Sigma}_{0}^{-1}(\rho)-\widehat{\mathbf{C}}^{-1}_{N}(\rho)\right)\mathbf{s_{0}}$
$\displaystyle+\sqrt{n}\>\mathbf{s^{*}_{0}}\left(\mathbf{\Sigma}_{0}^{-1}(\rho)-\widehat{\mathbf{C}}^{-1}_{N}(\rho)\right)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}.$

Similarly, using the resolvent identity, we can write

$\displaystyle\sqrt{n}\>\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}-\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)$
$\displaystyle=\sqrt{n}\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\left(\widehat{\mathbf{C}}_{N}(\rho)-\mathbf{\Sigma}_{0}(\rho)\right)\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}$
$\displaystyle+\sqrt{n}\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\left(\widehat{\mathbf{C}}_{N}(\rho)-\mathbf{\Sigma}_{0}(\rho)\right)\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}$
$\displaystyle=\sqrt{n}\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\left[\left(\mathbf{s_{0}^{t}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\right)\otimes\mathbf{I}_{N}\right]$
$\displaystyle\times\textnormal{vec}\left(\widehat{\mathbf{C}}_{N}(\rho)-\mathbf{\Sigma}_{0}(\rho)\right)+\sqrt{n}\>\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)$
$\displaystyle\times\left[\left(\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)^{t}\otimes\mathbf{I}_{N}\right]\textnormal{vec}\left(\widehat{\mathbf{C}}_{N}(\rho)-\mathbf{\Sigma}_{0}(\rho)\right).$

Also note that

$\displaystyle\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{\Sigma}_{N}\widehat{\mathbf{C}}^{-1}_{N}(\rho)\mathbf{s_{0}}\right)\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)}$
$\displaystyle\xrightarrow[n\to+\infty]{a.s}\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}$

Thus, by means of Slutsky’s theorem, it follows that

$\displaystyle Q_{2}\xrightarrow[n\to+\infty]{d}$
$\displaystyle\frac{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}{\left(\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{2}}\Biggl{[}\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)$
$\displaystyle\times\left[\left(\mathbf{s_{0}^{t}}\mathbf{\Sigma}_{0}^{-1}(\rho)\right)\otimes\mathbf{I}_{N}\right]+\mathbf{s^{*}_{0}}\mathbf{\Sigma}_{0}^{-1}(\rho)$
$\displaystyle\times\left[\left(\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{\Sigma}_{N}\mathbf{\Sigma}_{0}^{-1}(\rho)\mathbf{s_{0}}\right)^{t}\otimes\mathbf{I}_{N}\right]\Biggr{]}\mathbf{x}.$

Gathering the convergence results of $Q_{1}$ and $Q_{2}$, we thus obtain:

$\displaystyle\sqrt{n}\left(\widehat{\textnormal{SNR}}-\textnormal{SNR}_{0}\right)\xrightarrow[n\to+\infty]{d}\mathbf{c^{*}}\mathbf{x}.$

Noticing that

$\begin{split}\mathbf{c^{*}}\mathbf{x}&=\Re(\mathbf{c})^{t}\Re(\mathbf{x})+\Im(\mathbf{c})^{t}\Im(\mathbf{c})\\\
&=\widetilde{\mathbf{c}}^{t}\mathbf{v},\end{split}$ (21)

where $\mathbf{v}=\left[\Re(\mathbf{x})^{t}\Im(\mathbf{x})^{t}\right]^{t}$, it
suffices thus to derive the distribution of $\mathbf{v}$. This follows from
the following Lemma:

######  Lemma 2.

Let $\mathbf{x}=\left(x_{1},x_{2},\cdots,x_{k}\right)^{t}$ be a zero-mean
complex jointly-Gaussian random vector with covariance $\mathbf{M}_{1}$ and
pseudo-covariance $\mathbf{M}_{2}$ and let
$\mathbf{v}=\left[\Re(\mathbf{x})^{t}\Im(\mathbf{x})^{t}\right]^{t}$. Then,
following the results of [18],

$\mathbf{v}\sim\mathcal{N}\left(\mathbf{0},\Xi\right)$ (22)

Using Lemma 22, we conculde that $\mathbf{c^{*}}\mathbf{x}$ is normally
distributed with zero mean and variance
$\sigma_{n}^{2}=\widetilde{\mathbf{c}}^{t}\Xi\widetilde{\mathbf{c}}$. This
conculdes the proof of the theorem.

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