# Performance Analysis of the Matrix Pair Beamformer with Matrix Mismatch

Jianshu Chen, , Jian Wang, Xiu-Ming Shan,
Ning Ge, and Xiang-Gen Xia,
EDICS: SAM-PERF  This work was supported by the National Natural Science
Foundation of China under contract No. 60972019 and No. 60928001. Jianshu Chen
was with the Department of Electronic Engineering, Tsinghua University,
Beijing, P. R. China, 100084. He is with the Department of Electrical
Engineering, University of California, Los Angeles, CA 90095, USA. (e-mail:
<EMAIL_ADDRESS>Jian Wang, Xiu-Ming Shan and Ning Ge are with the
Department of Electronic Engineering, Tsinghua University, Beijing, P. R.
China, 100084. (e-mail: {jian-wang, shanxm, gening} @tsinghua.edu.cn) Xiang-
Gen Xia is with the Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA. (e-mail: xxia@ee.udel.edu). His
work was suppported in part by the Air Force Office of Scientific Research
(AFOSR) under Grant No. FA9550-08-1-0219, the National Science Foundation
(NSF) under Grant CCF-0964500, and the World Class University (WCU) Program
2008-000-20014-0, National Research Foundation, Korea.

###### Abstract

Matrix pair beamformer (MPB) is a blind beamformer. It exploits the temporal
structure of the signal of interest (SOI) and applies generalized eigen-
decomposition to a covariance matrix pair. Unlike other blind algorithms, it
only uses the second order statistics. A key assumption in the previous work
is that the two matrices have the same interference statistics. However, this
assumption may be invalid in the presence of multipath propagations or certain
“smart” jammers, and we call it as matrix mismatch. This paper analyzes the
performance of MPB with matrix mismatch. First, we propose a general framework
that covers the existing schemes. Then, we derive its normalized output SINR.
It reveals that the matrix mismatch leads to a threshold effect caused by
“steering vector competition”. Second, using matrix perturbation theory, we
find that, if there are generalized eigenvalues that are infinite, the
threshold will increase unboundedly with the interference power. This is
highly probable when there are multiple periodical interferers. Finally, we
present simulation results to verify our analysis.

###### Index Terms:

Adaptive beamforming, Matrix pair beamformer, Generalized eigen-decomposition,
Matrix mismatch

##  I Introduction

Beamforming is a spatial filter, which combines the outputs of multiple sensor
elements by an appropriately designed weight vector so as to pass a signal of
interest (SOI) while rejecting interfering signals. Since the pioneering work
of Howells[1], Applebaum[2] and Widrow[3], it has been intensively studied
during the past decades and widely applied in radar, sonar and wireless
communications [3, 4, 5, 6, 7, 8, 9, 10, 11] etc. For a comprehensive review,
we refer to [12, 13, 14, 15] and the references therein.

There are various forms of implementations for a beamformer. Some use the
direction of arrival (DOA) of the SOI and directly calculate the weight vector
by sample matrix inversion (SMI) [16]. Some employ a reference signal (e.g.
training signal[17] and decision feedback signal[6]) to iteratively calculate
the weight vector. And there are also blind beamformers which do not require
the DOA or the reference signal[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31].

To achieve blind beamforming, we need to exploit the special properties of the
SOI, such as constant modulus, non-Gaussianness, high order statistics etc.
One important property is the temporal structure of the SOI. Such kind of
blind beamformers are extensively studied for CDMA systems, where the inherent
structure of the spreading codes can be exploited[23, 24, 25, 26, 28, 29, 30,
31]. The advantage of this scheme is that it only relies on the second order
statistics of the covariance matrix pair. Although the implementation details
differ, the main idea of these approaches is to exploit a pair of array
covariance matrix and hence will be referred to as matrix pair beamformer
(MPB) in this paper. The MPB projects the discrete sequence in each antenna
onto the space spanned by the SOI’s signature vector (for a CDMA system, it is
the spreading code vector) and another carefully designed base vector. Then
two sets of array snapshots (i.e. signal snapshot and interference snapshot)
are acquired to calculate a pair of covariance matrices. With the processing
gain, the desired signal power in the signal snapshot is generally greater
than that in the interference snapshot. _A key and common assumption is that
the interference statistics in the two snapshots are identical._ These two
features enable the separation of the signal steering vector and the
interference covariance matrix from the two matices. And the weight vector
derived from the dominant eigenvector of the matrix pair will maximize the
output signal to interference plus noise ratio (SINR).

However, this key assumption that the two matrices share the same interference
statistics is not valid in many cases, which we refer to as _matrix mismatch_.
Matrix mismatch may occur when there are interferers with certain periodical
structure, such as multiple access interference (MAI) in CDMA systems, tones
and some other “smart jammers” in radar systems and it may lead to the failure
of a system. In practical applications, to avoid such an unexpected failure,
it is necessary to understand the detailed performance of a scheme when the
assumption/condition is not satisfied. To our best knowledge, little effort
has been devoted to analyzing the effect of matrix mismatch on the performance
of MPB. This paper aims to analyze the MPB’s performance under matrix
mismatch, and our contributions are:

  * •

proposing a general framework to model various existing MPB schemes;

  * •

deriving analytical expressions for the normalized output SINR as the
performance measure;

  * •

discovering a threshold effect for MPB, i.e., due to matrix mismatch, the
performance of MPB degrades rapidly when the input signal to noise ratio (SNR)
is below a predicted threshold, and the main beam points to the directions of
the interferers;

  * •

explaining how MPB works “blindly” by “steering vector competition”;

  * •

discussing various factors that have impact on the threshold, and showing that
when there is an generalized eigenvalue that is infinite which is called _the
noise-free covariance matrix pair_ , the threshold SNR increases unboundedly
with the interference power;

  * •

discussing several typical scenarios and showing that MPB is very vulnerable
to multiple periodical interferers.

The rest of the paper is organized as follows. Sec. II presents a general
framework of MPB to cover and reinterpret the basic ideas in [23, 24, 25, 26,
27, 28, 29, 30, 31], followed by a formulation of the matrix mismatch problem.
In Sec. III, we present the expressions for MPB’s weight vector and its
normalized output SINR, which uncovers the inherent threshold effect caused by
matrix mismatch. The discussion relies heavily on the approximation of the
generalized eigenvalue, which is derived in Appendix A using Gerschgorin
theorem[32, 33]. Sec. IV applies the Weyl-Lidskii type theorem in matrix
perturbation theory[32] to analyze MPB’s threshold SNR, and discusses it in
two typical scenarios. Finally, Sec. V presents simulation results to verify
our theoretical analysis and Sec. VI concludes the whole paper.

##  II Problem Formulation

###  II-A Signal Model

Figure 1: Block diagram of the matrix pair beamformer in a CDMA system. Data
corresponding to the $k$ symbol are blocked and projected onto two spaces to
produce signal channel snapshot $\mathbf{x}_{\mathcal{S}}(k)$ and interference
channel snapshot $\mathbf{X}_{\mathcal{I}}(k)$, respectively.

Consider an antenna array of $L$ isotropic elements that receives $D+1$
signals from a far field. After preprocessing (e.g. mixing, filtering, etc.)
and sampling, the output $L\times 1$ array vector can be written as

$\mathbf{x}(n)=\sum_{i=0}^{D}\sqrt{P_{i}}s_{i}(n)\mathbf{a}_{i}+\mathbf{v}(n),$
(1)

where $s_{i}(n)$ is the discrete sequence of the $i$th signal with normalized
power; $P_{i}$ is its power; $\mathbf{a}_{i}$ is the steering vector, which
depends on the DOA and the array geometry; $\mathbf{v}(n)$ is the additive
white Gaussian noise (AWGN) vector with zero mean and covariance matrix
$\sigma^{2}\mathbf{I}$. Assume $s_{0}(n)$ is the SOI, and
$s_{1}(n),s_{2}(n),\ldots,s_{D}(n)$ model all the possible interferers like
MAI and jammer etc. This paper considers the case when $s_{0}(n)$ has some
inherent temporal structure expressed as

$s_{0}(n)=\sum_{k=-\infty}^{+\infty}b_{0}(k)c_{0}[n-\tau(k)],$ (2)

where $c_{0}(n)$ is known and supported on $0\leq n\leq N-1$. Thus, $s_{0}(n)$
can be viewed as a train of pulses, with each of them being a delayed and
scaled replica of $c_{0}(n)$. And $\tau(k)$ and $b_{0}(k)$ are the
corresponding delay and amplitude of the $k$th pulse. This model is common in
modern communication systems. For instance, in CDMA system, $c_{0}(n)$ is the
spreading code of the desired user, $b_{0}(k)$ is its data bits and $\tau(k)$
is the delay of the $k$th symbol. Without loss of generality, let us consider
$b_{0}(k)\in\\{\pm 1\\}$, $c_{0}(n)\in\\{\pm 1\\}$ and $\tau(k)=kN+n_{0}$.
Then $N$ is the processing gain and $n_{0}$ is the propagation delay.

###  II-B A General Framework of the Matrix Pair Beamformer

We first propose a framework called _matrix pair beamformer_ (MPB) to cover
the common ideas in [23, 24, 25, 26, 27, 28, 29, 30, 31]. Our strategy is to
use orthogonal projection operation to model their ways of estimating the
covariance matrix pair. By this framework, we will find a common threshold
effect in these methods.

The steering vector $\mathbf{a}_{i}$ in (1) is a spatial signature of the
$i$th signal, which is different from other $\mathbf{a}_{j}$ so long as they
arrive from different directions. Beamformer is a spatial filter that exploits
such difference to pass the desired signal $s_{0}(n)$ while suppressing
$s_{1}(n)\ldots s_{D}(n)$ and $\mathbf{v}(n)$. A statistically optimum
beamformer [12, 15] generally requires at least, either explicitly or
implicitly, the information about the steering vector $\mathbf{a}_{0}$ and the
interference covariance matrix. The latter one may be replaced by the data
covariance matrix, so the remaining problem is how to acquire
$\mathbf{a}_{0}$. In DOA-based beamformer, it is calculated by the DOA and
array manifold information. As for training-based method or decision directed
method, it is inherent in the correlation vector between the reference signal
and the data vector.

To work “blindly”, i.e. without any explicit information of DOA, the methods
in [23, 24, 25, 26, 27, 28, 29, 30, 31] exploit the SOI’s temporal signature
$c_{0}(n)$ to acquire these spatial statistical information. Specifically, it
is implemented by two orthogonal projections and a generalized eigen-
decomposition of a covariance matrix pair. Hence, we refer to them as matrix
pair beamformer (MPB) in this literature. In Fig. 1, we summarize the common
structure of MPB. With the data segmentation, the array outputs corresponding
to the $k$th symbol of the desired user can be expressed in the following
matrix form:

$\displaystyle\mathbf{X}(k)\\!$
$\displaystyle\triangleq\\!\begin{bmatrix}\mathbf{x}(kN+n_{0})&\cdots&\mathbf{x}(kN+n_{0}+N-1)\end{bmatrix}=\\!\left[\sqrt{P_{0}}b_{0}(k)\right]\\!\mathbf{a}_{0}\mathbf{c}_{0}^{T}\\!\\!+\\!\\!\underbrace{\mathbf{A}_{I}\boldsymbol{\Theta}_{I}^{\frac{1}{2}}\mathbf{S}_{I}^{T}(k)\\!\\!+\\!\\!\mathbf{V}(k)}_{\mathbf{Z}(k)}$
(3)

where

$\displaystyle\mathbf{A}_{I}$
$\displaystyle\triangleq\big{[}\;\mathbf{a}_{1}\;\mathbf{a}_{2}\;\cdots\;\mathbf{a}_{D}\;\big{]}$
$\displaystyle\mathbf{c}_{0}$
$\displaystyle\triangleq\big{[}\;c_{0}(0)\;c_{0}(1)\;\cdots\;c_{0}(N-1)\;\big{]}^{T}$
$\displaystyle\mathbf{S}_{I}(k)$
$\displaystyle\triangleq\big{[}\;\mathbf{s}_{1}(k)\;\mathbf{s}_{2}(k)\;\cdots\;\mathbf{s}_{D}(k)\;\big{]}$
$\displaystyle\mathbf{s}_{i}(k)$
$\displaystyle\triangleq\big{[}\;s_{i}(kN+n_{0})\;\cdots\;s_{i}(kN+n_{0}+N-1)\;\big{]}^{T}$
$\displaystyle\mathbf{V}(k)$
$\displaystyle\triangleq\big{[}\;\mathbf{v}(kN+n_{0})\;\cdots\;\mathbf{v}(kN+n_{0}+N-1)\;\big{]}.$
$\displaystyle\boldsymbol{\Theta}_{I}$
$\displaystyle\triangleq\mathrm{diag}\\{P_{1},P_{2},\cdots,P_{D}\\}.$

Then, the $k$th data block in each antenna is projected onto two subspaces:
signal space $\mathcal{S}$ spanned by the SOI’s temporal signature vector
$\mathbf{h}_{\mathcal{S}}\\!=\\!\mathbf{c}_{0}/\sqrt{N}$, and a specifically
designed interference space
$\mathcal{I}\\!=\\!\mathcal{R}\\{\mathbf{H}_{\mathcal{I}}\\}$, respectively.
Without loss of generality, assume the columns of
$\mathbf{H}_{\mathcal{I}}\\!\\!\in\\!\\!\mathbb{C}^{N\times r_{\mathcal{I}}}$
are orthonormal. Then, the projections produce the signal snapshot
$\mathbf{x}_{\mathcal{S}}(k)\\!\\!=\\!\\!\mathbf{X}(k)\mathbf{h}_{\mathcal{S}}^{\ast}$
and the interference snapshot
$\mathbf{X}_{\mathcal{I}}(k)\\!\\!=\\!\\!\mathbf{X}(k)\mathbf{H}_{\mathcal{I}}^{\ast}$.
Define $\sigma_{\mathcal{S}_{0}}^{2}\\!\\!\triangleq\\!\\!NP_{0}$ and
$\sigma_{\mathcal{I}_{0}}^{2}\\!\\!\triangleq\\!\\!\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}^{H}\mathbf{c}_{0}\|^{2}$.
Then, the covariance matrices of $\mathbf{x}_{\mathcal{S}}(k)$ and
$\mathbf{X}_{\mathcal{I}}(k)$ are

$\displaystyle\mathbf{R}_{\mathcal{S}}$
$\displaystyle\triangleq\textsf{E}\\{\mathbf{x}_{\mathcal{S}}(k)\mathbf{x}_{\mathcal{S}}^{H}(k)\\}=\sigma_{\mathcal{S}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}^{H}_{0}+\mathbf{Q}_{\mathcal{S}}$
(4) $\displaystyle\mathbf{R}_{\mathcal{I}}$
$\displaystyle\triangleq\frac{1}{r_{\mathcal{I}}}\textsf{E}\\{\mathbf{X}_{\mathcal{I}}(k)\mathbf{X}_{\mathcal{I}}^{H}(k)\\}=\sigma_{\mathcal{I}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}^{H}_{0}+\mathbf{Q}_{\mathcal{I}}$
(5)

where $\sigma_{\mathcal{S}_{0}}^{2}$ and $\sigma_{\mathcal{I}_{0}}^{2}$ are
the SOI’s powers in the _signal channel_ and _interference channel_ (c.f. Fig.
1), respectively. $\mathbf{Q}_{\mathcal{S}}$ and $\mathbf{Q}_{\mathcal{I}}$
are the covariance matrices of the interference-plus-noise in them, defined by

$\displaystyle\mathbf{Q}_{\mathcal{S}}$
$\displaystyle=\textsf{E}\\{\mathbf{Z}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{Z}^{H}(k)\\}=\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}+\sigma^{2}\mathbf{I},$
(6) $\displaystyle\mathbf{Q}_{\mathcal{I}}$
$\displaystyle=\textsf{E}\\{\mathbf{Z}(k)\mathbf{H}_{\mathcal{S}}^{\ast}\mathbf{H}_{\mathcal{S}}^{T}\mathbf{Z}^{H}(k)\\}=\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}}\mathbf{A}_{I}^{H}+\sigma^{2}\mathbf{I},$
(7)

where
$\bm{\Phi}_{\mathcal{S}}\\!=\\!\sigma^{2}\textsf{INR}\cdot\bm{\Phi}_{\mathcal{S}_{0}}$,
$\bm{\Phi}_{\mathcal{I}}\\!=\\!\sigma^{2}\textsf{INR}\cdot\bm{\Phi}_{\mathcal{I}_{0}}$,
$\textsf{INR}\\!=\\!P_{1}/\sigma^{2}$, and

$\displaystyle\bm{\Phi}_{\mathcal{S}_{0}}=\bm{\Omega}_{I}^{\frac{1}{2}}\textsf{E}\left\\{\mathbf{S}_{I}^{T}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{S}_{I}^{\ast}(k)\right\\}\bm{\Omega}_{I}^{\frac{1}{2}},\qquad\bm{\Phi}_{\mathcal{I}_{0}}=\frac{1}{r_{\mathcal{I}}}\bm{\Omega}_{I}^{\frac{1}{2}}\textsf{E}\left\\{\mathbf{S}_{I}^{T}(k)\mathbf{H}_{\mathcal{I}}^{\ast}\mathbf{H}_{\mathcal{I}}^{T}\mathbf{S}_{I}^{\ast}(k)\right\\}\bm{\Omega}_{I}^{\frac{1}{2}},$
(8)

with $\bm{\Omega}_{I}\\!=\\!(\bm{\Theta}_{I}/\sigma^{2})/\textsf{INR}$. We can
see that $\bm{\Omega}_{I}$ is independent of INR and only depends on the
relative strength of the interferers. So are $\bm{\Phi}_{\mathcal{S}_{0}}$ and
$\bm{\Phi}_{\mathcal{I}_{0}}$, and we will use this conclusion in Sec. IV.

The MPB uses the eigenvector corresponding to the largest generalized
eigenvalue of the matrix pair
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$ as the weight vector
$\mathbf{w}$, which is the solution to the following equation

$\displaystyle\mathbf{R}_{\mathcal{S}}\mathbf{w}=\lambda_{\mathrm{max}}\mathbf{R}_{\mathcal{I}}\mathbf{w},$
(9)

where $\lambda_{\mathrm{max}}$ is the largest generalized eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$. $\mathbf{w}$ is applied
to $\mathbf{x}_{\mathcal{S}}(k)$ to yield the output

$\displaystyle
y_{o}(k)\\!\\!=\\!\\!\mathbf{w}^{H}\mathbf{x}_{\mathcal{S}}(k)\\!\\!=\\!\\!\underbrace{\left[\sigma_{\mathcal{S}_{0}}b_{0}(k)\right]\mathbf{w}^{H}\mathbf{a}_{0}}_{y_{S}(k)}\\!\\!+\\!\\!\underbrace{\mathbf{w}^{H}\mathbf{Z}(k)\mathbf{h}_{\mathcal{S}}^{\ast}}_{y_{I}(k)}$
(10)

where $y_{S}(k)$ and $y_{I}(k)$ are the output signal and interference-plus-
noise. Eq. (9) is also equivalent to

$\left(\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}\right)\mathbf{w}=(\lambda_{\mathrm{max}}-1)\mathbf{R}_{\mathcal{I}}\mathbf{w}.$
(11)

Using (4) and (5), we have

$\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}=(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}+\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}}.$
(12)

Since the columns of $\mathbf{H}_{\mathcal{I}}$ are orthonormal, i.e.
$\mathbf{H}_{\mathcal{I}}^{H}\mathbf{H}_{\mathcal{I}}=\mathbf{I}$, its
spectral norm is one and we can have

$\displaystyle\sigma_{\mathcal{I}_{0}}^{2}\triangleq\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}^{H}\mathbf{c}_{0}\|^{2}\leq\frac{P_{0}}{r_{\mathcal{I}}}\|\mathbf{H}_{\mathcal{I}}\|^{2}\|\mathbf{c}_{0}\|^{2}=\frac{NP_{0}}{r_{\mathcal{I}}}\leq\sigma_{\mathcal{S}_{0}}^{2}.$

The above inequality holds strictly when choosing
$\mathbf{H}_{\mathcal{I}}\neq\mathbf{h}_{\mathcal{S}}$ (either
$r_{\mathcal{I}}\\!>\\!1$ or
$\mathbf{H}_{\mathcal{I}}\\!\neq\\!\mathbf{h}_{\mathcal{S}}$ for
$r_{\mathcal{I}}\\!=\\!1$). Furthermore, it is commonly assumed [23, 24, 25,
26, 27, 28, 29, 30, 31] that
$\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$. Then,
$\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}}\\!=\\!\mathbf{O}$ in
(12), and the dominant eigenvector of
$(\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}},\mathbf{R}_{\mathcal{I}})=\left((\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H},\mathbf{R}_{\mathcal{I}}\right)$
is

$\displaystyle\mathbf{w}_{\mathrm{opt}}=\mu\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}=\mu^{\prime}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0},$
(13)

where $\mu$ and $\mu^{\prime}$ are scalars. Then $\mathbf{w}_{\mathrm{opt}}$
will maximize the output interference plus noise ratio (SINR) [15], and the
optimal SINR is

$\displaystyle\textsf{SINR}_{\textsf{opt}}=\left.\frac{\textsf{E}\\{|y_{S}(k)|^{2}\\}}{\textsf{E}\\{|y_{I}(k)|^{2}\\}}\right|_{\mathbf{w}_{\mathrm{opt}}}=\sigma_{\mathcal{S}_{0}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}$
(14)

All the methods in[23, 24, 25, 26, 27, 28, 29, 30, 31] share the structures
described above. They only differ in the dominant eigenvector searching
algorithm and the interference space $\mathcal{I}$ (i.e.
$\mathbf{H}_{\mathcal{I}}$). In most existing approaches, $\mathcal{I}$ is a
one dimension space ($r_{\mathcal{I}}=1$). The pre- and post-correlation
(PAPC) scheme[23, 24, 25, 26, 27] uses
$\mathbf{R}_{\mathcal{I}}\\!=\\!\textsf{E}\\{\mathbf{x}(n)\mathbf{x}^{H}(n)\\}$,
thus it is equivalent to choosing $\mathbf{H}_{\mathcal{I}}$ as

$\mathbf{H}_{\mathcal{I}}=\big{[}\;0\;\;\cdots\;\;0\;\;1\;\;0\;\;\cdots\;\;0\;\big{]}^{T}.$
(15)

where only one component in $\mathbf{H}_{\mathcal{I}}$ is nonzero. The Maximin
scheme in [28] and [29] employs a _monitor filter_ to isolate the
interference, which can be interpreted as

$\mathbf{H}_{\mathcal{I}}=\mathbf{c}_{0}\odot\left[\;1\;\;e^{j2\pi
f_{\mathrm{MF}}}\;\;\cdots\;\;e^{j2\pi f_{\mathrm{MF}}(N-1)}\;\right]^{T},$
(16)

where $f_{\mathrm{MF}}\in(0,1]$ is the normalized center frequency of monitor
filter, and $\odot$ is the Hadamard product.

###  II-C Matrix Mismatch

We see that the MPB relies heavily on the key assumption that
$\mathbf{Q}_{\mathcal{S}}=\mathbf{Q}_{\mathcal{I}}$, namely the interferers
have the same second order statistics in the two channels. By (6)–(8), we know
it is valid when each interferer is random enough in the temporal domain, say
directional white noise (c.f. Sec. IV-B1). However, it is generally not
satisfied, especially when there are multiple deterministic periodical
interferers, like tones and MAI (c.f. Sec. IV-B2). This is because when the
interferers are deterministic and periodical, the expectations in (8) can be
eliminated. Then $\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$
requires
$\mathbf{S}_{I}^{T}(k)\mathbf{h}_{\mathcal{S}}^{\ast}\\!=\\!\mathbf{S}_{I}^{T}(k)\mathbf{H}_{\mathcal{I}}^{\ast}$,
which is highly improbable when
$\mathbf{h}_{\mathcal{S}}\\!\neq\\!\mathbf{H}_{\mathcal{I}}$. We term this as
_“matrix mismatch”_. To our best knowledge, very little effort has been
devoted to analyze this problem. Therefore, we will investigate the
performance of MPB in this more general case. Before we proceed, we define the
normalized output SINR to measure performance degradation with respect to that
of no matrix mismatch.

###### Definition 1

The _normalized output SINR_ is defined as the actual output SINR of MPB
normalized by the optimal value, i.e.

$\displaystyle\textsf{G}\triangleq\left.\frac{\textsf{E}\\{|y_{S}(k)|^{2}\\}}{\textsf{E}\\{|y_{I}(k)|^{2}\\}}\right|_{\mathbf{w}}\cdot\frac{1}{\textsf{SINR}_{\textsf{{opt}}}}$
(17)

where $\textsf{SINR}_{\textsf{opt}}$ is given by (14), and $\mathbf{w}$ is the
solution to (11) without the assumption of
$\mathbf{Q}_{\mathcal{S}}=\mathbf{Q}_{\mathcal{I}}$.

G generally depends on the input SNR and the interference powers. So a
reasonable way to characterize the performance is to plot G against the input
$\textsf{SNR}\triangleq\sigma_{\mathcal{S}_{0}}^{2}/\sigma^{2}$, when fixing
the interference powers. In the following sections, we will base our analysis
on $\textsf{G}(\textsf{SNR})$, which we will refer to as _operating curve_.
Moreover, we assume infinite sample size so that the finite sample effect is
ignored.

##  III Performance Analysis of the Matrix Pair Beamformer with Matrix
Mismatch

In this section, we will derive the operating curve of MPB, and discuss how it
works blindly.

###  III-A Operating Curve of Matrix Pair Beamformer

We base our discussions on the following assumptions, and summarize the main
result in theorem 1.

###### Assumption 1

The spacing of DOA between any two signals is large enough (greater than a
mainlobe), so that $\\{\mathbf{a}_{i}\\}_{i=0}^{D}$ are linearly independent
and the projection of $\mathbf{a}_{0}$ onto
$\mathrm{span}\\{\\!\mathbf{a}_{i}\\}_{i=1}^{D}$ is much less than
$\|\mathbf{a}_{0}\|$.

###### Assumption 2

The steering vectors of all signals are normalized so that
$\|\mathbf{a}_{i}\|^{2}=L$, ($i=0,1,\ldots,D$).

######  Theorem 1 (operating curve)

The normalized output SINR of MPB with matrix mismatch is

$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{\displaystyle
P_{I}\\!+\\!1}{\displaystyle
P_{I}/\left[1\\!-\\!{\textsf{SNR}_{\textsf{T}0}}/{\textsf{SNR}}\right]^{2}\\!+\\!1}G_{U},&\textsf{SNR}>\textsf{SNR}_{\textsf{T2}}\\\
{\displaystyle\left[\frac{1\\!\\!+\\!\\!K_{0}}{1\\!\\!-\\!\\!{\textsf{SNR}}/{\textsf{SNR}_{\textsf{T}0}}\\!\\!+\\!\\!K_{0}\left({L\beta}\textsf{SNR}/{N}\\!\\!+\\!\\!1\right)}\right]^{2}G_{L},}&\textsf{SNR}<\textsf{SNR}_{\textsf{T1}}\end{array}\right.$
(20)

where $G_{U}$ and $G_{L}$ are the normalized output SINR when
$\textsf{SNR}\\!\\!=\\!\\!+\infty$ and $\textsf{SNR}\\!\\!=\\!\\!0$
($-\infty$dB), respectively.
$\beta\\!\\!\triangleq\\!\\!N\sigma_{\mathcal{I}_{0}}^{2}/\sigma_{\mathcal{S}_{0}}^{2}$
is the normalized power leakage ratio (PLR) in interference channel.
$P_{I}\\!\\!\triangleq\\!\\!1/G_{U}\\!\\!-\\!\\!1$ is the output interference
to noise ratio, and $\textsf{SNR}_{\textsf{T}0}$ is the empirical threshold
SNR. Their expressions are

$\displaystyle
G_{U}=\frac{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}}\cdot\frac{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{Q}_{\mathcal{S}}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}$
(21)
$\displaystyle\textsf{SNR}_{\textsf{T0}}=\frac{N}{L}\cdot\frac{1}{[{(N-\beta)}/{(\gamma_{1})^{+}}-\beta]^{+}}$
(22) $\displaystyle
K_{0}=\frac{\beta+\frac{N}{L}/\textsf{SNR}_{\textsf{T}0}}{N-\beta}\left(\gamma_{1}-\frac{N-\beta}{\beta}\right)^{+}$
(23)

where $(\cdot)^{+}=\max\\{\cdot,0\\}$, and $\gamma_{1}$ is the largest nonzero
generalized eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.111If
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
has less than $D$ nonzero eigenvalues, pad them with zeros up to $D$ and order
them decreasingly. The SNR at which $\textsf{G}(\textsf{SNR})$ is close to
$G_{L}$ and $G_{U}$ (within $3$dB), which are given by

$\displaystyle\textsf{SNR}_{\textsf{T}1}=\left(1-\sqrt{\frac{1}{2}}\right)\textsf{SNR}_{\textsf{T}0},\qquad\textsf{SNR}_{\textsf{T}2}=\left(1-\sqrt{\frac{P_{I}}{2P_{I}+1}}\right)^{-1}\\!\\!\\!\textsf{SNR}_{\textsf{T}0},$
(24)

respectively.

Though we will give expression for $G_{L}$ in Sec. III-D, its specific values
are of no interest to us. Fig. 2 shows a typical curve of
$\textsf{G}(\textsf{SNR})$. We plot the curve in failure area and operating
area given in (20), and connect their ends by a dashed line. We can see that
the performance of beamformer degrades rapidly when the input SNR is below
$\textsf{SNR}_{\textsf{T2}}$. And it fails completely after reaching
$\textsf{SNR}_{\textsf{T1}}$. Therefore, matrix mismatch causes a threshold
effect in MPB, and $\textsf{SNR}_{\textsf{T2}}$ is a critical parameter to be
optimized.

There are two special cases of $\textsf{G}(\textsf{SNR})$. The first one is
$\textsf{SNR}_{\textsf{T1}}\\!\\!=\\!\\!\textsf{SNR}_{\textsf{T2}}\\!\\!=\\!\\!0$
($-\infty$dB). This happens when there is no matrix mismatch, i.e.
$\mathbf{Q}_{\mathcal{S}}\\!\\!=\\!\\!\mathbf{Q}_{\mathcal{I}}$, so that
$\gamma_{1}\\!\\!=\\!\\!0$ and $\textsf{SNR}_{\textsf{T}i}$ given by (22) and
(24) are zero. Then, (20) implies the operating curve is a horizontal line as
shown in Fig. 2. Another interesting case happens in PAPC schemes mentioned in
II-B, whose PLR is $\beta\\!\\!=\\!\\!1\\!\\!>\\!\\!0$. If it further
satisfies $(N-\beta)/\gamma_{1}\\!\\!<\\!\\!\beta$, then by (22) and (24),
$\textsf{SNR}_{\textsf{T}1}\\!\\!=\\!\\!\textsf{SNR}_{\textsf{T}2}=+\infty$
and $\textsf{G}(\textsf{SNR})$ only has the failure area, which decreases in
the order of $\mathcal{O}(\textsf{SNR}^{-2})$ (c.f. Fig. 2). In the rest of
the section, we will derive $\textsf{G}(\textsf{SNR})$ and reveal how MPB
works blindly under matrix mismatch. The discussion of
$\textsf{SNR}_{\textsf{T0}}$ is left to Sec. IV.

\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Regular}
\put(36.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}$}}
\put(61.0,25.0){\small{$\textsf{SNR}_{\textsf{T2}}$}} \end{overpic}

\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Operating}
\put(20.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}=\textsf{SNR}_{\textsf{T2}}=-\infty$
dB}} \end{overpic}

\begin{overpic}[width=182.1196pt]{Fig_OperatingCurve_Failure}
\put(20.0,25.0){\small{$\textsf{SNR}_{\textsf{T1}}=\textsf{SNR}_{\textsf{T2}}=+\infty$
dB}} \put(66.0,57.0){\small{$\propto-20\log\textsf{SNR}$}} \end{overpic}

Figure 2: (a) A typical curve of $\textsf{G}(\textsf{SNR})$ has failure area,
threshold area and operating area, separated by $\textsf{SNR}_{\textsf{T1}}$
and $\textsf{SNR}_{\textsf{T2}}$. (b) Without matrix mismatch, the curve has
only operating area. (c) A curve has only failure area for some cases in PAPC.

###  III-B Derivation of the Weight Vector for MPB

As a step to prove Theorem 1, we first derive the expression of the weight
vector for MPB. By the arguments in Sec. II-B, it is the solution to (11).
Define
$\bm{\Phi}_{\Delta}\triangleq\bm{\Phi}_{\mathcal{S}}-\bm{\Phi}_{\mathcal{I}}$.
Then, by (6), (7) and (12), the problem becomes solving the following
generalized eigenequation

$\displaystyle\left[(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}+\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\right]\mathbf{w}=(\lambda_{\mathrm{max}}-1)\mathbf{R}_{\mathcal{I}}\mathbf{w},$
(25)

where $\lambda_{\mathrm{max}}$ is the largest eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$. Sec. II-B already gave
the result for $\mathbf{w}$ when there is no matrix mismatch, i.e.
$\bm{\Phi}_{\Delta}\\!\\!=\\!\\!\mathbf{O}$ (or
$\mathbf{Q}_{\mathcal{S}}\\!\\!=\\!\\!\mathbf{Q}_{\mathcal{I}}$). We concluded
that it is optimal in the sense of maximizing the output SINR. However, in the
presence of matrix mismatch, we have
$\bm{\Phi}_{\Delta}\\!\\!\neq\\!\\!\mathbf{O}$, which is the key challenge for
solving (25). To deal with this problem, we first have the following two
observations:

  * •

By left-multiplying
$\frac{1}{\lambda_{\mathrm{max}}-1}\mathbf{R}_{\mathcal{I}}^{-1}$ to both
sides of (25), we can see that $\mathbf{w}$ can be expressed as

$\displaystyle\mathbf{w}=\eta_{0}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\eta_{i}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{i}$
(26)

i.e. it is a linear combination of
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{1},\ldots,\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{D}$,
where
$\eta_{0}=\frac{\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2}}{\lambda_{\mathrm{max}}-1}\mathbf{a}_{0}^{H}\mathbf{w}$,
$\mathbf{a}_{i}$ is the $i$th column of $\mathbf{A}_{I}$, and $\eta_{i}$ is
the $i$th component of
$\bm{\eta}=\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\mathbf{w}$. To determine
$\eta_{0},\eta_{1},\ldots,\eta_{D}$, we only need to substitute (26) back into
(25) and solve a linear equation. However, we will immediately discover that
the solution is intractable for further analysis because
$\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I}$ and
$\bm{\Phi}_{\Delta}$ are not diagonal. To overcome this, we need the next
observation.

  * •

Suppose we can factorize $\bm{\Phi}_{\Delta}$ into diagonal form such that
$\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}=\mathbf{A}_{\epsilon}\bm{\Gamma}\mathbf{A}_{\epsilon}$
and
$\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}=\mathbf{I}$,
where $\bm{\Gamma}$ is a diagonal matrix. Then, by repeating the above
procedure except for replacing $\mathbf{a}_{1},\ldots,\mathbf{a}_{D}$ by
$\mathbf{a}_{\epsilon_{1}},\ldots,\mathbf{a}_{\epsilon_{D}}$, with
$\mathbf{a}_{\epsilon_{i}}$ being the $i$th column of $\mathbf{A}_{\epsilon}$,
we can solve $\mathbf{w}$ as

$\displaystyle\mathbf{w}=\eta_{0}\left[\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\frac{\gamma_{i}\tilde{\psi}_{T_{i}}}{\lambda_{\mathrm{max}}-1-\gamma_{i}}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{i}}\right],$
(27)

where $\gamma_{i}$ is the $i$th diagonal component of $\bm{\Gamma}$,
$\eta_{0}$ is an arbitrary constant, and
$\tilde{\psi}_{T_{i}}=\mathbf{a}_{\epsilon_{i}}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$.

So far, the only thing left is to figure out a way to factorize
$\bm{\Phi}_{\Delta}$ so that it meets the above requirement. Consider the
generalized eigen-decomposition of the matrix pair
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
By the simultaneous diagonalization theorem (c.f. [34], pp.133), there exists
a nonsingular matrix $\mathbf{T}$ such that

$\displaystyle\mathbf{T}^{H}\bm{\Phi}_{\Delta}\mathbf{T}=\bm{\Gamma},\qquad\mathbf{T}^{H}(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\mathbf{T}=\mathbf{I},$
(28)

where $\bm{\Gamma}=\mathrm{diag}(\gamma_{1},\gamma_{2},\ldots,\gamma_{D})$ is
a diagonal matrix whose diagonal terms are the generalized eigenvalues of
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
Define $\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$. Then, by
(28), we can verify that

$\displaystyle\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}\bm{\Gamma}\mathbf{T}^{-1}\mathbf{A}_{I}^{H}=\mathbf{A}_{\epsilon}\bm{\Gamma}\mathbf{A}_{\epsilon}^{H}$
$\displaystyle\mathbf{A}_{\epsilon}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}=\mathbf{T}^{-1}\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}=\mathbf{I}.$

Therefore, (28) is the exact decomposition of $\bm{\Phi}_{\Delta}$ we are
looking for. Furthermore, by
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$, we notice that
each column of $\mathbf{A}_{\epsilon}$ is in fact a linear combination of
$\mathbf{a}_{1},\ldots,\mathbf{a}_{D}$. This, together with (27), implies that
$\mathbf{w}$ is still a a linear combination of
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{1},\ldots,\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{D}$,
just as that in (26).

###  III-C How MPB Works Blindly

To fully understand the behavior of MPB given by (27) and how it works
blindly, we still need the expressions of $\lambda_{\mathrm{max}}$ and
$\gamma_{i}$. However, $\lambda_{\mathrm{max}}$ and $\gamma_{i}$ are the
solutions to

$\displaystyle\mathrm{det}\left\\{(\lambda-1)\mathbf{R}_{\mathcal{I}}-(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H}-\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\right\\}=0$
(29)
$\displaystyle\mathrm{det}\left\\{\lambda(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}-\bm{\Phi}_{\Delta}\right\\}=0$
(30)

which are polynomial equations of degree $D+1$ and $D$, respectively. It is
known that there are no general closed-form solutions if their degrees are
higher than four. Thus, our approach here is to derive approximate expressions
for $\lambda_{\mathrm{max}}$ and $\gamma_{i}$. The main idea is to transform
(29) and (30) into eigenvalue problems of a diagonal matrix perturbed by a
small term. Then we can approximate the eigenvalues by the diagonal entries,
and bound the error using matrix perturbation theory.

We first discuss $\lambda_{\mathrm{max}}$. Before this, let’s introduce the
following identity which will be used repeatedly in the analysis and can be
derived from Properties 16 and 17 in [34, pp.5].

$\displaystyle\det(\lambda\mathbf{I}-\mathbf{X}\mathbf{Y})=\lambda^{m-n}\cdot\det(\lambda\mathbf{I}-\mathbf{Y}\mathbf{X})$
(31)

where $\mathbf{X}\in\mathbb{C}^{m\times n}$ and
$\mathbf{Y}\in\mathbb{C}^{n\times m}$. Substituting the factorization of
$\bm{\Phi}_{\Delta}$ in (28) into (29) and using (31) as well as
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$, we can have the
equivalent form of (29) as below

$\displaystyle(\lambda-1)^{L-D-1}\cdot\det\mathbf{R}_{\mathcal{I}}\cdot\det\left\\{(\lambda-1)\mathbf{I}-\begin{bmatrix}\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2}&\mathbf{0}^{T}\\\
\mathbf{0}&\bm{\Gamma}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}&\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}\\\
\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}&\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{\epsilon}\end{bmatrix}\right\\}=0.$
(32)

Then, (32) implies that the solution of (29) is the same as the eigenvalues of
the following matrix

$\displaystyle\mathbf{M}=\begin{bmatrix}\gamma_{0}+1&\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\bm{\psi}_{T}^{H}\\\
\frac{\sqrt{\frac{\sigma^{2}}{L}}}{\frac{L\beta}{N}\textsf{SNR}+1}\bm{\Gamma}\bm{\psi}_{T}&\bm{\Gamma}+\mathbf{I}\end{bmatrix}$
(33)

except for the multiplicity of ones, where we have used the second equality of
(28). And $\gamma_{0}$ and $\bm{\psi}_{T}$ are

$\displaystyle\gamma_{0}\triangleq(\sigma_{\mathcal{S}_{0}}^{2}-\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0},\qquad\bm{\psi}_{T}\triangleq\left(\frac{L\beta}{N}\textsf{SNR}+1\right)\mathbf{A}_{\epsilon}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
(34)

Before proceeding on, we cite the following two lemmas. The first one is
summarized from Lemma 1 and Lemma 2 in [35], and the second one can be found
in [33, pp.344].

###### Lemma 1

The quantities $\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
and $\bm{\psi}_{T}$ have the following results

$\displaystyle\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\approx\frac{L}{\sigma^{2}}\frac{N}{{L\beta}\textsf{SNR}+N},\qquad\|\boldsymbol{\psi}_{T}\|\approx\sqrt{\frac{L}{\sigma^{2}}\kappa_{0}}\ll\sqrt{\frac{L}{\sigma^{2}}}.$

where $\beta$ is the PLR defined in Theorem 1, and $\kappa_{0}$ is a very
small positive number independent of SNR.

######  Lemma 2 (Gerschgorin)

Let matrix $\mathbf{A}\in\mathbb{C}^{n\times n}$ and $a_{ij}$ be its $i,j$th
element. Define

$\displaystyle R_{i}(\mathbf{A})\triangleq\sum_{j=1,j\neq
i}^{n}|a_{i,j}|,\quad 1\leq i\leq n$

as the _deleted absolute row sums_ of $\mathbf{A}$ . Then all the eigenvalues
of $\mathbf{A}$ are in the union of $n$ disks, i.e.

$\displaystyle\lambda(\mathbf{A})\in\bigcup_{i=1}^{n}G_{i}(\mathbf{A}),$

where
$G_{i}(\mathbf{A})\\!\\!\triangleq\\!\\!\\{z\\!\\!\in\\!\\!\mathbb{C}\\!\\!:\\!\\!|z\\!\\!-\\!\\!a_{ii}|\\!\\!\leq\\!\\!R_{i}(\mathbf{A})\\}$
is the $i$th Gerschgorin disk. Moreover, if a union of $k$ disks forms a
connected region disjoint from all the remaining disks, then there are $k$
eigenvalues of $\mathbf{A}$ in this region.

By Lemma 1, the off-diagonal terms of each row in (33) are much smaller than
the corresponding diagonal terms. Then, according to lemma 2, its eigenvalues
are approximately $\gamma_{0}\\!+\\!1,\ldots,\gamma_{D}\\!+\\!1$. We will give
a bound for the approximation error in the next subsection. Now we directly
use this approximation to discuss the behavior of MPB. Without loss of
generality, assume $\gamma_{1}>\ldots>\gamma_{D}$, then

$\displaystyle\lambda_{\mathrm{max}}\approx\max\\{\gamma_{0}+1,\gamma_{1}+1,\ldots,\gamma_{D}+1,1\\}=\max\\{\gamma_{0}+1,\gamma_{1}+1\\}.$
(35)

Substituting the expression of
$\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$ in Lemma 1
into its definition in (34), we can have

$\displaystyle\gamma_{0}=(N-\beta)\frac{L\textsf{SNR}}{L\beta\textsf{SNR}+N}$
(36)

which is a monotonically increasing function of SNR. Furthermore, we will show
later that $\gamma_{1}\\!\\!+\\!\\!1$ is approximate to one of the generalized
eigenvalue of $(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ and is
almost independent of SNR. Hence, there is a threshold
$\textsf{SNR}_{\textsf{T0}}$ such that $\lambda_{\mathrm{max}}$ switches from
$\gamma_{1}+1$ to $\gamma_{0}+1$ when SNR exceeds
$\textsf{SNR}_{\textsf{T0}}$. Its expression is in (22) and can be derived by
setting (36) to $\gamma_{1}$ and solving SNR. As a result,

  * •

When $\textsf{SNR}>\textsf{SNR}_{\textsf{T0}}$,
$\lambda_{\mathrm{max}}\approx\gamma_{0}+1$. If $\gamma_{0}+1$ can be much
greater than $\gamma_{1}+1$ as SNR increases, then (27) implies
$\mathbf{w}\approx\mu_{1}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$. Hence,
$\mathbf{a}_{0}$ is the dominant steering vector, and the beamformer can
operate properly by steering the mainlobe to the direction of the desired
signal.

  * •

When $\textsf{SNR}<\textsf{SNR}_{\textsf{T0}}$,
$\lambda_{\mathrm{max}}\approx\gamma_{1}+1$. Thus
$\mathbf{w}\approx\mu_{1}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{1}}$,
and $\mathbf{a}_{\epsilon_{1}}$ is the dominant steering vector in (27). By
$\mathbf{A}_{\epsilon}=\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$,
$\mathbf{a}_{\epsilon_{1}}$ is a linear combination of all the interference
steering vectors. Therefore, the beamformer fails for its mainlobe pointing to
the directions of the interferers. If $\beta\\!\\!\neq\\!\\!0$, then there
would be SOI in $\mathbf{R}_{\mathcal{I}}$, which makes MPB treat it as
interference and null it out. This explains the reason why
$\textsf{G}(\textsf{SNR})$ decreases in the order of
$\mathcal{O}(\textsf{SNR}^{-2})$ in Fig. 2.

Now, we further analyze how this threshold effect happens. By the
factorizations (28), we can rewrite
$\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}$ on the left hand side of
(25) as

$\displaystyle\mathbf{R}_{\mathcal{S}}-\mathbf{R}_{\mathcal{I}}=\gamma_{0}\cdot\mathbf{a}_{S_{0}}\mathbf{a}_{S_{0}}^{H}+\sum_{i=1}^{D}\gamma_{i}\cdot\mathbf{a}_{\epsilon_{i}}\mathbf{a}_{\epsilon_{i}}^{H},$
(37)

where
$\mathbf{a}_{S_{0}}\triangleq\mathbf{a}_{0}/[\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}]^{\frac{1}{2}}$.
It is obvious that $\gamma_{0}$ is the generalized eigenvalue of the matrix
pair
$\left((\sigma_{\mathcal{S}_{0}}^{2}\\!\\!-\\!\sigma_{\mathcal{I}_{0}}^{2})\mathbf{a}_{0}\mathbf{a}_{0}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
with $\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{S_{0}}$ being its “normalized”
eigenvector. By applying (31) to (30), we can also verify that
$\gamma_{1},\gamma_{2},\ldots,\gamma_{D}$ are the generalized eigenvalues of
$\left(\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
with the eigenvectors being
$\\{\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{i}}\\}_{i=1}^{D}$.
Comparing (37) with the left hand side of (25), we can find that $\gamma_{0}$
actually measures the mismatch of the desired signal between the signal
channel and the interference channel in Fig. 1, and
$\\{\gamma_{i}\\}_{i=1}^{D}$ measures that of the interferers. From the
previous discussion, we know that
$\lambda_{\mathrm{max}}=\max\\{\gamma_{0}+1,\gamma_{1}+1\\}$, and $\mathbf{w}$
takes $\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$ or
$\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{\epsilon_{1}}$ depending on which
one of $\gamma_{0}+1$ and $\gamma_{1}+1$ is larger. Therefore, MPB _blindly_
chooses the one with the largest mismatch metrix in (37) as its steering
vector. We term this as “ _steering vector competition_ ”. The threshold
happens when the SNR is large enough so that $\gamma_{0}$ exceeds
$\gamma_{1}$. If there is no mismatch, i.e. $\bm{\Phi}_{\Delta}=\mathbf{O}$ so
that $\gamma_{i}=0$, then the desired signal always wins out in the
competition ($\gamma_{0}>\gamma_{1}$), and MPB can work properly for all SNR.

###  III-D Proof of Theorem 1

Let us return to our original problem of operating curve for MPB and prove
Theorem 1. We first use the expression of $\mathbf{w}$ in (27) to derive
$\textsf{G}(\textsf{SNR})$ in term of $\lambda_{\mathrm{max}}$. The result is
given by the following lemma, with its proof in Appendix B.

###### Lemma 3

The normalized output SINR in (17) can be expressed as the following function
of $\lambda_{\mathrm{max}}$

$\displaystyle\textsf{G}(\textsf{SNR})=\frac{\displaystyle\left[1+\frac{N}{L\beta\textsf{SNR}+N}\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}}{\displaystyle\left[\psi_{I}(\lambda_{\mathrm{max}})-\frac{L\beta\textsf{SNR}}{L\beta\textsf{SNR}+N}\psi_{S}^{2}(\lambda_{\mathrm{max}})\right]+\left[1+\frac{N}{L\beta\textsf{SNR}+N}\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}}.$
(38)

where $\psi_{S}(\lambda_{\mathrm{max}})$ and
$\psi_{I}(\lambda_{\mathrm{max}})$ are in the following forms with
$\psi_{T_{i}}$ being the $i$th component of $\bm{\psi}_{T}$.

$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})\\!\\!\triangleq\\!\\!\frac{\sigma^{2}}{L}\sum_{i=1}^{D}\\!\\!\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}|\psi_{T_{i}}|^{2},\qquad\psi_{I}(\lambda_{\mathrm{max}})\\!\\!\triangleq\\!\\!\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}\\!(\gamma_{i}\\!\\!+\\!\\!1)\\!\left[\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}\right]^{2}\\!\\!\\!|\psi_{T_{i}}|^{2}.$
(39)

To further simplify (38), we need the expression of $\lambda_{\mathrm{max}}$.
In the previous subsection, we used the approximation (35) to analyze MPB. The
following lemma quantifies how precise it is by bounding the approximation
error. Besides checking the validity of (35), this is also critical in
deriving $\textsf{G}(\textsf{SNR})$.

###### Lemma 4

The largest generalized eigenvalue of
$(\mathbf{R}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}})$ defined in (4) and (5)
satisfies

$\displaystyle|\lambda_{\mathrm{max}}-(\lambda_{a}+1)|\leq\lambda_{a}\cdot
f\left(\frac{\lambda_{b}}{\lambda_{a}}\right)$ (40)

where $\lambda_{a}\triangleq\max\\{\gamma_{0},\gamma_{1}\\}$ and
$\lambda_{b}\triangleq\min\\{\gamma_{0},\gamma_{1}\\}$. $f(x)$ and $\delta$
are defined as

$\displaystyle f(x)$
$\displaystyle\triangleq\frac{1}{2}\left[1\\!\\!-\\!\\!x\\!\\!-\\!\\!\sqrt{(1\\!\\!-\\!\\!x)^{2}\\!\\!-\\!\\!4\delta|x|}\right],\quad\delta\triangleq\frac{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}{\frac{L\beta}{N}\textsf{SNR}+1}\ll
1,\quad
x\in(-\infty,1\\!-\\!2\gamma_{-}]\\!\cup\\![1\\!+\\!2\gamma_{+},+\infty)$

with
$\gamma_{\pm}\\!\triangleq\\!\sqrt{\delta^{2}\\!+\\!\delta}\\!\pm\\!\delta$.
Furthermore, $0\\!\leq\\!f(x)\\!\leq\\!\max\\{\delta,\gamma_{-}\\}\\!\ll\\!1$
when $x\in(-\infty,1\\!-\\!2\gamma_{-}]$.

###### Proof:

The key idea here is to design the appropriate similarity transform for
$\mathbf{M}$ in (33) and apply Lemma 2. Our detailed proof can be found in
Appendix A. ∎

Now we are ready to derive the expression of $\textsf{G}(\textsf{SNR})$. We
first consider the trivial case of
$-1\\!\\!\leq\\!\\!\gamma_{1}\\!\\!\leq\\!\\!0$.222Since $\gamma_{i}$ is the
eigenvalue of
$\left(\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H},\mathbf{R}_{\mathcal{I}}\right)$,
$\gamma_{i}\\!+\\!1$ is for
$\left(\sigma_{\mathcal{I}_{0}}^{2}\mathbf{a}_{0}\mathbf{a}_{0}^{H}\\!\\!+\\!\\!\mathbf{Q}_{\mathcal{S}},\mathbf{R}_{\mathcal{I}}\right)$,
a positive definite pair. Hence $\gamma_{i}\\!+\\!1\\!\\!>\\!\\!0$. Since
$\gamma_{D}\\!\\!<\\!\\!\ldots\\!\\!<\\!\\!\gamma_{1}$, we can immediately
have from (39) that
$0\\!\\!<\\!\\!\psi_{S}(\lambda_{\mathrm{max}})\\!\\!<\\!\\!\frac{\sigma^{2}}{L}\|\bm{\psi}_{T}\|^{2}\\!\\!\ll\\!\\!1$
and
$0\\!\\!<\\!\\!\psi_{I}(\lambda_{\mathrm{max}})\\!\\!<\\!\\!\frac{\sigma^{2}}{L}\|\bm{\psi}_{T}\|^{2}\\!\\!\ll\\!\\!1$.
Therefore, (38) becomes $\textsf{G}(\textsf{SNR})\\!\\!\approx\\!\\!1$ for all
SNR.

Next, we are going to discuss the case of $\gamma_{1}\\!\\!>\\!\\!0$ in two
separate cases.

####  III-D1 $\gamma_{0}>\gamma_{1}$

Then $\lambda_{a}\\!=\\!\gamma_{0}$, $\lambda_{b}\\!=\\!\gamma_{1}$ and
$\lambda_{\mathrm{max}}\\!\approx\\!\gamma_{0}\\!+\\!1$. So long as SNR is
slightly larger than $\textsf{SNR}_{\textsf{T}0}$ such that
$\gamma_{0}\\!\\!>\\!\\!\gamma_{1}/(1\\!\\!-\\!\\!\sqrt{{\sigma^{2}}/{L}}\|\bm{\psi}_{T}\|)\\!\\!\approx\\!\\!\gamma_{1}$,
we have the following approximation for $\psi_{S}$:

$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})\approx\frac{\sigma^{2}}{L}\sum_{i=1}^{D}\\!\\!\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{i}}|\psi_{T_{i}}|^{2}<\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{1}}\frac{\sigma^{2}}{L}\sum_{i=1}^{D}|\psi_{T_{i}}|^{2}<\sqrt{\frac{\sigma^{2}}{L}}\|\bm{\psi}_{T}\|\ll
1.$

Moreover, we approximate $\psi_{I}$ by the following bound, which is
asymptotically tight with respect to $\frac{\gamma_{0}}{\gamma_{1}}$.333In
fact, this approximation is precise enough when $\gamma_{0}$ is reasonably
larger than $\gamma_{1}$, say, $\gamma_{0}>2\gamma_{1}$.

$\displaystyle\psi_{I}(\lambda_{\mathrm{max}})\approx\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}\\!(\gamma_{i}\\!\\!+\\!\\!1)\\!\left(\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{i}}\right)^{2}\\!\\!\\!|\psi_{T_{i}}|^{2}\leq\left(\frac{\gamma_{0}}{\gamma_{0}\\!\\!-\\!\\!\gamma_{1}}\right)^{2}\frac{\sigma^{2}}{L}\\!\sum_{i=1}^{D}(\gamma_{i}\\!\\!+\\!\\!1)|\psi_{T_{i}}|^{2}.$

Substituting the above two approximations together with (36) and (22) into
(38), we can finally have

$\displaystyle\textsf{G}(\textsf{SNR})=\frac{P_{I}+1}{\displaystyle
P_{I}/\left[1-{\textsf{SNR}_{\textsf{T}0}}/{\textsf{SNR}}\right]^{2}+1}G_{U}$

where $G_{U}=1/(P_{I}+1)$ and $P_{I}$ is defined as

$\displaystyle
P_{I}\triangleq\left[\frac{(N-\beta)/\beta}{(N-\beta)/\beta-\gamma_{1}}\right]^{2}\frac{\sigma^{2}}{L}\sum_{i=1}^{D}(\gamma_{i}+1)|\psi_{T_{i}}|^{2}$

However, this expression is difficult to evaluate. Instead, we can compute
$G_{U}$ first and have $P_{I}\\!\\!=\\!\\!1/G_{U}\\!\\!-\\!\\!1$. Noticing
that $G_{U}$ is the output SINR as $\textsf{SNR}\\!\\!=\\!\\!\infty$, we can
ignore the term $\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}$ on the
left hand side of (25). Hence,
$\mathbf{w}_{\infty}\\!\\!=\\!\\!\mu\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\\!\\!=\\!\\!\mu^{\prime}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$,
and by (17) and (14), we have its expression in (21).

####  III-D2 $\gamma_{0}<\gamma_{1}$

Then $\lambda_{a}\\!=\\!\gamma_{1}$, $\lambda_{b}\\!=\\!\gamma_{0}$ and
$\lambda_{\mathrm{max}}\\!\approx\\!\gamma_{1}\\!+\\!1$. As a result, the term
corresponding to $\gamma_{1}$ would dominate $\psi_{S}$ and $\psi_{I}$ in
(39). To further evaluate these two terms, we need the bound in (40) to
measure how close $\lambda_{\mathrm{max}}$ is to $\gamma_{1}+1$, namely, we
use it to evaluate $\lambda_{\mathrm{max}}-(\gamma_{1}+1)$ in $\psi_{S}$ and
$\psi_{I}$:

$\displaystyle\psi_{S}(\lambda_{\mathrm{max}})$
$\displaystyle\\!\approx\\!\frac{\sigma^{2}}{L}\frac{\gamma_{1}}{\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{i}\\!\\!+\\!\\!1)}|\psi_{T_{1}}|^{2}\\!\approx\\!\frac{\sigma^{2}}{L}\frac{1}{f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)}|\psi_{T_{1}}|^{2}\\!=\\!\left(\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1\right)\frac{\gamma_{1}}{\gamma_{0}}\cdot
g\left(\frac{\gamma_{0}}{\gamma_{1}}\right)$
$\displaystyle\psi_{I}(\lambda_{\mathrm{max}})$
$\displaystyle\\!\approx\\!\frac{\sigma^{2}}{L}\frac{(\gamma_{1}\\!\\!+\\!\\!1)\gamma_{1}^{2}}{[\lambda_{\mathrm{max}}\\!\\!-\\!\\!(\gamma_{1}\\!\\!+\\!\\!1)]^{2}}|\psi_{T_{1}}|^{2}\\!\approx\\!\frac{\sigma^{2}}{L}\frac{\gamma_{1}+1}{\left[f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)\right]^{2}}|\psi_{T_{1}}|^{2}\\!=\\!\frac{\gamma_{1}+1}{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}\left[\psi_{S}(\lambda_{\mathrm{max}})\right]^{2}$

where
$g(x)=\frac{1}{2}[1\\!\\!-\\!\\!x\\!\\!+\\!\\!\sqrt{(1\\!\\!-\\!\\!x)^{2}\\!\\!-\\!\\!4\delta
x}]$. In fact, we can have the approximation that $g(x)\approx 1-x$, because

$\displaystyle\left|(1-x)-g(x)\right|=|f(x)|\ll 1$

Substituting the above expressions of $\psi_{S}$ and $\psi_{I}$ as well as the
approximation of $g(x)$ into (3), we get

$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle\\!\\!=\\!\\!\left[\frac{N}{L\beta\textsf{SNR}\\!\\!+\\!\\!N}\right]^{2}\\!\\!\left[\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\right]^{2}\\!\\!\left[\frac{\gamma_{1}\\!\\!+\\!\\!1}{\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}}\\!\\!-\\!\\!\frac{\frac{L\beta}{N}\textsf{SNR}}{\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1}\\!\\!+\\!\\!\left(\\!\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\\!\right)^{2}\right]^{-1}\\!\\!\\!\\!\\!\\!\\!\approx\\!\\!\left[\frac{N}{L\beta\textsf{SNR}\\!\\!+\\!\\!N}\right]^{2}\\!\\!\left[\frac{\gamma_{1}}{\gamma_{1}\\!\\!-\\!\\!\gamma_{0}}\right]^{2}\\!\\!G_{L},$

where
$G_{L}\triangleq\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}/(\gamma_{1}\\!\\!+\\!\\!1)$
is the output SINR when $\textsf{SNR}\\!\\!=\\!\\!0$ ($-\infty$dB). The last
approximation holds when
$\gamma_{0}\\!\\!<\\!\\!(1\\!\\!-\\!\\!\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|)\gamma_{1}$,
i.e. $\gamma_{0}$ is slightly smaller than $\gamma_{1}$, which implies
$\gamma_{1}^{2}/(\gamma_{1}\\!\\!-\\!\\!\gamma_{0})^{2}\\!\\!<\\!\\!1/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}<(\gamma_{1}+1)/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}$.
Furthermore, because $\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}\\!\\!\ll\\!\\!1$,
${\frac{L\beta}{N}\textsf{SNR}}/{(\frac{L\beta}{N}\textsf{SNR}\\!\\!+\\!\\!1)}\\!\\!<\\!\\!1\\!\\!\ll\\!\\!(\gamma_{1}+1)/\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}$.
There is, however, no easier way to evaluate $G_{L}$ than its definition.
Fortunately, we are not interested in its specific values but the threshold
$\textsf{SNR}_{\textsf{T}1}$ that G reaches this value, as discussed in Sec.
III-A. Noticing that $\textsf{SNR}_{\textsf{T}0}=\infty$ when
$(N-\beta)/\beta<\gamma_{1}$ (c.f. (22)), we have following fact regarding
$\gamma_{1}$ and $\textsf{SNR}_{\textsf{T}0}$,

$\displaystyle\gamma_{1}=\frac{L(N-\beta)\textsf{SNR}_{\textsf{T}0}}{L\beta\textsf{SNR}_{\textsf{T}0}+N}+\left(\gamma_{1}-\frac{N-\beta}{\beta}\right)^{+}.$

Combining the above expression and (36), $\textsf{G}(\textsf{SNR})$ can be
reduced to

$\displaystyle\textsf{G}(\textsf{SNR})=\left[\frac{1+K_{0}}{1-{\textsf{SNR}}/{\textsf{SNR}_{\textsf{T}0}}+K_{0}\left(\frac{L\beta}{N}\textsf{SNR}+1\right)}\right]^{2}G_{L}$
(41)

with $K_{0}$ defined in (23). Finally, the following lemma provides an easier
way to compute $\textsf{SNR}_{\textsf{T}0}$ from $\gamma_{1}$. The proof can
also be found in Appendix A.

###### Lemma 5

$\gamma_{1},\ldots,\gamma_{D}$ are approximate to all the nonzero generalized
eigenvalues of
$(\mathbf{Q}_{\mathcal{S}}\\!\\!-\\!\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
(padded up to $D$ with zeros if not enough), and is almost independent of SNR.

##  IV Discussion of the Threshold

From Theorem 1 in Sec. III-A, we know that the empirical threshold
$\textsf{SNR}_{\textsf{T}0}$ is a key parameter for MPB. By the discussion in
Sec. III-C, it is the intersection of the curves $\gamma_{0}\\!\\!+\\!\\!1$
and $\gamma_{1}\\!\\!+\\!\\!1$, as shown in Fig. 3(a). Therefore, we need to
investigate these two parameters to gain deeper insight of
$\textsf{SNR}_{\textsf{T}0}$.

###  IV-A General Results of $\gamma_{0}$ and $\gamma_{1}$

$\gamma_{0}$ has a simple expression of (36), from which we know that the
parameter $\beta$ is critical. Fig. 3(b) shows the curve of
$\gamma_{0}\\!+\\!1$ with different $\beta$. We see that, if $\beta\\!=\\!0$,
i.e. $\mathbf{H}_{\mathcal{I}}$ and $\mathbf{h}_{\mathcal{S}}$ are orthogonal,
then $\gamma_{0}\\!\\!=\\!\\!L\textsf{SNR}$ is unbounded as SNR goes to
infinity. Otherwise, there would be a limiting value of
$(N\\!-\\!\beta)/\beta$ so that $\gamma_{0}$ might never exceed $\gamma_{1}$
and $\textsf{SNR}_{\textsf{T}0}\\!\\!=\\!\\!+\infty$. Therefore,
$\beta\\!\\!=\\!\\!0$ is the best choice for $\gamma_{0}$.

(a) The curves of $\gamma_{0}\\!+\\!1$, $\gamma_{1}\\!\\!+\\!\\!1$ and
$\lambda_{\mathrm{max}}$ vs. SNR. ($\beta=0$)

(b) $\gamma_{0}\\!+\\!1$ against SNR with different $\beta$.

Figure 3: The curve of $\gamma_{0}+1$ and $\gamma_{1}+1$ against SNR. The
intersection of them is the empirical threshold $\textsf{SNR}_{\textsf{T}0}$.

$\gamma_{1}$ is another critical parameter that determines
$\textsf{SNR}_{\textsf{T}0}$. Fig. 3 shows that, as $\gamma_{1}$ increases,
its intersection with $\gamma_{0}$ moves rightward and
$\textsf{SNR}_{\textsf{T}0}$ increases. Therefore, knowing how to control
$\gamma_{1}$ is important in designing MPB. By Lemma 5, $\gamma_{1}$ is the
largest nonzero generalized eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.
(It is zero if there are less than $D$ nonzero eigenvalues and all of them are
negative.) To solve it directly, we need the roots of a polynomial eigen-
equation of order $L$, which has no general formula when $L\\!\\!>\\!\\!4$.
Instead, we resort to matrix perturbation theory again to derive a bound for
it. We first notice that the eigen-decomposition of
$(\mathbf{Q}_{\mathcal{S}}\\!-\\!\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
is equivalent to $(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ and
their eigenvalues only differ by one. By (6) and (7), the eigenvalue of
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ is further equivalent to
that of
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$,
where

$\displaystyle\mathbf{Y}_{\mathcal{S}}=\frac{1}{\sigma^{2}}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}=\textsf{INR}\cdot\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}_{0}}\mathbf{A}_{I}^{H},\qquad\mathbf{Y}_{\mathcal{I}}=\frac{1}{\sigma^{2}}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}}\mathbf{A}_{I}^{H}=\textsf{INR}\cdot\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}_{0}}\mathbf{A}_{I}^{H}.$
(42)

We term $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ as _noise free
pair_ , since it can be viewed as the covariance matrix pair of MPB without
noise. Our strategy here is to view
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$
as a perturbed version of
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ and apply the results in
matrix perturbation theory to derive a bound for $\gamma_{1}\\!\\!+\\!\\!1$.
Before we proceed on, we cite a more general definition of the generalized
eigenvalue of a matrix pair[32].

######  Definition 2 (Generalized eigenvalue)

The generalized eigenvalue of a matrix pair
$\left(\mathbf{A},\mathbf{B}\right)$ is a one dimensional space, denoted as
$\langle\nu,\mu\rangle\triangleq\left\\{\tau\cdot[\nu\;\mu]:\tau\in\mathbb{C}\right\\}$,
where $[\nu\;\mu]$ is a $1\times 2$ vector satisfying

$\displaystyle\mu\cdot\mathbf{A}\mathbf{x}=\nu\cdot\mathbf{B}\mathbf{x},$

with $\mathbf{x}$ being its eigenvector. If $\mathbf{B}$ is nonsingular, then
$\lambda\\!\\!=\\!\\!\nu/\mu$ becomes the conventional definition.

Comparing to the conventional definition of generalized eigenvalues, this one
includes the special case of $\mathbf{B}$ being singular so that
$\mu\\!\\!=\\!\\!0$ and $\nu\\!\\!\neq\\!\\!0$, namely,
$\lambda\\!\\!=\\!\\!+\infty$. We will see its importance later. Besides, we
also need the following definition and lemma from [32, pp.315–316].

######  Definition 3 (Definite pair)

If a matrix pair $(\mathbf{A},\mathbf{B})$ consists of two Hermitian matrices
and

$\displaystyle
C(\mathbf{A},\mathbf{B})\triangleq\min_{\|\mathbf{x}\|=1}\sqrt{\left(\mathbf{x}^{H}\mathbf{A}\mathbf{x}\right)^{2}+\left(\mathbf{x}^{H}\mathbf{B}\mathbf{x}\right)^{2}}>0,$

then $(\mathbf{A},\mathbf{B})$ is called definite pair, and
$C(\mathbf{A},\mathbf{B})$ is its Crawford number.

######  Theorem 2 (Weyl-Lidskii type)

Assume $\left(\mathbf{A},\mathbf{B}\right)$ is a definite pair and
$\left(\mathbf{A}\\!+\\!\mathbf{E},\mathbf{B}\\!+\\!\mathbf{F}\right)$ is its
perturbed version. Let $\langle\alpha_{i},\beta_{i}\rangle$ and
$\langle\widetilde{\alpha}_{i},\widetilde{\beta}_{i}\rangle$,
($i\\!\\!=\\!\\!1,\ldots,n$), be their ordered generalized eigenvalues,
respectively. If

$\displaystyle{\sqrt{\|\mathbf{E}\|_{S}^{2}+\|\mathbf{F}\|_{S}^{2}}}<{C(\mathbf{A},\mathbf{B})},$
(43)

where $\|\cdot\|_{S}$ denotes spectral norm of a matrix, then
$(\widetilde{\mathbf{A}},\widetilde{\mathbf{B}})$ is a definite pair, and

$\displaystyle\frac{|\alpha_{1}\widetilde{\beta}_{2}-\beta_{1}\widetilde{\alpha}_{2}|}{\sqrt{|\alpha_{1}|^{2}+|\beta_{1}|^{2}}\sqrt{|\widetilde{\alpha}_{2}|^{2}+|\widetilde{\beta}_{2}|^{2}}}\leq\frac{\sqrt{\|\mathbf{E}\|_{S}^{2}+\|\mathbf{F}\|_{S}^{2}}}{C(\mathbf{A},\mathbf{B})},$

We want to apply Theorem 2 to
$(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I})$
and $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ to derive a bound
for its eigenvalues. However,
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ is not a definite pair,
for the null spaces of $\mathbf{Y}_{\mathcal{S}}$ and
$\mathbf{Y}_{\mathcal{I}}$ may have nontrivial interset (larger than
$\\{0\\}$). Therefore, we need to transform
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ into a definite pair.
Let $\mathcal{N}_{\mathcal{S}}$ and $\mathcal{N}_{\mathcal{I}}$ be the null
spaces of $\mathbf{Y}_{\mathcal{S}}$ and $\mathbf{Y}_{\mathcal{I}}$,
respectively, and
$\mathcal{N}_{0}\\!\triangleq\\!\mathcal{N}_{\mathcal{S}}\bigcap\mathcal{N}_{\mathcal{I}}$.
Let $\mathbf{E}_{0}$ be a matrix whose columns are the orthonormal basis of
$\mathcal{N}_{0}^{\perp}$, with $\mathbf{E}_{0}^{H}\mathbf{E}_{0}=\mathbf{I}$.
Then, by the determinant identity in (31),
$\left(\mathbf{Y}_{\mathcal{S}}\\!+\\!\mathbf{I},\mathbf{Y}_{\mathcal{I}}\\!+\\!\mathbf{I}\right)$
has the same generalized eigenvalues as
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$,
except for the multiplicity of ones. Similarly,
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
has the same eigenvalue as
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ does except for the
multiplicity of zero. Therefore, instead, we can apply Theorem 2 to
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
to derive the bound. The Crawford number of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
is

$\displaystyle C_{Y}$
$\displaystyle=\min_{\begin{subarray}{c}\mathbf{w}\in\mathcal{N}_{0}^{\perp},\;\|\mathbf{w}\|=1\end{subarray}}\sqrt{\left(\mathbf{w}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{w}\right)^{2}\\!\\!+\\!\\!\left(\mathbf{w}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{w}\right)^{2}}>0,$
(44)

which means
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
is a positive pair. Before applying Theorem 2, we first analyze the dependency
of $C_{Y}$ on the interference power. Substituting (42) into (44), we have
$C_{Y}\\!=\\!\textsf{INR}\cdot C_{Y_{0}}$, where

$\displaystyle C_{Y_{0}}$
$\displaystyle=\min_{{\mathbf{w}\in\mathcal{N}_{0}^{\perp},\;\|\mathbf{w}\|=1}}\sqrt{\left(\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}_{0}}\mathbf{A}_{I}^{H}\mathbf{w}\right)^{2}\\!\\!+\\!\\!\left(\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{I}_{0}}\mathbf{A}_{I}^{H}\mathbf{w}\right)^{2}}.$

Since $\bm{\Phi}_{\mathcal{S}_{0}}$ and $\bm{\Phi}_{\mathcal{I}_{0}}$ are
independent of INR (c.f. Sec. II-B), $C_{Y_{0}}$ is also independent of INR.
Therefore, $C_{Y}$ is proportional to INR. Now, we are ready to use Theorem 2,
and we only consider the case of INR being large. Let
$\langle\nu_{0},\mu_{0}\rangle$ and $\langle\nu,\mu\rangle$ be the
corresponding generalized eigenvalues of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$,
respectively. Then, for large INR, (43) is satisfied and

$\displaystyle\frac{|\nu\mu_{0}-\mu\nu_{0}|}{\sqrt{|\nu|^{2}\\!\\!+\\!\\!|\mu|^{2}}\\!\sqrt{|\nu_{0}|^{2}\\!\\!+\\!\\!|\mu_{0}|^{2}}}\\!\\!<\\!\\!\frac{\sqrt{2}}{C_{Y_{0}}}\cdot\frac{1}{\textsf{INR}}.$
(45)

We now derive the bound for $\gamma_{1}\\!+\\!1$ in two separate cases.

####  IV-A1 $\mathcal{N}_{\mathcal{I}}\nsubseteq\mathcal{N}_{\mathcal{S}}$

There is a nonzero $\mathbf{w}_{0}$ such that
$\mathbf{Y}_{\mathcal{S}}\mathbf{w}_{0}\\!\neq\\!\mathbf{0}$ and
$\mathbf{Y}_{\mathcal{I}}\mathbf{w}_{0}\\!=\\!\mathbf{0}$. By Definition 2,
$\langle\nu_{0},0\rangle$ ($\forall\nu_{0}\\!\neq\\!0$) is a generalized
eigenvalue of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ and
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$.
This means the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ has an infinite
generalized eigenvalue. As a result, (45) becomes

$\displaystyle\frac{|\mu|}{\sqrt{|\nu|^{2}\\!\\!+\\!\\!|\mu|^{2}}}\\!\\!<\\!\\!\frac{\sqrt{2}}{C_{Y_{0}}}\cdot\frac{1}{\textsf{INR}}\quad\Leftrightarrow\quad\lambda\triangleq\frac{\nu}{\mu}>\sqrt{\frac{C_{Y_{0}}^{2}}{2}\textsf{INR}^{2}\\!\\!-\\!\\!1}\approx\frac{C_{Y_{0}}}{\sqrt{2}}\textsf{INR}.$

This means
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
would always have an eigenvalue that satisfies the above inequality. Since
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ has the same eigenvalue
except for the multiplicity of ones, $\gamma_{1}\\!+\\!1$ satisfies

$\displaystyle\gamma_{1}+1>\frac{C_{Y_{0}}}{\sqrt{2}}\textsf{INR},$ (46)

which gives a lower bound for $\gamma_{1}\\!+\\!1$. We can see that it goes up
unboundedly as INR increases. Hence, by (22) and (24), the threshold SNR also
increases unboundedly with INR.

####  IV-A2 $\mathcal{N}_{\mathcal{I}}\subseteq\mathcal{N}_{\mathcal{S}}$

Then, $\mathcal{N}_{0}\\!=\\!\mathcal{N}_{\mathcal{I}}$ and
$\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0}$ is nonsingular. As
a result, all eigenvalues of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{I}}\mathbf{E}_{0})$
and $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are bounded. When
INR is large, (45) becomes

$\displaystyle\frac{|\lambda-\lambda_{0}|}{\sqrt{1+\lambda^{2}}\sqrt{1+\lambda_{0}^{2}}}<\frac{C_{Y_{0}}}{\sqrt{2}}\cdot\frac{1}{\textsf{INR}}\quad\Longrightarrow\quad|\lambda-\lambda_{0}|<\frac{C_{Y_{0}}}{\sqrt{2}}(1+\lambda_{0}^{2})\cdot\frac{1}{\textsf{INR}}$

where $\lambda_{0}\\!=\\!\nu_{0}/\mu_{0}$ is the largest eigenvalue of
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0})$
and we used $\lambda\\!\approx\lambda_{0}$ in the second inequality. Since
$(\mathbf{Q}_{\mathcal{S}},\mathbf{Q}_{\mathcal{I}})$ has the same eigenvalue
as
$(\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I},\mathbf{E}_{0}^{H}\mathbf{Y}_{\mathcal{S}}\mathbf{E}_{0}\\!+\\!\mathbf{I})$
except for the multiplicity of ones, $\gamma_{1}+1$ is bounded around the
largest eigenvalue of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ or
one. Furthermore, from (42), we know that $\lambda_{0}$ is independent of INR.
Therefore, as all eigenvalues of noise free pair is finite,
$\gamma_{1}\\!+\\!1$ is bounded and independent of INR.

###  IV-B Two Typical Scenarios of $\gamma_{1}$

From the previous part, we know that whether the threshold of MPB is unbounded
is determined by the existence of infinite eigenvalue of the noise free pair.
Now, we discuss two typical classes of interferences. In the first case, all
eigenvalues are finite, and in the second case, there may be infinite
eigenvalues.

####  IV-B1 Directional White Noise

By uncorrelated directional white noise, we mean an interference that arrives
from a specific direction, with its samples in time domain being uncorrelated.
This means the entries of $\mathbf{S}_{I}(k)$ are independent identically
distributed random variables with zero mean and unit variance. (In section
II-A, we have already normalized the interference power in
$\mathbf{s}_{i}(k)$.) Then, by (8), we can immediately have
$\bm{\Phi}_{\mathcal{S}_{0}}\\!=\\!\bm{\Phi}_{\mathcal{I}_{0}}$, and all the
eigenvalue of the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are ones. Thus, in this
case, $\gamma_{1}\\!+\\!1$ is bounded and is independent of INR. In fact, we
can further have $\mathbf{Q}_{\mathcal{S}}\\!=\\!\mathbf{Q}_{\mathcal{I}}$
according to (6) and (7). Therefore, $\gamma_{1}\\!=\\!0$,
$\textsf{SNR}_{\textsf{T}0}\\!=\\!0$ and $\textsf{G}(\textsf{SNR})\\!=\\!1$.

####  IV-B2 Directional Periodical Interference

If an interference has periodical structure in time domain and arrives from
certain direction, then we term it as directional periodical interference. By
periodical, we mean the interfer is periodic with respect to the projection
basis, i.e.

$\displaystyle\mathbf{H}_{\mathcal{I}}^{H}\mathbf{S}_{I}(k)=e^{j\phi_{k}}\mathbf{H}_{\mathcal{I}}^{H}\mathbf{S}_{I},\qquad\mathbf{h}_{\mathcal{S}}^{H}\mathbf{S}_{I}(k)=e^{j\phi_{k}^{\prime}}\mathbf{h}_{\mathcal{S}}^{H}\mathbf{S}_{I}.$
(47)

A stronger condition would be $\mathbf{S}_{I}(k)\\!=\\!\mathbf{S}_{I}$.
However, (47) is good enough for our analysis. Now, we will discuss the
existence of infinite generalized eigenvalue in the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$. To do this, we need to
consider the relationship between their null spaces.

In practice, $\mathbf{R}_{\mathcal{S}}$ and $\mathbf{R}_{\mathcal{I}}$ are
estimated from sample average. Therefore, replacing the expectation in (8) by
sample average and using (47), we can get

$\displaystyle\bm{\Phi}_{\mathcal{S}_{0}}=\bm{\Omega}_{I}^{1/2}\mathbf{S}_{I}^{T}\mathbf{h}_{\mathcal{S}}^{\ast}\mathbf{h}_{\mathcal{S}}^{T}\mathbf{S}_{I}^{\ast}\bm{\Omega}_{I}^{1/2},\qquad\bm{\Phi}_{\mathcal{I}_{0}}=\frac{1}{r_{\mathcal{I}}}\bm{\Omega}_{I}^{1/2}\mathbf{S}_{I}^{T}\mathbf{H}_{\mathcal{I}}^{\ast}\mathbf{H}_{\mathcal{I}}^{T}\mathbf{S}_{I}^{\ast}\bm{\Omega}_{I}^{1/2}.$

Let
$\bm{\Pi}\\!\triangleq\\!\mathbf{S}_{I}\bm{\Omega}_{I}^{1/2}\mathbf{A}_{I}^{T}$.
Then, by (42), $\mathbf{Y}_{\mathcal{S}}$ and $\mathbf{Y}_{\mathcal{I}}$ in
this case can be expressed as

$\displaystyle\mathbf{Y}_{\mathcal{S}}=\textsf{INR}\cdot\left(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}\right)^{\ast},\qquad\mathbf{Y}_{\mathcal{I}}=\textsf{INR}\cdot\frac{1}{r_{\mathcal{I}}}\left(\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}\right)^{\ast}.$
(48)

Therefore, the noise free pair
$(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ has the same eigenvalue
as
$(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi},\frac{1}{r_{\mathcal{I}}}\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi})$
does, and we only have to check the null spaces of later matrix pair. Let
$\mathcal{V}_{I}\\!\triangleq\\!\mathcal{R}(\bm{\Pi})$ be the range space of
$\bm{\Pi}$. By definition, it is also the range space of $\mathbf{S}_{I}$,
namely the space spanned by all interference waveforms in one period. Then, we
can express the null spaces of
$\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}$ and
$\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}$ as

$\displaystyle\mathcal{N}_{\mathcal{S}}^{\prime}\triangleq\mathcal{N}\left(\bm{\Pi}^{H}\mathbf{h}_{\mathcal{S}}\mathbf{h}_{\mathcal{S}}^{H}\bm{\Pi}\right)=\left\\{\bm{\Pi}^{\dagger}\mathbf{x}:\;\mathbf{x}\in\mathcal{S}^{\perp}\cap\mathcal{V}_{I}\right\\}\oplus\mathcal{N}(\bm{\Pi})$
$\displaystyle\mathcal{N}_{\mathcal{I}}^{\prime}\triangleq\mathcal{N}\left(\bm{\Pi}^{H}\mathbf{H}_{\mathcal{I}}\mathbf{H}_{\mathcal{I}}^{H}\bm{\Pi}\right)=\left\\{\bm{\Pi}^{\dagger}\mathbf{x}:\;\mathbf{x}\in\mathcal{I}^{\perp}\cap\mathcal{V}_{I}\right\\}\oplus\mathcal{N}(\bm{\Pi}),$

where $\mathcal{S}\\!=\\!\mathcal{R}(\mathbf{h}_{\mathcal{S}})$,
$\mathcal{I}\\!=\\!\mathcal{R}(\mathbf{H}_{\mathcal{I}})$ as defined in Sec.
II-B, and $\dagger$ denotes Moore-Penrose pseudoinverse [34]. To see the
relationship between $\mathcal{N}_{\mathcal{S}}^{\prime}$ and
$\mathcal{N}_{\mathcal{I}}^{\prime}$, we only need to check the relationship
between $\mathcal{S}^{\perp}\cap\mathcal{V}_{I}$ and
$\mathcal{I}^{\perp}\cap\mathcal{V}_{I}$ which can be expressed as

$\displaystyle\mathcal{S}^{\perp}\cap\mathcal{V}_{I}=\\{\mathbf{x}:\;\mathbf{B}_{\mathcal{S}}^{H}\mathbf{x}=\mathbf{0}\\},\qquad\mathcal{I}^{\perp}\cap\mathcal{V}_{I}=\\{\mathbf{x}:\;\mathbf{B}_{\mathcal{I}}^{H}\mathbf{x}=\mathbf{0}\\},$
(49)

where

$\displaystyle\mathbf{B}_{\mathcal{I}}\triangleq\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{I}^{\perp}}&\mathbf{H}_{\mathcal{I}}\\\
\end{bmatrix},\qquad\mathbf{B}_{\mathcal{S}}\triangleq\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{I}^{\perp}}&\mathbf{h}_{\mathcal{S}}\\\
\end{bmatrix}.$

with $\mathbf{H}_{\mathcal{V}_{I}^{\perp}}$ being the matrix whose columns are
the orthonormal basis of $\mathcal{V}_{I}^{\perp}$. Then, by (49), whether the
eigenvalues of $(\mathbf{Y}_{\mathcal{S}},\mathbf{Y}_{\mathcal{I}})$ are
finite is equivalent to the validity of
$\mathcal{R}\\{\mathbf{B}_{\mathcal{I}}\\}\\!\\!\supseteq\\!\\!\mathcal{R}\\{\mathbf{B}_{\mathcal{S}}\\}$.
Define

$\displaystyle\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}\triangleq\begin{bmatrix}\mathbf{I}&-\mathbf{H}_{\mathcal{V}_{I}^{\perp}}^{H}\mathbf{H}_{\mathcal{I}}\\\
0&\mathbf{I}\end{bmatrix},\qquad\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}\triangleq\begin{bmatrix}\mathbf{I}&-\mathbf{H}_{\mathcal{V}_{I}^{\perp}}^{H}\mathbf{h}_{\mathcal{S}}\\\
0&\mathbf{I}\end{bmatrix}.$

Right-multiplying $\mathbf{B}_{\mathcal{I}}$ and $\mathbf{B}_{\mathcal{S}}$ by
$\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}$ and
$\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}$, respectively, we can have

$\displaystyle\mathbf{B}_{\mathcal{I}}\cdot\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}=\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{c}^{\perp}}&\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\end{bmatrix},\qquad\mathbf{B}_{\mathcal{S}}\cdot\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}=\begin{bmatrix}\mathbf{H}_{\mathcal{V}_{c}^{\perp}}&\mathbf{P}_{\mathcal{V}_{I}}\mathbf{h}_{\mathcal{S}}\end{bmatrix},$

where
$\mathbf{P}_{\mathcal{V}_{I}}\\!\triangleq\\!\mathbf{H}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{V}_{I}}^{H}$
is the projection matrix onto $\mathcal{V}_{I}$. Since
$\mathbf{T}_{\mathbf{B}_{\mathcal{I}}}$ and
$\mathbf{T}_{\mathbf{B}_{\mathcal{S}}}$ are nonsingular, the above two
matrices have the same range spaces as $\mathbf{B}_{\mathcal{S}}$ and
$\mathbf{B}_{\mathcal{I}}$ do, namely we can have

$\displaystyle\mathcal{R}\\{\mathbf{B}_{\mathcal{I}}\\}=\mathcal{V}_{I}^{\perp}\oplus\mathcal{R}\\{\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\\},\qquad\mathcal{R}\\{\mathbf{B}_{\mathcal{S}}\\}=\mathcal{V}_{I}^{\perp}\oplus\mathcal{R}\\{\mathbf{P}_{\mathcal{V}_{I}}\mathbf{h}_{\mathcal{S}}\\}$

Therefore, whether the generalized eigenvalues of the noise free pair are
bounded is finally equivalent to whether

$\displaystyle\mathcal{R}\big{\\{}\mathbf{P}_{\mathcal{V}_{I}}\mathbf{H}_{\mathcal{I}}\big{\\}}\\!\\!\supseteq\\!\\!\mathcal{R}\big{\\{}\mathbf{P}_{\mathcal{V}_{c}}\mathbf{h}_{\mathcal{S}}\big{\\}}.$
(50)

Figure 4:  Geometrical interpretation of (50).
$\mathbf{h}_{\mathcal{I}}^{(1)}$ and $\mathbf{h}_{\mathcal{I}}^{(2)}$ denote
the columns of $\mathbf{H}_{\mathcal{I}}$. For one interference channel,
$\mathbf{H}_{\mathcal{I}}\\!=\\!\mathbf{h}_{\mathcal{I}}$.

Fig. 4 shows the geometrical interpretation of (50). To see if $\gamma_{1}$ is
bounded, we can project $\mathbf{h}_{\mathcal{S}}$ and all the columns of
$\mathbf{H}_{\mathcal{I}}$ onto $\mathcal{V}_{I}$. If the former projection
lies in the space spanned by the later ones, then $\gamma_{1}$ is bounded.
Otherwise, there is an infinite eigenvalue in the noise free pair and
$\gamma_{1}$ will increase unboundedly with INR. Since
$\mathbf{H}_{\mathcal{I}}$ of the Maximin scheme and PAPC scheme has only one
column, this means the projections of $\mathbf{h}_{\mathcal{S}}$ and
$\mathbf{H}_{\mathcal{I}}$ should be on the same line, which is hardly valid.
Thus, these two approaches are sensitive to directional periodical
interferences, as we will see in later simulation results. Furthermore, Fig. 4
also shows the case of multiple interference channels, i.e. the dimension of
$\mathcal{I}$ is larger than one, then $\gamma_{1}$ is more likely to be
bounded. This issue is out of the scope of this paper and will be discussed in
in [36].

##  V Simulation Results

In this section, we simulate various scenarios and compare them to our
theoretical results. The matrix pair beamformers implemented include the
Maximin algorithm and PAPC algorithm, which have many kinds of adaptive
algorithms. However, we are only interested in their steady state performance.
The reason for this is that, if the methods suffer from poor steady state
performance, then it is meaningless to investigate their adaptive algorithms.
Therefore, we directly calculate their weight vectors by performing
generalized eigen-decompositions on the estimated matrix pairs. The
interference signals encountered in the simulation include BPSK signal, tones,
periodical noise and multiuser interference. The first one is the directional
white noise and the others are directional periodical interferences.

In all cases, we consider a uniform linear array (ULA) with eight isotropic
antennas ($L=8$) spaced half a wavelength apart. For each user, a $100$ kbps
DPSK signal is randomly generated and spreaded by a distinct 31-chip Gold
sequence in each simulation trial. Then it is modulated onto a $1$ GHz carrier
to form a RF signal with bandwidth $3.1$ MHz. In all the simulations, we
assume the SOI arrives from $0^{\circ}$ and the interferers have equal power.

Fig. 5(a)–Fig. 5(d) show the operating curve $\textsf{G}(\textsf{SNR})$ of the
Maximin and PAPC beamformers in the four scenarios. The simulated values are
obtained by using the simulated data and the theoretical ones are computed by
the piecewise function $\textsf{G}(\textsf{SNR})$ in (20). To eliminate the
randomness caused by finite sample effects, $K=10^{6}$ symbols are simulated
for each SNR and INR in every experiment.

(a) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under BPSK jammers.

(b) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under periodical noise.

(c) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under tone jammers.

(d) $\textsf{G}(\textsf{SNR})$ of Maximin and PAPC under multipath MAI.

Figure 5: Simulated and theoretical operating curve $\textsf{G}(\textsf{SNR})$
under different kinds of interference for Maximin and PAPC.

In Fig. 5(a), the interferers are three uncorrelated BPSK modulated signals,
i.e. random sequences of $\pm 1$. The rates of the interferers are all
3.1Mbps, which is the same as the chip rate of the SOI and covers its whole
bandwidth. Therefore, they belong to the type of directional white noise. We
can see that $\textsf{G}(\textsf{SNR})\approx 1$ and there is no threshold
effect, which is consistent with the analysis in Section IV-B1.

Fig. 5(b)–Fig. 5(d) show the results for three types of directional periodical
interferences: periodical noise, tones and multiple access interference (MAI)
with multipath. In Fig. 5(b), we consider two periodical noises, which arrives
from $30^{\circ}$ and $-40^{\circ}$, respectively, and each of them is
generated by repeating a segment of Gaussian white noise over times, with the
repeating frequency being $100$ kHz (same as that of SOI’s symbol rate). The
frequency offsets of the tones with respect to the carrier in Fig. 5(c) are
$100$kHz, $-300$kHz, $0$Hz, $400$kHz and $-100$kHz, respectively, and their
DOAs are $30^{\circ}$, $-50^{\circ}$, $-20^{\circ}$, $19^{\circ}$ and
$45^{\circ}$. In Fig. 5(d), there is one incident MAI signal with three-ray
multipath delays of $3$ chips, $5$ chips and $4$ chips arriving from
$30^{\circ}$, $-20^{\circ}$ and $-50^{\circ}$, respectively. We can see that
the theoretical values of $\textsf{G}(\textsf{SNR})$ match with the simulated
ones very well.

To verify the validity of the approximation given by Lemma 4, in Fig. 6(a) and
Fig. 6(b), we also show the curves of $\gamma_{0}\\!+\\!1$,
$\gamma_{1}\\!+\\!1$, and $\lambda_{\mathrm{max}}$. We can see that
$\max\\{\gamma_{0}+1,\gamma_{1}+1\\}$ can be an excellent approximation for
$\lambda_{\mathrm{max}}$. In Fig. 7(a) and Fig. 7(b), we show the array
patterns that are below and above the threshold. The results in these four
figures are simulated under two periodical noises, with the same parameter as
Fig. 5(b). The case for the tone jammers and the multiple access interference
cases are quite similar, and is thus omitted.

For the Maximin algorithm, all the curves of $\textsf{G}(\textsf{SNR})$ have
failure area, threshold area and operating area, which are consistent with the
typical curve in Fig. 2. We have also marked the predicted thresholds of
$\textsf{SNR}_{\textsf{T}1}$ and $\textsf{SNR}_{\textsf{T}2}$, computed by
(24), in the figures as well, which confirm our theoretical calculations.
Furthermore, they also show that the threshold SNR would increase with the
interference INR. This is because $\gamma_{1}+1$ (c.f. Fig. 6(a)) would moving
upward as INR increases, making the intersecting point of $\gamma_{0}+1$ and
$\gamma_{1}+1$ move rightwards. This is consistent with our claim in Section
IV-B2 that $\gamma_{1}$ would increase unboundedly with the INR. Fig. 7(a) and
Fig. 7(b) show its array pattern below and above threshold, respectively. We
can see that the mainlobe of the beamformer would mistakenly point to the
interferers once the SNR is below the threshold, as we have predicted in
Section III-C.

For the PAPC algorithm, Fig. 5(b)–Fig. 5(d) show that there are only failure
areas. This is because, no matter how large SNR is, $\gamma_{0}\\!\\!+\\!\\!1$
will never exceeds $\gamma_{1}\\!\\!+\\!\\!1$, as shown by Fig. 6(b).
Therefore, the threshold of PAPC in this case is infinity. The array pattern
shown in Fig. 7(a) also indicates that its mainlobe has pointed to the
interferers. We have stated in section III-C that the presence of SOI in
$\mathbf{R}_{\mathcal{I}}$ will make the beamformer mistakenly null the SOI.
The curves of PAPC in Fig. 5(b)–Fig. 5(d) confirms this observation, and Fig.
7(b) shows that PAPC beamformer has a deep null in the direction of the SOI.

(a) Maximin scheme

(b) PAPC scheme

Figure 6: The largest and second largest generalized eigenvalues of the
Maximin scheme vs SNR under two periodical noises.

(a) Case I: below threshold ($\textsf{SNR}\\!\\!=\\!\\!-10.1$dB,
$\textrm{INR}\\!\\!=\\!\\!30$dB)

(b) Case II: above threshold ($\textsf{SNR}\\!\\!=\\!\\!40.9$dB,
$\textrm{INR}\\!\\!=\\!\\!30$dB)

Figure 7: The array patterns corresponding to the Maximin and PAPC under two
periodical noises.

In summary, the conventional MPB like Maximin and PAPC work well in the
presence of directional white noise, even when the INR is large. However, they
are very vulnerable to multiple directional periodical interferers with
repeating structures in the time domain and arrive from different directions.

##  VI Conclusions

Matrix pair beamformer (MPB) is a general framework we proposed to model a
class of blind beamformers that exploit the temporal signature of the signal
of interest (SOI). It has the advantages only relying on the second order
statistics to achieve blind processing. In this paper, we have analyzed the
mechanism of MPB with matrix mismatch, and showed how it worked “blindly”. We
have discovered that there is a threshold effect in MPB, i.e. the beamformer
would fail completely if the SOI’s input SNR is below that threshold.
Meanwhile, its normalized output SINR has been derived as the performance
measure, and the threshold SNR has also been predicted. We have also observed
that the existence of infinite generalized eigenvalue in what is called _noise
free pair_ makes the threshold increase unboundedly with the interference
power. This is highly probable when there are multiple periodical interferers.
All our theoretical analysis matches with the simulation results very well.

Our analysis indicates that the conventional MPB is very vulnerable to
multiple periodical interferers. Moreover, it also implies the importance of
choosing the appropriate projection space for the interference channel. And we
will address this issue in another paper[36].

##  Appendix A Approximation of $\lambda_{\mathrm{max}}$ and $\gamma_{i}$

###  A-A Approximation of $\lambda_{\mathrm{max}}$

Since similarity transform does not change the eigenvalues, we can apply it to
$\mathbf{M}$ before using Lemma 2. As we will see later, the bounds derived in
this way can be surprisingly tight. Though any transform matrix can be used,
we prefer the following diagonal matrix that can preserve the diagonal terms
of $\mathbf{M}$

$\displaystyle\mathbf{F}=\mathrm{diag}\\{1,f_{1},f_{2},\ldots,f_{D}\\},\quad(f_{i}>0,\;i=1,2,\ldots,D)$

Applying it to $\mathbf{M}$ in (33) and using Lemma 2, we can have the
following Gerschgorin disks for $\mathbf{F}\mathbf{M}\mathbf{F}^{-1}$:

$\displaystyle G_{i}$
$\displaystyle=\\{\lambda:\;|\lambda-(\gamma_{i}+1)|<R_{i}(\mathbf{F})\\},\qquad
i=0,1,2,\ldots,D,$

where the radii of the disks are

$\displaystyle R_{0}(\mathbf{F})$
$\displaystyle=\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\sum_{i=1}^{D}|\psi_{T_{i}}|\frac{1}{f_{i}}$
(51) $\displaystyle R_{i}(\mathbf{F})$
$\displaystyle=\gamma_{i}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{i}}|\cdot\mathrm{sgn}(\gamma_{i})}{\frac{L\beta}{N}\textsf{SNR}+1}f_{i},\quad
1\leq i\leq D,$ (52)

with $\psi_{T_{i}}$ being the $i$th component of $\bm{\psi}_{T}$. Now, we are
going to optimize $f_{1},\ldots,f_{D}$. To derive an effective bound for
$\lambda_{\mathrm{max}}$, we should ensure the rightmost disk is separated
from the others (c.f. Lemma 2). Therefore, our criterion for finding the
optimal $\mathbf{F}$ is _minimizing the radius of the rightmost Gerschgorin
disk subject to the constraint that it is separated from all the remaining
ones_. Depending on $\gamma_{0}+1>\gamma_{1}+1$ or
$\gamma_{0}+1<\gamma_{1}+1$, the rightmost disk might be $G_{0}$ or $G_{1}$,
and we now discuss them separately.

If $\gamma_{0}+1>\gamma_{1}+1$, then $G_{0}$ is the rightmost disk, and we can
formulate the optimization of $\mathbf{F}$ as

$\displaystyle\min~{}$
$\displaystyle~{}R_{0}(\mathbf{F})=\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\sum_{i=1}^{D}|\psi_{T_{i}}|\frac{1}{f_{i}}$
(53) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{i}+1+R_{i}(F)\leq\gamma_{0}+1-R_{0}(\mathbf{F}),\quad
1\leq i\leq D.$ (54)

To minimize $R_{0}(\mathbf{F})$, we want $f_{1},\ldots,f_{D}$ to be as large
as possible. But this will increase the radius of $G_{1},\ldots,G_{D}$, and to
avoid connecting $G_{0}$, they cannot be too large. In fact,
$f_{1},\ldots,f_{D}$ have different importance in this tradeoff. Since
$\gamma_{0}>\ldots>\gamma_{D}$, $G_{2},\ldots,G_{D}$ are farther away from
$G_{0}$ than $G_{1}$. Therefore, $f_{2},\ldots,f_{D}$ can be reasonably larger
than $f_{1}$ when keeping separate from $G_{0}$. Thus, to simplify analysis,
we can ignore all the terms except $|\psi_{T_{1}}|/f_{1}$ in
$R_{0}(\mathbf{F})$, i.e.
$R_{0}(\mathbf{F})\approx\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}\frac{|\psi_{T_{1}}|}{f_{1}}$
in (53) and (54), and only the constraint for $i=1$ is effective in (54).
Then, the problem can be reduced to

$\displaystyle\min~{}$
$\displaystyle~{}R_{0}(\mathbf{F})\approx\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\frac{1}{f_{1}}$
$\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\cdot\mathrm{sgn}(\gamma_{1})}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}^{2}-(\gamma_{0}-\gamma_{1})f_{1}+\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\leq
0.$

This is a simple convex optimization problem, feasible when
$\frac{\gamma_{1}}{\gamma_{0}}\in(-\infty,1+2\delta-\sqrt{(1+2\delta)^{2}-1}]$,
where
$\delta\triangleq\frac{\sigma^{2}}{L}|\psi_{T_{1}}|^{2}/(\frac{L\beta}{N}\textsf{SNR}+1)\ll
1$. Thus, we can easily solve the above optimization and get

$\displaystyle|\lambda_{\mathrm{max}}-(\gamma_{0}+1)|<\gamma_{0}\cdot
f\left(\frac{\gamma_{1}}{\gamma_{0}}\right)$

where
$f(x)\triangleq\frac{1}{2}[1\\!-\\!x\\!-\\!\sqrt{(1\\!-\\!x)^{2}\\!-\\!4\delta|x|}]$
with
$x\in(-\infty,1\\!-\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta]\\!\cup\\![1\\!+\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta,+\infty)$.
Furthermore, by taking the derivative of $f(x)$, we can easily get
$0\\!\\!\leq\\!\\!f(x)\\!\\!\leq\\!\\!\max\\{\delta,\sqrt{\delta^{2}\\!\\!+\\!\\!\delta}\\!\\!-\\!\\!\delta\\}\\!\\!\ll\\!\\!1$
when $x\in(-\infty,1\\!-\\!2\sqrt{\delta\\!+\\!\delta^{2}}\\!+\\!2\delta]$.

If $\gamma_{0}+1<\gamma_{1}+1$, then $G_{1}$ is the rightmost disk and
$\gamma_{1}>0$. The optimization problem is

$\displaystyle\min~{}$
$\displaystyle~{}R_{1}(\mathbf{F})=\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}$
(55) $\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{0}+1+R_{0}(F)\leq\gamma_{1}+1-R_{1}(\mathbf{F}),$
(56)
$\displaystyle~{}\gamma_{i}+1+R_{i}(F)\leq\gamma_{1}+1-R_{1}(\mathbf{F}),\quad
2\leq i\leq D.$ (57)

According to (51) and (52), $f_{1}$ should be as small as possible to minimize
$R_{1}(\mathbf{F})$ in (55), (56) and (57). However, this will increase
$R_{0}(\mathbf{F})$ in (56), making $G_{0}$ connect with $G_{1}$. Therefore,
$f_{1}$ cannot be arbitrarily small. On the other hand, $f_{2},\ldots,f_{D}$
should be as large as possible to reduce $R_{0}(\mathbf{F})$ in (56) while
keeping (57) valid. As a result, $|\psi_{T_{1}}|/f_{1}$ is still the dominant
term in $R_{0}(\mathbf{F})$ (c.f. (51)) and the key point here remains the
tradeoff between $R_{1}(\mathbf{F})$ and $R_{0}(\mathbf{F})$. In other words,
the optimization becomes

$\displaystyle\min~{}$
$\displaystyle~{}R_{1}(\mathbf{F})=\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}$
$\displaystyle\mathrm{s.t.}~{}$
$\displaystyle~{}\gamma_{1}\frac{\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|}{\frac{L\beta}{N}\textsf{SNR}+1}f_{1}^{2}-(\gamma_{1}-\gamma_{0})f_{1}+\gamma_{0}\sqrt{\frac{\sigma^{2}}{L}}|\psi_{T_{1}}|\leq
0.$

The feasible region for this convex optimization is
$\frac{\gamma_{1}}{\gamma_{0}}\in[1+2\delta+\sqrt{(1+2\delta)^{2}-1},+\infty)$,
with $\delta$ defined in the previous case. By solving it, we can finally get
the bound for $\lambda_{\mathrm{max}}$ as

$\displaystyle|\lambda_{\mathrm{max}}-(\gamma_{1}+1)|\leq\gamma_{1}\cdot
f\left(\frac{\gamma_{0}}{\gamma_{1}}\right)$

where $f(x)$ is the same as in the previous case.

###  A-B Approximation of $\gamma_{i}$

Then, we discuss the approximation of $\gamma_{i}$. By substituting (5) into
(30) and using matrix inversion lemma, we can have the equivalent equation of
(30) as

$\displaystyle\det(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\cdot\det\left\\{\lambda\mathbf{I}-\left[\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}-\frac{\sigma_{\mathcal{I}}^{2}\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\cdot\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}}{1+\sigma_{\mathcal{I}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}\right]\bm{\Phi}_{\Delta}\right\\}=0$
(58)

To further reduce the above expression, we need factorizations of
$\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I}$ and
$\bm{\Phi}_{\Delta}$ like (28). Let $\sigma_{\mathcal{I}_{0}}^{2}=0$ in (28).
Then, $\mathbf{R}_{\mathcal{I}}=\mathbf{Q}_{\mathcal{I}}$, and (28) become

$\displaystyle\mathbf{T}^{H}_{0}\bm{\Phi}_{\Delta}\mathbf{T}_{0}=\bm{\Gamma}_{0},\qquad\mathbf{T}^{H}_{0}(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\mathbf{T}_{0}=\mathbf{I}.$

where $\mathbf{T}_{0}$ and $\bm{\Gamma}_{0}$ are the counterparts of
$\mathbf{T}$ and $\bm{\Gamma}$, respectively. Substituting them into (58), we
have

$\displaystyle\det(\mathbf{A}_{I}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\cdot\det\left\\{\lambda\mathbf{I}-\left[\mathbf{I}-\frac{\sigma_{\mathcal{I}}^{2}\tilde{\bm{\psi}}_{T_{0}}\tilde{\bm{\psi}}_{T_{0}}^{H}}{1+\sigma_{\mathcal{I}}^{2}\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}}\right]\bm{\Gamma}_{0}\right\\}=0$

where
$\tilde{\bm{\psi}}_{T_{0}}=\mathbf{T}_{0}^{-1}\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}$
is the counterpart of $\tilde{\bm{\psi}}_{T}$ when
$\sigma_{\mathcal{I}_{0}}^{2}=0$. By letting $\textsf{SNR}=0$ in Lemma 1,
$\mathbf{R}_{\mathcal{I}}$ becomes $\mathbf{Q}_{\mathcal{I}}$ and we can have
$\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\approx
L/\sigma^{2}$ and $\|\bm{\psi}_{T_{0}}\|^{2}=L\kappa_{0}/\sigma^{2}\ll
L/\sigma^{2}$. Therefore, by the similar argument of $\lambda_{\mathrm{max}}$
using Gerschgorin theorem, we can have $\gamma_{i}\approx\lambda_{i,0}$, where
$\lambda_{i,0}$ is the $i$th diagonal term of $\bm{\Gamma}_{0}$, namely, the
$i$th eigenvalue of
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$.
In fact, there is a correspondence between $\lambda_{i,0}$ and the generalized
eigenvalue of
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$.
By (31),

$\displaystyle\det\left\\{\lambda(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}-\bm{\Phi}_{\Delta}\right\\}$
$\displaystyle=\lambda^{D-L}\cdot\det(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1}\det\mathbf{Q}_{\mathcal{I}}^{-1}\cdot\det\\{\lambda\mathbf{Q}_{\mathcal{I}}-\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}\\}.$

Since
$\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}}=\mathbf{A}_{I}\bm{\Phi}_{\Delta}\mathbf{A}_{I}^{H}$,
the above expression implies that
$(\bm{\Phi}_{\Delta},(\mathbf{A}_{I}^{H}\mathbf{Q}_{\mathcal{I}}^{-1}\mathbf{A}_{I})^{-1})$
has the same eigenvalues as
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$
except for multiplicity of zeros. Thus, we can estimate $\gamma_{i}$ like
this: 1) take out all nonzero eigenvalues of
$(\mathbf{Q}_{\mathcal{S}}-\mathbf{Q}_{\mathcal{I}},\mathbf{Q}_{\mathcal{I}})$,
2) pad them up to $D$ eigenvalues with zeros, 3) order them decreasingly to
get $\lambda_{1,0},\ldots,\lambda_{D,0}$, and 4) let
$\gamma_{i}\approx\lambda_{i,0}$.

##  Appendix B Proof of Lemma 3

Substituting the definitions of $y_{S}(k)$ and $y_{I}(k)$ (c.f. (10)), (6) and
(14) into (17) , we can have

$\displaystyle\textsf{G}(\textsf{SNR})$
$\displaystyle=\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}\mathbf{w}+\sigma^{2}\|\mathbf{w}\|^{2}}\frac{1}{\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}}\approx\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\bm{\Phi}_{\mathcal{S}}\mathbf{A}_{I}^{H}\mathbf{w}+\sigma^{2}\|\mathbf{w}\|^{2}}\big{/}\frac{L}{\sigma^{2}},$
(59)

where we used
$\mathbf{a}_{0}^{H}\mathbf{Q}_{\mathcal{S}}^{-1}\mathbf{a}_{0}\\!\\!\approx\\!\\!{L}/{\sigma^{2}}$,
derived by replacing $\mathbf{R}_{\mathcal{I}}$ with
$\mathbf{R}_{\mathcal{S}}$ and letting
$\sigma_{\mathcal{S}_{0}}^{2}\\!\\!=\\!\\!0$ in Lemma 1.

To further derive G, we first need the expression for $\|\mathbf{w}\|^{2}$.
The key trick is to recognize that $\mathbf{w}\in\mathcal{R}(\mathbf{A})$ so
that its projection onto $\mathcal{R}(\mathbf{A})$ equals itself, where
$\mathcal{R}(\cdot)$ is the range space of a matrix and
$\mathbf{A}\triangleq[\mathbf{a}_{0}\;\mathbf{A}_{I}]$. This can be proved by
substituting (7) into (5), applying matrix inversion lemma and pluging it into
(27). Furthermore, $\mathcal{R}(\mathbf{A}_{I})$ is a subspace of
$\mathcal{R}(\mathbf{A})$ with the dimension lower by one. Thus,
$\mathcal{R}(\mathbf{A})\\!\\!=\\!\\!\mathcal{R}(\mathbf{A})\\!\oplus\\!\mathcal{R}(\hat{\mathbf{b}}_{0})$,
where $\oplus$ denotes the direct sum and $\hat{\mathbf{b}}_{0}$ is the unit
vector in $\mathcal{R}(\mathbf{A})$ that is orthogonal to
$\mathcal{R}(\mathbf{A}_{I})$. Then, the projection matrix of
$\mathcal{R}(\mathbf{A})$ can be written as

$\displaystyle\mathbf{P}_{A}=\mathbf{A}_{I}(\mathbf{A}_{I}^{H}\mathbf{A}_{I})^{-1}\mathbf{A}_{I}^{H}+\mathbf{\hat{b}}_{0}\mathbf{\hat{b}}_{0}^{H}.$

This together with $\mathbf{P}_{A}\mathbf{w}=\mathbf{w}$ and
$\mathbf{P}_{A}^{2}=\mathbf{P}_{A}$ yields

$\displaystyle\|\mathbf{w}\|^{2}=\mathbf{w}^{H}\mathbf{P}_{A}\mathbf{w}=\mathbf{w}^{H}\mathbf{A}_{I}(\mathbf{A}_{I}^{H}\mathbf{A}_{I})^{-1}\mathbf{A}_{I}^{H}\mathbf{w}+|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}$
(60)

Assumption 1 in Sec. III-A implies that $\mathbf{a}_{0}$ is almost orthogonal
to $\mathcal{R}(\mathbf{A}_{I})$. Thus, it is nearly aligned with
$\hat{\mathbf{b}}_{0}$, and intuitively,
$|\mathbf{w}^{H}\hat{\mathbf{b}}_{0}|\\!\\!\approx\\!\\!|\mathbf{w}^{H}\hat{\mathbf{a}}_{0}|$,
where
$\hat{\mathbf{a}}_{0}\\!\\!\triangleq\\!\\!\mathbf{a}_{0}/\|\mathbf{a}_{0}\|$
is the unit vector of $\mathbf{a}_{0}$. To prove it, let
$\mathcal{R}_{0}\\!\\!\triangleq\\!\\!\mathrm{span}\\{\hat{\mathbf{a}}_{0},\hat{\mathbf{b}}_{0}\\}$
and $\\{\hat{\mathbf{b}}_{0},\hat{\mathbf{b}}_{1}\\}$ be the orthonormal basis
of $\mathcal{R}_{0}$. Its projection matrix becomes
$\mathbf{P}_{\mathcal{R}_{0}}\\!\\!=\\!\\!\hat{\mathbf{b}}_{0}\hat{\mathbf{b}}_{0}^{H}\\!\\!+\\!\\!\hat{\mathbf{b}}_{1}\hat{\mathbf{b}}_{1}^{H}$,
which satisfies
$\mathbf{P}_{\mathcal{R}_{0}}\hat{\mathbf{a}}_{0}\\!\\!=\\!\\!\hat{\mathbf{a}}_{0}$
and
$\mathbf{P}_{\mathcal{R}_{0}}\hat{\mathbf{b}}_{0}\\!\\!=\\!\\!\hat{\mathbf{b}}_{0}$.
Define
$\mathbf{w}_{0}\\!\\!=\\!\\!\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}/\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|$.
Then,

$\displaystyle|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\left[|\mathbf{w}^{H}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\right]=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\left[|\mathbf{w}^{H}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}^{H}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{P}_{\mathcal{R}_{0}}\mathbf{\hat{a}}_{0}|^{2}\right]$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}\left[|\mathbf{w}_{0}^{H}\mathbf{\hat{b}}_{0}|^{2}\\!\\!-\\!\\!|\mathbf{w}_{0}^{H}\hat{\mathbf{b}}_{0}\cdot\hat{\mathbf{b}}_{0}^{H}\mathbf{\hat{a}}_{0}\\!\\!+\\!\\!\mathbf{w}_{0}^{H}\hat{\mathbf{b}}_{1}\cdot\hat{\mathbf{b}}_{1}^{H}\mathbf{\hat{a}}_{0}|^{2}\right]$
$\displaystyle=|\mathbf{w}^{H}\mathbf{\hat{a}}_{0}|^{2}\\!\\!+\\!\\!\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}\cdot\cos\phi_{I}\cdot\cos(\phi_{1}+2\phi_{I})$
(61)

where $\phi_{I}$ is the angle between $\hat{\mathbf{a}}_{0}$ and
$\mathcal{R}(\mathbf{A}_{I})$, and $\phi_{1}$ is the angle between
$\hat{\mathbf{a}}_{0}$ and $\hat{\mathbf{b}}_{0}$. And the derivation of the
last step in (61) involves some simple trigonometry identities like product-
to-sum formula. Let
$\boldsymbol{\Psi}_{I}\\!\\!\triangleq\\!\\!\mathbf{A}_{I}^{H}\mathbf{A}_{I}/L$.
Combining (60), (61),
$\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|\\!\\!\leq\\!\\!\|\mathbf{w}\|$ and
$|\cos\phi_{I}|\\!\\!\ll\\!\\!1$ (c.f. Assumption 1), we can have

$\displaystyle\|\mathbf{w}\|^{2}$
$\displaystyle=\frac{\frac{1}{L}\left[|\mathbf{a}_{0}^{H}\mathbf{w}|^{2}+\mathbf{w}^{H}\mathbf{A}_{I}\boldsymbol{\Psi}_{I}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}\right]}{1-\frac{\|\mathbf{P}_{\mathcal{R}_{0}}\mathbf{w}\|^{2}}{\|\mathbf{w}\|^{2}}\cdot\cos\phi_{I}\cdot\cos(\phi_{1}+2\phi_{I})}\approx\frac{1}{L}\left[|\mathbf{a}_{0}^{H}\mathbf{w}|^{2}+\mathbf{w}^{H}\mathbf{A}_{I}\boldsymbol{\Psi}_{I}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}\right],$

where we also used $\|\mathbf{a}_{0}\|^{2}\\!\\!=\\!\\!L$ in Assumption 2.
Substitute the above expression back to (59), and we get

$\displaystyle\textsf{G}(\textsf{SNR})\approx\
\frac{|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}{\mathbf{w}^{H}\mathbf{A}_{I}\\!\left[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\right]\\!\mathbf{A}_{I}^{H}\mathbf{w}\\!\\!+\\!\\!|\mathbf{w}^{H}\mathbf{a}_{0}|^{2}}.$
(62)

Next, we are going to evaluate $\mathbf{w}^{H}\mathbf{a}_{0}$ and
$\mathbf{w}^{H}\mathbf{A}_{I}[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}]\mathbf{A}_{I}^{H}\mathbf{w}$.
By the expression of $\mathbf{w}$ in (27),

$\displaystyle\mathbf{a}_{0}^{H}\mathbf{w}$
$\displaystyle=\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}+\sum_{i=1}^{D}\frac{\gamma_{i}}{\lambda_{\mathrm{max}}\\!-\\!(\gamma_{i}\\!+\\!1)}|\widetilde{\psi}_{T_{i}}|^{2}=\mathbf{a}_{0}^{H}\mathbf{R}_{\mathcal{I}}^{-1}\mathbf{a}_{0}\\!\\!-\\!\\!\|\widetilde{\boldsymbol{\psi}}_{T}\|^{2}\\!\\!+\\!\\!\sum_{i=1}^{D}\frac{\lambda_{\mathrm{max}}\\!\\!-\\!\\!1}{\lambda_{\mathrm{max}}\\!-\\!(\gamma_{i}+1)}|\widetilde{\psi}_{T_{i}}|^{2}$
$\displaystyle\approx\frac{L}{\sigma^{2}}\left(\frac{L\beta}{N}\textsf{SNR}+1\right)\left[1+\left(\frac{L\beta}{N}\textsf{SNR}+1\right)^{-1}\psi_{S}(\lambda_{\mathrm{max}})\right],$
(63)

where we used Lemma 1 in the approximation and
$\psi_{S}(\lambda_{\mathrm{max}})$ is defined as (39) in Lemma 3. Before the
derivation of
$\mathbf{w}^{H}\mathbf{A}_{I}[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}]\mathbf{A}_{I}^{H}\mathbf{w}$,
we first cite the following identity from Lemma 2 in [35].

$\displaystyle\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{I}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\\!\\!=\\!\\!\frac{L}{\sigma^{2}}(\mathbf{T}^{-1})^{H}\\!\\!\left[\mathbf{I}\\!-\\!\frac{{L\beta}\textsf{SNR}\cdot\frac{\sigma^{2}}{L}\boldsymbol{\psi}_{T}\boldsymbol{\psi}_{T}^{H}}{{L\beta}(1\\!\\!-\\!\\!\rho_{0})\textsf{SNR}\\!\\!+\\!\\!N}\right]\\!\\!\mathbf{T}^{-1}.$

Then, combining the above expression together with
$\bm{\Phi}_{\Delta}\\!\\!=\\!\\!\bm{\Phi}_{\mathcal{S}}\\!\\!-\\!\\!\bm{\Phi}_{\mathcal{I}}$,
(28) and
$\mathbf{A}_{\epsilon}\\!\\!\triangleq\\!\\!\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}$
we can have

$\displaystyle\mathbf{w}^{H}\mathbf{A}_{I}\left[\frac{L}{\sigma^{2}}\bm{\Phi}_{\mathcal{S}}\\!\\!+\\!\\!\boldsymbol{\Psi}_{I}^{-1}\right]\mathbf{A}_{I}^{H}\mathbf{w}$
$\displaystyle=\frac{L}{\sigma^{2}}\mathbf{w}^{H}\mathbf{A}_{I}(\mathbf{T}^{-1})^{H}\\!\\!\left[\mathbf{I}\\!+\\!\bm{\Gamma}\\!-\\!\frac{{L\beta}\textsf{SNR}\cdot\frac{\sigma^{2}}{L}\boldsymbol{\psi}_{T}\boldsymbol{\psi}_{T}^{H}}{{L\beta}(1\\!\\!-\\!\\!\rho_{0})\textsf{SNR}\\!\\!+\\!\\!N}\right]\\!\\!\mathbf{T}^{-1}\mathbf{A}_{I}^{H}\mathbf{w}$
$\displaystyle\approx\left[\frac{N{L}/{\sigma^{2}}}{{L\beta}\textsf{SNR}+N}\right]^{2}\left[\psi_{I}(\lambda_{\mathrm{max}})\\!-\\!\frac{{L\beta}\textsf{SNR}}{{L\beta}\textsf{SNR}+N}\psi_{S}^{2}(\lambda_{\mathrm{max}})\right],$
(64)

where $\psi_{I}(\lambda_{\mathrm{max}})$ is also given by (39) in Lemma 3, and
$\rho_{0}\\!\\!\ll\\!\\!1$ is a small number independent of SNR.

Finally, substituting (63) and (64) into (62), we can derive
$\textsf{G}(\textsf{SNR})$ in (38) immediately.

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