# Relaxed Zero-Forcing Beamformer under Temporally-Correlated Interference

Takehiro Konoa, Masahiro Yukawaa∗ Tomasz Piotrowskib

###### Abstract

The relaxed zero-forcing (RZF) beamformer is a quadratically-and-linearly
constrained minimum variance beamformer. The central question addressed in
this paper is whether RZF performs better than the widely-used minimum
variance distortionless response and zero-forcing beamformers under
temporally-correlated interference. First, RZF is rederived by imposing an
ellipsoidal constraint that bounds the amount of interference leakage for
mitigating the intrinsic gap between the output variance and the mean squared
error (MSE) which stems from the temporal correlations. Second, an analysis of
RZF is presented for the single-interference case, showing how the MSE is
affected by the spatio-temporal correlations between the desired and
interfering sources as well as by the signal and noise powers. Third,
numerical studies are presented for the multiple-interference case, showing
the remarkable advantages of RZF in its basic performance as well as in its
application to brain activity reconstruction from EEG data. The analytical and
experimental results clarify that the RZF beamformer gives near-optimal
performance in some situations.

###### keywords:

beamforming, temporal correlation, linear and quadratic constraints, uplink
communication, electroencephalography

††journal: Signal Processingmytitlenotemytitlenotefootnotetext: This work was
partially supported by JSPS Grants-in-Aid 18H01446. A preliminary version of
this paper was presented at 44th IEEE International Conference on Acoustics,
Speech, and Signal Processing, Brighton, UK, May 2019. The first and second
authors have an equal contribution.

organization=Department of Electronics and Electrical Engineering, Keio
University,addressline=Hiyoshi 3-14-1, Kohoku-ku, city=Yokohama,
postcode=223-8522, country=JAPAN

organization=Interdisciplinary Center for Modern Technologies and the Faculty
of Physics, Astronomy and Informatics, Nicolaus Copernicus
University,city=Torun, country=POLAND

##  1 Introduction

Beamforming is an attractive technology to filter out those interfering (or
jamming) signals which are spatially apart from the signal of interest. It has
a long history (see e.g., [1]) and has been applied to a wide range of
applications including radar, sonar, wireless communications, acoustics,
electroencephalography (EEG), among many others. For the upcoming fifth
generation (5G) wireless network and Internet of things (IoT) technologies,
for instance, beamforming is considered as a key component to revolutionize
connectivity as well as massive MIMO (multiple input multiple output) and
millimeter wave [2, 3, 4]. The minimum-variance distortionless response (MVDR)
beamformer minimizes the variance of the beamformer output under a linear
constraint to preserve the desired signal undistorted [5]. The MVDR beamformer
is known to achieve the optimal performance among all linear beamformers in
the absence of correlations among sources. In many applications, however, the
sources have temporal correlation, which makes MVDR no longer optimal [6].
Throughout the paper, unless otherwise stated explicitly, the term
“correlation” means a temporal correlation between the desired and interfering
sources. The zero-forcing (ZF) beamformer (aka. the nulling beamformer) [7, 6,
8, 9] minimizes the output variance under multiple linear constraints to
preserve the desired signal undistorted and, at the same time, to nullify the
interfering signals. Although it could attain better performance than MVDR in
the correlated-source settings with high signal to noise ratio (SNR), its
performance becomes even worse than MVDR in low SNR cases as it amplifies the
noise [8]. The MVDR and ZF beamformers are particular examples of the linearly
constrained minimum variance (LCMV) beamformer [10, 5], and can be implemented
by efficient adaptive algorithms such as the constrained normalized least mean
square (CNLMS) algorithm [11].

The studies of the temporal correlation between the desired and interfering
signals date back at least to 1970’s [12] in the communication area. The major
causes of the correlation are multipath propagation which may also happen in
other applications such as acoustics) and/or “smart” jammers which aim to
transmit antiphase replicas of the desired signal for jamming the
transmission. In the worst case, the desired and interfering signals would be
coherent; i.e., one source signal is an amplitude-scaled phase-shifted version
of another one, or in other words the correlation coefficient has a unit
amplitude. To alleviate this problem, spatial smoothing and its improved
versions have been proposed, and the impacts of the correlation coefficient on
the performance have been investigated in the literature (see [1, Section
6.12] and the references therein). This technique, however, has several
limitations. For instance, it is only applicable to a regular array geometry
such as uniform linear arrays, and/or it requires twice as many sensors as
signal sources. An efficient beamformer design that can manage the correlated
interference would be beneficial in the future wireless network systems to
enhance the connectivity.

Another primary application of beamformers is in inverse problems in EEG and
magnetoencephalography (MEG), which are imaging modalities that provide a
direct measure of the brain activity by measuring the voltage with a set of
sensors placed at various locations on the scalp of the subject. The EEG
inverse problem aims to localize and reconstruct the sources of brain
electrical activity from the EEG measurements. It is widely known that
neighboring brain activities (sources) have strong correlations typically, and
the signal correlation still remains a challenging issue to be addressed in
the brain signal processing area. Focusing on EEG, this is especially true for
applications requiring online processing of beamformer output such as brain-
computer interface (BCI) [13, 14] and source-space EEG neurofeedback, which is
currently another hot topic in brain signal processing [15, 16]. To date, the
dominant beamforming technique used in these fields has been the MVDR
beamformer, see, e.g., [17, 18, 19]. Therefore, an introduction of a
beamformer design which suppresses efficiently interfering activity, includes
as special cases the MVDR and ZF beamformers, and is amenable to efficient
adaptive implementation, should bring significant benefits to BCI and source-
space EEG neurofeedback.

Let us now turn our attention to the relaxed zero forcing (RZF) beamformer
[20, 21]. Despite its close link to the ZF beamformer, RZF has solely been
studied under the assumption that the source signals are mutually uncorrelated
in time domain. In this case, MVDR is the optimal linear beamformer, and hence
no other linear beamformer can outperform it.111 For clarification, the term
“linear” here means that the beamformer output is linear in terms of the
received vector (the input to the beamformer). RZF is a “linear beamformer”
involving a “nonlinear (quadratic) constraint” in its formulation. In fact,
the key point of the previous works of RZF lies in its adaptive
implementation. The central theme addressed therein is an efficient use of the
side information about the interference channel for enhancing the convergence
speed of adaptive algorithms. The side information is accommodated in the RZF
formulation as an ellipsoidal constraint that bounds the contribution of
interference in the output using the interference channel information. The RZF
beamformer contains MVDR and ZF as its two extremes: MVDR does not consider
the correlation explicitly, while ZF annihilates the interfering signals
completely. The RZF beamformer belongs to the class of linearly and
quadratically constrained minimum variance (LQCMV) beamformer due to the
presence of the quadratic constraint as well as the typical distortionless
constraint in its formulation. The classical adaptive algorithms such as CNLMS
or its projective extension using the projection onto a nonlinear constraint
set cannot be applied to RZF since the projection onto the ellipsoidal
constraint set admits no closed-form expression. Fortunately, however, it can
be implemented efficiently by the multi-domain adaptive learning method [22],
which implements the ellipsoidal constraint via the projection onto a closed
ball in a transform domain. The multi-domain method applied to RZF is referred
to as the dual-domain adaptive algorithm (DDAA) [22, 21], in which the side
information plays a role of guiding the update direction of adaptive
beamformer towards the optimal one and thereby yielding remarkably fast
convergence. As mentioned already, RZF has not yet been studied under the
correlated interference which permits some other linear beamformer to be
superior to MVDR. For clarification, the superiority is discussed here in
terms of the mean squared error (MSE) performance in the batch setting, not in
terms of convergence speed of adaptive algorithms. Our central research
question is whether the RZF beamformer can be superior to MVDR and ZF, and, if
the answer is positive, under what conditions it may happen.

The major contributions of the present paper are summarized below.

  1. 1.

A rederivation of the RZF beamformer is presented based on a decomposed form
of the MSE function in the presence of temporally correlated interference.

  2. 2.

A theoretical analysis of the RZF beamformer is presented for the tractable
case of a single interfering signal that is correlated with the desired
signal. Here, the analysis relies on the isomorphism between
${\mathbb{C}}^{2}$ and the two dimensional subspace spanned by the channel
vectors of the desired and interfering sources under some mapping preserving
the norms and inner products.

  3. 3.

The performance of the RZF beamformer is studied through extensive simulations
for the general case of multiple correlated interfering signals.

  4. 4.

Consequently, the RZF beamformer is shown to outperform the MVDR and ZF
beamformers, in the batch setting, in the presence of correlated interference
also in an EEG setup, where the beamformers are used for online reconstruction
of brain bioelectrical activity from realistically simulated EEG measurements.
It should be emphasized that this is a remarkable difference from the previous
studies [20, 21] for uncorrelated interference, which only allows RZF to
outperform MVDR in terms of the convergence speed of adaptive algorithms.

The MSE function can actually be decomposed into the following three terms:
(i) the output variance, (ii) the desired-signal-power dependent term, and
(iii) the interaction with interference (which will be referred to as the
interference term). The interference term here depends on the correlation
between the desired and interfering signals (as well as the beamformer outputs
of the interference channel vectors). No prior knowledge is assumed available
about the correlation, and the primitive idea is mitigating (to some
reasonable extent), but not completely annihilating, the interference term.
Although the interference term cannot be suppressed directly, it is mitigated
indirectly by a certain quadratic constraint that bounds the total power of
the beamformer outputs for the interference channel vectors. The resultant
beamformer coincides with the RZF beamformer, although its motivation as well
as the underlying assumption is totally different from the original work [20,
21]. The MSE of RZF for the single-interference case is analyzed, and its
explicit formula is derived in terms of the spatio-temporal correlations
between the desired and interference sources as well as the variances of noise
and interference. The analysis reveals that RZF achieves the minimum MSE
(MMSE) under certain conditions, and it also clarifies under what conditions
RZF outperforms MVDR and ZF, respectively. Numerical examples show that the
RZF beamformer achieves significant gains over the whole range of SNR and that
its performance is fairly close to the theoretical bound for high SNRs, and it
significantly outperforms the MVDR and ZF beamformers. To examine the
possibility of exploiting the temporal correlations to enhance the performance
further, we also investigate the approximate minimum MSE (A-MMSE) beamformer
relying on the availability of some erroneous estimates of the correlations
between the desired and interfering sources as well as the availability of an
estimate of the signal power. It turns out that A-MMSE is sensitive to the
errors in those estimates, while RZF is fairly insensitive to the choice of
the power-bounding parameter.

We emphasize that the present study is the first work that investigates the
RZF beamformer in the correlated-interference scenario, demonstrating its
efficacy in managing such interference. Compared to the preliminary version
[23], the new features of this paper include (i) treatment of complex signals,
(ii) comprehensive treatment of beamforming problems (rather than focusing on
the specific EEG application), (iii) theoretical studies of RZF (Section 4),
and (iv) extensive simulation studies including an EEG application, together
with considerations of the A-MMSE beamformer (Section 5).

##  2 System Model and Assumptions

Throughout, $\mathbb{R}$, ${\mathbb{C}}$, and $\mathbb{N}$ denote the sets of
real numbers, complex numbers, and nonnegative integers, respectively. Given a
matrix ${\boldsymbol{A}}$, ${\boldsymbol{A}}^{\sf T}$ and
${\boldsymbol{A}}^{\sf{H}}$ denote its transpose and Hermitian transpose,
respectively, and $\sigma_{{\rm max}}({\boldsymbol{A}})$ denotes its largest
singular value. The identity matrix is denoted by ${\boldsymbol{I}}$, and the
zero vector by ${\boldsymbol{0}}$. Given vectors
${\boldsymbol{x}},{\boldsymbol{y}}\in{\mathbb{C}}^{m}$ of arbitrary dimension
$m\in\mathbb{N}^{*}:=\mathbb{N}\setminus\\{0\\}$, define the inner product by
$\langle{{\boldsymbol{x}}},{\boldsymbol{y}}\rangle:={\boldsymbol{x}}^{\sf
H}{\boldsymbol{y}}$, and the induced norm by
$\|{\boldsymbol{x}}\|:=\langle{\boldsymbol{x}},{\boldsymbol{x}}\rangle^{\frac{1}{2}}$.
The expectation is denoted by $E(\cdot)$.

We consider the situation when $J+1$ spatially-separated sources transmit
signals towards a uniform linear array222 Different types of antenna arrays
have also been studied such as nonuniform linear arrays, nested array, coprime
arrays, or the maximum inter-element spacing constraint (MISC) arrays (see,
e.g., [24, 25, 26, 27, 28, 29]). An extension to those different arrays is
beyond the scope of the current work. with $N$ antenna sensors. The received
signal at sensors at time instant $k\in\mathbb{N}$ is modeled as

$\displaystyle{\boldsymbol{y}}(k)=\sum_{j=0}^{J}s_{j}(k){\boldsymbol{h}}_{j}+{\boldsymbol{n}}(k)\in{\mathbb{C}}^{N}.$
(1)

Here, ${\boldsymbol{h}}_{j}:=[h_{j,1},h_{j,2},\cdots,h_{j,N}]^{\sf
T}\in{\mathbb{C}}^{N}$ is the channel vector of the $j$th source such that
$\left\|{\boldsymbol{h}}_{j}\right\|=1$, $s_{j}(k)\in{\mathbb{C}}$ the $j$th
transmit signal, and
${\mathbb{C}}^{N}\ni{\boldsymbol{n}}(k):=[n_{1}(k),n_{2}(k),\cdots,n_{N}(k)]^{\sf
T}\sim\mathcal{CN}({\boldsymbol{0}},\sigma_{n}^{2}{\boldsymbol{I}})$ the noise
vector. Without loss of generality, $s_{0}(k)$ is supposed to be the desired
signal, and all the others are interference. The goal of beamforming is to
reconstruct the desired signal $s_{0}(k)$ from the measurements
${\boldsymbol{y}}(k)$ with the channel information. The assumptions that will
be used in the current study are listed below.

###### Assumption 1

  1. 1.

All the signals $s_{j}(k)$ and noise are zero-mean weakly-stationary
stochastic processes.

  2. 2.

The interfering signals $s_{j}(k)$, $j=1,2,\cdots,J$, are correlated with the
desired signal $s_{0}(k)$, i.e.,

$c_{j}:=E[s_{0}^{*}(k)s_{j}(k)]\neq 0,~{}~{}~{}j=1,2,\cdots,J.$ (2)

  3. 3.

The noise is uncorrelated with the signals, i.e.,
$E[s_{j}^{*}(k){\boldsymbol{n}}(k)]={\boldsymbol{0}}$ for $j=0,1,\cdots,J$.

  4. 4.

All the channel vectors ${\boldsymbol{h}}_{j}$ are available.333This is a
realistic assumption in uplink communication in wireless networks and also in
EEG applications.

  5. 5.

The correlations $c_{j}$ and the signal power
$\sigma_{0}^{2}:=E(\left|s_{0}(k)\right|^{2})$ are completely unknown.

We note that the model in (1) actually gives the forward model in the EEG/MEG
inverse problem, which aims to reconstruct the activity of the desired source
$s_{0}(k)$ from the EEG/MEG measurements ${\boldsymbol{y}}(k)$. In this
interpretation of the model in (1), for each source $j$, the channel vector
${\boldsymbol{h}}_{j}:={\boldsymbol{h}}({\boldsymbol{r}}_{j},{\boldsymbol{u}}_{j})\in{\mathbb{R}}^{N}$
corresponds to the so-called leadfield vector depending on the source position
${\boldsymbol{r}}_{j}$ and the orientation unit vector ${\boldsymbol{u}}_{j}$,
$s_{j}(k)\in{\mathbb{R}}$ is the electric/magnetic dipole moment at time
instant $k$, and the noise vector ${\boldsymbol{n}}(k)\in{\mathbb{R}}^{N}$
represents the background brain activity along with the noise recorded at the
sensor array. We will use this fact in Section 5.2, where the performance of
beamformers under consideration will be evaluated in this context.

##  3 Relaxed Zero-forcing Beamformer Rederived to Manage Temporally-
correlated Interference

Under Assumption 1, the MSE function can be written as

$\displaystyle J_{{\rm MSE}}({\boldsymbol{w}}):=$
$\displaystyle~{}E\left[\left|{\boldsymbol{w}}^{\sf
H}{\boldsymbol{y}}(k)-s_{0}(k)\right|^{2}\right]$ $\displaystyle=$
$\displaystyle~{}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{R}}{\boldsymbol{w}}+\sigma_{0}^{2}(1-{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}-{\boldsymbol{h}}_{0}^{\sf
H}{\boldsymbol{w}})-\sum_{j=1}^{J}(c_{j}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}+c_{j}^{*}{\boldsymbol{h}}_{j}^{\sf
H}{\boldsymbol{w}}),~{}{\boldsymbol{w}}\in{\mathbb{C}}^{N},$ (3)

where ${\boldsymbol{R}}:=E[{\boldsymbol{y}}(k){\boldsymbol{y}}(k)^{\sf H}]$.
Under the distortionless constraint

${\boldsymbol{w}}\in
C:=\\{{\boldsymbol{w}}\in{\mathbb{C}}^{N}\mid{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}=1\\},$ (4)

(3) reduces to

$J_{{\rm MSE}}({\boldsymbol{w}})=\underbrace{{\boldsymbol{w}}^{\sf
H}{\boldsymbol{R}}{\boldsymbol{w}}}_{\mbox{output
variance}}-\underbrace{\sigma_{0}^{2}}_{\mbox{signal
power}}-\underbrace{\sum_{j=1}^{J}(c_{j}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}+c_{j}^{*}{\boldsymbol{h}}_{j}^{\sf
H}{\boldsymbol{w}}).}_{\mbox{interaction with interference}}$ (5)

In the absence of correlated sources, the third term of (5) disappears, and
hence MSE (under the distortionless constraint) can be minimized by minimizing
the output variance. This is the reason why the MVDR beamformer, the minimizer
of (5) under (4), is optimal. In the presence of correlated sources, on the
other hand, the third term makes significant differences and MVDR is no longer
optimal. In this case, one may consider to annihilate the third term by
imposing the nulling constraints ${\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}=0$ for all $j=1,2,\cdots,J$. This is the so-called ZF
beamformer. Unfortunately, the ZF beamformer is known to amplify the noise,
and its performance degrades seriously under noisy environments. To circumvent
the noise amplification problem while exploiting the channel information of
the interfering signals, we impose a quadratic constraint, in addition to the
linear constraint (4), that bounds the amount of interference leakage by a
certain threshold $\varepsilon\geq 0$, rather than annihilating those signals
completely. The RZF beamformer is formally given as follows:

$\displaystyle{\rm minimize~{}~{}~{}}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{R}}{\boldsymbol{w}}$ (6a)
$\displaystyle{\rm~{}subject~{}to~{}}\left\\{\begin{array}[]{l}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}=1~{}~{}~{}(\Leftrightarrow{\boldsymbol{w}}\in C),\\\
\|{\boldsymbol{H}}_{\rm I}^{\sf
H}{\boldsymbol{w}}\|^{2}\leq\varepsilon~{}~{}~{}(\Leftrightarrow{\boldsymbol{H}}_{\rm
I}^{\sf H}{\boldsymbol{w}}\in
B_{\varepsilon}:=\\{{\boldsymbol{s}}\in{\mathbb{C}}^{J}\mid\|{\boldsymbol{s}}\|^{{\rm
2}}\leq\varepsilon\\}),\end{array}\right.$ (6d)

where ${\boldsymbol{H}}_{\rm
I}:=[{\boldsymbol{h}}_{1}~{}{\boldsymbol{h}}_{2}\cdots{\boldsymbol{h}}_{J}]$
is the interference channel matrix. Let
${\boldsymbol{R}}_{\varepsilon}:={\boldsymbol{R}}+\lambda_{\varepsilon}{\boldsymbol{H}}_{\rm
I}{\boldsymbol{H}}^{\sf H}_{\rm I}$ with the Lagrange multiplier
$\lambda_{\varepsilon}\geq 0$ satisfying $\left\|{\boldsymbol{H}}^{\sf H}_{\rm
I}{\boldsymbol{w}}_{\rm RZF}\right\|^{2}=\left\|\dfrac{{\boldsymbol{H}}_{\rm
I}^{\sf H}({\boldsymbol{R}}+\lambda_{\varepsilon}{\boldsymbol{H}}_{\rm
I}{\boldsymbol{H}}^{\sf H}_{\rm I})^{{\rm-1}}{{\boldsymbol{h}}}_{{\rm
0}}}{{\boldsymbol{h}}_{{\rm 0}}^{\sf
H}({\boldsymbol{R}}+\lambda_{\varepsilon}{\boldsymbol{H}}_{\rm
I}{\boldsymbol{H}}^{\sf H}_{\rm I})^{{\rm-1}}{\boldsymbol{h}}_{{\rm
0}}}\right\|^{2}=\varepsilon$,
${\boldsymbol{H}}:=[{\boldsymbol{h}}_{0}~{}{\boldsymbol{H}}_{\rm
I}]\in{\mathbb{C}}^{N\times(J+1)}$, and
${\boldsymbol{c}}:=[1,{\boldsymbol{0}}^{\sf T}]^{\sf T}\in\\{0,1\\}^{J+1}$.
The solution to the problem in (6) is then given by [21]

${\boldsymbol{w}}_{\rm
RZF}=\begin{cases}\dfrac{{\boldsymbol{R}}_{\varepsilon}^{{\rm-1}}{{\boldsymbol{h}}}_{{\rm
0}}}{{\boldsymbol{h}}_{{\rm 0}}^{\sf
H}{\boldsymbol{R}}_{\varepsilon}^{{\rm-1}}{\boldsymbol{h}}_{{\rm 0}}},&\mbox{
if }\varepsilon>0,\\\ {\boldsymbol{w}}_{{\rm
ZF}}:={\boldsymbol{R}}^{-1}{\boldsymbol{H}}\left({\boldsymbol{H}}^{\sf
H}{\boldsymbol{R}}^{-1}{\boldsymbol{H}}\right)^{-1}{\boldsymbol{c}},&\mbox{ if
}\varepsilon=0.\end{cases}$ (7)

For the particular choice of $\lambda_{\varepsilon}:=0$, RZF reduces to the
MVDR beamformer ${\boldsymbol{w}}_{\rm
MVDR}:=\dfrac{{\boldsymbol{R}}^{{\rm-1}}{{\boldsymbol{h}}}_{{\rm
0}}}{{\boldsymbol{h}}_{{\rm 0}}^{\sf
H}{\boldsymbol{R}}^{\rm-1}{\boldsymbol{h}}_{{\rm 0}}}$, of which the
corresponding $\varepsilon$ is $\varepsilon_{\rm
MVDR}:=\left\|{\boldsymbol{H}}_{\rm I}^{\sf H}{\boldsymbol{w}}_{\rm
MVDR}\right\|^{2}$. In contrast, as $\lambda_{\varepsilon}\rightarrow+\infty$,
the corresponding $\varepsilon$ vanishes. In fact,
$\lambda_{\varepsilon}\in[0,+\infty)$ is a strictly monotone decreasing
function of $\varepsilon\in(0,\varepsilon_{\rm MVDR}]$, and thus the
correspondence is one to one.

The RZF beamformer attains significantly better performance than MVDR and ZF
and could achieve (near) optimal performance in some cases, as shown in
Sections 4 and 5. For self-containedness, an adaptive implementation of RZF by
the dual-domain adaptive algorithm [22] is given in the appendix.

######  Remark 1 (On Unbiasedness)

Let $\mu_{j}\in{\mathbb{C}}$ denote the mean value of $s_{j}(k)$; i.e.,
$\mu_{j}:=E[s_{j}(k)]$, $j=0,1,\cdots,J$. Although $\mu_{j}:=0$,
$j=0,1,\cdots,J$, is assumed in Assumption 1, we also consider the general
case here to clarify the statistical assumptions required for unbiasedness. We
discuss the unbiasedness of a linear estimator
$\widehat{s}_{0}(k):={\boldsymbol{w}}^{\sf H}{\boldsymbol{y}}(k)$ of a random
variable $s_{0}(k)$ with a beamforming vector ${\boldsymbol{w}}$ subject to
the distortionless constraint ${\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}=1$. This includes the case of the RZF beamformer in
(6). Inserting (1) into $\widehat{s}_{0}(k)={\boldsymbol{w}}^{\sf
H}{\boldsymbol{y}}(k)$ yields

$\widehat{s}_{0}(k)={\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}s_{0}(k)+\sum_{j=1}^{J}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}s_{j}(k)+{\boldsymbol{w}}^{\sf H}{\boldsymbol{n}}(k).$
(8)

Since $s_{0}(k)$ is modeled as a random variable, the unbiasedness condition
of the estimator $\widehat{s}_{0}(k)$ is given by [30]:

$E[\widehat{s}_{0}(k)]=\mu_{0}.$

Due to the zero-mean assumption $E[{\boldsymbol{n}}(k)]={\boldsymbol{0}}$ for
noise as well as the distortionless constraint ${\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}=1$, it follows that

$E[\widehat{s}_{0}(k)]=\mu_{0}+\sum_{j=1}^{J}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}\mu_{j}.$ (9)

This means that $\widehat{s}_{0}(k)$ is an unbiased estimator of $s_{0}(k)$ if
$\sum_{j=1}^{J}{\boldsymbol{w}}^{\sf H}{\boldsymbol{h}}_{j}\mu_{j}=0$. We thus
obtain the following observations.

  1. 1.

Suppose that all interfering signals have zero mean, i.e., if
$\mu_{j}(:=E[s_{j}(k)])$ $=0$ for $j=1,2,\dots,n$. In this case, _any_
estimator of the form $\widehat{s}_{0}(k):={\boldsymbol{w}}^{\sf
H}{\boldsymbol{y}}(k)$ is unbiased under the distortionless constraint. This
applies to the present study, since the conditions are clearly satisfied under
Assumption 1. Note here that the only statistical assumptions required to draw
this conclusion is the zero-mean assumptions of the interfering signals and
the noise vector.

  2. 2.

Suppose that some interfering signal(s) has (have) non-zero mean. In this
case, the unbiasedness condition $\sum_{j=1}^{J}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}\mu_{j}=0$ can be achieved, e.g., by using the zero-
forcing constraints ${\boldsymbol{w}}^{\sf H}{\boldsymbol{h}}_{j}=0$ for
$j=1,\dots,n$, or only for those interfering signals with non-zero means (if
known).

######  Remark 2 (Tips for $\epsilon$-Parameter Design)

Below are some tips on how to choose the $\epsilon$ parameter.

  * 1.

In the high SNR regime, $\epsilon$ should be set to a small value, since the
impact of noise on MSE is small (see (11) in Lemma 1).

  * 2.

When one (or more) of the interfering signals, say the first one, is spatially
correlated to the desired signal, i.e.,
$\left|\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle\right|\approx
1$, $\epsilon$ should be set to a large value. The reason is the following: if
$\epsilon$ is small in such a situation, the beamforming vector
${\boldsymbol{w}}$ needs to be nearly orthogonal to the interference channel
vectors including ${\boldsymbol{h}}_{1}$, i.e.,
$\left\langle{{\boldsymbol{w}}},{{\boldsymbol{h}}_{1}}\right\rangle\approx 0$,
but at the same time it also needs to satisfy
$\left\langle{{\boldsymbol{w}}},{{\boldsymbol{h}}_{0}}\right\rangle=1$, and
those two requirements under
$\left|\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle\right|\approx
1$ make the norm $\left\|{\boldsymbol{w}}\right\|$ unacceptably large and thus
cause serious noise amplification.

Adaptation of $\epsilon$ is left as an interesting open issue.

##  4 Analytical Study of RZF Beamformer for Single Interference Case

After some general discussions about the MSEs of beamformers, we present the
MSE analysis of RZF for the single-interference case. The proofs are finally
given together with some remarks.

###  4.1 General Discussions

It is straightforward to verify the following lemma.

###### Lemma 1

Let $c_{l,j}:=E(s_{l}^{*}(k)s_{j}(k))$, $l,j=1,2,\cdots J$. Then, the MSE
defined in (3) can be rewritten as

$\displaystyle J_{\rm MSE}({\boldsymbol{w}})=$
$\displaystyle~{}\sigma_{0}^{2}\left(1+\left|{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}\right|^{2}-{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}-{\boldsymbol{h}}_{0}^{\sf
H}{\boldsymbol{w}}\right)+\sigma_{n}^{2}\left\|{\boldsymbol{w}}\right\|^{2}$
$\displaystyle+\sum_{j=1}^{J}c_{j}({\boldsymbol{h}}_{0}^{\sf
H}{\boldsymbol{w}}-1){\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}+\sum_{j=1}^{J}c_{j}^{*}({\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}-1){\boldsymbol{h}}_{j}^{\sf H}{\boldsymbol{w}}$
$\displaystyle+\sum_{l=1}^{J}\sum_{j=1}^{J}c_{l,j}{\boldsymbol{h}}_{l}^{\sf
H}{\boldsymbol{w}}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j},~{}~{}~{}{\boldsymbol{w}}\in{\mathbb{C}}^{N}.$ (10)

Under the distortionless constraint (4), MSE reduces to

$J_{\rm
MSE}({\boldsymbol{w}})=\sigma_{n}^{2}\left\|{\boldsymbol{w}}\right\|^{2}+\sum_{l=1}^{J}\sum_{j=1}^{J}c_{l,j}{\boldsymbol{h}}_{l}^{\sf
H}{\boldsymbol{w}}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j},~{}{\boldsymbol{w}}\in C.$ (11)

The minimizer of (11) is given by

${\boldsymbol{w}}_{\rm MMSE\mathchar 45\relax
DR}:=\frac{\widetilde{{\boldsymbol{R}}}^{-1}{\boldsymbol{h}}_{0}}{{\boldsymbol{h}}_{0}^{\sf
H}\widetilde{{\boldsymbol{R}}}^{-1}{\boldsymbol{h}}_{0}},$ (12)

which will be referred to shortly as MMSE-DR (MMSE under distortionless
response constraint), where
$\widetilde{{\boldsymbol{R}}}:=\sigma_{n}^{2}{\boldsymbol{I}}+\sum_{l=1}^{J}\sum_{j=1}^{J}c_{l,j}{\boldsymbol{h}}_{j}{\boldsymbol{h}}_{l}^{\sf
H}$.

In the analysis of the RZF beamformer, the following observation is useful:

${\boldsymbol{w}}_{\rm RZF}\in{\rm
span}\\{{\boldsymbol{h}}_{j}\\}_{j=0}^{J}:=\left\\{\sum_{j=0}^{J}\zeta_{j}{\boldsymbol{h}}_{j}\mid\zeta_{j}\in{\mathbb{C}}\right\\},$
(13)

which can easily be verified by applying the matrix inversion lemma [31] twice
in (7), first to ${\boldsymbol{R}}_{\varepsilon}^{-1}$ and then to
${\boldsymbol{R}}^{-1}$ which appears in the outcome of the first application
of the lemma. In the rest of this section, we restrict our attention to the
tractable case of $J:=1$. In this case, (13) implies that
${\boldsymbol{w}}_{\rm RZF}$ belongs to the two dimensional subspace ${\rm
span}\\{{\boldsymbol{h}}_{0},{\boldsymbol{h}}_{1}\\}$, and hence the problem
reduces completely to the two dimensional case of $N:=2$ by considering the
isomorphism between the two dimensional subspace ${\rm
span}\\{{\boldsymbol{h}}_{0},{\boldsymbol{h}}_{1}\\}$ of ${\mathbb{C}}^{N}$
and ${\mathbb{C}}^{2}$ under some mapping preserving the norms and inner
products. We shall discuss first about the real case, and then move to the
complex case.

###  4.2 Real Case for $J:=1$

We present analyses of RZF for the real-signal case under the presence of
single interference below.

###### Theorem 1

Let $J:=1$,
$\left\|{\boldsymbol{h}}_{0}\right\|=\left\|{\boldsymbol{h}}_{1}\right\|=1$,
$\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle=\sin\tau\in(-1,1)$,
the spatial-separation factor $\tau\in(-\pi/2,\pi/2)$, the powers
$\sigma_{0}^{2}>0$, $\sigma_{1}^{2}>0$, $\sigma_{n}^{2}>0$ of the desired
signal $s_{0}$, the interference $s_{1}$, and noise, respectively, and the
correlation factor
$c_{1}(:=E[s_{0}^{*}(k)s_{1}(k)])\in[-\sigma_{0}\sigma_{1},\sigma_{0}\sigma_{1}]$.444The
correlation coefficient satisfies
$\rho_{1}:=c_{1}/\sigma_{0}\sigma_{1}\in[-1,1]$ in general. Note here that the
trivial case of $\sin\tau=\pm 1$ is excluded, since there is no spatial
separation between ${\boldsymbol{h}}_{0}$ and ${\boldsymbol{h}}_{1}$ in this
case. The MSE of the RZF beamformer is given, as a function of the Lagrange
multiplier $\lambda_{\varepsilon}$, by

${\rm
MSE}(\lambda_{\varepsilon})=\frac{\delta^{2}(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})}{g^{2}(\lambda_{\varepsilon})}-\frac{2\sigma_{n}^{2}\delta\tan\tau}{g(\lambda_{\varepsilon})}+\sigma_{n}^{2}(\tan^{2}\tau+1),$
(14)

where

$\displaystyle\delta:=$ $\displaystyle~{}\sigma_{n}^{2}\tan\tau-
c_{1}\cos\tau,$ (15) $\displaystyle g(\lambda_{\varepsilon}):=$
$\displaystyle~{}\cos^{2}\tau\lambda_{\varepsilon}+\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}>0.$
(16)

  1. 1.

Assume that $\delta=0$. Then, the MSE reduces to
$\sigma_{n}^{2}(\tan^{2}\tau+1)$, which is constant in
$\lambda_{\varepsilon}$.

  2. 2.

Assume that $\delta\neq 0$. If

$\gamma:=\dfrac{\sigma_{n}^{2}\tan\tau}{\delta}\leq 0,$ (17)

${\rm MSE}(\lambda_{\varepsilon})$ is monotonically decreasing, and the ZF
beamformer (corresponding to the limit of
$\lambda_{\varepsilon}\rightarrow+\infty$) is optimal in the MSE sense. If
$\gamma>0$ ($\Rightarrow\sin\tau\neq 0$), ${\rm MSE}(\lambda_{\varepsilon})$
is minimized by

$\lambda_{\varepsilon}:=\begin{cases}-\dfrac{c_{1}(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})}{\sigma_{n}^{2}\sin\tau}>0,&\mbox{if
}\gamma\in(0,1),\\\ 0,&\mbox{if }\gamma\geq 1.\end{cases}$ (18)

Note here by (15) and (17) that
$0<(\gamma^{-1}-1)/\cos^{2}\tau=-c_{1}/(\sigma_{n}^{2}\sin\tau)$ for
$\gamma\in(0,1)$.

######  Theorem 2 (Achieved MSE in real case)

Assume that $\delta(:=\sigma_{n}^{2}\tan\tau-c_{1}\cos\tau)=0$. Then, it holds
that $\mbox{MSE}_{\rm RZF}=\mbox{MSE}_{\rm ZF}=\mbox{MSE}_{\rm MVDR}$ for any
$\lambda_{\varepsilon}\geq 0$. Assume now that $\delta\neq 0$. Then, with the
optimal Lagrange multiplier $\lambda_{\varepsilon}$, it holds that

$\mbox{MSE}_{\rm RZF}=\begin{cases}\mbox{MSE}_{\rm ZF},&\mbox{if
}\gamma\left(:=(\sigma_{n}^{2}\tan\tau)/\delta\right)\leq 0,\\\
\mbox{MSE}_{\rm MMSE\mathchar 45\relax DR},&\mbox{if }\gamma\in(0,1),\\\
\mbox{MSE}_{\rm MVDR},&\mbox{if }\gamma\geq 1,\end{cases}$ (19)

where

$\displaystyle\mbox{MSE}_{\rm MVDR}=$
$\displaystyle~{}\frac{\sigma_{n}^{2}(\sigma_{1}^{2}+\sigma_{n}^{2})+\left|c_{1}\right|^{2}\cos^{2}\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}},$
(20) $\displaystyle\mbox{MSE}_{\rm ZF}=$
$\displaystyle~{}\sigma_{n}^{2}(\tan^{2}\tau+1),$ (21)
$\displaystyle\mbox{MSE}_{\rm MMSE\mathchar 45\relax DR}=$
$\displaystyle~{}\frac{\sigma_{n}^{2}(\sigma_{1}^{2}+\sigma_{n}^{2})}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}$
(22) $\displaystyle=$ $\displaystyle~{}\mbox{MSE}_{\rm
MVDR}-\frac{\left|c_{1}\right|^{2}\cos^{2}\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}$
(23) $\displaystyle=$ $\displaystyle~{}\mbox{MSE}_{\rm
ZF}-\frac{\sigma_{n}^{4}\tan^{2}\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}.$
(24)

The expression in (20) implies that, when $c_{1}\neq 0$ (in the presence of
temporal correlations between the desired and interfering sources), the MSE of
MVDR does not decrease monotonically in general and may even increase as the
noise power $\sigma_{n}^{2}$ decreases. This is also true for the case of many
users as shown in Section 5.

###  4.3 Complex Case for $J:=1$

In an analogous way to the real case, the following theorems can be verified.

###### Theorem 3

Let $J:=1$,
$\left\|{\boldsymbol{h}}_{0}\right\|=\left\|{\boldsymbol{h}}_{1}\right\|=1$,
$\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle=\sin\tau
e^{i\phi_{z}}$, the spatial-separation factor $\tau\in[0,\pi/2)$,
$\phi_{z}\in[0,2\pi)$, $\sigma_{0}^{2}>0$, $\sigma_{1}^{2}>0$,
$\sigma_{n}^{2}>0$, and the correlation factor
$c_{1}:=\left|c_{1}\right|e^{i\phi_{c}}\in\\{c\in{\mathbb{C}}\mid\left|c\right|\leq\sigma_{0}\sigma_{1}\\}$,
$\phi_{c}\in[0,2\pi)$.555The range $[0,\pi/2)$ of $\tau$ is a half of the
range $(-\pi/2,\pi/2)$ in Theorem 1 as the other half $(-\pi/2,0)$ is covered
by letting $\phi_{z}:=\pi$. Here, the trivial case of $\sin\tau=1$ is excluded
as in Theorem 1. Then, the MSE is given, as a function of the Lagrange
multiplier $\lambda_{\varepsilon}$, by

${\rm
MSE}(\lambda_{\varepsilon})=\frac{\left|\delta_{2}\right|^{2}(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})}{g^{2}(\lambda_{\varepsilon})}-\frac{2\sigma_{n}^{2}\delta_{1}\tan\tau}{g(\lambda_{\varepsilon})}+\sigma_{n}^{2}(\tan^{2}\tau+1).$
(25)

Here, $g(\lambda_{\varepsilon})$ is defined in (16) and

$\displaystyle\delta_{1}:=$
$\displaystyle~{}\sigma_{n}^{2}\tan\tau-\left|c_{1}\right|\cos\tau\cos(\phi_{c}+\phi_{z}),$
(26) $\displaystyle\delta_{2}:=$
$\displaystyle~{}\sigma_{n}^{2}\tan\tau-\left|c_{1}\right|\cos\tau\exp[i(\phi_{c}+\phi_{z})].$
(27)

  1. 1.

Assume that $\delta_{2}=0$ ($\Rightarrow\delta_{1}=0$). Then, the MSE reduces
to $\sigma_{n}^{2}(\tan^{2}\tau+1)$, which is constant in
$\lambda_{\varepsilon}$.

  2. 2.

Assume that $\delta_{2}\neq 0$.

    1. (a)

If

$\gamma:=\dfrac{\delta_{1}\sigma_{n}^{2}\tan\tau}{\left|\delta_{2}\right|^{2}}\leq
0~{}(\Leftrightarrow\delta_{1}\tan\tau\leq 0),$ (28)

${\rm MSE}(\lambda_{\varepsilon})$ is monotonically decreasing, and the ZF
beamformer is optimal in the MSE sense.

    2. (b)

If $\gamma\geq 1$, ${\rm MSE}(\lambda_{\varepsilon})$ is monotonically
increasing, and thus is minimized by $\lambda_{\varepsilon}:=0$.

    3. (c)

If $\gamma\in(0,1)$, ${\rm MSE}(\lambda_{\varepsilon})$ is minimized by

$\lambda_{\varepsilon}:=\dfrac{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}{\cos^{2}\tau}\left(\frac{1-\gamma}{\gamma}\right)>0.$
(29)

######  Theorem 4 (Superiority of RZF)

Under the same settings as in Theorem 3, the RZF beamformer with the optimal
Lagrange multiplier $\lambda_{\varepsilon}$ gives a strictly smaller MSE than
the MVDR and ZF beamformers (and achieves the minimum MSE under the
distortionless constraint) if and only if
$0<\delta_{1}\tan\tau<\left|\delta_{2}\right|^{2}$. Moreover, the same is true
if (but not only if) $c_{1}\neq 0$ and $\sin\tau\cos(\phi_{c}+\phi_{z})<0$.

######  Theorem 5 (Achieved MSE in complex case)

Assume the same settings as in Theorem 3. If
$\delta_{2}(:=\sigma_{n}^{2}\tan\tau-\left|c_{1}\right|\cos\tau\exp[i(\phi_{c}+\phi_{z})])=0$,
it holds that $\mbox{MSE}_{\rm RZF}=\mbox{MSE}_{\rm ZF}=\mbox{MSE}_{\rm MVDR}$
for any $\lambda_{\varepsilon}\geq 0$. Assume that $\delta_{2}\neq 0$. Then,
with the optimal Lagrange multiplier $\lambda_{\varepsilon}$, it holds that

$\mbox{MSE}_{\rm RZF}=\begin{cases}\mbox{MSE}_{\rm ZF},&\mbox{if
}\gamma\left(:=\dfrac{\delta_{1}\sigma_{n}^{2}\tan\tau}{\left|\delta_{2}\right|^{2}}\right)\leq
0,\\\ \mbox{MSE}_{\rm MMSE\mathchar 45\relax
DR}+\left(1-\left|\dfrac{\delta_{1}}{\delta_{2}}\right|^{2}\right)\dfrac{\sigma_{n}^{4}\tan^{2}\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}},&\mbox{if
}\gamma\in(0,1),\\\ \mbox{MSE}_{\rm MVDR},&\mbox{if }\gamma\geq 1,\end{cases}$
(30)

where the MSEs of the MVDR, ZF, and MMSE-DR beamformers are given by the same
expressions as in Theorem 2.

######  Remark 3 (On Theorem 5: single-interference case)

  1. 1.

Suppose that $\delta_{1}\neq 0$ ($\Rightarrow\delta_{2}\neq 0$),
$\tau\in(0,\pi/2)$, $\sigma_{n}^{2}>0$, and $c_{1}\neq 0$. Then, it can be
shown that MSE${}_{\rm RZF}=$ MSEMMSE-DR (i.e., RZF is optimal) if and only if
$\cos(\phi_{c}+\phi_{z})=-1$ ($\Leftrightarrow\delta_{1}=\delta_{2}$ and
$\gamma\in(0,1)$).

  2. 2.

The ratio between ZF and MMSE-DR is given by

$\frac{\mbox{MSE}_{\rm MMSE\mathchar 45\relax DR}}{\mbox{MSE}_{\rm
ZF}}=1-\frac{1}{[1+(\sigma_{1}^{2}/\sigma_{n}^{2})\cos^{2}\tau](1+\tan^{-2}\tau)},$

which approaches zero as $\cos^{2}\tau\rightarrow 0$.

  3. 3.

The ratio between MVDR and MMSE-DR is given by

$\frac{\mbox{MSE}_{\rm MMSE\mathchar 45\relax DR}}{\mbox{MSE}_{\rm
MVDR}}=\frac{1}{1+\dfrac{\left|c_{1}\right|^{2}\cos^{2}\tau}{\sigma_{n}^{2}(\sigma_{1}^{2}+\sigma_{n}^{2})}}\in\Bigg{(}\frac{\sigma_{n}^{2}}{\sigma_{n}^{2}+1},1\Bigg{]},$

where the lower bound corresponds to the extreme case of
$\left|c_{1}\right|^{2}=\sigma_{1}^{2}$ and $\cos^{2}\tau=1$ in the limit of
$\sigma_{1}^{2}\rightarrow\infty$, and this lower bound approaches zero as
$\sigma_{n}^{2}\rightarrow 0$. The ratio increases as
$(\sigma_{1}^{2}/\sigma_{n}^{2})\left|c_{1}/\sigma_{1}\right|^{2}\cos^{2}\tau$
increases.

  4. 4.

The MSE of MVDR increases as the amplitude $\left|c_{1}\right|$ of the
correlation does, while those of MMSE-DR and ZF are independent of the
correlation. The performance of RZF depends on $\left|c_{1}\right|$ in general
but is less sensitive than that of MVDR, as shown in Section 5.

  5. 5.

It holds that $\mbox{MSE}_{\rm RZF}=\mbox{MSE}_{\rm MVDR}$ when $c_{1}=0$
(there is no correlation between the desired and the interfering signals),
$\sigma_{n}^{2}\rightarrow+\infty$ (the noise power is huge), or
$\cos^{2}\tau=0$ (${\boldsymbol{h}}_{0}$ and ${\boldsymbol{h}}_{1}$ are
completely aligned). This is consistent with Theorems 1 and 3. In sharp
contrast, $\mbox{MSE}_{\rm RZF}=\mbox{MSE}_{\rm ZF}$ when $\sigma_{n}^{2}=0$
(the signal is noise free), $\sin^{2}\tau=0$ (${\boldsymbol{h}}_{1}$ is
orthogonal to ${\boldsymbol{h}}_{0}$), or $\sigma_{1}^{2}\rightarrow+\infty$
(the interference power is huge).

###  4.4 Proofs

The proofs of Theorems 1 – 5 are given below.

####  4.4.1 Proof of Theorem 1

In the case of $J=1$, (11) reduces to

$J_{\rm
MSE}({\boldsymbol{w}})=\sigma_{n}^{2}\left\|{\boldsymbol{w}}\right\|^{2}+\sigma_{1}^{2}|{\boldsymbol{h}}_{1}^{\sf
H}{\boldsymbol{w}}|^{2},~{}{\boldsymbol{w}}\in C.$ (31)

Now, applying the matrix inversion lemma to (7), one can easily verify that
$\left\|{\boldsymbol{w}}_{\rm RZF}\right\|^{2}$ and
$\left|\left\langle{{\boldsymbol{h}}_{1}},{{\boldsymbol{w}}_{\rm
RZF}}\right\rangle\right|^{2}$ depend only on the norms
($\left\|{\boldsymbol{h}}_{0}\right\|$ and
$\left\|{\boldsymbol{h}}_{1}\right\|$) and the inner product
$\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle$. This
simple observation implies that the MSE of the RZF beamformer in the theorem
can be expressed analytically via analyzing the two-dimensional case666The
dimensionality reduction can also be understood through the isomorphism
between ${\rm
span}\\{{\boldsymbol{h}}_{0},{\boldsymbol{h}}_{1}\\}\subset{\mathbb{C}}^{N}$
and ${\mathbb{C}}^{2}$. Indeed, the pair
$({\boldsymbol{h}}_{0},{\boldsymbol{h}}_{1})$ of vectors in Theorem 1 has the
same geometric structure as in Lemma 2. with such unit vectors
${\boldsymbol{h}}_{0}\in{\mathbb{C}}^{2}$ and
${\boldsymbol{h}}_{1}\in{\mathbb{C}}^{2}$ that satisfy
$\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{1}}\right\rangle=\sin\tau$.
The proof is thus completed by proving the following lemma.

###### Lemma 2

Let $N:=2$, $J:=1$, ${\boldsymbol{h}}_{0}:=[0,1]^{\sf T}$,
${\boldsymbol{h}}_{1}:=[\cos\tau,\sin\tau]^{\sf T}$, $\tau\in(-\pi/2,\pi/2)$,
$\sigma_{0}^{2}>0$, $\sigma_{1}^{2}>0$, $\sigma_{n}^{2}>0$, and
$c_{1}\in[-\sigma_{0}\sigma_{1},\sigma_{0}\sigma_{1}]$. The RZF beamformer is
then given by

${\boldsymbol{w}}_{\rm
RZF}=\left[\begin{array}[]{c}-\dfrac{\cos\tau\left[(\sigma_{1}^{2}+\lambda_{\varepsilon})\sin\tau+c_{1}\right]}{(\sigma_{1}^{2}+\lambda_{\varepsilon})\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right],$ (32)

for which the MSE is given by (14).

  1. (i)

Assume that $\delta=0$. Then, MSE is given by
$\sigma_{n}^{2}(\tan^{2}\tau+1)$, which is constant in
$\lambda_{\varepsilon}$.

  2. (ii)

Assume that $\delta\neq 0$. If
$\gamma:=\dfrac{\sigma_{n}^{2}\tan\tau}{\delta}\leq 0$, ${\rm
MSE}(\lambda_{\varepsilon})$ is monotonically decreasing, and the ZF
beamformer (corresponding to the limit of
$\lambda_{\varepsilon}\rightarrow+\infty$) is optimal in the MSE sense. If
$\gamma>0$ ($\Rightarrow\sin\tau\neq 0$), ${\rm MSE}(\lambda_{\varepsilon})$
is minimized by (18). Note here that
$0<(\delta/(\sigma_{n}^{2}\tan\tau)-1)/\cos^{2}\tau=-c_{1}/(\sigma_{n}^{2}\sin\tau)$.

Proof: For the sake of simple notation, we let $c:=\cos\tau$ and
$s:=\sin\tau$. By
${\boldsymbol{R}}=\sigma_{0}^{2}{\boldsymbol{h}}_{0}{\boldsymbol{h}}_{0}^{\sf
T}+\sigma_{1}^{2}{\boldsymbol{h}}_{1}{\boldsymbol{h}}_{1}^{\sf
T}+c_{1}({\boldsymbol{h}}_{0}{\boldsymbol{h}}_{1}^{\sf
T}+{\boldsymbol{h}}_{1}{\boldsymbol{h}}_{0}^{\sf
T})+\sigma_{n}^{2}{\boldsymbol{I}}$, we can verify that

${\boldsymbol{R}}_{\varepsilon}=\left[\begin{array}[]{cc}(\sigma_{1}^{2}+\lambda_{\varepsilon})c^{2}+\sigma_{n}^{2}&(\sigma_{1}^{2}+\lambda_{\varepsilon})cs+c_{1}c\\\
(\sigma_{1}^{2}+\lambda_{\varepsilon})cs+c_{1}c&~{}(\sigma_{1}^{2}+\lambda_{\varepsilon})s^{2}+2c_{1}s+\sigma_{n}^{2}+\sigma_{0}^{2}\\\
\end{array}\right],$ (33)

from which it follows that

${\boldsymbol{R}}_{\varepsilon}^{-1}{\boldsymbol{h}}_{0}\sim\left[\begin{array}[]{c}-c\left[(\sigma_{1}^{2}+\lambda_{\varepsilon})s+c_{1}\right]\\\
~{}(\sigma_{1}^{2}+\lambda_{\varepsilon})c^{2}+\sigma_{n}^{2}\\\
\end{array}\right].$ (34)

Hence, we obtain

${\boldsymbol{w}}_{\rm
RZF}=\dfrac{{\boldsymbol{R}}_{\varepsilon}^{{\rm-1}}{{\boldsymbol{h}}}_{{\rm
0}}}{{\boldsymbol{h}}_{{\rm 0}}^{\sf
T}{\boldsymbol{R}}_{\varepsilon}^{{\rm-1}}{\boldsymbol{h}}_{{\rm
0}}}=\left[\begin{array}[]{c}-\dfrac{c\left[(\sigma_{1}^{2}+\lambda_{\varepsilon})s+c_{1}\right]}{(\sigma_{1}^{2}+\lambda_{\varepsilon})c^{2}+\sigma_{n}^{2}}\\\
1\end{array}\right].$ (35)

Substituting (35) into (31) yields (14).

In the rest of the proof, we solely consider the case of $\delta\neq 0$, as
the case of $\delta=0$ is clear. In this case, define
$x:=1/g(\lambda_{\varepsilon})\in(0,1/(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})]$
for $\lambda_{\varepsilon}\in[0,+\infty)$. Then, MSE can be seen as a
quadratic function of $x$, and its minimum over ${\mathbb{R}}$ is achieved by

$x^{*}:=\frac{1}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\times\frac{\sigma_{n}^{2}\tan\tau}{\underbrace{\sigma_{n}^{2}\tan\tau-c\cos\tau}_{=\delta}}.$
(36)

If $\gamma(:=(\sigma_{n}^{2}\tan\tau)/\delta)\leq 0$, $x^{*}\leq 0$ and hence
MSE is a monotonically increasing function of $x$ within the actual range
$x\in(0,1/(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})]$. If, on the other
hand, $\dfrac{\sigma_{n}^{2}\tan\tau}{\delta}\geq 1$, $x^{*}\geq
1/(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})$ and hence MSE is a
monotonically decreasing function of $x$ within the range. In this case, the
minimum is clearly achieved by
$x:=1/(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})\Leftrightarrow\lambda_{\varepsilon}:=0$.
If $\dfrac{\sigma_{n}^{2}\tan\tau}{\delta}\in(0,1)$, $x^{*}$ lies in the
range, and its corresponding $\lambda_{\varepsilon}$ is given by
$\lambda_{\varepsilon}:=-c_{1}\left[(\sigma_{1}^{2}/\sigma_{n}^{2})\cos^{2}\tau+1\right]/\sin\tau$.
$\Box$

####  4.4.2 Proof of Theorem 2

We first prove the following proposition.

###### Proposition 1

Under the same settings as Lemma 2, the MMSE beamformer under
${\boldsymbol{w}}\in C$ are given respectively by

$\displaystyle{\boldsymbol{w}}_{\rm MMSE\mathchar 45\relax DR}=$
$\displaystyle~{}\left[\begin{array}[]{c}-\dfrac{\sigma_{1}^{2}\sin\tau\cos\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right],$ (39) $\displaystyle{\boldsymbol{w}}_{\rm MVDR}=$
$\displaystyle~{}\left[\begin{array}[]{c}-\dfrac{\cos\tau(\sigma_{1}^{2}\sin\tau+c_{1})}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right]$ (42) $\displaystyle=$
$\displaystyle~{}{\boldsymbol{w}}_{\rm MMSE\mathchar 45\relax
DR}-\left[\begin{array}[]{c}\dfrac{c_{1}\cos\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
0\end{array}\right].$ (45)

Proof: Equation (39) can be verified by substituting the following equations
into (12):

$\displaystyle\widetilde{{\boldsymbol{R}}}=$
$\displaystyle~{}{\boldsymbol{Q}}{\boldsymbol{\Lambda}}{\boldsymbol{Q}}^{\sf
T},$ $\displaystyle{\boldsymbol{h}}_{0}=$
$\displaystyle~{}{\boldsymbol{Q}}{\boldsymbol{h}}_{1}^{\perp},$ (46)

where

$\displaystyle{\boldsymbol{Q}}:=$
$\displaystyle~{}\left[{\boldsymbol{h}}_{1}~{}{\boldsymbol{h}}_{1}^{\perp}\right]:=\left[\begin{array}[]{cc}\cos\tau&\sin\tau\\\
\sin\tau&-\cos\tau\end{array}\right],$ (49)
$\displaystyle{\boldsymbol{\Lambda}}:=$
$\displaystyle~{}\left[\begin{array}[]{cc}\sigma_{1}^{2}+\sigma_{n}^{2}&0\\\
0&\sigma_{n}^{2}\end{array}\right].$ (52)

Note here that the matrix ${\boldsymbol{Q}}$ is symmetric; i.e.,
${\boldsymbol{Q}}^{\sf T}={\boldsymbol{Q}}$. Equation (45) can readily be
verified by substituting $\lambda_{\varepsilon}:=0$ into (35). $\Box$

Proof of Theorem 2: Substituting (39) and (45) into (31) yields (20) and (22).
The MSE (21) of ZF is verified by considering the limit of
$\lambda_{\varepsilon}\rightarrow+\infty$ in (14). The rest to prove is (19),
in which the cases of $\gamma\leq 0$ and $\gamma\geq 1$ can be verified
straightforwardly from Theorem 1. The case of $\gamma\in(0,1)$ can be verified
by substituting (15) – (18) into (14).

(a) $c_{1}:=0.99$, $\tau:=\pi/6$ ($\gamma=-$2.0619) (b) $c_{1}:=-0.2$,
$\tau:=\pi/6$ ($\gamma=$0.7692) (c) $c_{1}:=0.1$, $\tau:=\pi/6$
($\gamma=$1.1765)

Figure 1: The $w_{1}$ coordinate for $N:=2$ and $J:=1$ with
${\boldsymbol{h}}_{0}:=[0,1]^{\sf T}$ under
$\sigma_{1}^{2}:=\sigma_{n}^{2}:=1.0$. For RZF,
$\lambda_{\varepsilon}:=10^{\alpha}$, where $\alpha$ changes from $-2$ to $3$
at interval $0.1$.

Figure 2: A possible situation in two-interference case with
${\boldsymbol{h}}_{0}:=[0,0,1]^{\sf T}$ ($N:=3$, $J:=2$): (a) a schematic
illustration and (b) the projections of the beamformers on $C$ onto the
$w_{1}$-$w_{2}$ plane for $\sigma_{1}^{2}:=\sigma_{n}^{2}:=1.0$,
$c_{1}:=c_{2}:=c_{1,2}:=0.6$,
${\boldsymbol{h}}_{1}:=[0,\cos(3\pi/5),\sin(3\pi/5)]$, and
${\boldsymbol{h}}_{2}:=[\cos(5\pi/6),\sin(5\pi/6)\cos(5\pi/6),\sin(5\pi/6)\sin(5\pi/6)]$.
For RZF, $\lambda_{\varepsilon}:=10^{\alpha}$ with $\alpha$ changing from $-2$
to $3$ at interval $0.05$.

###### Remark 4

Figure 1 illustrates the three cases of $\gamma$ in Theorem 1 for $N:=2$: (a)
$\gamma\leq 0$, (b) $\gamma\in(0,1)$, and (c) $\gamma\geq 1$. If $c_{1}:=0$
for $J:=1$, the $w_{1}$ components of the MVDR, ZF, RZF, and MMSE-DR
beamformers have the same sign as $-\sin\tau$; note here that $\cos\tau>0$.
This is clear from the nulling constraint for ZF; see Lemma 2 and Proposition
1 for the other beamformers. If $c_{1}\neq 0$, on the other hand, the MVDR,
RZF, and MMSE-DR beamformers may have the opposite sign from $-\sin\tau$ (see
Figure 1(b)), while only the ZF beamformer sticks to the same sign as
$-\sin\tau$. This means that RZF may have a smaller norm than MVDR (and also
than ZF), implying that it may suppress the noise better than MVDR. This also
applies to the case of $J\geq 2$ as illustrated in Figure 2, in which some of
the RZF beamformers have a smaller norm than MVDR. The MSE expression given in
(11) implies that the MSE (under the distortionless constraint) depends on the
norm $\left\|{\boldsymbol{w}}\right\|^{2}$ of the beamformer and the
interference leakage
$\left\langle{{\boldsymbol{w}}},{{\boldsymbol{h}}_{j}}\right\rangle$. In the
particular case of $J:=1$, MSE is given by the simple sum of
$\sigma_{n}^{2}\left\|{\boldsymbol{w}}\right\|^{2}$ and
$\sigma_{1}^{2}\left|\left\langle{{\boldsymbol{w}}},{{\boldsymbol{h}}_{1}}\right\rangle\right|^{2}$.
To attain a small MSE (under the distortionless constraint),
${\boldsymbol{w}}$ needs to be as close as possible to the origin, and
orthogonal to ${\boldsymbol{h}}_{1}$ as much as possible, simultaneously. The
optimal beamformer has the best balance between those two terms, while ZF
over-weights the second one. Indeed, RZF may attain better noise-and-
interference suppression performance than MVDR for multiple interfering
sources, as shown in Section 5.

####  4.4.3 Proof of Theorem 3

The claim can be verified by proving the following lemma due to the same
arguments as given in the proof of Theorem 1.

###### Lemma 3

Let $N:=2$, $J:=1$, ${\boldsymbol{h}}_{0}:=[0,1]^{\sf T}$,
${\boldsymbol{h}}_{1}:=[\cos\tau,z]^{\sf T}:=[\cos\tau,\sin\tau
e^{i\phi_{z}}]^{\sf T}$ with $\tau\in[0,\pi/2)$ and $\phi_{z}\in[0,2\pi)$,
$\sigma_{0}^{2}>0$, $\sigma_{1}^{2}>0$, $\sigma_{n}^{2}>0$, and
$c_{1}:=\left|c_{1}\right|e^{i\phi_{c}}\in\\{c\in{\mathbb{C}}\mid\left|c\right|\leq\sigma_{0}\sigma_{1}\\}$,
$\phi_{c}\in[0,2\pi)$.777One may consider the more general case of $\cos\tau
e^{i\phi}$ for some $\phi\in[0,2\pi)$, instead of $\cos\tau$ of zero phase.
However, the results in the theorem (excluding the expression of
${\boldsymbol{w}}_{\rm RZF}$) are independent of the choice of $\phi$, and we
can thus assume that $\phi:=0$ without any essential loss of generality. The
RZF beamformer is then given by

${\boldsymbol{w}}_{\rm
RZF}=\left[\begin{array}[]{c}-\dfrac{\cos\tau\left[(\sigma_{1}^{2}+\lambda_{\varepsilon})z^{*}+c_{1}\right]}{(\sigma_{1}^{2}+\lambda_{\varepsilon})\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right],$ (53)

for which the MSE is given by (25).

  1. (i)

Assume that $\delta_{2}=0$ ($\Rightarrow\delta_{1}=0$), where $\delta_{1}$ and
$\delta_{2}$ are defined in (26) and (27), respectively. Then, the MSE is
given by $\sigma_{n}^{2}(\tan^{2}\tau+1)$, which is constant in
$\lambda_{\varepsilon}$.

  2. (ii)

Assume that $\delta_{2}\neq 0$.

    1. (a)

If
$\gamma:=\dfrac{\delta_{1}\sigma_{n}^{2}\tan\tau}{\left|\delta_{2}\right|^{2}}\leq
0$ ($\Leftrightarrow\delta_{1}\tan\tau\leq 0$), ${\rm
MSE}(\lambda_{\varepsilon})$ is monotonically decreasing, and the ZF
beamformer is optimal in the MSE sense.

    2. (b)

If $\gamma\geq 1$, ${\rm MSE}(\lambda_{\varepsilon})$ is monotonically
increasing, and thus is minimized by $\lambda_{\varepsilon}:=0$.

    3. (c)

If $\gamma\in(0,1)$, ${\rm MSE}(\lambda_{\varepsilon})$ is minimized by (29).

Proof: The claim can be verified in an analogous way to the proof of Lemma 2.
$\Box$

####  4.4.4 Proof of Theorem 4

The former part is clear from the proof of Lemma 2; note that the necessary
and sufficient condition $0<\delta_{1}\tan\tau<\left|\delta_{2}\right|^{2}$
implies $\delta_{2}\neq 0$ and
$\dfrac{\delta_{1}\sigma_{n}^{2}\tan\tau}{\left|\delta_{2}\right|^{2}}\in(0,1)$.
To verify the latter part (the sufficient condition), we assume that
$c_{1}\neq 0$ and $\sin\tau\cos(\phi_{c}+\phi_{z})<0$. Then, we immediately
have

$\displaystyle\delta_{1}\tan\tau=$
$\displaystyle~{}(\sigma_{n}^{2}\tan\tau-\left|c_{1}\right|\cos\tau\cos(\phi_{c}+\phi_{z}))\tan\tau$
$\displaystyle=$
$\displaystyle~{}\sigma_{n}^{2}\tan^{2}\tau-\left|c_{1}\right|\sin\tau\cos(\phi_{c}+\phi_{z})>0.$
(54)

Observing that

$\left|\delta_{2}\right|^{2}=\delta_{1}^{2}+(\left|c_{1}\right|\cos\tau\sin(\phi_{c}+\phi_{z}))^{2},$
(55)

we can verify that

$\displaystyle
0<-\left|c_{1}\right|\cos^{2}\tau\frac{\cos(\phi_{c}+\phi_{z})}{\sin\tau}+\frac{(\left|c_{1}\right|\cos\tau\sin(\phi_{c}+\phi_{z}))^{2}}{\delta_{1}\tan\tau}$
$\displaystyle\Leftrightarrow\sigma_{n}^{2}<\frac{\delta_{1}}{\tan\tau}+\frac{(\left|c_{1}\right|\cos\tau\sin(\phi_{c}+\phi_{z}))^{2}}{\delta_{1}\tan\tau}$
$\displaystyle\Leftrightarrow\sigma_{n}^{2}<\frac{\left|\delta_{2}\right|^{2}}{\delta_{1}\tan\tau},$
(56)

from which it follows readily that $\delta_{2}\neq 0$ and
$\dfrac{\delta_{1}\sigma_{n}^{2}\tan\tau}{\left|\delta_{2}\right|^{2}}<1$.

####  4.4.5 Proof of Theorem 5

In an analogous way to the proof of Theorem 2, the claim is verified with the
following proposition.

###### Proposition 2

Under the same settings as in Lemma 3, the MMSE-DR and the MVDR beamformers
are given respectively by

$\displaystyle{\boldsymbol{w}}_{\rm MMSE\mathchar 45\relax DR}=$
$\displaystyle~{}\left[\begin{array}[]{c}-\dfrac{\sigma_{1}^{2}z^{*}\cos\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right],$ (59) $\displaystyle{\boldsymbol{w}}_{\rm MVDR}=$
$\displaystyle~{}\left[\begin{array}[]{c}-\dfrac{\cos\tau(\sigma_{1}^{2}z^{*}+c_{1})}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
1\end{array}\right]$ (62) $\displaystyle=$
$\displaystyle~{}{\boldsymbol{w}}_{\rm MMSE\mathchar 45\relax
DR}-\left[\begin{array}[]{c}\dfrac{c_{1}\cos\tau}{\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2}}\\\
0\end{array}\right].$ (65)

Proof: The claim can be verified in an analogous way to the proof of
Proposition 1. $\Box$

###### Remark 5

In the case of $c_{1}:=0$, the $w_{1}$ components of the MVDR, ZF, RZF, and
MMSE-DR beamformers have the same “phase” as $-\cos\tau z^{*}$ (see Lemma 3
and Proposition 2; cf. Remark 4). If $c_{1}\neq 0$, on the other hand, the
MVDR, RZF, and MMSE-DR beamformers have different phases in general from
$-\cos\tau z^{*}$, while the ZF beamformer sticks to the same phase as
$-\cos\tau z^{*}$. Hence, RZF, which resides between ZF and MVDR, possibly has
a strictly smaller norm than those two ends, as in the real case. This implies
that RZF has a potential to suppress the noise, as well as the interference,
more efficiently than MVDR, because the impact of noise on the MSE is
proportional to $\left\|{\boldsymbol{w}}\right\|^{2}$ (see (11) in Lemma 1).
Compared to ZF, RZF with an appropriate $\lambda_{\varepsilon}$ can achieve a
significantly better noise suppression performance while suppressing the
interference at the same level approximately, as shown in Section 5.

|
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(a) $N=16$, $J=7$ (b) $N=64$, $J=19$

Figure 3: Performances across $\rho_{j}$ for $\beta:=0.8$,
$\varepsilon_{\rho}:=0.1$, and $\varepsilon_{\phi}:=\pi/12$ under SNR 0 dB and
SIR 0 dB.

|
---|---
(a) $N=16$, $J=7$ (b) $N=64$, $J=19$

Figure 4: Sensitivity of RZF to the parameter $\varepsilon$ ($\rho_{j}:=0.6$,
$\beta:=0.8$, $\varepsilon_{\rho}:=0.1$, $\varepsilon_{\phi}:=\pi/12$, SNR 0
dB, SIR 0 dB).

|
---|---

Figure 5: Sensitivity of the A-MMSE beamformer to the errors
$\varepsilon_{\rho}$ and $\beta$ ($\rho_{j}:=0.6$, SNR 0 dB, SIR 0 dB).

##  5 Simulation Study of RZF Beamformer for Multiple Interference Case

The basic performance of the RZF beamformer is studied with toy models, and
its efficacy is then shown in application to brain activity reconstruction
with EEG measurements. We define SNR and the signal to interference ratio
(SIR) as the ratios of the power of desired signal projected onto sensors to
the power of noise and interference counterparts, respectively; i.e., SNR
$:=\sum_{i=1}^{N}E(\left|h_{0,i}s_{0}(k)\right|^{2})/\sum_{i=1}^{N}E(\left|n_{i}(k)\right|^{2})$,
and ${\rm SIR}:=\sum_{i=1}^{N}E(\left|h_{0,i}s_{0}(k)\right|^{2})/$
$\sum_{i=1}^{N}E(|\sum_{j=1}^{J}h_{j,i}s_{j}(k)|^{2})$. The parameter
$\epsilon$ is tuned manually by grid search for each setting in all
simulations but the one which studies the sensitivity of RZF to the value of
$\epsilon$ to show the potential performance of RZF.

###  5.1 Basic Performance with Toy Model

The uniform linear array is considered with the array response
${\boldsymbol{h}}_{j}:={\boldsymbol{h}}(\theta_{j}):=\frac{1}{\sqrt{N}}[1,e^{2\pi
i\frac{d}{\lambda}\cos\theta_{j}},\cdots,e^{2\pi
i(N-1)\frac{d}{\lambda}\cos\theta_{j}}]^{\sf T}\in{\mathbb{C}}^{N}$ at the
receiver, where $\theta_{j}\in[0,\pi]$ is the direction of arrival (DOA) of
the $j$th signal, $\lambda$ is the carrier wavelength, and $d:=\lambda/2$ is
the antenna spacing [1]. Two scenarios are considered regarding the array
size: (a) $N=16$ and $J=7$ (middle array size), and (b) $N=64$ and $J=19$
(large array size). In each setting, we let
$\\{\theta_{j}\\}_{j=0}^{J}=\\{(j+1)\pi/(J+2)\\}_{j=0}^{J}$ with
$\theta_{0}:=\lceil(J+1)/3\rceil\pi/(J+2)$. The desired signal is drawn from
the i.i.d. standard complex Gaussian distribution, i.e.,
$s_{0}(k)\sim\mathcal{CN}(0,1)$. For convenience in controlling the
correlation between $s_{0}(k)$ and $s_{j}(k)$, the interfering signals are
generated as

$s_{j}(k):=\dfrac{\sigma_{j}e^{i\phi_{j}}}{\sqrt{1+\sigma_{v}^{2}}}(s_{0}(k)+v_{j}(k)),$
(66)

where $\sigma_{j}^{2}:=E(\left|s_{j}(k)\right|^{2})$,
$v_{j}(k)\sim\mathcal{CN}(0,\sigma_{v}^{2})$, and the relative phase of the
$j$th source is set to $\phi_{j}:=0$. In this case, the correlation
coefficient between $s_{0}(k)$ and $s_{j}(k)$ is given by

$\rho_{j}:=c_{j}/(\sigma_{0}\sigma_{j})=e^{i\phi_{j}}/\sqrt{1+\sigma_{v}^{2}}\in\\{c\in{\mathbb{C}}\mid\left|c\right|\in(0,1]\\}.$
(67)

The desired-signal power is fixed to $\sigma_{0}^{2}:=1$, and the powers of
the interference and noise are changed according to the SIR and SNR,
respectively.

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(a) $N=16$, $J=7$, SIR $=$ 0 dB (b) $N=64$, $J=19$, , SIR $=$ 0 dB

|
---|---
(c) $N=16$, $J=7$, SIR $=$ 10 dB (d) $N=64$, $J=19$, SIR $=$ 10 dB

Figure 6: Performances across SNR for $\rho_{j}:=0.6$, $\beta:=0.8$,
$\varepsilon_{\rho}:=0.1$, $\varepsilon_{\phi}:=\pi/12$.

|
---|---
(a) $N=16$, $J=7$, SNR $=$ 0 dB (b) $N=64$, $J=19$, SNR $=$ 0 dB

|
---|---
(c) $N=16$, $J=7$, SNR $=$ 10 dB (d) $N=64$, $J=19$, SNR $=$ 10 dB

Figure 7: Performances across SIR for $\rho_{j}:=0.6$, $\beta:=0.8$,
$\varepsilon_{\rho}:=0.1$, $\varepsilon_{\phi}:=\pi/12$.

A fresh look at the MSE expression in (5) may suggest that one might “exploit”
the temporal correlations to reduce the MSE further, rather than bounding the
leakage of interference. To see whether this active approach works, we
consider an approximate MMSE (A-MMSE) beamformer under the assumption that
some erroneous estimates $\hat{c}_{j}\in{\mathbb{C}}$ and
$\hat{\sigma}_{0}^{2}>0$ of the correlation $c_{j}(:=E[s_{0}^{*}(k)s_{j}(k)])$
and the signal power $\sigma_{0}^{2}(:=E[\left|s_{0}(k)\right|^{2}])$ are
available. To be precise, the erroneous estimates $\hat{\sigma}_{0}^{2}$ and
$\hat{c}_{j}=\left|\hat{c}_{j}\right|e^{i\hat{\phi}_{j}}$ are generated,
respectively, by $\hat{\sigma}_{0}^{2}:=\beta\sigma_{0}^{2}$ and
$\left|\hat{c}_{j}\right|:=\left|c_{j}\right|+\varepsilon_{\rho}\sigma_{0}\sigma_{j}$,
and $\hat{\phi}_{j}:=\phi_{j}+\varepsilon_{\phi}$, where
$\beta\in(0,+\infty)$, $\varepsilon_{\rho}\in(-\rho_{j},1-\rho_{j})$, and
$\varepsilon_{\phi}\in(-\pi,\pi]$. In this case, being free from the
distortionless constraint, the MSE $J_{{\rm MSE}}({\boldsymbol{w}})$ in (3)
can be approximated by $\hat{J}_{{\rm
MSE}}({\boldsymbol{w}}):={\boldsymbol{w}}^{\sf
H}{\boldsymbol{R}}{\boldsymbol{w}}+\hat{\sigma}_{0}^{2}(1-{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{0}-{\boldsymbol{h}}_{0}^{\sf
H}{\boldsymbol{w}})-\sum_{j=1}^{J}(\hat{c}_{j}{\boldsymbol{w}}^{\sf
H}{\boldsymbol{h}}_{j}+\hat{c}_{j}^{*}{\boldsymbol{h}}_{j}^{\sf
H}{\boldsymbol{w}})$, of which the minimizer is given by
${\boldsymbol{w}}_{\rm A\mathchar 45\relax
MMSE}:={\boldsymbol{R}}^{-1}\left(\hat{\sigma}_{0}^{2}{\boldsymbol{h}}_{0}+\sum_{j=1}^{J}\hat{c}_{j}{\boldsymbol{h}}_{j}\right)$.
The matrix ${\boldsymbol{R}}$ is computed from 8,000 samples for all the
beamformers.

Figure 3 depicts the performance of the beamformers under different temporal-
correlations for SNR 0 dB and SIR 0 dB. It is seen that RZF exhibits
remarkably better performances than MVDR in the strong correlation cases (cf.
Remark 3.5). See Section 5.3 for discussions about the gap between the MSEs of
RZF and MMSE-DR. Figure 4 shows how the performance of RZF changes according
to the choice of $\varepsilon$ for SNR 0 dB and SIR 0 dB. It is seen that a
wide range of the $\varepsilon$ parameter yields lower MSEs than the
conventional MVDR and ZF beamformers. Figure 5 shows how the performance of
A-MMSE changes due to the estimation errors in $\hat{c}_{j}$ and
$\hat{\sigma}_{0}^{2}$ for $N=16$, SNR 0 dB, and SIR 0 dB. In Figure 5(a),
$\varepsilon_{\phi}:=0$ and $\beta:=1$. In Figure 5(b),
$\varepsilon_{\rho}:=\varepsilon_{\phi}:=0$. In contrast to the case of RZF,
the performance of A-MMSE degrades by slight errors in the estimates. Figures
6 and 7 depict the performances for $\rho_{j}:=0.6$ under different SNR
conditions with SIR 0 dB and 10 dB, and under different SIR conditions with
SNR 0 dB and 10 dB, respectively. The results show that RZF yields near
optimal performances in the case of $N:=16$ (excluding the low-SIR case in
Figure 7(c)), and that it outperforms MVDR and ZF significantly over the whole
settings in the case of $N:=64$. Note here that the performance of RZF for
$N:=64$ is suboptimal because the large $J$ makes the minimal angular
separation between the desired interfering sources be rather small. Comparing
the performances of RZF for SIR $=$ 0 dB and 10 dB in Figure 6, the tendency
is nearly the same; precisely, for high SNR, closer performance to MMSE is
achieved in the case of SIR $=$ 10 dB than the case of SIR $=$ 0 dB. Turning
our attention to Figure 7, we observe that (i) the maximum gain to MVDR is 10
dB approximately for SNR $=$ 0 dB while it is 20 dB approximately for SNR $=$
10 dB, and (ii) MSE of RZF increases towards that of ZF as SIR decreases, more
clearly for SNR $=$ 10 dB. The first observation is due to the fact that RZF
as well as ZF benefits significantly from the reduction of noise. The second
one is also reasonable since the zero-forcing strategy (i.e., $\epsilon:=0$)
will be optimal as the interference power approaches infinity.

###  5.2 EEG Application

Using the interpretation of the model in (1) as the forward model used to
solve the EEG inverse problem (_cf._ Section 2), we consider the situation
when there are $J+1$ dipole sources $(J:=29)$ of brain activity and the
signals are measured by an array of $N:=128$ EEG sensors. HydroCel Geodesic
Sensor Net model is used as a realistic EEG cap layout. The source signals and
forward model were generated using SupFunSim library [32], which utilizes
FieldTrip toolbox [33] for generation of volume conduction model and
leadfields. The activity $s_{0}(k)$ of the desired source is generated by an
autoregressive (AR) model of order $6$, where the coefficients for each order
are set to $0.2$. The interfering activities are generated by (66) with
$\phi_{j}:=0$. The source activities are supposed to change in time, while
their positions and orientations are assumed known and remain the same during
the measurement period. The following two cases are considered: the low
correlation case ($\rho_{j}:=0.5$) and the high correlation case
($\rho_{j}:=0.95$). Throughout the experiments, the power of the desired
signal is fixed to $\sigma_{0}^{2}:=1$, and those of the interference and
noise are changed depending on SNR and SIR, respectively. The relaxation
parameter $\varepsilon$ of RZF is optimized for each SNR and SIR.

|
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(a) $\rho_{j}:=0.5$ (b) $\rho_{j}:=0.95$

Figure 8: Performances for EEG application under low and high correlations
(SIR $0$ dB).

Figure 9: Performance in EEG application under low correlation $\rho_{j}:=0.5$
(SNR $0$ dB).

Figure 8 shows the results for SIR $0$ dB under different SNR conditions in
the cases of (a) low correlation ($\rho_{j}:=0.5$) and (b) high correlation
($\rho_{j}:=0.95$). Figure 9 shows the results for SNR $0$ dB under different
SIR conditions in the case of low correlation ($\rho_{j}:=0.5$). (The results
for $\rho_{j}:=0.95$ are omitted as the results are similar.) It is seen that
the RZF beamformer attains significant gains compared to the MVDR and ZF
beamformers. It is also seen that its performance is fairly close to the
theoretical bound (MMSE-DR) in the low correlation case with low SIR. See
Section 5.3 for discussions about the MSE gap between RZF and MMSE-DR together
with that for the wireless-network case.

Figure 10 shows the powers of noise and interfering activities remaining in
the beamformer output for SNR $-2$ dB and SIR $0$ dB. It is seen that RZF
attains an excellent tradeoff; it attains reasonably small noise-leakage by
allowing some slight (invisible) leakage of the interfering activities. In
Figure 10(a), the total leakage of RZF is slightly lower than the noise
leakage of MVDR (see Remark 4). Figure 11(a) plots the MSE performance of RZF
for each relaxation parameter $\varepsilon$ under SNR $-2$ dB, SIR $0$ dB, and
$\rho_{j}:=0.5$. Within the range of $1.0<\varepsilon<1.0\times 10^{3}$, the
MSE of RZF is below $-7.5$ dB. This implies that RZF is reasonably insensitive
to the choice of $\varepsilon$. Figure 11(b) provides more precise
information, plotting the power of noise/interference leakage contained in the
beamformer output for different $\varepsilon$ values. It is seen that both
noise and interference are suppressed simultaneously by RZF for an
appropriately chosen $\varepsilon$. This supports the results of Figure 10(a).

(a) $\rho_{j}:=0.5$ (b) $\rho_{j}:=0.95$

Figure 10: Powers of noise/interference leakage for EEG application under low
and high correlations (SNR $-2$ dB, SIR $0$ dB).

|
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Figure 11: Sensitivity of RZF to the parameter $\varepsilon$ for EEG
application ($\rho_{j}:=0.5$, SNR $-2$ dB, SIR $0$ dB).

We now consider the adaptive implementations: RZF implemented by DDAA, and
MVDR and ZF implemented by CNLMS [11]. The step sizes for all online
algorithms are set to $0.1$. The parameter of the DDAA is set to
$\alpha_{k}:=0.5$, $\forall k\in{\mathbb{N}}$ (see the appendix). Figure 12(a)
plots the steady-state MSE of each algorithm for SIR $0$ dB and
$\rho_{j}:=0.5$ under different SNR conditions. One can see that each
beamformer is successfully implemented adaptively by each adaptive algorithm.
Figure 12(b) plots the learning curves for the specific case of SNR $0$ dB,
where each point is computed by averaging the squared errors over 300
independent trials. (For visual clarity, the MSE values are further averaged
over the previous 30 iterations.) It can be seen that the MSEs of the adaptive
algorithms for the RZF and ZF beamformers converge reasonably fast to those of
the analytical solutions, respectively. In contrast, MVDR implemented by CNLMS
converges slowly due to no use of the channel information of the interfering
activities.

|
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(a) steady-state performance (b) learning curves under SNR $0$ dB

Figure 12: Performances of adaptive implementations for EEG application under
low correlation $\rho_{j}:=0.5$ (SIR $0$ dB). RZF implemented with DDAA, and
MVDR and ZF implemented both with CNLMS.

We remark finally that the RZF beamformer may be also used when surface
Laplacian-processed signals are considered instead of the raw EEG signal to
enhance spatial resolution [34, 35]. In such a case, the channel matrix is
replaced by the surface Laplacian matrix and the covariance matrix
${\boldsymbol{R}}$ is replaced by the covariance matrix of the Laplacian-
treated signal.

###  5.3 Discussions and Remarks

Figure 13: An example of the $\gamma$ value for each interfering source in the
EEG application under low correlation $\rho_{j}:=0.5$ (SNR $0$ dB, SIR $0$
dB).

Figure 13 plots the $\gamma$ values for different sources in the setting
corresponding to Figure 9 with SIR 0 dB. It is seen that only 9 (out of 28)
sources have the $\gamma$ values within the range $(0,1)$; see (19) in Theorem
2. To see whether those interfering sources which have $\gamma$ outside the
range $(0,1)$ cause the MSE gaps between RZF and MMSE-DR, we removed all such
interfering sources and evaluated the MSE performances only with the remaining
sources having $\gamma\in(0,1)$ under $\rho_{j}:=0.5$, SNR $-5$ dB, and SIR
$0$ dB. The gap then droped, from $3.4$ dB, down to $1.5$ dB approximately. We
also tested with the same number of randomly picked sources removed. In this
case, the gap was no less than $2.8$ dB. This indicates that those sources
having $\gamma\not\in(0,1)$ would be a main cause of the MSE gaps. In the
simulation results presented in Section 5.1, there are some visible gaps
between the MSEs of RZF and MMSE-DR in Figure 3(b). In the case of complex
signals, Theorem 5 suggests the existence of another factor
$\left(1-\left|\delta_{1}/\delta_{2}\right|^{2}\right)\sigma_{n}^{4}\tan^{2}\tau/(\sigma_{1}^{2}\cos^{2}\tau+\sigma_{n}^{2})$
that may cause the MSE gap.

Indeed, $\gamma$ is a function of $\delta_{1}$, as well as the others.
Referring to (26), the sign of $\delta_{1}$ depends on the phase $\phi_{c}$ of
the temporal correlation $c_{1}$. This implies that the MSE of RZF may change
significantly due to the phase (or the sign in particular) of $c_{1}$.
Nevertheless, the plots for RZF are nearly symmetric for the positive and
negative values of $\rho_{j}(:=c_{j}/(\sigma_{0}\sigma_{j}))$. In addition, we
actually tested different values of $\phi_{j}$s, which yielded no visible
differences in MSE. This would be because the phase $\phi_{z_{j}}$ of the
spatial correlation
$z_{j}:=\left\langle{{\boldsymbol{h}}_{0}},{{\boldsymbol{h}}_{j}}\right\rangle$
is quite random among different sources.

##  6 Conclusion

We studied the RZF beamformer, which minimizes the output variance under the
constraints of bounded interference leakage and undistorted desired signal, in
the presence of temporally correlated interference. The (quadratic) relaxed
zero-forcing constraint alleviates the gap between the output variance
(available) and MSE (unavailable) without amplifying the noise. An adaptive
implementation of the RZF beamformer based on the dual-domain adaptive
algorithm was also presented. We analyzed the MSE of RZF for the single-
interference case and derived the formula of MSE in terms of the spatio-
temporal correlations between the desired and interfering signals as well as
the variances of noise and interference, clarifying that RZF achieves the
minimum MSE under the conditions. The presented analysis gave useful insights
into the multiple-interference case, as discussed in the experimental section.
Numerical examples showed the remarkable advantages of RZF over the MVDR and
ZF beamformers. The RZF beamformer also achieved near-optimal performance in
some cases. We conclude the present study by stating that the RZF beamformer
will be useful particularly for the EEG brain activity reconstruction because
(i) it enjoys superior performance by mitigating the interfering sources
efficiently, (ii) it includes the MVDR and ZF beamformers as special cases,
and (iii) it can be implemented efficiently by DDAA.

##  Appendix A Dual-Domain Adaptive Algorithm: An Implementation of RZF
Beamformer

Given any closed convex subset $K$ of ${\mathbb{C}}^{m}$ of dimension $m$, the
metric projection $P_{K}({\boldsymbol{x}})$ of a point
${\boldsymbol{x}}\in{\mathbb{C}}^{m}$ onto $K$ is defined by
$P_{K}({\boldsymbol{x}}):=\operatornamewithlimits{argmin}_{{\boldsymbol{y}}\in{\mathbb{C}}^{m}}\left\|{\boldsymbol{x}}-{\boldsymbol{y}}\right\|$.
The set $\\{{\boldsymbol{w}}\in{\mathbb{C}}^{N}\mid\|{\boldsymbol{H}}_{\rm
I}^{\sf H}{\boldsymbol{w}}\|^{2}\leq\varepsilon\\}$ is a (degenerate)
ellipsoid onto which the projection has no closed-form expression.
Fortunately, the projection of the dual-domain vector ${\boldsymbol{H}}_{\rm
I}^{\sf H}{\boldsymbol{w}}\in{\mathbb{C}}^{J}$ onto the closed ball
$B_{\varepsilon}$ is easy to compute, thus used in the dual-domain algorithm.
Define the normalized channel matrix $\widetilde{{\boldsymbol{H}}}_{\rm
I}:={\boldsymbol{H}}_{\rm I}/\sigma_{1}({\boldsymbol{H}}_{\rm I})$. Given an
initial beamforming vector ${\boldsymbol{w}}_{0}\in C$, the DDAA update
equation is then given as follows:

${\boldsymbol{w}}_{k+1}:={\boldsymbol{w}}_{k}+\lambda_{{\boldsymbol{w}},k}\eta_{k}{\boldsymbol{g}}_{k},$

where $\lambda_{{\boldsymbol{w}},k}\in(0,2)$ and

$\displaystyle{\boldsymbol{g}}_{k}:=$
$\displaystyle~{}\alpha_{k}{\boldsymbol{g}}^{{\rm(1)}}_{k}+(1-\alpha_{k}){\boldsymbol{Q}}\widetilde{{\boldsymbol{H}}}_{\rm
I}{\boldsymbol{g}}^{{\rm(2)}}_{k},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}{\boldsymbol{Q}}:={\boldsymbol{I}}-{\boldsymbol{h}}_{0}{\boldsymbol{h}}_{0}^{\sf
H}/({\boldsymbol{h}}_{0}^{\sf
H}{\boldsymbol{h}}_{0}),~{}~{}~{}\alpha_{k}\in[0,1],$
$\displaystyle{\boldsymbol{g}}^{{\rm(1)}}_{k}:=$ $\displaystyle~{}P_{C\cap
H_{k}}({\boldsymbol{w}}_{k})-{\boldsymbol{w}}_{k}=-\frac{{\boldsymbol{w}}_{k}^{\sf
H}{\boldsymbol{y}}(k)}{{\boldsymbol{y}}(k)^{\sf
H}{\boldsymbol{Q}}{\boldsymbol{y}}(k)}{\boldsymbol{Q}}{\boldsymbol{y}}(k),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}H_{k}:=\\{{\boldsymbol{w}}\in{\mathbb{C}}^{N}\mid{{\boldsymbol{w}}}^{\sf
H}{\boldsymbol{y}}(k)=0\\},$ $\displaystyle{\boldsymbol{g}}^{{\rm(2)}}_{k}:=$
$\displaystyle~{}P_{B_{\varepsilon}}\left(\widetilde{{\boldsymbol{H}}}_{\rm
I}^{\sf H}{\boldsymbol{w}}_{k}\right)-\widetilde{{\boldsymbol{H}}}_{\rm
I}^{\sf H}{\boldsymbol{w}}_{k}$ $\displaystyle=$
$\displaystyle~{}\begin{cases}\Bigg{(}\dfrac{\sqrt{\varepsilon}}{\left\|\widetilde{{\boldsymbol{H}}}_{\rm
I}^{\sf
H}{\boldsymbol{w}}_{k}\right\|}-1\Bigg{)}\widetilde{{\boldsymbol{H}}}_{\rm
I}^{\sf H}{\boldsymbol{w}}_{k},&\mbox{if
}\left\|\widetilde{{\boldsymbol{H}}}_{\rm I}^{\sf
H}{\boldsymbol{w}}_{k}\right\|^{2}>\varepsilon,\\\ {\boldsymbol{0}},&\mbox{if
}\left\|\widetilde{{\boldsymbol{H}}}_{\rm I}^{\sf
H}{\boldsymbol{w}}_{k}\right\|^{2}\leq\varepsilon,\end{cases}$
$\displaystyle\eta_{k}:=$
$\displaystyle~{}\begin{cases}\dfrac{\alpha_{k}{\big{\|}}{\boldsymbol{g}}^{{\rm(1)}}_{k}{\big{\|}}^{{\rm
2}}+(1-\alpha_{k})\left\|{\boldsymbol{g}}_{k}^{(2)}\right\|^{2}}{\left\|{\boldsymbol{g}}_{k}\right\|^{2}},&\mbox{if
}{\boldsymbol{g}}_{k}\neq{\boldsymbol{0}},\\\ 1,&\mbox{otherwise.}\end{cases}$

Here, the vector ${\boldsymbol{g}}_{k}^{{\rm(1)}}$ contributes to reducing the
output variance in (6a), while ${\boldsymbol{g}}_{k}^{{\rm(2)}}$ contributes
to reducing the violation of the quadratic constraint of (6d). The DDAA method
has been proven to generate a sequence convergent to an asymptotically optimal
point under certain mild conditions. See [22, 21] for the detailed properties
of the algorithm.

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