\documentstyle[open,preprint]{spawc}
\newcommand{\dtest}{\stackrel{\scriptstyle <}{\scriptstyle >}}
\title{AN ADAPTIVE SPATIAL DIVERSITY RECEIVER FOR NON-GAUSSIAN
INTERFERENCE AND NOISE }
\author{	
Rick S. Blum\and Richard J. Kozick\and Brian M. Sadler}
\affiliation{ Electrical Engineering and Computer Science Department, 
Lehigh University, Bethlehem, PA 18015\\
Electrical Engineering Department, Bucknell University, Lewisburg, PA  17837\\
Army Research Laboratory, Information Sciences and Technology Directorate, 
Adelphi, MD  20783} 

\begin{document}
\maketitle
\begin{abstract}
Standard {\em linear\/} diversity combining techniques are
not effective in combating fading in the presence
of non-Gaussian noise.  An adaptive spatial diversity receiver is developed
for wireless communication channels with
slow, flat fading and additive non-Gaussian noise.
The noise is modeled as a mixture of Gaussian distributions, and
the expectation-maximization (EM) algorithm is used to
derive estimates for the model parameters.
The parameter estimates are used in a generalized likelihood
ratio test to reproduce the transmitted signals.
The new receiver is shown to be relatively insensitive to
errors in the parameter estimates as well as to 
errors in modeling the actual noise distribution.
\end{abstract}

\section{Introduction}

Wireless communication systems often operate 
in environments with severe fading due to multipath
propagation. 
To mitigate the effects of fading, which limits system
performance~\cite{weer},
diversity techniques using multiple antennas
have been studied and implemented for signals received in Gaussian noise. 
Significant performance improvements are obtained with
these schemes for
channels with additive Gaussian noise and slow Rayleigh 
fading \cite{wint94}.
Impulsive noise occurs in indoor and outdoor environments due to a variety
of sources \cite{rapp93,midnew}.  In general,
optimum reception schemes designed for Gaussian noise 
environments perform very poorly when impulsive noise is present \cite{rob}. 
In particular, it has been demonstrated in \cite{dretar}
that the standard diversity combining methods (maximal ratio
combining, equal gain combining, selection diversity) are
not effective in impulsive noise environments. 

As an alternative to using 
schemes that are optimum only for Gaussian noise cases, we study 
diversity schemes that adapt to the actual 
noise distribution encountered. 
The adaptive diversity receiver  we have developed 
is a generalized likelihood ratio detector that is based on
modeling the noise as a mixture of Gaussian distributions.
Maximum likelihood estimates of the model parameters are obtained
using the  expectation-maximization (EM) algorithm.
The Gaussian mixture distribution noise model and EM algorithm
for parameter estimation fit together well in this application.
Middleton and others \cite{mid1,mid2} have shown that
the Gaussian mixture distribution  model
is accurate for many impulsive noise environments, 
while Redner and Walker \cite{EMmix} and others \cite{zabin6}
have shown that the EM algorithm has attractive computational
features when applied to Gaussian mixture distributions.
Thus the Gaussian mixture distribution and EM algorithm 
combine to provide accurate noise modeling with relatively
simple and reliable computations.
The adaptive receiver achieves significant improvements over
the more common linear diversity approaches, and it appears to
be relatively insensitive to mismatch between the actual noise
distribution and the Gaussian mixture distribution model.

As an example of
the bit error rate (BER) performance of a maximal ratio combiner
versus our new receiver,
a channel with flat, Rayleigh fading and
impulsive noise was simulated.
The details of the signal and noise environment are
explained in \cite{dretar}, but the key result is that
the BER of the maximal ratio combiner 
does not improve,  
and thus remains poor, 
as more antennas are added, while
the new receiver performs better with each new antenna
that is added.
Typical numerical BER values with four antennas are 0.1
for the maximal ratio combiner and 0.0025 for the new receiver
at a signal to noise ratio of $-14$ dB.

\section{Model and Optimum Receiver}
\label{sec:RecModel}

Consider the following model for the complex envelope of
the received signals
in a communication system with multiple antennas,
slow and flat fading, and additive non-Gaussian noise and 
interference.
Let ${\bf s}_1, \ldots, {\bf s}_{J}$ denote the vectors in a
$J$-ary signal constellation, where each 
signal vector ${\bf s}_j = [s_{1j}, \ldots, s_{mj}]^T$
has dimension $m \times 1$.
(The superscripts $T$, $H$, and $*$ will denote
the transpose, conjugate transpose, and complex conjugate operations,
respectively.)
The elements of ${\bf s}_j$ may be taken as $m$ time samples of the 
transmitted waveform that encodes signal $j$.
If the system contains $N$ receiving antennas, then the
$m \times 1$ vector of observations for one received symbol
at antenna $k$ is modeled as 
\begin{equation}
{\bf x}_k = r_k {\bf s} + {\bf w}_k , \;\;\; k=1, \ldots, N,
\label{xkDef}
\end{equation}
where $\bf s$ is one of the transmitted signal vectors
${\bf s}_1, \ldots, {\bf s}_{J}$, 
$r_k$ is the complex fading coefficient at antenna $k$, 
and ${\bf w}_k$ is the $m \times 1$ vector of additive noise samples.
The observations at all $N$ antennas can be combined into a
single $mN \times 1$ vector 
$ {\bf x} $ of the form
\begin{equation}
{\bf x} = 
\left[ \begin{array}{c} {\bf x}_1 \\
\vdots \\ {\bf x}_N \end{array} \right] =
{\bf r} \otimes {\bf s} + {\bf w} ,
\label{xDef}
\end{equation}
where ${\bf r} = [ r_1, \ldots, r_N ]^T$ is an $N \times 1$
vector of fading coefficients, $\otimes$ denotes Kronecker
product, and ${\bf w}$ is formed by stacking the ${\bf w}_k$
vectors 
in the same manner as ${\bf x}$ is formed by stacking the ${\bf x}_k$
vectors.
The objective is to determine which signal ${\bf s}_j$ was transmitted
by processing the observations ${\bf x}$ in (\ref{xDef}).

We will assume that
the observations from $T$ known 
training symbols are available at the $N$ antennas.
The received training symbols will be denoted as follows.
Define $s_i(t) = s_{i, j(t)}$ as the sample at time $i$ of training
symbol $t$, and ${\bf s}(t) = \left[ s_1(t), \ldots, s_m(t) \right]^T$.
Then the model in (\ref{xkDef}) is extended to 
\begin{equation}
{\bf x}_k(t) = r_k {\bf s}(t) + {\bf w}_k(t) , \;\;\; k=1, \ldots, N,
\; t=1, \ldots, T
\label{xktDef}
\end{equation}
to describe the observations from all $T$ training symbols. 
In (\ref{xktDef}) $ {\bf w}_k(t) $ represents the vector of noise samples 
observed during the reception of training symbol $ t $.  
Equation (\ref{xktDef}) describes the $T \cdot N \cdot m$
observations that are available in the training set.
Note that the $m \times 1$ signal vectors ${\bf s} (t)$ 
are known for each $t=1,\ldots,T$.
Also note that the fading coefficients $r_k$ are modeled as
being fixed over the entire training set, i.e.\ they do not vary with $t$.  

Some observations and further development of the model are as follows.
\begin{enumerate}
\item The fading is {\em flat\/} or frequency non-selective, so it is modeled
by a single complex coefficient $r_k$ that describes the amplitude
scaling and phase shift experienced by the transmitted signal.
The fading is {\em slow\/}, which means that the fading coefficient
is constant for several symbols to allow estimation of the fading 
coefficients.
\item The vectors ${\bf r}, {\bf s}, {\bf w}$ in (\ref{xDef})
are realizations of complex-valued random vectors ${\bf R}, {\bf S}, {\bf W}$.
\item $\bf S$ is a discrete random vector that takes the values
${\bf s}_1, \ldots, {\bf s}_J$ with probabilities
$\pi_1, \ldots, \pi_J$.
\item The fading coefficients, the components of $\bf R$, are modeled as
independent and identically distributed (iid) random variables. 
This implies that the antennas are spaced sufficiently far apart so
that the fading is independent from antenna to antenna.
Although the probability density function (pdf) of $\bf R$ affects the
system performance, our processing is not based on any particular
distribution for $\bf R$.
Simulation examples we ran typically assumed Rayleigh fading. 
\item The additive noise vector ${\bf W}$ has components that are
iid, with each component modeled as an $L$-term mixture of
Gaussian pdfs.
The pdf of ${\bf W}$ is
\begin{equation} 
f_{\bf W}( {\bf w } ) = \prod_{i=1}^{m} \prod_{k=1}^{N} 
\sum_{\ell=1}^{L} \lambda_{\ell} 
\frac{1}{2 \pi \sigma_{\ell}^2} 
\exp{\left( - \frac{ |w_{ki}|^2 }{ 2 \sigma_{\ell}^2 } \right)}  .
\label{pdfW} 
\end{equation} 
A straightforward extension of (\ref{pdfW}) allows the noise
parameters to vary from antenna to antenna. 
\item The model is for a single user scheme, with any multiuser
interference considered part of the additive non-Gaussian noise.
\end{enumerate}

The model in 
(\ref{pdfW}) includes cases that are similar to 
a truncated-sum version of Middleton's canonical class A noise model 
\cite{midnew,mid1,mid2},
which has received extensive study.
The truncated version of Middleton's 
model  with $L=2$ or $3$ terms has been shown to be a
good approximation for 
some cases of interest \cite{Vast84,mult}, including highly impulsive
noise.  
In order to model impulsive noise, 
a common convention for the noise pdf parameters \cite{AazPoor1}
is to take $L=2$, 
$\lambda_1 > \lambda_2$ and $\sigma_1^2 < \sigma_2^2$, so 
that large noise samples are produced every so often by the larger variance 
term.

If the fading coefficient vector $\bf r$ and the noise pdf
parameters are known exactly,
then detection based on 
the model in (\ref{xDef}) reduces to detecting one of $J$
known signals in additive noise.
The decision rule that maximizes the probability of correct decisions
is the maximum {\em a posteriori\/} (MAP) detector, which chooses
the signal ${\bf s}_j$ with index $j$ that maximizes the
posterior probability
$ {\rm Pr} ({\bf S} = {\bf s}_j \mid {\bf R} = {\bf r}, {\bf X} = {\bf
x}) $.
The MAP detector
is equivalent to choosing the index $j \in \{ 1, \ldots, J \} $ that maximizes
$ \pi_j \; f_{\bf W} ({\bf x} - {\bf r} \otimes {\bf s}_j ) $,
which can also be determined 
by using likelihood ratio (LR)  
test statistics. 
An explicit form for this test is obtained 
easily using the model we have outlined. 
The resulting test is optimum with regard to minimizing the 
probability of incorrect decisions only when the fading coefficients
$\bf r$ and the noise pdf parameters $L, \lambda_1, \ldots, \lambda_L,
\sigma_1^2, \ldots, \sigma_L^2$ in (\ref{pdfW}) are known.
In practice, these quantities must be estimated from the data and
updated over time as the communication environment changes.
Our approach is to apply the MAP test just described with the Gaussian
mixture pdf in (\ref{pdfW}), using the {\em estimates\/}
for $\bf r$ and $L, \lambda_1, \ldots, \lambda_L,
\sigma_1^2, \ldots, \sigma_L^2$ in place of the true values for these 
parameters.
A procedure for obtaining maximum likelihood
estimates for these  parameters using the EM algorithm
is presented in the next section. 
The use of maximum likelihood parameter estimates in a
MAP test is a common procedure, and the resulting detector
is known as a {\em generalized\/} likelihood ratio test.

An alternative detector structure, the MD-MMSE detector, 
is based on minimizing the 
distance between each possible signal ${\bf s}_j$ 
and a minimum mean-squared error (MMSE) estimate
of the transmitted signal.
We have proven that the MD-MMSE detector
is equivalent to the MAP
detector in many cases of interest, including equal-energy binary signaling 
cases.  
The MD-MMSE detector has the structure of
a class of neural networks called Gaussian mixture basis function
networks (GMBFNs) 
\cite{ChaICASSP},\cite{ChaIP},\cite{ChaNL},\cite{ChaDiss}.

\section{Estimation of Model Parameters with EM Algorithm}
\label{sec:Est}

In this section we present the update equations used 
for recursive estimation of the fading coefficients $r_1, \ldots, r_N$
and the noise pdf parameters $\lambda_1, \ldots, \lambda_L,
\sigma_1^2, \ldots, \sigma_L^2$.
These equations are new, 
they provide extremely good estimates with sufficient training data,  
and they have interesting interpretations which will be discussed in 
Section~\ref{sec:Rob}  If the fading coefficients are already available, 
from a pilot signal for example, then fewer quantities need to be
estimated.

The observations of $T$ training symbols as modeled by 
(\ref{xktDef}) are assumed to be available.
The EM algorithm is an iterative procedure for obtaining maximum
likelihood parameter estimates, and its
application to mixture pdfs is reviewed in \cite{EMmix}.
The number of terms $L$ in the mixture pdf (\ref{pdfW}) can be
estimated. However, 
we have taken the simpler approach of fixing $L$ equal to 2, 3 or 4.
This simplifies the processing, and in addition
other studies \cite{Vast84,mult} have demonstrated that using  
$L = 2, 3$ or $4$ frequently produces a 
good approximation for cases of interest.

The general structure of the
EM algorithm is as follows.
Beginning with current estimates $r'_k, \lambda'_{l}, 
{\sigma'}_{l}^2$ of the parameters, new estimates $r_k, \lambda_{l},
\sigma_{l}^2$ are computed by processing the current estimates
along with the training data.
Explicit formulas for this processing are given below.
Then the new estimates are assigned to the current
estimates,
and the same training
data is processed again to improve the estimates.
This process is repeated until the change in parameter estimates
is small from one iteration to the next iteration.

The EM update equations 
to estimate the model parameters
are listed below, where
first
\begin{equation}
p'_{lki}(t) = 
\frac{\lambda'_{l}}{2 \pi
{\sigma'}_{l}^2} \exp \left( -
\frac{\left| x_{ki}(t) - r'_k s_i(t) \right|^2}
{2 {\sigma'}_{l}^2} \right) 
\label{PlkiDefL}
\end{equation}
is defined for $ l=1,\ldots,L, k=1,\ldots,N, i=1,\ldots,m, t=1,\ldots,T $.
Then the parameter estimates are updated as follows:
\begin{equation}
\lambda_{l} = \frac{1}{mTN} \sum_{k=1}^K \sum_{t=1}^T \sum_{i=1}^m
\frac{p'_{lki}(t)}
{\sum_{q=1}^L p'_{qki}(t)}, 
\label{EMlaml}
\end{equation}
\begin{equation}
r_k = \frac{
\sum_{t=1}^T \sum_{i=1}^m \sum_{l=1}^L
s_i^*(t) x_{ki}(t)  \cdot 
\frac{p'_{lki}(t)}
{ {\sigma'}_l^2 \sum_{q=1}^L p'_{qki}(t)}} {
\sum_{t=1}^T \sum_{i=1}^m  \sum_{l=1}^L
|s_i(t)|^2 \cdot \frac{p'_{lki}(t)}
{ {\sigma'}_l^2 \sum_{q=1}^L p'_{qki}(t)}}  , 
\label{EMrk}
\end{equation}
\begin{equation}
\sigma_{l}^2 = 
\frac{
\sum_{k=1}^N \sum_{t=1}^T \sum_{i=1}^m
\left| x_{ki}(t) - r'_k s_i(t) \right|^2
\cdot 
\frac{p'_{lki}(t)}
{ \sum_{q=1}^L p'_{qki}(t)}} {
2 \sum_{k=1}^N \sum_{t=1}^T \sum_{i=1}^m
\frac{p'_{lki}(t)}
{\sum_{q=1}^L p'_{qki}(t)}} , 
\label{EMvarl}
\end{equation}
for $ l=1,\ldots,L, k=1,\ldots, N $. 
Initial values for the parameters are required for the first
iteration of the EM algorithm.
We have developed rules of thumb for selecting initial values.
Preliminary investigations of the EM algorithm for parameter
estimation followed by MAP detection
are very encouraging. 

\section{Convergence and accuracy of estimates}

The EM algorithm is known to have good global convergence properties
\cite{EMmix}, and we have observed this in our simulation tests.
In most cases convergence is achieved in about 6 iterations.
Speed of convergence is discussed in \cite{EMmix}, with the result that
the EM algorithm converges quickly when the component densities
in the Gaussian mixture pdf are ``widely separated'' so that
\begin{equation}
\frac{
\left( \lambda_i / \sigma_i^2 \right) 
\exp \left( - \frac{|x|^2}{2 \sigma_i^2} \right)
}
{
\sum_{q=1}^L
\left( \lambda_q / \sigma_q^2 \right) 
\exp \left( - \frac{|x|^2}{2 \sigma_q^2} \right)
}
\cdot
\frac{
\left( \lambda_l / \sigma_l^2 \right) 
\exp \left( - \frac{|x|^2}{2 \sigma_l^2} \right)
}
{
\sum_{q=1}^L
\left( \lambda_q / \sigma_q^2 \right) 
\exp \left( - \frac{|x|^2}{2 \sigma_q^2} \right)
} \approx 0, 
\label{eq:EMsep}
\end{equation}
for $ i \neq l $ and all complex $x$.
Impulsive noise cases typically satisfy (\ref{eq:EMsep}),
which explains the fast convergence that we typically observed.

An important issue with any estimator is its accuracy with
a finite amount of data. 
We have derived a formula to approximate the number of
training symbols $T$ required to obtain accurate estimates
for the fading coefficients.
The result is that $T$ should be slightly more than 10 times the
reciprocal of the average signal to noise ratio at one antenna. 
Simulated results appear to agree with this approximation and show 
that these $T$ also provide accurate estimates of the noise pdf. 

\subsection{Robust behavior of EM estimates and MAP detector}
\label{sec:Rob}

The EM estimates in (\ref{PlkiDefL})-(\ref{EMvarl}) and the
MAP detector 
are derived using a Gaussian mixture pdf that can describe impulsive noise. 
Thus the estimates and the detector will properly
handle large noise samples and will not be adversely affected by them.
We have quantified this robust behavior 
with the main result that the factors
$ p'_{lki}(t) / \sum_{q=1}^L p'_{qki}(t) $
in the EM update equations
(\ref{PlkiDefL})-(\ref{EMvarl}) 
tend to ``classify'' each observation according to the likelihood
that the observation is produced by the $l^{th}$ term in
the Gaussian mixture pdf (\ref{pdfW}).  Thus in an $L=2$ impulsive noise 
case with $ \lambda_1 $ near $1$ and $ \lambda_2 $ 
near $0$ and $ \sigma_2^2 >> \sigma_1^2 $ we find that small observations 
are classified as coming from the smaller variance term.  Thus smaller 
observations are used to update the estimate of $ \lambda_1 $ and 
$ \sigma_1^2 $.  Larger observations are used to update $ \lambda_2 $ 
and $ \sigma_2^2 $. 
Similar processing occurs in the MAP detector.
We have shown that for an $L=2$ impulsive noise case each observation
is processed by a memoryless nonlinear function that limits the influence
of large observations.   Such processing is known to be robust to 
mismatch in the exact noise model assumed \cite{rob}.  

\subsection{Detector sensitivity to modeling errors}
\label{sec:Sens}

We have studied the performance of the adaptive receiver 
for cases where the noise and fading coefficient 
estimates obtained by the receiver 
are different from the true noise and fading parameters.  
The main result is that the detector is robust to impulses
as long as the estimated noise parameters correspond to a
pdf that is at least slightly impulsive.
In this case large observations are limited, as described in
the previous subsections.
Note that Gaussian noise is a special case of the Gaussian
mixture pdf, so our receiver is capable of adapting to Gaussian
noise and performing the linear processing that is optimum.
Preliminary simulations have also been performed to investigate
other types of mismatch between the true noise and the noise
model in the receiver.
For example, when the number
of mixture terms $L$ is mismatched,
the EM algorithm still yields reasonable estimates and the
detector performance is only slightly degraded.

The mixture model we have considered is known to be useful for 
representing a large number of noise pdfs which occur in practical 
situations \cite{kasbook}.  
The relationship to the 
GMBFN with its  universal approximation capabilities further justifies 
using the mixture model to characterize unknown noise.

We have investigated mismatch 
cases in which the non-Gaussian noise is outside the 
mixture class.  The results have been quite good. 
For noise 
with in-phase and quadrature components
that are circularly symmetric with Cauchy marginal pdfs
\cite{ModNingo}, the
simulated BER performance results are presented in Table
\ref{tab:cauchy}.
The optimum detector is a likelihood ratio test using the
true noise pdf, and
the other detectors in Table \ref{tab:cauchy} are the maximal ratio
combiner and our new adaptive diversity combiner using
$L=2, 3$, and 4 terms in the Gaussian mixture pdf to model the noise.
The EM algorithm is used to estimate the mixture pdf parameters 
from 100 training symbols for each value of $L$.
For $N=1$ antenna, all methods perform poorly due to
fading.
However, the adaptive receiver with $L=3$ and $L=4$ performs very
close to the optimum receiver, indicating that very few terms in
the Gaussian mixture pdf provide a good approximation to the optimum
detector.
\begin{table}
\begin{center}
\begin{tabular}{c|lllll}
 & \multicolumn{5}{c}{Type of Detector} \\
$N$ & Optimum  & Max ratio & $L=2$ & $L=3$ &
$L=4$ \\ \hline 
1 & 0.0764   &    0.2821   &    0.1270   &   0.0864  &    0.0786 \\
4 &  0.0012  &     0.2254  &     0.0074 &     0.0020 &     0.0014
\end{tabular}
\end{center}
\caption{
BER performance of optimum, maximum ratio, and
adaptive combiner in narrowband
noise with Cauchy in-phase and quadrature components.
The adaptive combiner results are presented for the cases of $L=2,3$,
and 4 terms in the Gaussian mixture pdf.
}
\label{tab:cauchy}
\end{table}

\section{Conclusion}

An adaptive spatial diversity receiver is developed
for wireless communication channels with
slow, flat fading and additive non-Gaussian noise. 
The adaptive receiver provides considerable improvement 
over the commonly employed linear diversity combining schemes 
when impulsive noise or interference is present.  
Derivations, proofs, and results from an extensive set of 
simulations have not been given here due to page limitations. 
A number of important issues have also not been discussed. 
For example, decision directed 
schemes for on-line training are easily constructed.  
The adaptive receiver studied appears promising and 
deserves further study.   For example, cases with more 
complicated observation models are being studied. 

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\end{document}