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\begin{document}
\title{Adaptive Microphone Array Employing Calibration Signals. An
Analytical Evaluation.}
\author{Sven Nordholm Ingvar Claesson}
\maketitle
\date{today}
\begin{abstract}
This report evaluates an adaptive microphone array
which facilitates a simple built-in calibration to the
environment and instrumentation. The method has been
suggested for use in hands-free mobile telephones for both speech
enhancement and acoustic echo-cancellation as well as speech
recognition. The scheme has several advantages such as a simple
calibration procedure, directed jammer suppression, versatile
robust beamforming and reduced target signal distortion. The
analysis employs noncausal Wiener filters yielding compact and
effective theoretical suppression limits.
\end{abstract}
\tableofcontents
\chapter{Introduction}
Methods to increase the signal-to-noise ratio when using mobile telephones in
hands-free operation are spectral subtraction \cite{proakdellhans,boll,yang},
temporal filtering (high-pass) and array techniques \cite{kan86}.
We have previously presented
\cite{veh93} an adaptive microphone array solution based on a spatial filtering.
In the European car industry
DRIVE-II project, one of the main priorities has been to replace many
of the hand-controlled functions in a car by voice control.
We have also evaluated array noise reduction for speech
recognition in this application \cite{Nor93,snusa}.
The method analysed in this report employs a calibrated adaptive microphone array.
The method was
evaluated, and outperformed the spatial filtering approach both in mobile telephony
and speech enhancement in cars \cite{impr94,hkr-2}. In car applications
it is necessary to consider the near field in an enclosure, and the use
of low cost, low precision, microphones and equipment. This
aspect raises new problems which are difficult to describe in detail with a priori
modeling. A simple beamformer with reduced inherent
theoretical modeling is likely to outperform an ingenious
array based on erroneous a priori information. The scheme presented in this report
calibrates the array to the speaker and jammer locations, the microphone elements and their
positions and lobe gains, amplifiers. The array is also calibrated for the acoustic situation in
the car. The main idea behind this development is the calibration which provides the
adaptive algorithm with excellent ``desired'' and ``undesired'' signals.
A feature of vital interest both for telephony and for speech recognition system
is speech distortion.
The discrepancy between the total transfer function in training
and operation should be
small. This can be controlled in the proposed signal beamformer by means of
amplification of training signals. A vital problem for
hands-free mobile telephony is to avoid acoustic feedback and maintain good speech
quality. This means that the filtering must suppress the loudspeaker as well as the
background noise and room reverberation without causing severe speech distortion.
In the analytical evaluation below, point sources and free field are considered.
We do not try to model the interior of a car. The main subject for this study
was to find the parameters that affect the suppression of the hands-free
loudspeaker and the distortion of the speaker. The model is, however, also
well-suited to include real measured transfer functions between the hands-free
loudspeaker and the microphone elements. In this way it is possible to include a
fairly good model of the car situation and also make theoretical calculations in this
environment. This will, however, be discussed in a later report.
\chapter{Working Scheme for the Adaptive Beamformer}
The performance and theory of most adaptive arrays rely on high-precision channel
matching
\cite{veh93,ap,Veen88}. This demands a careful calibration of the
array elements, which can be an expensive and difficult task, in particular for
broadband arrays. Another aspect is that the microphone elements that are likely to be
used are of standard quality with a considerable spread in performance. This motivates a
new type of adaptive beamformer which incorporates the calibration phase.
The main idea is to record calibration sequences from both jammer and target in the
real situation through the actual system,
with all its imperfections, and when no car noise is present. The recordings are gathered in a
memory and are later used as training signals in the adaptive phase. This approach
gives an inherent calibration where it is also possible to average and weigh
interesting frequency bands, microphones and spatial points. This approach does not rely
on any geometric a priori information on array element similarities or accurate
positioning. We obtain a system that is tailored for the actual situation.
\section{Description of the Calibration Phase}
The adaptive beamformer can be calibrated on site in a {\underline parked} car by using the existing
handsfree loudspeaker and letting the human speaker read a representative sequence
either directly or via a loudspeaker in the correct position. The sequences are gathered into a memory, see
Fig. \ref{fig:phs1}. This
means that the total channels from the (loud)speakers to A/D converters are included.
The environmental noise level in a parked car,
is very low providing a good signal-to-noise ratio
during recording. If
the situation and equipment can be regarded as time invariant, the array system
possesses calibration signals, which can be combined to form inputs and a
suitable desired signal for an adaptive system.
The microphone elements and placement can be chosen arbitrarily. In order to obtain a more
robust system the (loud)speaker position can be varied during calibration in the
vicinity of the speakers' normal position with varying power. By adding the
recorded signals, weighted average training signals are obtained. These signals,
gathered in the memory are later used as
training input signals, and are also combined to form a desired signal during adaptation.
\section{Operation/Adapting Phase}
In the adapting phase, while the telephone is in use, the speaker is silent (approximately half the time) and
the memory signals are utilized instead, see Fig.\ref{fig:phs2}. The signals are combined to form
a desired signal for the adaptive filters, and are also added to the incoming microphone signals, which
otherwise contain environmental noise only. This produces memory speech signals plus jammer and noise at the
lower beamformer, and a known desired signal which has passed through the same electronic equipment when no
noise was present. This is a conventional situation for adaptive filters, which now possess all the information
needed to adapt to the correct filter coefficients in the least square sense. For this purpose we used a
normalized LMS algorithm, which converges within fractions of a second and behaves very robustly in this
application. The coefficients are hence updated
only while the speaker is silent.
The coefficients are copied and used in the filtering upper beamformer. Two beamformers must be used since the
calibration signals are added at the filter inputs in the lower beamformer. When the speaker is active the
adaptation is switched off. This is done to avoid echo-effects and also to yield a more robust system in the
sense that the adaptive filters can not operate on the real speech signal. Note that the speech signal is
filtered through the upper beamformer only.
\section{Choice of Reference Signal}
The construction of the reference signal is a vital factor in determining speech quality.
As previously explained, the calibrations signals are affected by room reverberation,
the
microphones,amplifiers etc.. This implies that the sequences gathered from the speakers position
are affected. In order to achieve high speech quality from the beamforming scheme, it is vital
that room reverberation is limited, and that undesired signals are rejected from the speech signal.
The task for the adaptive filters is to minimise the difference between $y(t)$ and the
reference signal $y_{r}(t)$ in a least-squares
sense, implying that if the reference signal is chosen in an improper way it will seriously
affect $y(t)$. Assuming that the channels from the speakers position are time invariant, the task for
the beamformer is to perform a blind restoration of the original sequence, and at the same time, suppress
any background noise. It is thus essential that the reference signal $y_{r}(t)$
closely represents the original sequence, optionally with some spectral weighting.
\begin{figure}[htb]
\centerline{\scaledpicturee 163mm by 142mm (FAS1PILOTBFic.E scaled 750)}
\begin{picture}(14.8,0)(-1.0,0)
%\put(12.8,11.1){\makebox(0,0)[lb]{\large$\varepsilon$}}
%\put(12.8,11.12){\makebox(0,0)[lb]{\large $\varepsilon$}}
%\thicklines
%\put(-1.0,0){\framebox(14.8,13){~}}
\end{picture}
\caption{ Data Gathering/Recording}
\label{fig:phs1}
\end{figure}
\begin{figure}[p]
\centerline{\scaledpicturee 125mm by 159mm (fas2pilotbfsn.e scaled 750)}
\begin{picture}(14.8,0)(-1.0,0)
\put(12.4,9.2){\makebox(0,0)[lb]{ ${\large\varepsilon}(t)$}}
%\put(12.8,11.12){\makebox(0,0)[lb]{\large $\varepsilon$}}
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\end{picture}
\caption{Adaptive Microphone Array/Operation phase}
\label{fig:phs2}
\end{figure}
\clearpage
\chapter{Signal Model}
The signal model is
general in the sense that microphone elements and sources
can be placed arbitrarily with any spectral content. The $M$ different point signal
sources $s_{m}(t), \ m=1...M$ with spectral densities $R_{s_{m}s_{m}}(\omega)$
are assumed to be mutually uncorrelated, i.e.
the cross power spectral density $R_{s_{l}s_{m}}(\omega)$ is zero if $l \neq m$.
All sources impinge on an array of $N$ microphone elements, each corrupted with
mutually uncorrelated noise $n_{l}(t)$. The transfer
function between source no. $m$ and an array element no. $n$ is denoted
$G_{m,n}(\omega)$ and is either measured, or modeled. In the model, a spherical source
in a free field and homogeneous medium has been assumed.
In the evaluation the situation is the adapting phase
assuming that the desired signal $y_{r}(t)$ in Fig. \ref{fig:phs2} is combined from
the recorded memory signals from the
calibration phase.
The input signals, $x_{n}(t)$ in Fig. \ref{fig:phs2} are
\begin{equation}
x_{n}(t)=\sum_{m=1}^{M}s_{m}(t)*g_{m,n}(t)+n_{n}(t) \ \ n=1....N
\end{equation}
where $*$ denotes convolution.
The desired signal to the adaptive scheme $y_{r}(t)=s_{1}(t)*f(t)$ is a filtered
version of the target signal. This filter can be formed more generally as a fix
beamformer, or formed from an electrical signal taken just before the loudspeaker
where one is used.
In this way the filter $f(t)$
can be used for weighing the target signal both in the spatial and frequency domains.
Alternatively, another high quality microphone may be used
which is placed close to the mouth during the calibration phase.
In the sequel we assume that all input signals are sampled and correctly band-limited.
Hence, the power spectral density matrix for the discrete-time signals is given by
\begin{equation}
{\bf R}_{\bf xx}(\Omega)=
\sum_{m=1}^{M}R_{s_{m}}(\Omega){\bf G}_{s_{m}}(\Omega){\bf G}_{s_{m}}^{H}(\Omega)+{\bf R_{nn}}(\Omega)
\end{equation} where
${\bf G}_{s_{m}}(\Omega)$ denotes a column vector with dimension $N$ containing
all the corresponding digitised signal frequency transfer functions from a source $m$
to all elements.
The noncausal Wiener solution minimises the error spectral
density with respect to the row filter vector ${\bf W}(\Omega)$ and can be found
by expressing the orthogonality
between the error and the inputs
\begin{equation}
{\bf R}_{\varepsilon {\bf x}}(\Omega)=
{\bf R}_{y_{r}{\bf x}}(\Omega)-{\bf W}(\Omega){\bf R}_{{\bf xx}}(\Omega)=0
\end{equation}
yielding
\begin{equation}
{\bf W}_{opt}(\Omega)={\bf R}_{y_{r}{\bf x}}(\Omega){\bf R}_{{\bf xx}}^{-1}(\Omega)
\label{eq:wopt}
\end{equation}
The corresponding minimum error power spectral density is given by
\begin{equation}
{\bf R}_{\varepsilon\varepsilon, opt}(\Omega)=
{\bf R}_{y_{r}y_{r}}(\Omega)-{\bf R}_{y_{r}{\bf x}}(\Omega){\bf R}_{{\bf xx}}^{-1}(\Omega)
{\bf R}_{{\bf x}y_{r}}(\Omega)
\label{eq:optou}
\end{equation}
By expressing the Wiener solution, the optimum performance
for the adaptive beamformer can be studied.
\begin{figure}[htb]
\begin{center}
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\put(0,0){.}
\thinlines \put(88,156){$ W_{1}(\Omega)$}
\put(88,116){$ W_{2}(\Omega)$}
\put(88,61){$ W_{N}(\Omega)$}
\put(93,8){$ F(\Omega)$}
\put(19,166){$x_{1}[k]$}
\put(19,126){$x_{2}[k]$}
\put(19,71){$x_{N}[k]$}
\put(19,18){$s_{1}[k]$}
\put(323,125){$\varepsilon[k]$}
\put(226,125){$y[k]$}
\put(251,125){$-$}
\put(251,86){$y_{r}[k]$}
\put(273,114.5){$+$}
\put(180,113){{\Large $\Sigma$}}
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\put(127,120){\vector(1,0){38}}
\put(127,160){\vector(1,0){38}}
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\thicklines \put(86,104){\framebox(41,35){}}
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\thicklines \put(86,-4){\framebox(41,35){}}
\thicklines \put(165,49){\framebox(41,136){}}
\end{picture}}
\end{center}
\caption{Figure describing the beamforming scheme}
\label{fig:bfsc}
\end{figure}
The power spectral density matrix ${\bf R}_{{\bf xx}}$ consists of target signal and
jamming signals corrupted with noise. We can separate ${\bf R}_{{\bf xx}}$
in two parts, one with desired signals and one with undesired
\begin{equation}
{\bf R}_{{\bf xx}}(\Omega)=R_{s_{1}}(\Omega){\bf G}_{s_{1}}
(\Omega){\bf G}_{s_{1}}^{H}(\Omega)+{\bf R_{\nu\nu}}(\Omega)
\end{equation}
The matrix ${\bf R_{\nu\nu}}$ consists of all undesired signals, including measurement noise.
Hence, it is assumed to
be non-singular, and the matrix inversion lemma can be employed in order to
express ${\bf R}_{\bf xx}^{-1}$ in closed form.
Using this expression in the eq. (\ref{eq:wopt}) and
finding an expression for ${\bf R}_{{\bf x}y_{r}}$ yields a compact formula for the
solution.
\begin{equation}
{\bf W}_{opt}(\Omega)=\frac{R_{s_{1}}(\Omega)
F(\Omega){\bf G}^{H}_{s_{1}}(\Omega){\bf R}_{\nu\nu}^{-1}(\Omega)}{R_{s_{1}}(\Omega)
{\bf G}_{s_{1}}^{H}(\Omega){\bf R}_{\nu\nu}^{-1}(\Omega){\bf G}_{s_{1}}(\Omega)+1}
\label{eq:wopt1}
\end{equation}
The corresponding expression for the error power spectral density is
\begin{equation}
R_{\varepsilon\varepsilon, opt}(\Omega)=
\frac{R_{s_{1}}(\Omega)
|F(\Omega)|^{2}}{R_{s_{1}}(\Omega)
{\bf G}_{s_{1}}^{H}(\Omega){\bf R_{\nu\nu}}^{-1}(\Omega){\bf G}_{s_{1}}(\Omega)+1}
\label{eq:optou1}
\end{equation}
The array output signal $y(k)$ is closely related to the desired
signal $y_{r}(k)$, i.e. $F(\Omega)$.
Once the optimum filters ${\bf W}_{opt}(\Omega)$ are found for a given
situation, we can investigate the total transfer functions from any spatial point,
provided the transfer function is known from the point to each element. In particular,
the transfer function from each source to the output of the array can be determined. The
total transfer function $H_{m}(\Omega)$ expresses how each signal is affected by the
optimum beamformer and is given by
\begin{equation}
H_{m}(\Omega)={\bf W}_{opt}(\Omega){\bf G}_{s_{m}}(\Omega)=
\frac{F(\Omega)R_{s_{1}}(\Omega){\bf G}_{s_{1}}^{H}(\Omega){\bf
R}_{\nu\nu}^{-1}(\Omega) {\bf G}_{s_{m}}(\Omega)}{R_{s_{1}}(\Omega){\bf
G}_{s_{1}}^{H}(\Omega) {\bf R}_{\nu\nu}^{-1}(\Omega){\bf G}_{s_{1}}(\Omega)+1}
\label{eq:trpwo}
\end{equation}
\section{Two Directional Sources}
The normal situation for acoustic echo-cancellation for
hands-free mobile telephones is a single
target (speaker) and a dominating jammer (hands-free loudspeaker).
In this situation, it is possible to derive somewhat
more explicit expressions on the transfer functions.
These expressions can
be used to evaluate parameters and features that affect the echo
suppression, such as the number of microphones, signal levels, loudspeaker placement and
microphone configuration. The construction of the desired signal $y_{r}[k]$
is crucial, since it
provides two important quantities, an output which matches the
original speech and a frequency weighting of the input signal in order to
obtain better speech quality.
A single jamming signal together with noise constitutes only
two terms in the matrix ${\bf R_{\nu\nu}}$ representing all unwanted signals
\begin{equation}
{\bf R_{\nu\nu}}(\Omega)=R_{s_{2}}(\Omega){\bf G}_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega)
+\sigma_{n}^{2}{\bf I}
\end{equation}
In order to derive a more explicit expression of \ref{eq:trpwo} the matrix inversion
lemma is again used to yield the inverse of the matrix ${\bf R_{\nu\nu}}(\Omega)$
\begin{equation}
{\bf R}_{\nu\nu}^{-1}(\Omega)=\frac{1}{\sigma_{n}^{2}}\left({\bf
I}-\frac{R_{s_{2}}(\Omega) {\bf G}_{s_{2}}(\Omega){\bf
G}_{s_{2}}^{H}(\Omega)}{R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega) {\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right) \end{equation}
Inserting this expression in eq. (\ref{eq:trpwo}) finally yields
\begin{equation}
H_{m}(\Omega)=
\frac{F(\Omega)\frac{\displaystyle R_{s_{1}}(\Omega)}{\displaystyle
\sigma_{n}^{2}} \left({\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{m}}(\Omega)-
\frac{\displaystyle R_{s_{2}}(\Omega){\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)
{\bf G}_{s_{2}}^{H}(\Omega){\bf G}_{s_{m}}(\Omega)} {\displaystyle
R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)} {\frac{\displaystyle
R_{s_{1}}(\Omega)}{\displaystyle \sigma_{n}^{2}} \left({\bf G}_{s_{1}}^{H}(\Omega)
{\bf G}_{s_{1}}(\Omega)- \frac{\displaystyle R_{s_{2}}(\Omega)|{\bf
G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)|^{2}} {\displaystyle
R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)+1}
\label{eq:trpwo1}
\end{equation}
This expression is used below to investigate different parameters that
affect the total transfer function from source to
array output. By evaluating the expression from the target direction we
can quantify the impact on the target signal, and analogously, an evaluation from the jammer
direction yields the echo suppression of the loudspeaker. A similar evaluation can also be performed
for realistic situations in a car using measured transfer functions.
\chapter{Target Distortion and Jammer Suppression}
This evaluation
assumes a simple model, point sources and spherical wave propagation in a homogeneous
medium. A simple situation can still give guidelines for different parameter
choices, such as element placing, and quality and spread in microphones and
loudspeakers in a more complex environment.
From the simple model we obtain transfer functions from a point source to each
element
\begin{equation}
G_{m,l}(\omega)=\frac{e^{-j\omega\tau_{m,l}}}{r_{m,l}}
\end{equation}
where $\tau_{m,l}$ is the delay of the signal and $r_{m,l}$ is the distance from
source to array element. In the far-field $r_{m,l}$ is much larger than the
array aperture and a linear array simplifies the array response vector to
\begin{equation}
{\bf G}_{s_{m}}(\omega,\theta)=
\frac{e^{-j\omega\tau_{bulk}}}{r_{m}}\left( 1, e^{-j\omega \frac{d \sin(\theta)}{c}}
\cdots e^{-j\omega(N-1)\frac{d \sin(\theta)}{c}}\right)^{T}
\end{equation}
where $e^{-j\omega\tau_{bulk}}$ is an initial bulk delay, $d$ the distance between
adjacent elements and $c$ the sound propagation velocity in the medium. Below, we again work
only with sampled and bandlimited signals.
\section{Target Direction}
The digital target direction total transfer function is given by inserting ${\bf G}_{s_{1}}$ in eq.
(\ref{eq:trpwo1}). We obtain
\begin{equation}
H_{1}(\Omega)=
\frac{F(\Omega)\frac{ \displaystyle R_{s_{1}}(\Omega)}{\displaystyle
\sigma_{n}^{2}}\left({\bf G}_{s_{1}}^{H}(\Omega) {\bf G}_{s_{1}}(\Omega)-\frac{
\displaystyle R_{s_{2}}(\Omega){\bf G}_{s_{1}}^{H}(\Omega){\displaystyle \bf
G}_{s_{2}}(\Omega) {\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{1}}(\Omega)}{\displaystyle R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega) {\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)}
{\frac{ \displaystyle R_{s_{1}}(\Omega)}{ \displaystyle \sigma_{n}^{2}}\left({\bf
G}_{s_{1}}^{H}(\Omega) {\bf G}_{s_{1}}(\Omega)-\frac{\displaystyle
R_{s_{2}}(\Omega)|{\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)|^{2}}
{\displaystyle R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)+1}
\label{eq:trpwo11}
\end{equation}
In
this direction
\begin{equation}
\mbox{SINR}_{eq}(\Omega)=\frac{R_{s_{1}}(\Omega)}{\sigma_{n}^{2}}\left({\bf G}_{s_{1}}^{H}(\Omega)
{\bf G}_{s_{1}}(\Omega)-\frac{R_{s_{2}}(\Omega)|{\bf G}_{s_{1}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)|^{2}} {R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega) {\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)
\end{equation}
can be viewed as an Equivalent Signal-to-Interference-Noise Ratio, $\mbox{SINR}_{eq}(\Omega)$
simplifying $H_{1}(\Omega)$ to the conventional Wiener expression
\begin{equation}
H_{1}(\Omega)=\frac{ F(\Omega) \displaystyle \mbox{SINR}_{eq}(\Omega)}{ 1+\displaystyle \mbox{SINR}_{eq}(\Omega)}=
\frac{F(\Omega)}{1+\frac{1}{ \displaystyle \mbox{SINR}_{eq}(\Omega)}}
\end{equation}
The expression
${\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)$ can be interpreted as a
scalar product and can be
rewritten as $ {\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)=||{\bf
G}_{s_{1}}(\Omega)|| ||{\bf G}_{s_{2}}(\Omega)|| \rho(\Omega)
$
where $\rho(\Omega)$ is a measure of closeness between ${\bf G}_{s_{1}}(\Omega)$ and
${\bf G}_{s_{2}}(\Omega)$. From Schwarz inequality it follows that
$|\rho(\Omega)|\leq 1$. An alternative expression of
$\mbox{SINR}_{eq}$ is obtained by using the relation above
$$
\mbox{SINR}_{eq}(\Omega)=\frac{R_{s_{1}}(\Omega)||{\bf
G}_{s_{1}}(\Omega)||^{2}}{\sigma_{n}^{2}}\left(\frac{R_{s_{2}}(\Omega)
||{\bf G}_{s_{2}}(\Omega)||^2(1-|\rho(\Omega)|^{2})+\sigma_{n}^{2}}
{R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^2 +\sigma_{n}^{2}}\right)
$$
The maximum effective signal-to-noise ratio is obtained if $\rho(\Omega)=0$ i.e.
the vectors ${\bf G}_{s_{1}}(\Omega)$ and
${\bf G}_{s_{2}}(\Omega)$ are orthogonal.
We also define a Signal to Noise Ratio at each element given as the target signal power
to the measurement noise
\begin{equation}
\mbox{SNR}=\frac{\frac{1}{2 \pi }\int_{-\pi}^{\pi} R_{s_{1}}(\Omega) d\Omega}
{\sigma_{n}^{2}}
\end{equation}
and a Signal to Interference Ratio is also defined
\begin{equation}
\mbox{SIR}=\frac{\frac{1}{2 \pi }\int_{-\pi}^{\pi} R_{s_{1}}(\Omega) d\Omega}
{\frac{1}{2 \pi }\int_{-\pi}^{\pi} R_{s_{2}}(\Omega) d\Omega}
\end{equation}
\subsection{Examples}
In order to obtain low distortion of the target signal, $\mbox{SINR}_{eq}$ must be
large. A parameter of interest is the correlation coefficient $ \rho(\Omega)$
which should be kept small, i.e. the target and jammer array responses should be
orthogonal and the
target signal-to-noise ratio large, see
Fig. \ref{fig:H1G1}. However, when the array responses are similar
${\bf G}_{s_{1}}(\Omega)\approx {\bf G}_{s_{2}}(\Omega)$ over some frequency
range, the distortion of the target signal might become large and
highly dependent on the jammer spectral density $R_{s_{2}}(\Omega)$.
When the noise level is small
compared to $R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^2$, an approximative
expression with close array responses is given by
\begin{equation}
H_{1}(\Omega)\approx
\frac{F(\Omega)}{1+\frac{\displaystyle R_{s_{2}}(\Omega)}{\displaystyle
R_{s_{1}}(\Omega)}} \label{eq:h1a}
\end{equation}
In Fig. \ref{fig:H1G1} the target and jammer have identical
spectral densities, and are approximately flat within 600 Hz - 3600 Hz. When the
signal to noise ratio is improved, see Fig. \ref{fig:H1G14}, the jammer affects
the target signal less, provided they are well separated in directions. Employing more
array elements gives also a possibility to separate the target and
jammer for smaller angular differences, compare figures \ref{fig:H1G1} and\ref{fig:H1G11}. When the jammer
is stronger, see Fig. \ref{fig:H1G12}, the transfer function is determined by the
target to jammer ratio for small angular
differences, as
predicted by Eq. (\ref{eq:h1a}).
Finally, Fig. \ref{fig:H1G13} illustrates the effect of
spatial aliasing when the array elements are too far apart.
\begin{figure}
[l] \centerline{ \scaledpicturee
170 mm by 150 mm (h1c10.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
\put(13.5,0.0){\makebox(0,0)[lb]{ $\theta \ [^{\circ }]$}}
\put(2.2,7.8){\makebox(0,0)[l]{$[dB]$}}
%\thicklines
\put(6.4,4.6){\makebox(0,0)[l]{\vector(-1,1){0.4}}}
\put(5.2,8.1){\makebox(0,0)[l]{\vector(1,-1){0.4}}}
\put(2,8.4){\makebox(0,0)[l]{$|H_{1}(\Omega)|$}}
\put(4.5,8.7){\makebox(0,0)[l]{$f=3220$Hz}}
\put(6.5,4.3){\makebox(0,0)[l]{$f=720$Hz}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Target transfer function $H_{1}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.05m, SNR=10dB, SIR=0dB}
\label{fig:H1G1}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (H1c16.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
\put(13.5,0.0){\makebox(0,0)[lb]{ $\theta \ [^{\circ }]$}}
\put(2.2,7.8){\makebox(0,0)[l]{$[dB]$}}
\put(5.0,7.4){\makebox(0,0)[l]{\vector(-1,1){0.4}}}
\put(3.9,8.4){\makebox(0,0)[l]{\vector(1,-1){0.4}}}
\put(2,8.4){\makebox(0,0)[l]{$|H_{1}(\Omega)|$}}
\put(3.7,8.9){\makebox(0,0)[l]{$f=3220$Hz}}
\put(5.0,7.1){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
%\put(4.4,0.2){\makebox(0,0)[l]{$0.8$}}
%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Target transfer function $H_{1}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.05m, SNR=30dB, SIR=0dB}
\label{fig:H1G14}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee 170 mm by 150 mm (h1c11.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
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\put(5.3,6.2){\makebox(0,0)[l]{\vector(-1,1){0.4}}}
\put(4.2,8.2){\makebox(0,0)[l]{\vector(1,-1){0.4}}}
\put(2,8.4){\makebox(0,0)[l]{$|H_{1}(\Omega)|$}}
\put(4,8.7){\makebox(0,0)[l]{$f=3220$Hz}}
\put(5.4,6){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
%\put(4.4,0.2){\makebox(0,0)[l]{$0.8$}}
%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Target transfer function $H_{1}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=7, d=0.05m, SNR=10dB, SIR=0dB}
\label{fig:H1G11}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h1c12.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\put(4.1,8.4){\makebox(0,0)[l]{\vector(1,-1){0.4}}}
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\put(3.6,8.8){\makebox(0,0)[l]{$f=3220$Hz}}
\put(5.4,7){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
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%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Target transfer function $H_{1}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=7, d=0.05m, SNR=10dB, SIR=-10dB}
\label{fig:H1G12}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h1c14.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
\put(13.5,0.0){\makebox(0,0)[lb]{ $\theta \ [^{\circ }]$}}
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\put(3.5,8.9){\makebox(0,0)[l]{$f=3220$Hz}}
\put(4.85,7.3){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
%\put(4.4,0.2){\makebox(0,0)[l]{$0.8$}}
%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Target transfer function $H_{1}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=7, d=0.2m, SNR=10dB, SIR=-10dB}
\label{fig:H1G13}
\end{figure}
\newpage
\section{Jammer Direction}
In the jammer direction the transfer function is given by Eq.
(\ref{eq:trpwo1}) with $G_{s_{2}}$ inserted yielding
\begin{equation}
H_{2}(\Omega)=
\frac{F(\Omega)\frac{\displaystyle R_{s_{1}}(\Omega)}{\displaystyle
\sigma_{n}^{2}}\left({\bf G}_{s_{1}}^{H}(\Omega) {\bf
G}_{s_{2}}(\Omega)-\frac{\displaystyle R_{s_{2}}(\Omega){\bf
G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega) {\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)}{\displaystyle R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega)
{\bf G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)} {\frac{\displaystyle
R_{s_{1}}(\Omega)}{\displaystyle \sigma_{n}^{2}} \left({\bf G}_{s_{1}}^{H}(\Omega)
{\bf G}_{s_{1}}(\Omega)- \frac{\displaystyle R_{s_{2}}(\Omega)|{\bf
G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)|^{2}} {\displaystyle
R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)+1}
\label{eq:trpwo2}
\end{equation}
This transfer function can also be expressed using the $\mbox{SINR}_{eq}$
and inner product interpretation
\begin{equation}
H_{2}(\Omega)=
\frac{F(\Omega) R_{s_{1}}(\Omega)||{\bf G}_{s_{1}}(\Omega)||||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} {\left(R_{s_{2}}(\Omega)||{\bf
G}_{s_{2}}(\Omega)||^{2} +\sigma_{n}^{2} \right)
\left( \mbox{SINR}_{eq}(\Omega)+1\right)}
\label{eq:trpwo3}
\end{equation}
which is close to zero when target and jammer are well separated
\subsection{Examples}
In Figs. \ref{fig:H2G2} to \ref{fig:H2G23}, the target and
jammer have identical spectral densities and are approximately flat within
600 Hz - 3600 Hz.
We observe that it is essential that the scalar product between ${\bf
G}_{s_{2}}(\Omega)$ and ${\bf G}_{s_{1}}(\Omega)$ is kept small, i.e. the two sources
must be well separated, see Fig. \ref{fig:H2G2}, or more
elements must be used, see Fig. \ref{fig:H2G21}.
Other essential parameters are the noise
level at the array elements, see Fig. \ref{fig:H2G24}, and the strength of
the jamming signal, see Fig. \ref{fig:H2G25}. Figure \ref{fig:H2G23}
again shows the effect of spatial undersampling.
Observe that the function ${\bf G}_{s_{2}}(\Omega)$ is
actually sampled twice i.e. it is periodic both in $\Omega$ and $\theta$ \cite{oc94}.
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h2c3.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
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\put(2.2,7.8){\makebox(0,0)[l]{$[dB]$}}
\put(13.5,0.0){\makebox(0,0)[lb]{ $\theta $}}
\put(8,7.5){\makebox(0,0)[l]{\vector(-1,-1){0.4}}}
\put(6.5,5.2){\makebox(0,0)[l]{\vector(1,1){0.4}}}
\put(2,8.4){\makebox(0,0)[l]{$|H_{2}(\Omega)|$}}
\put(4.5,4.6){\makebox(0,0)[l]{$f=3220$Hz}}
\put(7.2,8){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
%\put(4.4,0.2){\makebox(0,0)[l]{$0.8$}}
%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Jammer transfer function $H_{2}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.05m, SNR=10dB, SIR=0dB}
\label{fig:H2G2}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h2c2.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\put(4.5,2.3){\makebox(0,0)[l]{$f=3220$Hz}}
\put(7.2,6.6){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
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%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
%\put(1.8,7.1){\makebox(0,0)[r]{$-20$}}
%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Jammer transfer function $H_{2}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=7, d=0.05m, SNR=10dB, SIR=0dB}
\label{fig:H2G21}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h2c6.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
\put(0,0){\makebox(0,0)[lb]{.}}
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\put(7.2,6.4){\makebox(0,0)[l]{$f=720$Hz}}
%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
%\thicklines
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%\put(12.4,0.2){\makebox(0,0)[l]{$3.2$}}
%\put(8.4,0.2){\makebox(0,0)[l]{$2.0$}}
%\put(1.8,9.3){\makebox(0,0)[r]{$0$}}
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%\put(1.8,4.9){\makebox(0,0)[r]{$-40$}}
%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Jammer transfer function $H_{2}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.05m, SNR=30dB, SIR=0dB}
\label{fig:H2G24}
\end{figure}
\begin{figure}
[htb] \centerline{ \scaledpicturee
170 mm by 150 mm (h2c7.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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%\put(14.5,0.0){\makebox(0,0)[lb]{ $f [kHz] $}}
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%\put(1.8,2.7){\makebox(0,0)[r]{$-60$}}
%\put(1.8,0.5){\makebox(0,0)[r]{$-80$}}
%\multiput(2.2,0.37)(0.0,0.20){29}{\line(1,0){13.0}}
\end{picture}
\caption{Jammer transfer function $H_{2}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.05m, SNR=30dB, SIR=-10dB}
\label{fig:H2G25}
\end{figure}
\begin{figure}
[l] \centerline{ \scaledpicturee
170 mm by 150 mm (h2c5.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\end{picture}
\caption{Jammer transfer function $H_{2}(\Omega)$ for varying jamming directions at
10 different frequencies varying from 720 Hz
to 3220 Hz. Linear array with N=3, d=0.2m, SNR=30dB, SIR=-10dB}
\label{fig:H2G23}
\end{figure}
\chapter{Varying Signal Levels during Adaptation}
In the adaptive mode, see Fig. \ref{fig:abfsc}, we also want to vary the signal
levels of the gathered calibration signals in order to control the desired performance of
the beamformer. This mean that level controls for the
target calibration signal as well as for the jamming calibration signal are included. These levels
can be adjusted independently.
\begin{figure}[htb]
\begin{center}
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\put(144,96){target}
\put(53,65){signals}
\put(44,80){calibration}
\put(54,96){jammer}
\put(350,218){$y[k]$}
\put(16,180){$x_{N}[k]$}
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\put(360,172){$y_{r}[k]$}
\put(377,183){+}
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\put(37,59){\framebox(74,50){}}
\put(14,173){\circle*{8}}
\put(14,209){\circle*{8}}
\put(14,234){\circle*{8}}
\put(14,186){\circle*{1}}
\put(14,191){\circle*{1}}
\put(14,196){\circle*{1}}
\end{picture}}
\end{center}
~\vspace{-2.5cm}\\
\caption{A description of the adaptive mode}
\label{fig:abfsc}
\end{figure}
\section{Target Distortion}
The memory
target signal is amplified a factor $\alpha$, implying that the spectrum
$R_{s_{1}}(\Omega)$ is multiplied by $\alpha^{2}$ and the calibrated jamming signal
is analogously amplified with a factor $\beta$.
This is inserted in eq. (\ref{eq:trpwo11}) yielding
\begin{equation}
H_{1}(\Omega)=
\frac{F(\Omega)\frac{ \displaystyle \alpha^{2}R_{s_{1}}(\Omega)}{\displaystyle
\sigma_{n}^{2}}\left({\bf G}_{s_{1}}^{H}(\Omega) {\bf G}_{s_{1}}(\Omega)-\frac{
\displaystyle \beta^{2}R_{s_{2}}(\Omega){\bf G}_{s_{1}}^{H}(\Omega){\displaystyle \bf
G}_{s_{2}}(\Omega) {\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{1}}(\Omega)}{\displaystyle \beta^{2}R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega) {\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)}
{\frac{ \displaystyle \alpha^{2}R_{s_{1}}(\Omega)}{ \displaystyle \sigma_{n}^{2}}\left({\bf
G}_{s_{1}}^{H}(\Omega) {\bf G}_{s_{1}}(\Omega)-\frac{\displaystyle
\beta^{2}R_{s_{2}}(\Omega)|{\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)|^{2}}
{\displaystyle \beta^{2}R_{s_{2}}(\Omega){\bf G}_{s_{2}}^{H}(\Omega){\bf
G}_{s_{2}}(\Omega)+\sigma_{n}^{2}}\right)+1}
\label{eq:trpwab}
\end{equation}
and the $\mbox{SINR}_{eq}(\Omega)$ is given by
\begin{equation}
\mbox{SINR}_{eq}(\Omega)=\frac{\alpha^{2}R_{s_{1}}(\Omega)||{\bf
G}_{s_{1}}(\Omega)||^{2}}{\sigma_{n}^{2}}\left(\frac{\beta^{2}R_{s_{2}}(\Omega)
||{\bf G}_{s_{2}}(\Omega)||^2(1-|\rho(\Omega)|^{2})+\sigma_{n}^{2}}
{\beta^{2}R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^2 +\sigma_{n}^{2}}\right)
\end{equation}
The transfer function $H_{1}(\Omega)$ can be used to calculate the target signal
distortion ratio, which we define as
\begin{equation}
D=\frac{\frac{1}{2 \pi }\int_{-\pi}^{\pi} R_{s_{1}}(\Omega)|H_{1}(\Omega)-F(\Omega)|^{2}d\Omega}
{\frac{1}{2 \pi }\int_{-\pi}^{\pi} R_{s_{1}}(\Omega)d\Omega}
\end{equation}
Observe that the actual real target signal, jammer signals and background noise are not affected by
these amplifications, since they are only performed
on the memory signals in order to control the total transfer function.
Since
\begin{equation}
H_{1}(\Omega)=
\frac{F(\Omega)}{1+\frac{1}{ \displaystyle \mbox{SINR}_{eq}(\Omega)}}
\end{equation}
larger $\mbox{SINR}_{eq}$ gives reduced target distortion.
The distortion is strongly dependent on the amplitude of the
calibration signal, see Figs. \ref{fig:lalph1} and \ref{fig:lalph2}. These figures present
the distortion ratio $D$ versus the amplifying parameter
$\alpha$ for different jamming
directions. If $\alpha$ is small, $\mbox{SINR}_{eq}(\Omega)$ will also be small causing
$H_{1}(\Omega)$, to deviate more from $F(\Omega)$ which in turn yields high distortion.
For larger $\alpha$, the distortion is lower.
In Fig. \ref{fig:lalph2}
more microphone elements are used, which give better source
resolution and smaller $\rho(\Omega)$. The actual dependence is given by a
sind-function(for a plane wave and a linear array case). This can easily be verified
by evaluating ${\bf G}_{s_{1}}^{H}(\Omega){\bf G}_{s_{2}}(\Omega)$.
% Mer forklaring om detta skall vara med
%In this example plane
%waves and a linear array have been used.
In Figs. \ref{fig:lbeta1} and \ref{fig:lbeta2}
the distortion function $D$ is presented for varying memory jammer amplification $\beta$.
A small $\beta$ i.e. when $\beta^{2}R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^2$
is of the
same order as $\sigma_{n}^{2}$, makes the target distortion almost independent
of the jamming signal.
For a large $\beta$ the main influence will be via $\rho(\Omega)$.
For well-separated sources, an expression for the
equivalent signal-to-noise-ratio is
\begin{equation}
\mbox{SINR}_{eq}(\Omega)\approx \frac{\alpha^{2}R_{s_{1}}(\Omega)||{\bf
G}_{s_{1}}(\Omega)||^{2}}{\sigma_{n}^{2}}(1-|\rho(\Omega)|^{2})
\end{equation}
%detta aer obegripligt, formulera om om det skall vara med
%This behaviour is even more accentuated when the number of array elements is increased
%, see Fig. \ref{fig:lbeta2}.
%If $\rho(\Omega)$ is close to one i.e. the depence is high
%see for $\theta=10$ in figures \ref{fig:lbeta1} and \ref{fig:lbeta2}.
A good approximation with close target and jammer will be
\begin{equation}
\mbox{SINR}_{eq}(\Omega)\approx \frac{\alpha^{2}R_{s_{1}}(\Omega)||{\bf
G}_{s_{1}}(\Omega)||^{2}}
{\beta^{2}R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^2 +\sigma_{n}^{2}}
\end{equation}
and the distortion depends directly on the target to jammer ratio.
\section{Jammer Suppression}
The suppression of a jammer for
varying $\alpha$and $\beta$ has also been investigated
in relation to
the jammer level at a single array element. We define the suppression ratio as
\begin{equation}
S=\frac{\frac{1}{2 \pi }\int_{-\pi}^{\pi}
R_{s_{2}}(\Omega)|H_{2}(\Omega)|^{2}d\Omega}{\frac{1}{2 \pi } \int_{-\pi}^{\pi} R_{s_{2}}(\Omega) d\Omega}
\end{equation}
The jammer transfer function is given by
\begin{equation}
H_{2}(\Omega)=
\frac{F(\Omega) \alpha^{2}R_{s_{1}}(\Omega)||{\bf G}_{s_{1}}(\Omega)||||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} {\left(\beta^{2}R_{s_{2}}(\Omega)||{\bf
G}_{s_{2}}(\Omega)||^{2} +\sigma_{n}^{2} \right)
\left( \mbox{SINR}_{eq}(\Omega)+1\right)}
\label{eq:trpwoab}
\end{equation}
corresponding to eq. (\ref{eq:trpwo3}) with
$\alpha$ and $\beta$ included.
In Figs. \ref{fig:jalph1} and \ref{fig:jalph2},
the suppression ratio $S$ is presented for different jamming directions with $\alpha$ as
parameter. As the figures show, the true jammer suppression is strongly dependent on the
amplitude of the calibration memory target signal. The interaction of Eq.
(\ref{eq:trpwoab}) is, however, rather involved since the parameter $\alpha$ affects both
nominator and denominator.
For small $\alpha$,
i.e. when $\alpha^{2}R_{s_{1}}(\Omega)||{\bf G}_{s_{1}}(\Omega)||^{2}$,
the jammer transfer function $H_{2}(\Omega)$ will be small,
i.e. good suppression is obtained and the transfer function can
be approximated by
\begin{equation}
H_{2}(\Omega)\approx
\frac{F(\Omega) \alpha^{2}R_{s_{1}}(\Omega)||{\bf G}_{s_{1}}(\Omega)||||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} {\beta^{2}R_{s_{2}}(\Omega)||{\bf
G}_{s_{2}}(\Omega)||^{2} +\sigma_{n}^{2}}
\label{eq:trpwoaba}
\end{equation}
For large
$\alpha$:s, the transfer function can be approximated by
\begin{equation}
H_{2}(\Omega)\approx
\frac{F(\Omega)||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} {\left(\beta^{2}R_{s_{2}}(\Omega)||{\bf
G}_{s_{2}}(\Omega)||^{2} +\sigma_{n}^{2} \right)
\left( \frac{\displaystyle ||{\bf
G}_{s_{1}}(\Omega)||}{\displaystyle \sigma_{n}^{2}}(1-|\rho(\Omega)|^{2}) \right)}
\end{equation}
In
Figs. \ref{fig:jbeta1} and
\ref{fig:jbeta2}, the strength of the calibration jammer is varied.
A small $\beta$ implies that
the transfer function can be approximated by
\begin{equation}
H_{2}(\Omega)\approx
\frac{F(\Omega)||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} { ||{\bf
G}_{s_{1}}(\Omega)||}
\end{equation}
Here $\beta^{2}R_{s_{2}}(\Omega)||{\bf G}_{s_{2}}(\Omega)||^{2}$ is assumed to be of the
same strength or weaker than $\sigma_{n}^{2}$, and accordingly, $H_{2}(\Omega)$ will
not be dependent on
the calibration memory jammer or target strength. The direction is the dominating parameter.
This is the classic situation for a data
independent beamformer, and this is the performance the beamformer will show when
there is only target signal and uncorrelated noise at each array element.
When investigating larger $\beta$:s, the
calibration memory jammer rules behaviour.
A good approximation of the true jammer transfer function is
\begin{equation}
H_{2}(\Omega)\approx
\frac{F(\Omega)||{\bf
G}_{s_{2}}(\Omega)||\rho(\Omega)} {\left(\beta^{2}R_{s_{2}}(\Omega)||{\bf
G}_{s_{2}}(\Omega)||^{2} \right)
\left( \frac{\displaystyle ||{\bf
G}_{s_{1}}(\Omega)||}{\displaystyle \sigma_{n}^{2}}(1-|\rho(\Omega)|^{2}) \right)}
\end{equation}
The suppression ratio is proportional to $\beta^{4}$, see
Figs. \ref{fig:jbeta1} and \ref{fig:jbeta2}.
\subsection{ Multipath Jammer Situation}
In the above examples, the jamming signal was assumed to arrive as a plane wave. We conclude
with some
examples with multipath jamming, which are more closely related to a realistic
situation.
The jammer transfer function is formed by a sum of plane wave array
response vectors
\begin{equation}
{\bf G}_{s_{2}}(\omega)=\frac{1}{K}\sum_{k=1}^{K} {\bf d}(\omega,\theta_{k})
\end{equation}
where
$$
{\bf d}(\omega,\theta_{k})=\left( 1, e^{-j\omega \frac{d \sin(\theta_{k})}{c}}
\cdots e^{-j\omega(M-1)\frac{d \sin(\theta_{k})}{c}}\right)^{T}
$$
Some examples have been calculated for
two different multipath situations. In the first situation the jamming signal angle
$\theta_{k}$ was more concentrated and closer to the target signal,
see Figs. \ref{fig:jbetm1} and \ref{fig:jbetm2}.
In the second situation, $\theta_{k}$:s
were more spread out and further away from the target, see
Figs. \ref{fig:jbetm3} and \ref{fig:jbetm4}.
We observe that the ability to suppress a multipath jammer is as good
as for a single plane wave. The reason is that since the jamming
signal originates from a single source only, it only increases the rank of the
power spectral
density matrix by one. This means that the adaptive beamformer can treat the
signal as if it comes from "one" direction.
\begin{figure}
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170 mm by 150 mm (svlssta1.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\put(6.2,7){\makebox(0,0)[l]{$\theta=10$}}
\put(13.5,0.0){\makebox(0,0)[lb]{ $\alpha [$dB$]$}}
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\caption{Target distortion $D$ versus $\alpha$.
When $\alpha=1$ (0 dB) it corresponds to SNR=30dB and SIR=0dB. Linear array with
d=0.05 m, N=3. Jamming directions $[10,45,60]$.}
\label{fig:lalph1}
\end{figure}
\begin{figure}
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170 mm by 150 mm (svlssta2.e scaled 600)}
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\caption{Target distortion $D$ versus $\alpha$.
When $\alpha=1$ (0 dB) it corresponds to SNR=30dB and SIR=0dB. Linear array with
d=0.05 m, N=3. Jamming directions $[10,45,60]$.}
\label{fig:lalph2}
\end{figure}
\begin{figure}
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170 mm by 150 mm (svlsstb2.e scaled 600)}
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\caption{Target distortion $D$ versus $\beta$.
When $\beta=1$ (0dB) it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=3. Jamming directions $[10,45,60]$.}
\label{fig:lbeta1}
\end{figure}
\begin{figure}
[l] \centerline{ \scaledpicturee
170 mm by 150 mm (svlsstb3.e scaled 600)}
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\caption{Target distortion $D$ versus $\beta$.
When $\beta=1$ (0dB) it corresponds to SIR=0dB, SNR=30dB. Linear array with
d=0.05 m, N=7. Jamming directions $[10,45,60]$.}
\label{fig:lbeta2}
\end{figure}
\begin{figure} [l]
\centerline{\scaledpicturee 170 mm by 150 mm (jasua1.e scaled 600)}
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\end{picture}
\caption{Jammer suppression $S$ versus $\alpha$.
When $\alpha=1$ (0 dB) it corresponds to SNR=30dB and SIR=0dB. Linear
array with d=0.05 m, N=3. Jamming directions $[10,45,60]$.}
\label{fig:jalph1}
\end{figure}
\begin{figure} [l]
\centerline{\scaledpicturee 170 mm by 150 mm (jasua2.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\caption{Jammer suppression $S$ versus $\alpha$.
When $\alpha=1$ (0 dB) it corresponds to SNR=30dB and SIR=0dB. Linear
array with d=0.05 m, N=7. Jamming directions $[10,45,60]$.}
\label{fig:jalph2}
\end{figure}
\begin{figure}[l]
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\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=3. Jamming directions $[10,45,60]$.}
\label{fig:jbeta1}
\end{figure}
\begin{figure}[l]
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\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=7. Jamming directions $[10,45,60]$.}
\label{fig:jbeta2}
\end{figure}
\begin{figure}[l] \centerline{ \scaledpicturee170 mm by 150 mm (jasbmth5.e scaled
600)}
\begin{picture}(14.8,0)(1.2,0)
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\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=3. Jamming directions $[9,10,11,12,13]$.}
\label{fig:jbetm1}
\end{figure}
\begin{figure}[l]
\centerline{\scaledpicturee 170 mm by 150 mm (jasbmth7.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
%\put(0,0){\makebox(0,0)[lb]{.}}
\put(2,8.4){\makebox(0,0)[l]{$S [$dB$]$}}
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\end{picture}
\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=7. Jamming directions $[9,10,11,12,13]$.}
\label{fig:jbetm2}
\end{figure}
\begin{figure}
[l] \centerline{ \scaledpicturee
170 mm by 150 mm (jasbmth4.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
%\put(0,0){\makebox(0,0)[lb]{.}}
\put(2,8.4){\makebox(0,0)[l]{$S [$dB$]$}}
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\end{picture}
\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB, SNR=30dB. Linear array
with d=0.05 m, N=3. Jamming directions $[10,30,45,60,80]$.}
\label{fig:jbetm3}
\end{figure}
\begin{figure}
[l] \centerline{ \scaledpicturee
170 mm by 150 mm (jasbmth6.e scaled 600)}
\begin{picture}(14.8,0)(1.2,0)
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\caption{Jammer suppression $S$ versus $\beta$.
When $\beta=1$ it corresponds to SIR=0dB,SNR=30dB. Linear array
with d=0.05 m, N=7. Jamming directions $[10,30,45,60,80]$.}
\label{fig:jbetm4}
\end{figure}
\chapter{Conclusions and Future Work}
A cumbersome part of the microphone array realizations is the
calibration of the microphones and analog channels at the inputs. A
self-calibrating realization has thus been developed. This beamformer has been
studied from an analytical point of view by using noncausal Wiener solutions. The study has
gained insight into which parameters affect the jammer suppression and give target
signal distortion.
An interesting application is when the jammer is a hands-free loudspeaker.
Placing and design of the hands-free loudspeaker provide different
performance.
The study have shown that in order to achieve good echo suppression and small
target distortion, the target signal and jammer signals should be well-separated in
spatial domain in order to make the inner product between their transfer functions small.
A method to improve spatial resolution at low frequencies is
to consider a sub-band beamformer with unequal element spacing in different frequency
bands.
Further work is
also needed on this basic beamformer type. We will employ
measured transfer functions, and study finite length FIR filters, as well as
combine the beamformer with a conventional echo canceller. The design
and placing of microphones and loudspeakers in a real environment will also be studied.
\include{teorcbfbib}
\end{document}