<s>
A	O
sensor	B-Algorithm
array	I-Algorithm
is	O
a	O
group	O
of	O
sensors	O
,	O
usually	O
deployed	O
in	O
a	O
certain	O
geometry	O
pattern	O
,	O
used	O
for	O
collecting	O
and	O
processing	O
electromagnetic	O
or	O
acoustic	O
signals	O
.	O
</s>
<s>
The	O
advantage	O
of	O
using	O
a	O
sensor	B-Algorithm
array	I-Algorithm
over	O
using	O
a	O
single	O
sensor	O
lies	O
in	O
the	O
fact	O
that	O
an	O
array	O
adds	O
new	O
dimensions	O
to	O
the	O
observation	O
,	O
helping	O
to	O
estimate	O
more	O
parameters	O
and	O
improve	O
the	O
estimation	O
performance	O
.	O
</s>
<s>
For	O
example	O
an	O
array	O
of	O
radio	O
antenna	O
elements	O
used	O
for	O
beamforming	B-Algorithm
can	O
increase	O
antenna	O
gain	O
in	O
the	O
direction	O
of	O
the	O
signal	O
while	O
decreasing	O
the	O
gain	O
in	O
other	O
directions	O
,	O
i.e.	O
,	O
increasing	O
signal-to-noise	O
ratio	O
(	O
SNR	O
)	O
by	O
amplifying	O
the	O
signal	O
coherently	O
.	O
</s>
<s>
Another	O
example	O
of	O
sensor	B-Algorithm
array	I-Algorithm
application	O
is	O
to	O
estimate	O
the	O
direction	B-Algorithm
of	I-Algorithm
arrival	I-Algorithm
of	O
impinging	O
electromagnetic	O
waves	O
.	O
</s>
<s>
A	O
third	O
examples	O
includes	O
chemical	B-Algorithm
sensor	I-Algorithm
arrays	I-Algorithm
,	O
which	O
utilize	O
multiple	O
chemical	O
sensors	O
for	O
fingerprint	O
detection	O
in	O
complex	O
mixtures	O
or	O
sensing	O
environments	O
.	O
</s>
<s>
Application	O
examples	O
of	O
array	O
signal	O
processing	O
include	O
radar/sonar	O
,	O
wireless	O
communications	O
,	O
seismology	O
,	O
machine	O
condition	O
monitoring	O
,	O
astronomical	O
observations	O
fault	O
diagnosis	O
,	O
etc	O
.	O
</s>
<s>
Using	O
array	O
signal	O
processing	O
,	O
the	O
temporal	O
and	O
spatial	O
properties	O
(	O
or	O
parameters	O
)	O
of	O
the	O
impinging	O
signals	O
interfered	O
by	O
noise	O
and	O
hidden	O
in	O
the	O
data	O
collected	O
by	O
the	O
sensor	B-Algorithm
array	I-Algorithm
can	O
be	O
estimated	O
and	O
revealed	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
sensor	B-Algorithm
array	I-Algorithm
is	O
assumed	O
to	O
be	O
in	O
the	O
far-field	B-Algorithm
of	O
a	O
signal	O
source	O
so	O
that	O
it	O
can	O
be	O
treated	O
as	O
planar	O
wave	O
.	O
</s>
<s>
The	O
delays	O
are	O
closely	O
related	O
to	O
the	O
incident	O
angle	O
and	O
the	O
geometry	O
of	O
the	O
sensor	B-Algorithm
array	I-Algorithm
.	O
</s>
<s>
The	O
process	O
of	O
time-shifting	O
signals	O
using	O
a	O
well	O
selected	O
set	O
of	O
delays	O
for	O
each	O
channel	O
of	O
the	O
sensor	B-Algorithm
array	I-Algorithm
so	O
that	O
the	O
signal	O
is	O
added	O
constructively	O
is	O
called	O
beamforming	B-Algorithm
.	O
</s>
<s>
Sensor	B-Algorithm
arrays	I-Algorithm
have	O
different	O
geometrical	O
designs	O
,	O
including	O
linear	O
,	O
circular	O
,	O
planar	O
,	O
cylindrical	O
and	O
spherical	O
arrays	O
.	O
</s>
<s>
There	O
are	O
sensor	B-Algorithm
arrays	I-Algorithm
with	O
arbitrary	O
array	O
configuration	O
,	O
which	O
require	O
more	O
complex	O
signal	O
processing	O
techniques	O
for	O
parameter	O
estimation	O
.	O
</s>
<s>
Digital	B-Algorithm
antenna	I-Algorithm
array	I-Algorithm
,	O
this	O
is	O
smart	O
antenna	O
with	O
multi	O
channels	O
digital	B-Algorithm
beamforming	I-Algorithm
,	O
usually	O
by	O
using	O
FFT	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
delay-and-sum	O
beamforming	B-Algorithm
.	O
</s>
<s>
For	O
direction	B-Algorithm
of	I-Algorithm
arrival	I-Algorithm
(	O
DOA	O
)	O
estimation	O
,	O
one	O
can	O
iteratively	O
test	O
time	O
delays	O
for	O
all	O
possible	O
directions	O
.	O
</s>
<s>
The	O
trial	O
angle	O
that	O
maximizes	O
the	O
mean	O
output	O
is	O
an	O
estimation	O
of	O
DOA	O
given	O
by	O
the	O
delay-and-sum	O
beamformer	B-Algorithm
.	O
</s>
<s>
Adding	O
an	O
opposite	O
delay	O
to	O
the	O
input	O
signals	O
is	O
equivalent	O
to	O
rotating	O
the	O
sensor	B-Algorithm
array	I-Algorithm
physically	O
.	O
</s>
<s>
Delay	O
and	O
sum	O
beamforming	B-Algorithm
is	O
a	O
time	O
domain	O
approach	O
.	O
</s>
<s>
It	O
is	O
simple	O
to	O
implement	O
,	O
but	O
it	O
may	O
poorly	O
estimate	O
direction	B-Algorithm
of	I-Algorithm
arrival	I-Algorithm
(	O
DOA	O
)	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
transforms	O
the	O
signal	O
from	O
the	O
time	O
domain	O
to	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
Frequency	O
domain	O
beamforming	B-Algorithm
algorithms	O
use	O
the	O
spatial	O
covariance	O
matrix	O
,	O
represented	O
by	O
.	O
</s>
<s>
This	O
model	O
is	O
of	O
central	O
importance	O
in	O
frequency	O
domain	O
beamforming	B-Algorithm
algorithms	O
.	O
</s>
<s>
Some	O
spectrum-based	O
beamforming	B-Algorithm
approaches	O
are	O
listed	O
below	O
.	O
</s>
<s>
The	O
Bartlett	O
beamformer	B-Algorithm
is	O
a	O
natural	O
extension	O
of	O
conventional	O
spectral	O
analysis	O
(	O
spectrogram	O
)	O
to	O
the	O
sensor	B-Algorithm
array	I-Algorithm
.	O
</s>
<s>
Though	O
the	O
MVDR/Capon	O
beamformer	B-Algorithm
can	O
achieve	O
better	O
resolution	O
than	O
the	O
conventional	O
(	O
Bartlett	O
)	O
approach	O
,	O
this	O
algorithm	O
has	O
higher	O
complexity	O
due	O
to	O
the	O
full-rank	O
matrix	O
inversion	O
.	O
</s>
<s>
Technical	O
advances	O
in	O
GPU	B-Architecture
computing	I-Architecture
have	O
begun	O
to	O
narrow	O
this	O
gap	O
and	O
make	O
real-time	O
Capon	O
beamforming	B-Algorithm
possible	O
.	O
</s>
<s>
MUSIC	O
(	O
MUltiple	O
SIgnal	O
Classification	O
)	O
beamforming	B-Algorithm
algorithm	O
starts	O
with	O
decomposing	O
the	O
covariance	O
matrix	O
as	O
given	O
by	O
Eq	O
.	O
</s>
<s>
Therefore	O
MUSIC	O
beamformer	B-Algorithm
is	O
also	O
known	O
as	O
subspace	O
beamformer	B-Algorithm
.	O
</s>
<s>
Compared	O
to	O
the	O
Capon	O
beamformer	B-Algorithm
,	O
it	O
gives	O
much	O
better	O
DOA	O
estimation	O
.	O
</s>
<s>
SAMV	B-Algorithm
beamforming	B-Algorithm
algorithm	O
is	O
a	O
sparse	O
signal	O
reconstruction	O
based	O
algorithm	O
which	O
explicitly	O
exploits	O
the	O
time	O
invariant	O
statistical	O
characteristic	O
of	O
the	O
covariance	O
matrix	O
.	O
</s>
<s>
It	O
achieves	O
superresolution	B-Algorithm
and	O
robust	O
to	O
highly	O
correlated	O
signals	O
.	O
</s>
<s>
One	O
of	O
the	O
major	O
advantages	O
of	O
the	O
spectrum	O
based	O
beamformers	B-Algorithm
is	O
a	O
lower	O
computational	O
complexity	O
,	O
but	O
they	O
may	O
not	O
give	O
accurate	O
DOA	O
estimation	O
if	O
the	O
signals	O
are	O
correlated	O
or	O
coherent	O
.	O
</s>
<s>
An	O
alternative	O
approach	O
are	O
parametric	O
beamformers	B-Algorithm
,	O
also	O
known	O
as	O
maximum	O
likelihood	O
(	O
ML	O
)	O
beamformers	B-Algorithm
.	O
</s>
<s>
One	O
example	O
of	O
a	O
maximum	O
likelihood	O
method	O
commonly	O
used	O
in	O
engineering	O
is	O
the	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
ML	O
beamformers	B-Algorithm
the	O
quadratic	O
penalty	O
function	O
is	O
used	O
to	O
the	O
spatial	O
covariance	O
matrix	O
and	O
the	O
signal	O
model	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
maximum	O
likelihood	O
beamformer	B-Algorithm
is	O
to	O
find	O
the	O
DOA	O
,	O
the	O
independent	O
variable	O
of	O
matrix	O
,	O
so	O
that	O
the	O
penalty	O
function	O
in	O
Eq	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
there	O
are	O
two	O
major	O
categories	O
of	O
maximum	O
likelihood	O
beamformers	B-Algorithm
:	O
Deterministic	O
ML	O
beamformers	B-Algorithm
and	O
stochastic	O
ML	O
beamformers	B-Algorithm
,	O
corresponding	O
to	O
a	O
deterministic	O
and	O
a	O
stochastic	O
model	O
,	O
respectively	O
.	O
</s>
<s>
In	O
order	O
to	O
simplify	O
the	O
optimization	O
algorithm	O
,	O
logarithmic	O
operations	O
and	O
the	O
probability	O
density	O
function	O
(	O
PDF	O
)	O
of	O
the	O
observations	O
may	O
be	O
used	O
in	O
some	O
ML	O
beamformers	B-Algorithm
.	O
</s>
<s>
If	O
the	O
Newton-Raphson	O
search	O
method	O
is	O
employed	O
to	O
minimize	O
the	O
beamforming	B-Algorithm
penalty	O
function	O
,	O
the	O
resulting	O
beamformer	B-Algorithm
is	O
called	O
Newton	O
ML	O
beamformer	B-Algorithm
.	O
</s>
<s>
Several	O
well-known	O
ML	O
beamformers	B-Algorithm
are	O
described	O
below	O
without	O
providing	O
further	O
details	O
due	O
to	O
the	O
complexity	O
of	O
the	O
expressions	O
.	O
</s>
<s>
In	O
deterministic	O
maximum	O
likelihood	O
beamformer	B-Algorithm
(	O
DML	O
)	O
,	O
the	O
noise	O
is	O
modeled	O
as	O
a	O
stationary	O
Gaussian	O
white	O
random	O
processes	O
while	O
the	O
signal	O
waveform	O
as	O
deterministic	O
(	O
but	O
arbitrary	O
)	O
and	O
unknown	O
.	O
</s>
<s>
In	O
stochastic	O
maximum	O
likelihood	O
beamformer	B-Algorithm
(	O
SML	O
)	O
,	O
the	O
noise	O
is	O
modeled	O
as	O
stationary	O
Gaussian	O
white	O
random	O
processes	O
(	O
the	O
same	O
as	O
in	O
DML	O
)	O
whereas	O
the	O
signal	O
waveform	O
as	O
Gaussian	O
random	O
processes	O
.	O
</s>
<s>
Method	O
of	O
direction	O
estimation	O
(	O
MODE	O
)	O
is	O
subspace	O
maximum	O
likelihood	O
beamformer	B-Algorithm
,	O
just	O
as	O
MUSIC	O
,	O
is	O
the	O
subspace	O
spectral	O
based	O
beamformer	B-Algorithm
.	O
</s>
<s>
Subspace	O
ML	O
beamforming	B-Algorithm
is	O
obtained	O
by	O
eigen-decomposition	O
of	O
the	O
sample	O
covariance	O
matrix	O
.	O
</s>
