<s>
In	O
signal	O
processing	O
,	O
multidimensional	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
covers	O
all	O
signal	O
processing	O
done	O
using	O
multidimensional	B-Algorithm
signals	I-Algorithm
and	O
systems	O
.	O
</s>
<s>
While	O
multidimensional	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
is	O
a	O
subset	O
of	O
signal	O
processing	O
,	O
it	O
is	O
unique	O
in	O
the	O
sense	O
that	O
it	O
deals	O
specifically	O
with	O
data	O
that	O
can	O
only	O
be	O
adequately	O
detailed	O
using	O
more	O
than	O
one	O
dimension	O
.	O
</s>
<s>
In	O
m-D	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
useful	O
data	O
is	O
sampled	O
in	O
more	O
than	O
one	O
dimension	O
.	O
</s>
<s>
Examples	O
of	O
this	O
are	O
image	B-Algorithm
processing	I-Algorithm
and	O
multi-sensor	O
radar	O
detection	O
.	O
</s>
<s>
Processing	O
in	O
multi-dimension	O
(	O
m-D	O
)	O
requires	O
more	O
complex	O
algorithms	O
,	O
compared	O
to	O
the	O
1-D	O
case	O
,	O
to	O
handle	O
calculations	O
such	O
as	O
the	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
due	O
to	O
more	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
Typically	O
,	O
multidimensional	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
is	O
directly	O
associated	O
with	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
because	O
its	O
complexity	O
warrants	O
the	O
use	O
of	O
computer	O
modelling	O
and	O
computation	O
.	O
</s>
<s>
A	O
multidimensional	B-Algorithm
signal	I-Algorithm
is	O
similar	O
to	O
a	O
single	O
dimensional	O
signal	O
as	O
far	O
as	O
manipulations	O
that	O
can	O
be	O
performed	O
,	O
such	O
as	O
sampling	B-Algorithm
,	O
Fourier	O
analysis	O
,	O
and	O
filtering	O
.	O
</s>
<s>
Multidimensional	O
sampling	B-Algorithm
requires	O
different	O
analysis	O
than	O
typical	O
1-D	O
sampling	B-Algorithm
.	O
</s>
<s>
Single	O
dimension	O
sampling	B-Algorithm
is	O
executed	O
by	O
selecting	O
points	O
along	O
a	O
continuous	O
line	O
and	O
storing	O
the	O
values	O
of	O
this	O
data	O
stream	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
multidimensional	O
sampling	B-Algorithm
,	O
the	O
data	O
is	O
selected	O
utilizing	O
a	O
lattice	O
,	O
which	O
is	O
a	O
"	O
pattern	O
"	O
based	O
on	O
the	O
sampling	B-Algorithm
vectors	O
of	O
the	O
m-D	O
data	O
set	O
.	O
</s>
<s>
Multidimensional	O
sampling	B-Algorithm
is	O
similar	O
to	O
classical	O
sampling	B-Algorithm
as	O
it	O
must	O
adhere	O
to	O
the	O
Nyquist	O
–	O
Shannon	O
sampling	B-Algorithm
theorem	O
.	O
</s>
<s>
It	O
is	O
affected	O
by	O
aliasing	B-Error_Name
and	O
considerations	O
must	O
be	O
made	O
for	O
eventual	O
Multidimensional	B-Algorithm
Signal	I-Algorithm
Reconstruction	O
.	O
</s>
<s>
A	O
multidimensional	B-Algorithm
signal	I-Algorithm
can	O
be	O
represented	O
in	O
terms	O
of	O
sinusoidal	O
components	O
.	O
</s>
<s>
This	O
is	O
typically	O
done	O
with	O
a	O
type	O
of	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
m-D	O
Fourier	B-Algorithm
transform	I-Algorithm
transforms	O
a	O
signal	O
from	O
a	O
signal	O
domain	O
representation	O
to	O
a	O
frequency	O
domain	O
representation	O
of	O
the	O
signal	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
digital	O
processing	O
,	O
a	O
discrete	O
Fourier	B-Algorithm
Transform	I-Algorithm
(	O
DFT	O
)	O
is	O
utilized	O
to	O
transform	O
a	O
sampled	O
signal	O
domain	O
representation	O
into	O
a	O
frequency	O
domain	O
representation	O
:	O
</s>
<s>
where	O
X	O
stands	O
for	O
the	O
multidimensional	O
discrete	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
x	O
stands	O
for	O
the	O
sampled	O
time/space	O
domain	O
signal	O
,	O
m	O
stands	O
for	O
the	O
number	O
of	O
dimensions	O
in	O
the	O
system	O
,	O
n	O
are	O
sample	O
indices	O
and	O
k	O
are	O
frequency	O
samples	O
.	O
</s>
<s>
Computational	O
complexity	O
is	O
usually	O
the	O
main	O
concern	O
when	O
implementing	O
any	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
For	O
multidimensional	B-Algorithm
signals	I-Algorithm
,	O
the	O
complexity	O
can	O
be	O
reduced	O
by	O
a	O
number	O
of	O
different	O
methods	O
.	O
</s>
<s>
The	O
computation	O
may	O
be	O
simplified	O
if	O
there	O
is	O
independence	O
between	O
variables	O
of	O
the	O
multidimensional	B-Algorithm
signal	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
fast	O
Fourier	B-Algorithm
transforms	I-Algorithm
(	O
FFTs	O
)	O
,	O
reduce	O
the	O
number	O
of	O
computations	O
by	O
a	O
substantial	O
factor	O
.	O
</s>
<s>
Typically	O
,	O
a	O
prototype	B-Application
filter	O
is	O
designed	O
in	O
a	O
single	O
dimension	O
and	O
that	O
filter	O
is	O
extrapolated	O
to	O
m-D	O
using	O
a	O
mapping	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Both	O
FIR	O
and	O
IIR	O
filters	O
can	O
be	O
transformed	O
to	O
m-D	O
,	O
depending	O
on	O
the	O
application	O
and	O
the	O
mapping	B-Algorithm
function	I-Algorithm
.	O
</s>
