<s>
In	O
pulsed	B-Algorithm
radar	I-Algorithm
and	O
sonar	B-Application
signal	O
processing	O
,	O
an	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
a	O
two-dimensional	O
function	O
of	O
propagation	O
delay	O
and	O
Doppler	O
frequency	O
,	O
.	O
</s>
<s>
It	O
represents	O
the	O
distortion	O
of	O
a	O
returned	O
pulse	B-Algorithm
due	O
to	O
the	O
receiver	O
matched	O
filter	O
(	O
commonly	O
,	O
but	O
not	O
exclusively	O
,	O
used	O
in	O
pulse	B-Algorithm
compression	I-Algorithm
radar	B-Application
)	O
of	O
the	O
return	O
from	O
a	O
moving	O
target	O
.	O
</s>
<s>
The	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
defined	O
by	O
the	O
properties	O
of	O
the	O
pulse	B-Algorithm
and	O
of	O
the	O
filter	O
,	O
and	O
not	O
any	O
particular	O
target	O
scenario	O
.	O
</s>
<s>
Many	O
definitions	O
of	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
exist	O
;	O
some	O
are	O
restricted	O
to	O
narrowband	O
signals	O
and	O
others	O
are	O
suitable	O
to	O
describe	O
the	O
delay	O
and	O
Doppler	O
relationship	O
of	O
wideband	O
signals	O
.	O
</s>
<s>
Often	O
the	O
definition	O
of	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
given	O
as	O
the	O
magnitude	O
squared	O
of	O
other	O
definitions	O
(	O
Weiss	O
)	O
.	O
</s>
<s>
The	O
matched	O
filter	O
output	O
as	O
a	O
function	O
of	O
time	O
(	O
the	O
signal	O
one	O
would	O
observe	O
in	O
a	O
radar	B-Application
system	I-Application
)	O
is	O
a	O
Doppler	O
cut	O
,	O
with	O
the	O
constant	O
frequency	O
given	O
by	O
the	O
target	O
's	O
Doppler	O
shift	O
:	O
.	O
</s>
<s>
Pulse-Doppler	B-Algorithm
radar	I-Algorithm
equipment	O
sends	O
out	O
a	O
series	O
of	O
radio	O
frequency	O
pulses	O
.	O
</s>
<s>
Each	O
pulse	B-Algorithm
has	O
a	O
certain	O
shape	O
(	O
waveform	O
)	O
—	O
how	O
long	O
the	O
pulse	B-Algorithm
is	O
,	O
what	O
its	O
frequency	O
is	O
,	O
whether	O
the	O
frequency	O
changes	O
during	O
the	O
pulse	B-Algorithm
,	O
and	O
so	O
on	O
.	O
</s>
<s>
If	O
the	O
waves	O
reflect	O
off	O
a	O
single	O
object	O
,	O
the	O
detector	O
will	O
see	O
a	O
signal	O
which	O
,	O
in	O
the	O
simplest	O
case	O
,	O
is	O
a	O
copy	O
of	O
the	O
original	O
pulse	B-Algorithm
but	O
delayed	O
by	O
a	O
certain	O
time	O
—	O
related	O
to	O
the	O
object	O
's	O
distance	O
—	O
and	O
shifted	O
by	O
a	O
certain	O
frequency	O
—	O
related	O
to	O
the	O
object	O
's	O
velocity	O
(	O
Doppler	O
shift	O
)	O
.	O
</s>
<s>
If	O
the	O
original	O
emitted	O
pulse	B-Algorithm
waveform	O
is	O
,	O
then	O
the	O
detected	O
signal	O
(	O
neglecting	O
noise	O
,	O
attenuation	O
,	O
and	O
distortion	O
,	O
and	O
wideband	O
corrections	O
)	O
will	O
be	O
:	O
</s>
<s>
This	O
motivates	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Different	O
pulse	B-Algorithm
shapes	O
(	O
waveforms	O
)	O
have	O
different	O
ambiguity	B-Algorithm
functions	I-Algorithm
,	O
and	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
relevant	O
when	O
choosing	O
what	O
pulse	B-Algorithm
to	O
use	O
.	O
</s>
<s>
The	O
ambiguity	B-Algorithm
function	I-Algorithm
plays	O
a	O
key	O
role	O
in	O
the	O
field	O
of	O
time	B-Algorithm
–	I-Algorithm
frequency	I-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
as	O
it	O
is	O
related	O
to	O
the	O
Wigner	O
–	O
Ville	O
distribution	O
by	O
a	O
2-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
This	O
relationship	O
is	O
fundamental	O
to	O
the	O
formulation	O
of	O
other	O
time	O
–	O
frequency	O
distributions	O
:	O
the	O
bilinear	O
time	O
–	O
frequency	O
distributions	O
are	O
obtained	O
by	O
a	O
2-dimensional	O
filtering	O
in	O
the	O
ambiguity	O
domain	O
(	O
that	O
is	O
,	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
of	O
the	O
signal	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
ambiguity	O
distribution	O
can	O
be	O
seen	O
as	O
the	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
of	O
a	O
signal	O
using	O
the	O
signal	O
itself	O
as	O
the	O
window	O
function	O
.	O
</s>
<s>
This	O
remark	O
has	O
been	O
used	O
to	O
define	O
an	O
ambiguity	O
distribution	O
over	O
the	O
time-scale	O
domain	O
instead	O
of	O
the	O
time-frequency	B-Algorithm
domain	I-Algorithm
.	O
</s>
<s>
The	O
wideband	O
ambiguity	B-Algorithm
function	I-Algorithm
of	O
is	O
:	O
</s>
<s>
When	O
the	O
wave	O
speed	O
in	O
the	O
medium	O
is	O
sufficiently	O
faster	O
than	O
the	O
target	O
speed	O
,	O
as	O
is	O
common	O
with	O
radar	B-Application
,	O
this	O
compression	O
in	O
frequency	O
is	O
closely	O
approximated	O
by	O
a	O
shift	O
in	O
frequency	O
Δf	O
=	O
fc*	O
v/c	O
(	O
known	O
as	O
the	O
doppler	O
shift	O
)	O
.	O
</s>
<s>
For	O
a	O
narrow	O
band	O
signal	O
,	O
this	O
approximation	O
results	O
in	O
the	O
narrowband	O
ambiguity	B-Algorithm
function	I-Algorithm
given	O
above	O
,	O
which	O
can	O
be	O
computed	O
efficiently	O
by	O
making	O
use	O
of	O
the	O
FFT	O
algorithm	O
.	O
</s>
<s>
An	O
ambiguity	B-Algorithm
function	I-Algorithm
of	O
interest	O
is	O
a	O
2-dimensional	O
Dirac	O
delta	O
function	O
or	O
"	O
thumbtack	O
"	O
function	O
;	O
that	O
is	O
,	O
a	O
function	O
which	O
is	O
infinite	O
at	O
(	O
0	O
,	O
0	O
)	O
and	O
zero	O
elsewhere	O
.	O
</s>
<s>
An	O
ambiguity	B-Algorithm
function	I-Algorithm
of	O
this	O
kind	O
would	O
be	O
somewhat	O
of	O
a	O
misnomer	O
;	O
it	O
would	O
have	O
no	O
ambiguities	O
at	O
all	O
,	O
and	O
both	O
the	O
zero-delay	O
and	O
zero-Doppler	O
cuts	O
would	O
be	O
an	O
impulse	O
.	O
</s>
<s>
This	O
is	O
not	O
usually	O
desirable	O
(	O
if	O
a	O
target	O
has	O
any	O
Doppler	O
shift	O
from	O
an	O
unknown	O
velocity	O
it	O
will	O
disappear	O
from	O
the	O
radar	B-Application
picture	O
)	O
,	O
but	O
if	O
Doppler	O
processing	O
is	O
independently	O
performed	O
,	O
knowledge	O
of	O
the	O
precise	O
Doppler	O
frequency	O
allows	O
ranging	O
without	O
interference	O
from	O
any	O
other	O
targets	O
which	O
are	O
not	O
also	O
moving	O
at	O
exactly	O
the	O
same	O
velocity	O
.	O
</s>
<s>
This	O
type	O
of	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
produced	O
by	O
ideal	O
white	O
noise	O
(	O
infinite	O
in	O
duration	O
and	O
infinite	O
in	O
bandwidth	O
)	O
.	O
</s>
<s>
There	O
is	O
no	O
pulse	B-Algorithm
that	O
will	O
produce	O
from	O
the	O
definition	O
of	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
matched	O
filter	O
output	O
is	O
distorted	O
into	O
a	O
sinc	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
greater	O
the	O
Doppler	O
shift	O
,	O
the	O
smaller	O
the	O
peak	O
of	O
the	O
resulting	O
sinc	B-Algorithm
,	O
</s>
<s>
In	O
general	O
,	O
the	O
square	O
pulse	B-Algorithm
is	O
not	O
a	O
desirable	O
waveform	O
from	O
a	O
pulse	B-Algorithm
compression	I-Algorithm
standpoint	O
,	O
because	O
the	O
autocorrelation	O
function	O
is	O
too	O
short	O
in	O
amplitude	O
,	O
making	O
it	O
difficult	O
to	O
detect	O
targets	O
in	O
noise	O
,	O
and	O
too	O
wide	O
in	O
time	O
,	O
making	O
it	O
difficult	O
to	O
discern	O
multiple	O
overlapping	O
targets	O
.	O
</s>
<s>
A	O
commonly	O
used	O
radar	B-Application
or	O
sonar	B-Application
pulse	B-Algorithm
is	O
the	O
linear	O
frequency	O
modulated	O
(	O
LFM	O
)	O
pulse	B-Algorithm
(	O
or	O
"	O
chirp	O
"	O
)	O
.	O
</s>
<s>
It	O
has	O
the	O
advantage	O
of	O
greater	O
bandwidth	O
while	O
keeping	O
the	O
pulse	B-Algorithm
duration	O
short	O
and	O
envelope	O
constant	O
.	O
</s>
<s>
A	O
constant	O
envelope	O
LFM	O
pulse	B-Algorithm
has	O
an	O
ambiguity	B-Algorithm
function	I-Algorithm
similar	O
to	O
that	O
of	O
the	O
square	O
pulse	B-Algorithm
,	O
except	O
that	O
it	O
is	O
skewed	O
in	O
the	O
delay-Doppler	O
plane	O
.	O
</s>
<s>
The	O
ambiguity	B-Algorithm
function	I-Algorithm
can	O
be	O
extended	O
to	O
multistatic	O
radars	B-Application
,	O
which	O
comprise	O
multiple	O
non-colocated	O
transmitters	O
and/or	O
receivers	O
(	O
and	O
can	O
include	O
bistatic	B-Algorithm
radar	I-Algorithm
as	O
a	O
special	O
case	O
)	O
.	O
</s>
<s>
For	O
these	O
types	O
of	O
radar	B-Application
,	O
the	O
simple	O
linear	O
relationship	O
between	O
time	O
and	O
range	O
that	O
exists	O
in	O
the	O
monostatic	O
case	O
no	O
longer	O
applies	O
,	O
and	O
is	O
instead	O
dependent	O
on	O
the	O
specific	O
geometry	O
–	O
i.e.	O
</s>
<s>
Therefore	O
,	O
the	O
multistatic	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
mostly	O
usefully	O
defined	O
as	O
a	O
function	O
of	O
two	O
-	O
or	O
three-dimensional	O
position	O
and	O
velocity	O
vectors	O
for	O
a	O
given	O
multistatic	O
geometry	O
and	O
transmitted	O
waveform	O
.	O
</s>
<s>
Just	O
as	O
the	O
monostatic	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
naturally	O
derived	O
from	O
the	O
matched	O
filter	O
,	O
the	O
multistatic	O
ambiguity	B-Algorithm
function	I-Algorithm
is	O
derived	O
from	O
the	O
corresponding	O
optimal	O
multistatic	O
detector	O
–	O
i.e.	O
</s>
<s>
Such	O
systems	O
are	O
sometimes	O
described	O
as	O
MIMO	O
radars	B-Application
due	O
to	O
the	O
information	O
theoretic	O
similarities	O
to	O
MIMO	O
communication	O
systems	O
.	O
</s>
<s>
The	O
Ambiguity	B-Algorithm
function	I-Algorithm
(	O
AF	O
)	O
is	O
the	O
operators	O
that	O
are	O
related	O
to	O
the	O
WDF	B-Algorithm
.	O
</s>
<s>
For	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
:	O
</s>
