<s>
Pulse	B-Algorithm
compression	I-Algorithm
is	O
a	O
signal	O
processing	O
technique	O
commonly	O
used	O
by	O
radar	B-Application
,	O
sonar	B-Application
and	O
echography	B-Application
to	O
either	O
increase	O
the	O
range	O
resolution	O
when	O
pulse	O
length	O
is	O
constrained	O
or	O
increase	O
the	O
signal	O
to	O
noise	O
ratio	O
when	O
the	O
peak	O
power	O
and	O
the	O
bandwidth	O
(	O
or	O
equivalently	O
range	O
resolution	O
)	O
of	O
the	O
transmitted	O
signal	O
are	O
constrained	O
.	O
</s>
<s>
This	O
is	O
achieved	O
by	O
modulating	B-Application
the	O
transmitted	O
pulse	O
and	O
then	O
correlating	O
the	O
received	O
signal	O
with	O
the	O
transmitted	O
pulse	O
.	O
</s>
<s>
The	O
ideal	O
model	O
for	O
the	O
simplest	O
,	O
and	O
historically	O
first	O
type	O
of	O
signals	O
a	O
pulse	O
radar	B-Application
or	O
sonar	B-Application
can	O
transmit	O
is	O
a	O
truncated	O
sinusoidal	O
pulse	O
(	O
also	O
called	O
a	O
CW	O
--carrier	O
wave--	O
pulse	O
)	O
,	O
of	O
amplitude	O
and	O
carrier	O
frequency	O
,	O
,	O
truncated	O
by	O
a	O
rectangular	O
function	O
of	O
width	O
,	O
.	O
</s>
<s>
This	O
is	O
achieved	O
by	O
convolving	B-Language
the	O
incoming	O
signal	O
with	O
a	O
conjugated	O
and	O
time-reversed	O
version	O
of	O
the	O
transmitted	O
signal	O
.	O
</s>
<s>
This	O
is	O
where	O
pulse	B-Algorithm
compression	I-Algorithm
enters	O
the	O
picture	O
.	O
</s>
<s>
this	O
signal	O
is	O
designed	O
so	O
that	O
after	O
matched	O
filtering	O
,	O
the	O
width	O
of	O
the	O
intercorrelated	O
signals	O
is	O
smaller	O
than	O
the	O
width	O
obtained	O
by	O
the	O
standard	O
sinusoidal	O
pulse	O
,	O
as	O
explained	O
above	O
(	O
hence	O
the	O
name	O
of	O
the	O
technique	O
:	O
pulse	B-Algorithm
compression	I-Algorithm
)	O
.	O
</s>
<s>
In	O
radar	B-Application
or	O
sonar	B-Application
applications	O
,	O
linear	O
chirps	B-Application
are	O
the	O
most	O
typically	O
used	O
signals	O
to	O
achieve	O
pulse	B-Algorithm
compression	I-Algorithm
.	O
</s>
<s>
The	O
chirp	B-Application
definition	O
above	O
means	O
that	O
the	O
phase	O
of	O
the	O
chirped	O
signal	O
(	O
that	O
is	O
,	O
the	O
argument	O
of	O
the	O
complex	O
exponential	O
)	O
,	O
is	O
the	O
quadratic	O
:	O
</s>
<s>
The	O
relation	O
of	O
phase	O
to	O
frequency	O
is	O
often	O
used	O
in	O
the	O
other	O
direction	O
,	O
starting	O
with	O
the	O
desired	O
and	O
writing	O
the	O
chirp	B-Application
phase	O
via	O
the	O
integration	O
of	O
frequency	O
:	O
</s>
<s>
Around	O
0	O
,	O
this	O
function	O
behaves	O
as	O
the	O
sinc	B-Algorithm
(	O
or	O
cardinal	B-Algorithm
sine	I-Algorithm
)	O
term	O
,	O
defined	O
here	O
as	O
.	O
</s>
<s>
The	O
−3dB	O
temporal	O
width	O
of	O
that	O
cardinal	B-Algorithm
sine	I-Algorithm
is	O
more	O
or	O
less	O
equal	O
to	O
.	O
</s>
<s>
For	O
the	O
common	O
values	O
of	O
,	O
is	O
smaller	O
than	O
,	O
hence	O
the	O
pulse	B-Algorithm
compression	I-Algorithm
name	O
.	O
</s>
<s>
Since	O
the	O
cardinal	B-Algorithm
sine	I-Algorithm
can	O
have	O
annoying	O
sidelobes	O
,	O
a	O
common	O
practice	O
is	O
to	O
filter	O
the	O
result	O
by	O
a	O
window	O
(	O
Hamming	O
,	O
Hann	O
,	O
etc	O
.	O
)	O
.	O
</s>
<s>
In	O
practice	O
,	O
this	O
can	O
be	O
done	O
at	O
the	O
same	O
time	O
as	O
the	O
adapted	O
filtering	O
by	O
multiplying	O
the	O
reference	O
chirp	B-Application
with	O
the	O
filter	O
.	O
</s>
<s>
Result	O
2	O
The	O
distance	O
resolution	O
reachable	O
with	O
a	O
linear	O
frequency	O
modulation	B-Application
of	O
a	O
pulse	O
on	O
a	O
bandwidth	O
is	O
:	O
where	O
is	O
the	O
speed	O
of	O
the	O
wave	O
.	O
</s>
<s>
Definition	O
Ratio	O
is	O
the	O
pulse	B-Algorithm
compression	I-Algorithm
ratio	O
.	O
</s>
<s>
+	O
Example	O
(	O
chirped	O
pulse	O
)	O
:	O
transmitted	O
signal	O
in	O
red	O
(	O
carrier	O
10	O
hertz	O
,	O
modulation	B-Application
on	O
16	O
hertz	O
,	O
amplitude	O
1	O
,	O
duration	O
1	O
second	O
)	O
and	O
two	O
echoes	O
(	O
in	O
blue	O
)	O
.	O
</s>
<s>
As	O
we	O
have	O
seen	O
above	O
,	O
things	O
are	O
written	O
so	O
that	O
the	O
energy	O
of	O
the	O
signal	O
does	O
not	O
vary	O
during	O
pulse	B-Algorithm
compression	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
now	O
located	O
in	O
the	O
main	O
lobe	O
of	O
the	O
cardinal	B-Algorithm
sine	I-Algorithm
,	O
whose	O
width	O
is	O
approximately	O
.	O
</s>
<s>
which	O
yields	O
an	O
increase	O
in	O
power	O
after	O
pulse	B-Algorithm
compression	I-Algorithm
:	O
</s>
<s>
In	O
the	O
spectral	O
domain	O
,	O
the	O
power	O
spectrum	O
of	O
the	O
chirp	B-Application
has	O
a	O
nearly	O
constant	O
spectral	O
density	O
in	O
interval	O
and	O
zero	O
elsewhere	O
,	O
so	O
that	O
energy	O
is	O
equivalently	O
expressed	O
as	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
net	O
effect	O
of	O
pulse	B-Algorithm
compression	I-Algorithm
is	O
that	O
,	O
compared	O
to	O
the	O
equivalent	O
CW	O
pulse	O
,	O
the	O
signal-to-noise	O
ratio	O
(	O
SNR	O
)	O
has	O
improved	O
by	O
a	O
factor	O
because	O
the	O
signal	O
is	O
amplified	O
but	O
not	O
the	O
noise	O
.	O
</s>
<s>
Result	O
3	O
After	O
pulse	B-Algorithm
compression	I-Algorithm
,	O
the	O
signal-to-noise	O
ratio	O
can	O
be	O
considered	O
as	O
being	O
amplified	O
by	O
as	O
compared	O
to	O
the	O
baseline	O
situation	O
of	O
a	O
continuous-wave	O
pulse	O
of	O
duration	O
and	O
the	O
same	O
amplitude	O
as	O
the	O
chirp-modulated	O
signal	O
before	O
compression	O
,	O
where	O
the	O
received	O
signal	O
and	O
noise	O
have	O
(	O
implicitly	O
)	O
undergone	O
a	O
bandpass	O
filtering	O
on	O
.	O
</s>
<s>
This	O
additional	O
gain	O
can	O
be	O
injected	O
into	O
the	O
radar	B-Application
equation	O
.	O
</s>
<s>
However	O
during	O
baseband	B-Algorithm
shifting	O
the	O
signal	O
undergoes	O
a	O
bandpass	O
filtering	O
on	O
which	O
has	O
the	O
same	O
net	O
effect	O
on	O
the	O
noise	O
as	O
the	O
correlation	O
,	O
so	O
the	O
overall	O
reasoning	O
remains	O
the	O
same	O
(	O
that	O
is	O
,	O
the	O
SNR	O
makes	O
only	O
sense	O
for	O
noise	O
defined	O
on	O
a	O
given	O
bandwidth	O
,	O
here	O
being	O
that	O
of	O
the	O
signal	O
)	O
.	O
</s>
<s>
If	O
the	O
peak	O
transmitting	O
power	O
and	O
the	O
bandwidth	O
are	O
constrained	O
,	O
pulse	B-Algorithm
compression	I-Algorithm
thus	O
achieves	O
a	O
better	O
peak	O
power	O
(	O
but	O
same	O
resolution	O
)	O
by	O
transmitting	O
a	O
longer	O
pulse	O
(	O
that	O
is	O
,	O
more	O
energy	O
)	O
,	O
compared	O
to	O
an	O
equivalent	O
CW	O
pulse	O
of	O
same	O
peak	O
power	O
and	O
bandwidth	O
,	O
and	O
squeezing	O
the	O
pulse	O
by	O
correlation	O
.	O
</s>
<s>
While	O
pulse	B-Algorithm
compression	I-Algorithm
can	O
ensure	O
good	O
SNR	O
and	O
fine	O
range	O
resolution	O
in	O
the	O
same	O
time	O
,	O
digital	O
signal	O
processing	O
in	O
such	O
a	O
system	O
can	O
be	O
difficult	O
to	O
implement	O
because	O
of	O
the	O
high	O
instantaneous	O
bandwidth	O
of	O
the	O
waveform	O
(	O
can	O
be	O
hundreds	O
of	O
megahertz	O
or	O
even	O
exceed	O
1GHz	O
.	O
)	O
</s>
<s>
If	O
the	O
transmitted	O
waveform	O
is	O
the	O
chirp	B-Application
waveform	O
:	O
</s>
<s>
Result	O
4	O
Through	O
stretch	O
processing	O
,	O
the	O
bandwidth	O
at	O
the	O
receiver	O
output	O
is	O
less	O
than	O
the	O
original	O
signal	O
bandwidth	O
if	O
,	O
thereby	O
facilitating	O
the	O
implementation	O
of	O
DSP	O
system	O
in	O
a	O
linear-frequency-modulation	O
radar	B-Application
system	I-Application
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
Fourier	O
transform	O
of	O
y(t )	O
is	O
actually	O
a	O
sinc	B-Algorithm
function	I-Algorithm
with	O
Rayleigh	O
resolution	O
.	O
</s>
<s>
which	O
is	O
the	O
same	O
as	O
the	O
resolution	O
of	O
the	O
original	O
linear	O
frequency	O
modulation	B-Application
waveform	O
.	O
</s>
<s>
Although	O
stretch	O
processing	O
can	O
reduce	O
the	O
bandwidth	O
of	O
received	O
baseband	B-Algorithm
signal	I-Algorithm
,	O
all	O
of	O
the	O
analog	O
components	O
in	O
RF	O
front-end	O
circuitry	O
still	O
must	O
be	O
able	O
to	O
support	O
an	O
instantaneous	O
bandwidth	O
of	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
effective	O
wavelength	O
of	O
the	O
electromagnetic	O
wave	O
changes	O
during	O
the	O
frequency	O
sweep	O
of	O
a	O
chirp	B-Application
signal	O
,	O
and	O
therefore	O
the	O
antenna	O
look	O
direction	O
will	O
be	O
inevitably	O
changed	O
in	O
a	O
Phased	O
array	O
system	O
.	O
</s>
<s>
The	O
baseband	B-Algorithm
signal	I-Algorithm
can	O
be	O
expressed	O
as	O
:	O
</s>
<s>
To	O
demonstrate	O
stepped-frequency	O
waveform	O
preserves	O
range	O
resolution	O
,	O
it	O
should	O
be	O
noticed	O
that	O
is	O
a	O
sinc-like	O
function	O
,	O
and	O
therefore	O
it	O
has	O
a	O
Rayleigh	O
resolution	O
of	O
.	O
</s>
<s>
which	O
is	O
the	O
same	O
of	O
the	O
resolution	O
of	O
the	O
original	O
linear-frequency-modulation	O
waveform	O
.	O
</s>
<s>
Phase	B-Device
modulation	I-Device
is	O
a	O
commonly	O
used	O
technique	O
;	O
in	O
this	O
case	O
,	O
the	O
pulse	O
is	O
divided	O
in	O
time	O
slots	O
of	O
duration	O
for	O
which	O
the	O
phase	O
at	O
the	O
origin	O
is	O
chosen	O
according	O
to	O
a	O
pre-established	O
convention	O
.	O
</s>
<s>
The	O
precise	O
way	O
of	O
choosing	O
the	O
sequence	O
of	O
phases	O
can	O
be	O
done	O
according	O
to	O
a	O
technique	O
known	O
as	O
Barker	B-Protocol
codes	I-Protocol
.	O
</s>
<s>
The	O
advantages	O
of	O
the	O
Barker	B-Protocol
codes	I-Protocol
are	O
their	O
simplicity	O
(	O
as	O
indicated	O
above	O
,	O
a	O
de-phasing	O
is	O
a	O
simple	O
sign	O
change	O
)	O
,	O
but	O
the	O
pulse	B-Algorithm
compression	I-Algorithm
ratio	O
is	O
lower	O
than	O
in	O
the	O
chirp	B-Application
case	O
and	O
the	O
compression	O
is	O
very	O
sensitive	O
to	O
frequency	O
changes	O
due	O
to	O
the	O
Doppler	O
effect	O
if	O
that	O
change	O
is	O
larger	O
than	O
.	O
</s>
<s>
Other	O
pseudorandom	O
binary	O
sequences	O
have	O
nearly	O
optimal	O
pulse	B-Algorithm
compression	I-Algorithm
properties	O
,	O
such	O
as	O
Gold	O
codes	O
,	O
JPL	B-Protocol
codes	I-Protocol
or	O
Kasami	B-Protocol
codes	I-Protocol
,	O
because	O
their	O
autocorrelation	O
peak	O
is	O
very	O
narrow	O
.	O
</s>
<s>
As	O
with	O
a	O
linear	O
chirp	B-Application
,	O
pulse	B-Algorithm
compression	I-Algorithm
is	O
achieved	O
through	O
intercorrelation	O
.	O
</s>
