<s>
The	O
chirp	B-Application
pulse	B-Algorithm
compression	I-Algorithm
process	O
transforms	O
a	O
long	O
duration	O
frequency-coded	O
pulse	O
into	O
a	O
narrow	O
pulse	O
of	O
greatly	O
increased	O
amplitude	O
.	O
</s>
<s>
It	O
is	O
a	O
technique	O
used	O
in	O
radar	B-Application
and	O
sonar	B-Application
systems	O
because	O
it	O
is	O
a	O
method	O
whereby	O
a	O
narrow	O
pulse	O
with	O
high	O
peak	O
power	O
can	O
be	O
derived	O
from	O
a	O
long	O
duration	O
pulse	O
with	O
low	O
peak	O
power	O
.	O
</s>
<s>
The	O
basics	O
of	O
the	O
method	O
for	O
radar	B-Application
applications	O
were	O
developed	O
in	O
the	O
late	O
1940s	O
and	O
early	O
1950s	O
,	O
but	O
it	O
was	O
not	O
until	O
1960	O
,	O
following	O
declassification	O
of	O
the	O
subject	O
matter	O
,	O
that	O
a	O
detailed	O
article	O
on	O
the	O
topic	O
appeared	O
the	O
public	O
domain	O
.	O
</s>
<s>
Briefly	O
,	O
the	O
basic	O
pulse	B-Algorithm
compression	I-Algorithm
properties	O
can	O
be	O
related	O
as	O
follows	O
.	O
</s>
<s>
For	O
a	O
chirp	B-Application
waveform	O
that	O
sweeps	O
over	O
a	O
frequency	O
range	O
F1	O
to	O
F2	O
in	O
a	O
time	O
period	O
T	O
,	O
the	O
nominal	O
bandwidth	O
of	O
the	O
pulse	O
is	O
B	O
,	O
where	O
B	O
=	O
F2	O
–	O
F1	O
,	O
and	O
the	O
pulse	O
has	O
a	O
time-bandwidth	O
product	O
of	O
T×B	O
.	O
</s>
<s>
Following	O
pulse	B-Algorithm
compression	I-Algorithm
,	O
a	O
narrow	O
pulse	O
of	O
duration	O
τ	O
is	O
obtained	O
,	O
where	O
τ	O
≈	O
1/B	O
,	O
together	O
with	O
a	O
peak	O
voltage	O
amplification	O
of	O
.	O
</s>
<s>
In	O
order	O
to	O
compress	O
a	O
chirp	B-Application
pulse	O
of	O
duration	O
T	O
seconds	O
,	O
which	O
sweeps	O
linearly	O
in	O
frequency	O
from	O
F1	O
Hz	O
to	O
F2	O
Hz	O
,	O
a	O
device	O
with	O
the	O
characteristics	O
of	O
a	O
dispersive	O
delay	O
line	O
is	O
required	O
.	O
</s>
<s>
Such	O
a	O
delay	O
characteristic	O
ensures	O
that	O
all	O
frequency	O
components	O
of	O
the	O
chirp	B-Application
pass	O
through	O
the	O
device	O
,	O
to	O
arrive	O
at	O
the	O
detector	O
at	O
the	O
same	O
time	O
instant	O
and	O
so	O
augment	O
one	O
another	O
,	O
to	O
produce	O
a	O
narrow	O
high	O
amplitude	O
pulse	O
,	O
as	O
shown	O
in	O
the	O
figure	O
:	O
</s>
<s>
A	O
chirp	B-Application
pulse	O
,	O
however	O
generated	O
,	O
can	O
be	O
considered	O
as	O
the	O
output	O
of	O
one	O
of	O
a	O
pair	O
of	O
filters	O
,	O
which	O
have	O
dispersive	O
characteristics	O
.	O
</s>
<s>
In	O
principle	O
,	O
the	O
transmitted	O
pulses	O
can	O
be	O
generated	O
by	O
applying	O
impulses	O
to	O
the	O
input	O
of	O
the	O
dispersive	O
transmit	O
filter	O
,	O
with	O
the	O
resultant	O
output	O
chirp	B-Application
amplified	O
as	O
necessary	O
for	O
transmission	O
.	O
</s>
<s>
Alternatively	O
,	O
a	O
voltage	O
controlled	O
oscillator	O
may	O
be	O
used	O
to	O
generate	O
the	O
chirp	B-Application
signal	O
.	O
</s>
<s>
To	O
achieve	O
maximum	O
transmitted	O
power	O
(	O
and	O
so	O
achieve	O
maximum	O
range	O
)	O
it	O
is	O
normal	O
for	O
a	O
radar	B-Application
system	I-Application
to	O
transmit	O
chirp	B-Application
pulses	O
at	O
constant	O
amplitude	O
from	O
a	O
transmitter	O
run	O
in	O
a	O
near-limiting	O
condition	O
.	O
</s>
<s>
The	O
chirp	B-Application
signals	O
reflected	O
from	O
targets	O
are	O
amplified	O
in	O
the	O
receiver	O
and	O
then	O
processed	O
by	O
the	O
compression	O
filter	O
to	O
give	O
narrow	O
pulses	O
of	O
high	O
amplitude	O
,	O
as	O
previously	O
described	O
.	O
</s>
<s>
For	O
the	O
compression	O
filter	O
to	O
be	O
matched	O
to	O
the	O
radiated	O
chirp	B-Application
signal	O
,	O
its	O
response	O
is	O
the	O
complex	O
conjugate	O
of	O
the	O
time	O
inverse	O
of	O
the	O
transmit	O
filter	O
's	O
impulse	O
response	O
.	O
</s>
<s>
So	O
,	O
the	O
output	O
of	O
this	O
matched	O
filter	O
is	O
given	O
by	O
the	O
convolution	B-Language
of	O
the	O
signal	O
h(t )	O
with	O
the	O
conjugate	O
impulse	O
response	O
h*( 	O
-t	O
)	O
:	O
</s>
<s>
The	O
waveform	O
of	O
this	O
spectrum	O
is	O
obtained	O
from	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
i.e.	O
</s>
<s>
For	O
the	O
case	O
of	O
a	O
linear	O
chirp	B-Application
,	O
with	O
constant	O
amplitude	O
and	O
time	O
duration	O
T	O
,	O
compression	O
by	O
the	O
matched	O
filter	O
gives	O
a	O
waveform	O
with	O
the	O
sinc	B-Algorithm
characteristic	O
,	O
with	O
duration	O
2T	O
,	O
as	O
shown	O
later	O
.	O
</s>
<s>
The	O
sinc	B-Algorithm
characteristic	O
of	O
a	O
compressed	O
pulse	O
is	O
a	O
direct	O
consequence	O
of	O
the	O
spectrum	O
of	O
the	O
linear	O
chirp	B-Application
pulse	O
having	O
a	O
rectangular	O
profile	O
.	O
</s>
<s>
By	O
modifying	O
the	O
spectrum	O
to	O
have	O
a	O
bell-shaped	O
profile	O
,	O
by	O
means	O
of	O
a	O
weighting	B-General_Concept
(	O
or	O
windowing	B-Language
,	O
or	O
apodization	B-Algorithm
)	O
function	O
,	O
lower	O
level	O
sidelobes	O
are	O
obtained	O
.	O
</s>
<s>
When	O
windowing	B-Language
is	O
implemented	O
,	O
some	O
signal	O
attenuation	O
occurs	O
and	O
there	O
is	O
a	O
broadening	O
of	O
the	O
main	O
pulse	O
,	O
so	O
both	O
signal-to-noise	O
ratio	O
and	O
range	O
resolution	O
are	O
impaired	O
by	O
the	O
process	O
.	O
</s>
<s>
Preferably	O
,	O
the	O
transmitted	O
and	O
received	O
pulses	O
should	O
be	O
modified	O
in	O
equal	O
measure	O
,	O
but	O
when	O
this	O
is	O
impractical	O
,	O
windowing	B-Language
in	O
the	O
compression	O
filter	O
alone	O
is	O
still	O
beneficial	O
.	O
</s>
<s>
When	O
the	O
frequency	O
sweep	O
of	O
a	O
chirp	B-Application
is	O
linear	O
,	O
the	O
compression	O
process	O
is	O
found	O
to	O
be	O
very	O
tolerant	O
of	O
Doppler	O
frequency	O
shifts	O
on	O
target	O
returns	O
,	O
for	O
a	O
wide	O
range	O
of	O
time-bandwidth	O
products	O
.	O
</s>
<s>
In	O
these	O
situations	O
a	O
chirp	B-Application
with	O
a	O
hyperbolic	O
frequency	O
law	O
can	O
be	O
used	O
,	O
as	O
it	O
has	O
been	O
shown	O
to	O
be	O
fully	O
tolerant	O
to	O
Doppler	O
shifts	O
.	O
</s>
<s>
Windowing	B-Language
techniques	O
can	O
still	O
be	O
applied	O
to	O
the	O
compressed	O
pulse	O
spectra	O
,	O
to	O
lower	O
sidelobe	O
levels	O
,	O
in	O
a	O
similar	O
manner	O
to	O
linear	O
chirps	B-Application
.	O
</s>
<s>
When	O
T×B	O
is	O
less	O
than	O
about	O
75	O
,	O
the	O
windowing	B-Language
process	O
is	O
not	O
altogether	O
successful	O
,	O
especially	O
when	O
it	O
is	O
applied	O
only	O
within	O
the	O
compressor	O
.	O
</s>
<s>
There	O
are	O
techniques	O
available	O
that	O
will	O
reduce	O
the	O
amplitude	O
of	O
the	O
spectral	O
ripple	O
(	O
see	O
chirp	B-Algorithm
spectrum	I-Algorithm
)	O
and	O
so	O
reduce	O
the	O
amplitude	O
of	O
these	O
far-out	O
sidelobes	O
,	O
but	O
they	O
are	O
not	O
very	O
effective	O
when	O
T×B	O
.	O
</s>
<s>
This	O
is	O
a	O
more	O
efficient	O
way	O
of	O
achieving	O
the	O
required	O
spectral	O
shape	O
than	O
by	O
applying	O
amplitude	O
weighting	B-General_Concept
to	O
the	O
spectrum	O
of	O
the	O
linear	O
chirp	B-Application
because	O
no	O
attenuation	O
of	O
signal	O
power	O
is	O
necessary	O
in	O
order	O
to	O
achieve	O
it	O
.	O
</s>
<s>
As	O
the	O
mathematics	O
of	O
non-linear	O
chirps	B-Application
is	O
more	O
complicated	O
than	O
that	O
of	O
linear	O
chirps	B-Application
,	O
many	O
early	O
workers	O
resorted	O
to	O
stationary	O
phase	O
methods	O
to	O
design	O
them	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
time	O
sidelobe	O
levels	O
of	O
the	O
compressed	O
pulses	O
were	O
high	O
with	O
these	O
early	O
systems	O
,	O
even	O
after	O
spectral	O
weighting	B-General_Concept
,	O
with	O
results	O
no	O
better	O
than	O
those	O
achieved	O
by	O
phase	O
coding	O
or	O
chip	O
coding	O
at	O
that	O
time	O
.	O
</s>
<s>
Matching	O
the	O
chirp	B-Application
characteristic	O
from	O
a	O
VCO	O
to	O
a	O
dispersive	O
delay	O
line	O
proved	O
difficult	O
and	O
,	O
in	O
addition	O
,	O
achieving	O
adequate	O
temperature	O
compensation	O
proved	O
challenging	O
.	O
</s>
<s>
A	O
major	O
improvement	O
in	O
performance	O
of	O
chirp	B-Application
pulse	O
generation	O
and	O
compression	O
systems	O
was	O
achieved	O
with	O
the	O
development	O
of	O
SAW	B-Algorithm
filters	I-Algorithm
.	O
</s>
<s>
These	O
allowed	O
much	O
more	O
precision	O
in	O
the	O
synthesis	O
of	O
filter	O
characteristics	O
and	O
consequently	O
in	O
the	O
radar	B-Application
performance	O
.	O
</s>
<s>
The	O
increased	O
precision	O
offered	O
by	O
SAW	O
technology	O
,	O
enabled	O
time	O
sidelobe	O
levels	O
approaching	O
−30dB	O
to	O
become	O
achievable	O
by	O
radar	B-Application
systems	I-Application
.	O
</s>
<s>
SAW	O
technology	O
still	O
continues	O
to	O
be	O
relevant	O
to	O
radar	B-Application
systems	I-Application
and	O
is	O
particularly	O
useful	O
for	O
systems	O
using	O
very	O
wideband	O
sweeps	O
,	O
where	O
digital	O
technology	O
(	O
see	O
below	O
)	O
may	O
not	O
always	O
be	O
appropriate	O
or	O
may	O
be	O
difficult	O
to	O
implement	O
.	O
</s>
<s>
The	O
compression	O
of	O
the	O
chirps	B-Application
and	O
additional	O
signal	O
processing	O
is	O
all	O
performed	O
by	O
a	O
digital	O
computer	O
,	O
which	O
has	O
stored	O
within	O
it	O
the	O
chirp	B-Application
pulse	O
data	O
needed	O
to	O
carry	O
out	O
the	O
compression	O
process	O
numerically	O
.	O
</s>
<s>
In	O
practice	O
,	O
in	O
a	O
radar	B-Application
system	I-Application
,	O
for	O
example	O
,	O
it	O
is	O
not	O
just	O
a	O
chirp	B-Application
pulse	O
sequence	O
that	O
is	O
to	O
be	O
compressed	O
,	O
but	O
a	O
long	O
data	O
sequence	O
of	O
returns	O
from	O
a	O
given	O
range	O
spoke	O
,	O
within	O
which	O
the	O
returning	O
chirp	B-Application
pulse	O
is	O
located	O
.	O
</s>
<s>
By	O
applying	O
the	O
Overlap-save	B-Algorithm
method	I-Algorithm
,	O
reconstruction	O
of	O
the	O
full-duration	O
compressed	O
signal	O
is	O
achieved	O
.	O
</s>
<s>
Some	O
writers	O
favour	O
using	O
the	O
ambiguity	B-Algorithm
function	I-Algorithm
as	O
a	O
way	O
of	O
assessing	O
the	O
Doppler	O
tolerance	O
of	O
chirps	B-Application
.	O
</s>
<s>
Others	O
causes	O
of	O
signal	O
impairment	O
include	O
amplitude	O
ripple	O
and	O
slope	O
across	O
the	O
passband	O
,	O
phase	O
ripple	O
across	O
the	O
passband	O
,	O
large	O
band-edge	O
phase	O
shifts	O
caused	O
by	O
band-limiting	O
filters	O
,	O
phase	B-Device
modulation	I-Device
due	O
to	O
poorly	O
regulated	O
power	O
supplies	O
,	O
all	O
of	O
which	O
lead	O
to	O
higher	O
sidelobe	O
levels	O
.	O
</s>
<s>
Another	O
type	O
of	O
waveform	O
degradation	O
is	O
caused	O
by	O
eclipsing	O
,	O
where	O
part	O
of	O
a	O
returning	O
chirp	B-Application
pulse	O
is	O
missing	O
.	O
</s>
<s>
which	O
is	O
an	O
integral	O
which	O
has	O
been	O
evaluated	O
in	O
chirp	B-Algorithm
spectrum	I-Algorithm
.	O
</s>
<s>
Here	O
it	O
is	O
more	O
convenient	O
to	O
find	O
from	O
the	O
convolution	B-Language
of	O
the	O
two	O
time	O
domain	O
responses	O
,	O
i.e.	O
</s>
<s>
[	O
Note	O
:	O
the	O
same	O
transform	O
can	O
also	O
be	O
found	O
in	O
Fourier	B-Algorithm
transforms	I-Algorithm
,	O
no	O
.	O
</s>
<s>
With	O
y(t )	O
determined	O
,	O
the	O
output	O
sout(t )	O
can	O
be	O
obtained	O
from	O
the	O
convolution	B-Language
of	O
s1(t )	O
and	O
y(t )	O
,	O
i.e.	O
</s>
<s>
So	O
,	O
for	O
a	O
unit	O
amplitude	O
linear	O
chirp	B-Application
,	O
with	O
a	O
pulse	O
duration	O
T	O
seconds	O
and	O
a	O
frequency	O
sweep	O
B	O
Hz	O
(	O
i.e.	O
</s>
<s>
which	O
has	O
the	O
form	O
of	O
the	O
familiar	O
sinc	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
product	O
T.B	O
gives	O
the	O
compression	O
ratio	O
of	O
the	O
system	O
and	O
it	O
equates	O
,	O
approximately	O
,	O
to	O
the	O
improvement	O
in	O
signal	O
to	O
noise	O
ratio	O
of	O
the	O
main	O
lobe	O
of	O
the	O
compressed	O
pulse	O
relative	O
to	O
that	O
of	O
the	O
original	O
chirp	B-Application
.	O
</s>
<s>
In	O
the	O
closed	O
form	O
solution	O
,	O
just	O
presented	O
,	O
the	O
compressed	O
waveform	O
has	O
the	O
standard	O
sinc	B-Algorithm
function	I-Algorithm
response	O
,	O
because	O
a	O
rectangular	O
shape	O
was	O
assumed	O
for	O
the	O
amplitude	O
of	O
the	O
pulse	O
spectrum	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
spectrum	O
of	O
a	O
linear	O
chirp	B-Application
has	O
a	O
rectangular	O
profile	O
only	O
when	O
the	O
time-bandwidth	O
product	O
of	O
a	O
pulse	O
is	O
large	O
,	O
i.e.	O
</s>
<s>
When	O
the	O
product	O
is	O
small	O
,	O
the	O
spectral	O
profile	O
of	O
the	O
chirp	B-Application
pulse	O
is	O
seriously	O
degraded	O
by	O
Fresnel	O
ripples	O
as	O
shown	O
in	O
chirp	B-Algorithm
spectrum	I-Algorithm
and	O
so	O
is	O
that	O
of	O
the	O
matched	O
filter	O
.	O
</s>
<s>
To	O
investigate	O
fully	O
the	O
consequences	O
of	O
these	O
ripples	O
it	O
is	O
advisable	O
to	O
consider	O
each	O
situation	O
individually	O
,	O
either	O
by	O
evaluating	O
convolution	B-Language
integrals	O
,	O
or	O
more	O
conveniently	O
,	O
by	O
means	O
of	O
FFTs	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
,	O
at	O
high	O
values	O
of	O
TB	O
,	O
the	O
plots	O
match	O
the	O
sinc	B-Algorithm
characteristic	O
closely	O
,	O
but	O
at	O
low	O
values	O
,	O
significant	O
differences	O
can	O
be	O
seen	O
.	O
</s>
<s>
As	O
the	O
sinc-like	O
characteristics	O
of	O
a	O
compressed	O
pulse	O
are	O
due	O
to	O
the	O
near-rectangular	O
profile	O
of	O
its	O
spectrum	O
,	O
then	O
by	O
modifying	O
that	O
characteristic	O
into	O
a	O
bell	O
shape	O
,	O
for	O
example	O
,	O
it	O
is	O
possible	O
to	O
reduce	O
the	O
sidelobe	O
levels	O
considerably	O
.	O
</s>
<s>
So	O
,	O
for	O
example	O
,	O
in	O
the	O
case	O
of	O
antennas	O
,	O
the	O
spatial	O
sidelobes	O
on	O
the	O
beam	O
pattern	O
are	O
improved	O
by	O
applying	O
a	O
weighting	B-Algorithm
function	I-Algorithm
to	O
the	O
array	O
elements	O
,	O
and	O
in	O
the	O
case	O
of	O
digital	O
signal	O
processing	O
,	O
window	B-Language
functions	I-Language
are	O
used	O
to	O
reduce	O
the	O
amplitude	O
of	O
unwanted	O
sidelobes	O
on	O
the	O
sampled	O
functions	O
.	O
</s>
<s>
In	O
an	O
example	O
of	O
the	O
process	O
,	O
the	O
spectrum	O
of	O
a	O
chirp	B-Application
pulse	O
,	O
with	O
a	O
time-bandwidth	O
product	O
of	O
250	O
,	O
is	O
shown	O
on	O
the	O
left	O
,	O
and	O
is	O
approximately	O
rectangular	O
in	O
profile	O
.	O
</s>
<s>
Below	O
this	O
plot	O
,	O
also	O
on	O
the	O
left	O
,	O
is	O
shown	O
the	O
waveform	O
after	O
compression	O
of	O
the	O
chirp	B-Application
by	O
its	O
matched	O
filter	O
and	O
is	O
similar	O
to	O
the	O
sinc	B-Algorithm
function	I-Algorithm
,	O
as	O
expected	O
.	O
</s>
<s>
The	O
top	O
plot	O
,	O
on	O
the	O
right	O
,	O
is	O
the	O
spectrum	O
after	O
Hamming	O
weighting	B-General_Concept
.	O
</s>
<s>
(	O
This	O
was	O
achieved	O
by	O
applying	O
a	O
root-Hamming	O
characteristic	O
to	O
both	O
the	O
chirp	B-Algorithm
spectrum	I-Algorithm
and	O
to	O
the	O
compressor	O
spectrum	O
.	O
)	O
</s>
<s>
Although	O
the	O
sidelobe	O
level	O
has	O
been	O
much	O
reduced	O
,	O
there	O
are	O
some	O
undesirable	O
consequences	O
of	O
the	O
weighting	B-General_Concept
process	O
.	O
</s>
<s>
In	O
,	O
say	O
,	O
a	O
radar	B-Application
system	I-Application
,	O
these	O
effects	O
would	O
cause	O
loss	O
of	O
range	O
and	O
reduced	O
range	O
resolution	O
,	O
respectively	O
.	O
</s>
<s>
However	O
,	O
the	O
various	O
windowing	B-Language
functions	O
do	O
perform	O
differently	O
from	O
one	O
other	O
,	O
with	O
some	O
giving	O
main	O
lobes	O
which	O
are	O
unnecessarily	O
broad	O
for	O
the	O
sidelobe	O
levels	O
achieved	O
.	O
</s>
<s>
The	O
most	O
efficient	O
function	O
is	O
the	O
Dolph	O
–	O
Chebyshev	O
window	O
(	O
see	O
window	B-Language
functions	I-Language
)	O
as	O
this	O
gives	O
the	O
narrowest	O
pulse	O
at	O
a	O
given	O
sidelobe	O
level	O
.	O
</s>
<s>
A	O
selection	O
of	O
the	O
better	O
performing	O
windowing	B-Language
functions	O
are	O
shown	O
in	O
the	O
graph	O
of	O
Beamwidth	O
×	O
Bandwidth	O
as	O
sidelobe	O
level	O
.	O
</s>
<s>
The	O
lowest	O
full	O
line	O
on	O
the	O
graph	O
is	O
for	O
Dolph-Chebyshev	O
weighting	B-General_Concept
which	O
,	O
as	O
already	O
mentioned	O
,	O
sets	O
the	O
narrowest	O
lobe	O
possible	O
for	O
a	O
given	O
sidelobe	O
level	O
.	O
</s>
<s>
Thus	O
a	O
chirp	B-Application
sweeping	O
over	O
a	O
20MHz	O
frequency	O
band	O
will	O
have	O
a	O
compressed	O
pulsewidth	O
of	O
60	O
nanoseconds	O
(	O
at	O
least	O
)	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
from	O
the	O
diagram	O
,	O
Taylor	O
weighting	B-General_Concept
is	O
particularly	O
efficient	O
,	O
and	O
Hamming	O
and	O
the	O
three	O
and	O
four	O
term	O
Blackman-Harris	O
functions	O
all	O
give	O
good	O
results	O
,	O
too	O
.	O
</s>
<s>
The	O
example	O
of	O
a	O
chirp	B-Application
with	O
TB	O
=	O
250	O
and	O
Hamming	O
weighting	B-General_Concept
,	O
given	O
earlier	O
,	O
illustrates	O
the	O
benefits	O
of	O
weighting	B-General_Concept
but	O
is	O
not	O
representative	O
of	O
a	O
normal	O
situation	O
because	O
the	O
results	O
there	O
were	O
achieved	O
by	O
applying	O
weighting	B-General_Concept
equally	O
to	O
both	O
the	O
signal	O
chirp	B-Application
and	O
to	O
its	O
compressor	O
.	O
</s>
<s>
However	O
,	O
in	O
a	O
typical	O
radar	B-Application
system	I-Application
the	O
chirp	B-Application
pulse	O
is	O
usually	O
transmitted	O
by	O
an	O
amplifier	O
operating	O
in	O
or	O
near	O
to	O
compression	O
,	O
to	O
maximise	O
transmitter	O
efficiency	O
.	O
</s>
<s>
In	O
such	O
a	O
case	O
,	O
amplitude	O
modulation	O
of	O
the	O
chirp	B-Application
waveform	O
or	O
its	O
spectrum	O
is	O
not	O
possible	O
,	O
so	O
the	O
full	O
window	O
characteristic	O
has	O
to	O
be	O
incorporated	O
into	O
the	O
compressor	O
response	O
.	O
</s>
<s>
Unfortunately	O
,	O
this	O
arrangement	O
has	O
undesirable	O
consequences	O
for	O
the	O
far-out	O
sidelobes	O
of	O
the	O
compressed	O
pulse	O
,	O
especially	O
when	O
the	O
time-bandwidth	O
of	O
the	O
chirp	B-Application
is	O
small	O
.	O
</s>
<s>
For	O
this	O
result	O
no	O
weighting	B-General_Concept
has	O
been	O
applied	O
to	O
the	O
transmit	O
pulse	O
,	O
but	O
the	O
full	O
Hamming	O
weighting	B-General_Concept
has	O
been	O
applied	O
in	O
the	O
compressor	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
,	O
the	O
close-in	O
sidelobe	O
levels	O
are	O
consistent	O
with	O
the	O
Hamming	O
weighting	B-General_Concept
( −42dB	O
)	O
,	O
but	O
further	O
out	O
,	O
the	O
sidelobe	O
levels	O
rise	O
to	O
a	O
peak	O
value	O
of	O
−45dB	O
at	O
T/2	O
each	O
side	O
of	O
the	O
main	O
lobe	O
.	O
</s>
<s>
Best	O
results	O
seem	O
to	O
be	O
obtained	O
when	O
the	O
window	B-Language
function	I-Language
is	O
applied	O
in	O
the	O
time	O
domain	O
to	O
the	O
compressor	O
waveform	O
(	O
as	O
amplitude	O
modulation	O
)	O
rather	O
than	O
in	O
the	O
frequency	O
domain	O
to	O
its	O
spectrum	O
.	O
</s>
<s>
Several	O
of	O
the	O
methods	O
are	O
featured	O
in	O
chirp	B-Algorithm
spectrum	I-Algorithm
.	O
</s>
<s>
A	O
chirp	B-Application
with	O
slow	O
rise	O
and	O
fall	O
times	O
has	O
reduced	O
ripple	O
on	O
its	O
spectrum	O
(	O
see	O
chirp	B-Algorithm
spectrum	I-Algorithm
)	O
,	O
so	O
will	O
result	O
in	O
lower	O
time	O
sidelobes	O
on	O
the	O
compressed	O
pulse	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
consider	O
first	O
the	O
figure	O
shows	O
the	O
compressed	O
spectrum	O
of	O
a	O
linear	O
chirp	B-Application
,	O
which	O
has	O
fast	O
rise	O
and	O
fall	O
times	O
,	O
with	O
T×B	O
=	O
100	O
and	O
where	O
Blackman-Harris	O
weighting	B-General_Concept
has	O
been	O
applied	O
.	O
</s>
<s>
The	O
procedure	O
is	O
most	O
effective	O
when	O
both	O
the	O
signal	O
chirp	B-Application
and	O
the	O
compressor	O
chirp	B-Application
have	O
modified	O
rise-times	O
,	O
when	O
sidelobe	O
levels	O
can	O
be	O
lowered	O
by	O
15	O
–	O
20dB	O
.	O
</s>
<s>
The	O
precise	O
manner	O
in	O
which	O
the	O
rise	O
and	O
fall	O
times	O
are	O
made	O
less	O
severe	O
is	O
not	O
very	O
critical	O
,	O
so	O
the	O
technique	O
of	O
adding	O
cosine	O
tapers	O
to	O
the	O
compressed	O
pulse	O
spectrum	O
(	O
as	O
with	O
the	O
Tukey	O
weighting	B-Algorithm
function	I-Algorithm
)	O
gives	O
a	O
similar	O
improvement	O
–	O
of	O
several	O
dBs	O
.	O
</s>
<s>
An	O
alternative	O
form	O
of	O
waveform	O
'	O
tweaking	O
 '	O
is	O
where	O
frequency	O
modulation	O
distortion	O
is	O
applied	O
to	O
the	O
chirps	B-Application
,	O
instead	O
of	O
amplitude	O
modulation	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
a	O
pulse	O
considered	O
earlier	O
,	O
with	O
T×B	O
=	O
100	O
and	O
Blackman	O
–	O
Harris	O
weighting	B-General_Concept
,	O
is	O
modified	O
with	O
phase	O
tweaking	O
and	O
the	O
results	O
are	O
shown	O
.	O
</s>
<s>
The	O
spectral	O
response	O
of	O
the	O
matched	O
filter	O
has	O
a	O
magnitude	O
which	O
is	O
the	O
mirror	O
image	O
of	O
that	O
of	O
the	O
expanded	O
pulse	O
,	O
when	O
a	O
chirp	B-Algorithm
spectrum	I-Algorithm
has	O
symmetry	O
about	O
its	O
centre	O
frequency	O
,	O
so	O
the	O
Fresnell	O
ripples	O
on	O
the	O
spectrum	O
are	O
augmented	O
by	O
the	O
compression	O
process	O
.	O
</s>
<s>
where	O
H( )	O
is	O
the	O
spectrum	O
of	O
the	O
signal	O
,	O
C’( 	O
)	O
is	O
the	O
target	O
spectrum	O
for	O
the	O
compressed	O
pulse	O
and	O
has	O
the	O
low	O
sidelobes	O
of	O
the	O
chosen	O
weighting	B-Algorithm
function	I-Algorithm
and	O
K( )	O
is	O
the	O
spectrum	O
of	O
the	O
compression	O
filter	O
which	O
has	O
the	O
reciprocal	O
ripple	O
properties	O
.	O
</s>
<s>
As	O
an	O
example	O
of	O
the	O
procedure	O
consider	O
a	O
linear	O
chirp	B-Application
with	O
T×B	O
=	O
100	O
.	O
</s>
<s>
The	O
left-hand	O
figures	O
shows	O
(	O
one	O
half	O
of	O
)	O
the	O
spectrum	O
of	O
the	O
chirp	B-Application
,	O
and	O
the	O
right-hand	O
figure	O
shows	O
the	O
waveform	O
after	O
compression	O
.	O
</s>
<s>
In	O
the	O
next	O
figure	O
,	O
Blackman-Harris	O
weighting	B-General_Concept
has	O
been	O
applied	O
to	O
the	O
compressed	O
pulse	O
spectrum	O
.	O
</s>
<s>
When	O
an	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
carried	O
out	O
on	O
this	O
spectrum	O
,	O
in	O
order	O
to	O
determine	O
the	O
characteristics	O
of	O
its	O
waveform	O
,	O
it	O
is	O
found	O
that	O
the	O
waveform	O
is	O
of	O
extremely	O
long	O
duration	O
,	O
typically	O
exceeding	O
10T	O
.	O
</s>
<s>
Even	O
assuming	O
the	O
waveform	O
is	O
no	O
longer	O
than	O
10T	O
,	O
it	O
means	O
that	O
the	O
total	O
time	O
needed	O
to	O
process	O
one	O
chirp	B-Application
will	O
be	O
at	O
least	O
11T	O
,	O
in	O
total	O
,	O
a	O
length	O
of	O
time	O
unacceptable	O
in	O
most	O
circumstances	O
.	O
</s>
<s>
Whenever	O
the	O
radial	O
distance	O
between	O
a	O
moving	O
target	O
and	O
the	O
radar	B-Application
changes	O
with	O
time	O
,	O
the	O
reflected	O
chirp	B-Application
returns	O
will	O
exhibit	O
a	O
frequency	O
shift	O
(	O
Doppler	O
shift	O
)	O
.	O
</s>
<s>
In	O
a	O
typical	O
radar	B-Application
system	I-Application
,	O
the	O
Doppler	O
frequency	O
is	O
a	O
small	O
fraction	O
of	O
the	O
swept	O
frequency	O
range	O
(	O
i.e.	O
</s>
<s>
the	O
system	O
bandwidth	O
)	O
of	O
the	O
chirp	B-Application
,	O
so	O
the	O
range	O
errors	O
due	O
to	O
Doppler	O
are	O
found	O
to	O
be	O
minor	O
.	O
</s>
<s>
and	O
where	O
fd	O
is	O
the	O
Doppler	O
frequency	O
,	O
B	O
is	O
the	O
frequency	O
sweep	O
of	O
the	O
chirp	B-Application
,	O
T	O
is	O
the	O
duration	O
of	O
the	O
chirp	B-Application
,	O
fm	O
is	O
the	O
mid	O
(	O
centre	O
)	O
frequency	O
of	O
the	O
chirp	B-Application
,	O
Vr	O
is	O
the	O
radial	O
velocity	O
of	O
the	O
target	O
and	O
c	O
is	O
the	O
velocity	O
of	O
light	O
(=	O
3×108	O
m/s	O
)	O
.	O
</s>
<s>
Consider	O
as	O
an	O
example	O
,	O
a	O
chirp	B-Application
centered	O
on	O
10GHz	O
,	O
with	O
pulse	O
duration	O
of	O
10μs	O
and	O
a	O
bandwidth	O
of	O
10MHz	O
.	O
</s>
<s>
Linear	O
chirps	B-Application
with	O
a	O
time-bandwidth	O
product	O
of	O
less	O
than	O
2000	O
,	O
say	O
,	O
are	O
found	O
to	O
be	O
very	O
tolerant	O
of	O
Doppler	O
frequency	O
shifts	O
,	O
so	O
main	O
pulse	O
width	O
and	O
the	O
time	O
sidelobe	O
levels	O
show	O
little	O
change	O
for	O
Doppler	O
frequencies	O
up	O
to	O
several	O
percent	O
of	O
system	O
bandwidth	O
.	O
</s>
<s>
In	O
addition	O
,	O
linear	O
chirps	B-Application
which	O
use	O
phase	O
pre-distortion	O
to	O
lower	O
sidelobe	O
levels	O
,	O
as	O
described	O
in	O
an	O
earlier	O
section	O
,	O
are	O
found	O
to	O
be	O
tolerant	O
of	O
Doppler	O
.	O
</s>
<s>
Only	O
when	O
chirp	B-Application
time-bandwidth	O
products	O
are	O
very	O
high	O
,	O
say	O
well	O
over	O
2000	O
,	O
is	O
it	O
necessary	O
consider	O
a	O
sweep-frequency	O
law	O
other	O
than	O
linear	O
,	O
to	O
cope	O
with	O
Doppler	O
frequency	O
shifts	O
.	O
</s>
<s>
It	O
is	O
not	O
possible	O
to	O
determine	O
a	O
precise	O
Doppler	O
frequency	O
at	O
which	O
r-r	O
fails	O
because	O
the	O
Fresnell	O
ripples	O
on	O
chirp	B-Application
spectra	O
do	O
not	O
have	O
a	O
single	O
dominant	O
component	O
.	O
</s>
<s>
With	O
linear	O
chirp	B-Application
pulses	O
this	O
can	O
be	O
achieved	O
by	O
applying	O
a	O
window	B-Language
function	I-Language
either	O
in	O
the	O
time	O
domain	O
or	O
in	O
the	O
frequency	O
domain	O
,	O
i.e.	O
</s>
<s>
by	O
amplitude	O
modulating	O
the	O
chirp	B-Application
waveforms	O
or	O
by	O
applying	O
weighting	B-General_Concept
to	O
the	O
compressed	O
pulse	O
spectra	O
.	O
</s>
<s>
An	O
alternative	O
way	O
to	O
obtain	O
the	O
required	O
spectral	O
shape	O
is	O
to	O
use	O
a	O
non-linear	O
frequency	O
sweep	O
in	O
the	O
chirp	B-Application
.	O
</s>
<s>
Consider	O
,	O
as	O
an	O
example	O
,	O
the	O
frequency	O
versus	O
time	O
plot	O
that	O
achieves	O
the	O
Blackman-Harris	O
windowing	B-Language
profile	O
.	O
</s>
<s>
As	O
this	O
technique	O
does	O
not	O
take	O
account	O
of	O
the	O
Fresnel	O
ripples	O
,	O
these	O
have	O
to	O
be	O
dealt	O
with	O
in	O
additional	O
ways	O
,	O
as	O
was	O
the	O
case	O
with	O
linear	O
chirps	B-Application
.	O
</s>
<s>
In	O
order	O
to	O
achieve	O
the	O
required	O
spectral	O
shape	O
for	O
low	O
time	O
sidelobes	O
,	O
linear	O
chirps	B-Application
require	O
amplitude	O
weighting	B-General_Concept
and	O
consequently	O
incur	O
a	O
mismatch	O
loss	O
.	O
</s>
<s>
Non-linear	O
chirps	B-Application
,	O
however	O
,	O
have	O
the	O
advantage	O
that	O
by	O
achieving	O
the	O
spectral	O
shaping	O
directly	O
,	O
close-in	O
sidelobe	O
levels	O
can	O
be	O
made	O
low	O
with	O
negligible	O
mismatch	O
loss	O
(	O
typically	O
less	O
than	O
0.1dB	O
)	O
.	O
</s>
<s>
Another	O
benefit	O
is	O
that	O
the	O
far	O
out	O
sidelobes	O
,	O
due	O
to	O
Fresnel	O
ripples	O
on	O
the	O
spectrum	O
,	O
tend	O
to	O
be	O
lower	O
than	O
for	O
a	O
linear	O
chirp	B-Application
with	O
the	O
same	O
T×B	O
product	O
(	O
4	O
to	O
5dB	O
lower	O
with	O
large	O
T×B	O
)	O
.	O
</s>
<s>
However	O
,	O
for	O
chirps	B-Application
where	O
the	O
T×B	O
product	O
is	O
low	O
,	O
the	O
far-out	O
sidelobe	O
levels	O
of	O
the	O
compressed	O
pulse	O
can	O
still	O
be	O
disappointingly	O
high	O
,	O
because	O
of	O
high	O
amplitude	O
Fresnel	O
ripples	O
on	O
the	O
spectrum	O
.	O
</s>
<s>
As	O
with	O
linear	O
chirps	B-Application
,	O
results	O
can	O
be	O
improved	O
by	O
means	O
of	O
reciprocal	O
ripple	O
correction	O
but	O
,	O
as	O
previously	O
,	O
truncation	O
of	O
the	O
compression	O
waveform	O
results	O
in	O
the	O
appearance	O
of	O
gating	O
sidelobes	O
.	O
</s>
<s>
The	O
left	O
hand	O
figure	O
shows	O
the	O
spectrum	O
of	O
a	O
non-linear	O
chirp	B-Application
,	O
with	O
a	O
time	O
bandwidth	O
product	O
of	O
40	O
,	O
aiming	O
to	O
have	O
a	O
Blackman-Harris	O
profile	O
.	O
</s>
<s>
A	O
major	O
disadvantage	O
of	O
non-linear	O
chirps	B-Application
is	O
their	O
sensitivity	O
to	O
Doppler	O
frequency	O
shifts	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
non-linear	O
chirp	B-Application
pulse	O
and	O
the	O
effects	O
of	O
Doppler	O
are	O
shown	O
.	O
</s>
<s>
The	O
non-linear	O
characteristic	O
is	O
chosen	O
to	O
achieve	O
−50dB	O
sidelobes	O
using	O
Taylor	O
weighting	B-General_Concept
.	O
</s>
<s>
The	O
first	O
figure	O
shows	O
the	O
compressed	O
pulse	O
for	O
a	O
non-linear	O
chirp	B-Application
,	O
with	O
bandwidth	O
10MHz	O
,	O
pulse	O
duration	O
10usec	O
,	O
so	O
T×B	O
=	O
100	O
,	O
and	O
with	O
no	O
Doppler	O
shift	O
.	O
</s>
<s>
The	O
final	O
figure	O
shows	O
the	O
effect	O
of	O
100kHz	O
Doppler	O
on	O
a	O
linear	O
chirp	B-Application
which	O
has	O
had	O
amplitude	O
weighting	B-General_Concept
applied	O
to	O
give	O
the	O
same	O
spectral	O
shape	O
as	O
that	O
of	O
the	O
non-linear	O
chirp	B-Application
.	O
</s>
<s>
A	O
way	O
of	O
reducing	O
the	O
susceptibility	O
of	O
non-linear	O
chirps	B-Application
to	O
Doppler	O
is	O
to	O
use	O
a	O
‘	O
hybrid’	O
scheme	O
,	O
where	O
part	O
of	O
the	O
spectral	O
shaping	O
is	O
achieved	O
by	O
a	O
non-linear	O
sweep	O
,	O
but	O
with	O
additional	O
spectral	O
shaping	O
achieved	O
by	O
amplitude	O
weighting	B-General_Concept
.	O
</s>
<s>
In	O
the	O
two	O
examples	O
below	O
,	O
the	O
chirps	B-Application
have	O
a	O
non-linear	O
sweep	O
characteristic	O
which	O
gives	O
a	O
spectrum	O
with	O
Taylor	O
weighting	B-General_Concept
which	O
,	O
used	O
alone	O
,	O
will	O
achieve	O
a	O
sidelobe	O
level	O
of	O
−20dB	O
on	O
its	O
compressed	O
pulses	O
.	O
</s>
<s>
To	O
achieve	O
lower	O
level	O
sidelobes	O
,	O
this	O
spectral	O
shape	O
is	O
augmented	O
by	O
amplitude	O
weighting	B-General_Concept
so	O
that	O
the	O
final	O
target	O
sidelobe	O
level	O
for	O
the	O
compressed	O
pulses	O
is	O
−50dB	O
.	O
</s>
<s>
However	O
,	O
the	O
mismatch	O
loss	O
has	O
increased	O
from	O
0.1dB	O
to	O
0.6dB	O
,	O
but	O
this	O
is	O
still	O
better	O
than	O
the	O
1.6dB	O
figure	O
for	O
linear	O
chirps	B-Application
.	O
</s>
<s>
The	O
amplitude	O
of	O
random	O
noise	O
is	O
not	O
changed	O
by	O
the	O
compression	O
process	O
,	O
so	O
the	O
signal	O
to	O
noise	O
ratios	O
of	O
received	O
chirp	B-Application
signals	O
are	O
increased	O
in	O
the	O
process	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
high	O
power	O
search	O
radars	B-Application
,	O
this	O
extends	O
the	O
range	O
performance	O
of	O
the	O
system	O
,	O
while	O
for	O
stealth	O
systems	O
the	O
property	O
will	O
permit	O
lower	O
transmitter	O
powers	O
to	O
be	O
used	O
.	O
</s>
<s>
As	O
an	O
illustration	O
,	O
a	O
possible	O
received	O
noise	O
sequence	O
is	O
shown	O
,	O
which	O
contains	O
a	O
low	O
amplitude	O
chirp	B-Application
signal	O
obscured	O
within	O
it	O
.	O
</s>
<s>
When	O
pulse	B-Algorithm
compression	I-Algorithm
is	O
carried	O
out	O
in	O
digital	O
signal	O
processing	O
,	O
after	O
the	O
incoming	O
signals	O
are	O
digitised	O
by	O
A/D	O
converters	O
,	O
it	O
is	O
important	O
that	O
level	O
of	O
the	O
noise	O
floor	O
is	O
correctly	O
set	O
.	O
</s>
<s>
If	O
the	O
noise	O
level	O
is	O
too	O
low	O
,	O
Nyquist	O
will	O
not	O
be	O
satisfied	O
,	O
and	O
any	O
embedded	O
chirp	B-Application
will	O
not	O
be	O
recovered	O
correctly	O
.	O
</s>
<s>
For	O
systems	O
using	O
digital	O
processing	O
,	O
it	O
is	O
important	O
to	O
carry	O
out	O
the	O
chirp	B-Algorithm
compression	I-Algorithm
in	O
the	O
digital	O
domain	O
,	O
after	O
the	O
A/D	O
converters	O
.	O
</s>
<s>
If	O
the	O
compression	O
process	O
is	O
carried	O
out	O
in	O
the	O
analogue	O
domain	O
before	O
digitization	O
(	O
by	O
a	O
SAW	B-Algorithm
filter	I-Algorithm
,	O
for	O
example	O
)	O
,	O
the	O
resulting	O
high-amplitude	O
pulses	O
will	O
place	O
excessive	O
demands	O
on	O
the	O
dynamic	O
range	O
of	O
the	O
A/D	O
converters	O
.	O
</s>
<s>
The	O
transmitter	O
and	O
receiver	O
subsystems	O
of	O
a	O
radar	B-Application
are	O
not	O
distortion	O
free	O
.	O
</s>
<s>
This	O
is	O
not	O
difficult	O
to	O
implement	O
in	O
radars	B-Application
using	O
digital	O
look-up	O
tables	O
,	O
since	O
these	O
tables	O
can	O
be	O
easily	O
amended	O
to	O
include	O
compensation	O
data	O
.	O
</s>
<s>
where	O
,	O
as	O
before	O
,	O
H( )	O
is	O
the	O
initial	O
chirp	B-Algorithm
spectrum	I-Algorithm
and	O
C'( 	O
)	O
is	O
the	O
target	O
spectrum	O
,	O
such	O
as	O
a	O
Taylor	O
window	O
,	O
but	O
now	O
additional	O
terms	O
have	O
been	O
included	O
,	O
namely	O
,	O
(( )	O
)	O
and	O
A( )	O
which	O
are	O
the	O
phase	O
and	O
amplitude	O
characteristics	O
that	O
require	O
compensation	O
.	O
</s>
<s>
A	O
compressor	O
chirp	B-Application
waveform	O
that	O
includes	O
phase	O
correction	O
data	O
will	O
have	O
additional	O
ripple	O
components	O
present	O
at	O
each	O
end	O
of	O
the	O
waveform	O
(	O
pre-shoots	O
and	O
after-shoots	O
)	O
.	O
</s>
<s>
Time	O
shift	O
can	O
be	O
useful	O
to	O
position	O
the	O
main	O
lobes	O
of	O
compressed	O
pulses	O
at	O
a	O
standard	O
location	O
,	O
regardless	O
of	O
chirp	B-Application
pulse	O
length	O
.	O
</s>
<s>
There	O
has	O
been	O
a	O
growth	O
in	O
interest	O
in	O
adaptive	O
filters	O
for	O
pulse	B-Algorithm
compression	I-Algorithm
,	O
made	O
possible	O
by	O
the	O
availability	O
of	O
small	O
fast	O
computers	O
,	O
and	O
some	O
relevant	O
articles	O
are	O
mentioned	O
in	O
the	O
next	O
section	O
.	O
</s>
<s>
The	O
growth	O
in	O
digital	O
processing	O
and	O
methods	O
had	O
a	O
significant	O
influence	O
in	O
the	O
field	O
of	O
chirp	B-Application
pulse	B-Algorithm
compression	I-Algorithm
.	O
</s>
<s>
An	O
introduction	O
to	O
these	O
techniques	O
is	O
provided	O
in	O
a	O
chapter	O
of	O
the	O
Radar	B-Application
Handbook	O
(	O
3rd	O
ed	O
.	O
</s>
<s>
The	O
main	O
aims	O
of	O
most	O
investigations	O
into	O
pulse	B-Algorithm
compression	I-Algorithm
has	O
been	O
to	O
obtain	O
narrow	O
main	O
lobes	O
,	O
with	O
low	O
sidelobe	O
levels	O
,	O
a	O
tolerance	O
to	O
Doppler	O
frequency	O
shifts	O
and	O
to	O
incur	O
low	O
system	O
losses	O
.	O
</s>
<s>
A	O
number	O
of	O
other	O
works	O
,	O
with	O
a	O
variety	O
of	O
approaches	O
to	O
pulse	B-Algorithm
compression	I-Algorithm
,	O
are	O
listed	O
below	O
:	O
</s>
<s>
Further	O
studies	O
of	O
Hyperbolic	O
chirps	B-Application
have	O
been	O
carried	O
out	O
by	O
Kiss	O
,	O
Readhead	O
,	O
Nagajyothi	O
and	O
Rajarajeswari	O
and	O
Yang	O
and	O
Sarkar	O
.	O
</s>
<s>
Convolution	B-Language
windows	O
have	O
been	O
investigated	O
by	O
Sahoo	O
and	O
Panda	O
who	O
show	O
that	O
they	O
can	O
result	O
in	O
very	O
low	O
sidelobes	O
yet	O
be	O
Doppler	O
tolerant	O
,	O
but	O
may	O
suffer	O
from	O
some	O
pulse	O
broadening	O
.	O
</s>
<s>
Wen	O
and	O
his	O
co-workers	O
have	O
also	O
discussed	O
convolution	B-Language
windows	O
.	O
</s>
<s>
Some	O
new	O
window	B-Language
functions	I-Language
have	O
been	O
proposed	O
by	O
Samad	O
and	O
Sinha	O
and	O
Ferreira	O
,	O
which	O
claim	O
improved	O
performance	O
over	O
the	O
familiar	O
functions	O
.	O
</s>
<s>
Several	O
techniques	O
to	O
lower	O
the	O
sidelobe	O
levels	O
of	O
the	O
compressed	O
pulses	O
for	O
non-linear	O
FM	O
chirps	B-Application
are	O
compared	O
by	O
Varshney	O
and	O
Thomas	O
.	O
</s>
<s>
In	O
a	O
paper	O
by	O
Vizitui	O
,	O
sidelobe	O
reduction	O
is	O
considered	O
where	O
phase	O
pre-distortion	O
is	O
applied	O
to	O
non-linear	O
FM	O
chirps	B-Application
,	O
rather	O
than	O
to	O
linear	O
chirps	B-Application
.	O
</s>
<s>
There	O
have	O
been	O
extensive	O
investigations	O
of	O
phase	B-Device
modulation	I-Device
for	O
pulse	B-Algorithm
compression	I-Algorithm
schemes	O
,	O
such	O
as	O
biphase	O
(	O
binary	O
phase-shift	B-General_Concept
keying	I-General_Concept
)	O
and	O
polyphase	O
coding	O
methods	O
,	O
but	O
this	O
work	O
is	O
not	O
considered	O
here	O
.	O
</s>
