<s>
The	O
spectrum	O
of	O
a	O
chirp	B-Application
pulse	O
describes	O
its	O
characteristics	O
in	O
terms	O
of	O
its	O
frequency	O
components	O
.	O
</s>
<s>
This	O
frequency-domain	O
representation	O
is	O
an	O
alternative	O
to	O
the	O
more	O
familiar	O
time-domain	O
waveform	O
,	O
and	O
the	O
two	O
versions	O
are	O
mathematically	O
related	O
by	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
when	O
a	O
chirp	B-Application
pulse	O
is	O
compressed	O
by	O
its	O
matched	O
filter	O
,	O
the	O
resulting	O
waveform	O
contains	O
not	O
only	O
a	O
main	O
narrow	O
pulse	O
but	O
,	O
also	O
,	O
a	O
variety	O
of	O
unwanted	O
artifacts	O
many	O
of	O
which	O
are	O
directly	O
attributable	O
to	O
features	O
in	O
the	O
chirp	B-Application
's	O
spectral	O
characteristics	O
.	O
</s>
<s>
The	O
simplest	O
way	O
to	O
derive	O
the	O
spectrum	O
of	O
a	O
chirp	B-Application
,	O
now	O
that	O
computers	O
are	O
widely	O
available	O
,	O
is	O
to	O
sample	B-Algorithm
the	O
time-domain	O
waveform	O
at	O
a	O
frequency	O
well	O
above	O
the	O
Nyquist	O
limit	O
and	O
call	O
up	O
an	O
FFT	O
algorithm	O
to	O
obtain	O
the	O
desired	O
result	O
.	O
</s>
<s>
These	O
early	O
methods	O
still	O
remain	O
helpful	O
,	O
however	O
,	O
as	O
they	O
give	O
additional	O
insight	O
into	O
the	O
behavior	O
and	O
properties	O
of	O
chirps	B-Application
.	O
</s>
<s>
The	O
frequency	O
spectrum	O
of	O
this	O
waveform	O
is	O
obtained	O
by	O
calculating	O
the	O
Fourier	B-Algorithm
Transform	I-Algorithm
of	O
,	O
i.e.	O
</s>
<s>
In	O
a	O
few	O
special	O
cases	O
,	O
the	O
integral	O
can	O
be	O
solved	O
to	O
give	O
an	O
analytical	O
expression	O
,	O
but	O
often	O
the	O
characteristics	O
of	O
and	O
(	O
t	O
)	O
are	O
such	O
that	O
the	O
integral	O
can	O
only	O
be	O
evaluated	O
by	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
or	O
by	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
In	O
the	O
special	O
case	O
where	O
s(t )	O
is	O
constrained	O
to	O
be	O
an	O
up-chirp	O
,	O
flat	O
topped	O
pulse	O
with	O
its	O
instantaneous	O
frequency	O
varying	O
as	O
a	O
linear	O
function	O
of	O
time	O
,	O
then	O
an	O
analytical	O
solution	O
is	O
possible	O
.	O
</s>
<s>
This	O
product	O
,	O
T.F	O
,	O
is	O
often	O
referred	O
to	O
as	O
the	O
time-bandwidth	O
product	O
of	O
the	O
chirp	B-Application
.	O
</s>
<s>
In	O
addition	O
,	O
a	O
number	O
of	O
mathematical	O
software	O
programs	O
,	O
such	O
as	O
Mathcad	B-Application
,	O
MATLAB	B-Language
and	O
Mathematica	B-Language
have	O
built-in	O
routines	O
to	O
evaluate	O
the	O
integrals	O
,	O
either	O
as	O
standard	O
functions	O
or	O
in	O
extension	O
packages	O
.	O
</s>
<s>
They	O
have	O
a	O
phase	O
angle	O
close	O
to	O
a	O
value	O
of	O
/4	O
within	O
the	O
chirp	B-Application
range	O
and	O
they	O
only	O
start	O
to	O
change	O
significantly	O
for	O
frequencies	O
beyond	O
this	O
range	O
.	O
</s>
<s>
Consequently	O
,	O
for	O
frequencies	O
within	O
the	O
sweep	O
range	O
of	O
the	O
chirp	B-Application
,	O
it	O
is	O
the	O
square-law	O
phase	O
term	O
1( )	O
and	O
its	O
group	O
delay	O
function	O
(	O
=	O
-d1/d	O
(	O
)	O
)	O
that	O
are	O
of	O
most	O
interest	O
.	O
</s>
<s>
Fortunately	O
,	O
the	O
energy	O
content	O
of	O
the	O
chirp	B-Algorithm
spectrum	I-Algorithm
is	O
very	O
small	O
at	O
these	O
frequencies	O
(	O
as	O
is	O
demonstrated	O
in	O
a	O
later	O
section	O
)	O
.	O
</s>
<s>
When	O
the	O
Frequency-Time	O
characteristic	O
is	O
non-linear	O
,	O
the	O
Fourier	B-Algorithm
integral	I-Algorithm
is	O
difficult	O
to	O
evaluate	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
it	O
is	O
possible	O
to	O
resort	O
to	O
an	O
approximation	O
method	O
such	O
as	O
the	O
stationary	O
phase	O
approximation	O
,	O
or	O
to	O
use	O
numerical	B-General_Concept
methods	I-General_Concept
.	O
</s>
<s>
Often	O
(	O
as	O
in	O
radar	B-Application
applications	O
)	O
a(t )	O
is	O
a	O
slowly	O
varying	O
function	O
of	O
time	O
and	O
the	O
phase	O
(	O
t	O
)	O
is	O
oscillatory	O
and	O
varies	O
rapidly	O
,	O
over	O
the	O
range	O
of	O
integration	O
.	O
</s>
<s>
The	O
method	O
relies	O
on	O
the	O
fact	O
that	O
the	O
major	O
contributions	O
to	O
the	O
Fourier	B-Algorithm
integral	I-Algorithm
come	O
from	O
the	O
region	O
where	O
the	O
rate	O
of	O
change	O
of	O
phase	O
is	O
minimal	O
,	O
i.e.	O
</s>
<s>
This	O
is	O
a	O
very	O
useful	O
expression	O
linking	O
,	O
as	O
it	O
does	O
,	O
the	O
spectrum	O
profile	O
to	O
the	O
amplitude	O
and	O
phase	O
characteristics	O
of	O
the	O
chirp	B-Application
.	O
</s>
<s>
to	O
find	O
the	O
time	O
domain	O
function	O
s(t )	O
given	O
frequency	O
domain	O
data	O
,	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
derived	O
.	O
</s>
<s>
Typically	O
,	O
in	O
the	O
radar	B-Application
case	O
,	O
a(t )	O
is	O
a	O
constant	O
over	O
the	O
duration	O
of	O
the	O
signal	O
and	O
,	O
for	O
convenience	O
,	O
is	O
assumed	O
here	O
to	O
be	O
unity	O
.	O
</s>
<s>
The	O
two	O
filters	O
with	O
these	O
characteristics	O
can	O
be	O
used	O
as	O
the	O
transmitter	O
and	O
receiver	O
filters	O
of	O
a	O
radar	B-Application
system	I-Application
and	O
are	O
interchangeable	O
.	O
</s>
<s>
The	O
linear	O
chirp	B-Application
is	O
just	O
one	O
special	O
case	O
which	O
,	O
in	O
any	O
case	O
,	O
can	O
be	O
calculated	O
more	O
precisely	O
by	O
the	O
methods	O
of	O
the	O
earlier	O
section	O
.	O
</s>
<s>
In	O
such	O
cases	O
the	O
spectral	O
response	O
can	O
be	O
shaped	O
to	O
meet	O
some	O
desired	O
design	O
criteria	O
,	O
for	O
example	O
,	O
low	O
side-lobes	O
when	O
a	O
chirp	B-Application
is	O
compressed	O
.	O
</s>
<s>
Although	O
these	O
cosine	O
functions	O
are	O
amenable	O
to	O
mathematical	O
manipulation	O
,	O
they	O
are	O
rarely	O
chosen	O
to	O
define	O
the	O
spectral	O
characteristics	O
of	O
a	O
chirp	B-Application
,	O
in	O
practice	O
,	O
because	O
when	O
compressed	O
they	O
give	O
broad	O
main	O
pulses	O
with	O
high	O
side-lobe	O
levels	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
figure	O
below	O
shows	O
a	O
chirp	B-Algorithm
spectrum	I-Algorithm
with	O
T.F	O
=	O
250	O
obtained	O
for	O
a	O
non-linear	O
chirp	B-Application
aiming	O
to	O
match	O
the	O
Hamming	O
window	O
,	O
using	O
the	O
methods	O
described	O
above	O
.	O
</s>
<s>
Whenever	O
a	O
Fourier	B-Algorithm
integral	I-Algorithm
cannot	O
be	O
evaluated	O
by	O
analytical	O
means	O
,	O
an	O
approximate	O
solution	O
is	O
usually	O
possible	O
by	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
(	O
desired	O
)	O
baseband	O
spectrum	O
,	O
additional	O
versions	O
of	O
the	O
spectrum	O
occur	O
,	O
centered	O
on	O
multiples	O
of	O
the	O
sampling	B-Algorithm
frequency	I-Algorithm
.	O
</s>
<s>
no	O
aliasing	B-Error_Name
)	O
the	O
Nyquist	O
sampling	O
theorem	O
must	O
be	O
satisfied	O
.	O
</s>
<s>
A	O
straightforward	O
way	O
to	O
approximate	O
an	O
integral	O
,	O
such	O
as	O
a	O
Fourier	B-Algorithm
integral	I-Algorithm
,	O
is	O
to	O
use	O
the	O
standard	O
'	O
rectangle	O
rule	O
 '	O
for	O
numerical	B-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
The	O
method	O
assumes	O
the	O
signal	O
value	O
taken	O
at	O
a	O
sample	B-Algorithm
instant	O
remains	O
constant	O
for	O
one	O
sampling	O
interval	O
,	O
until	O
the	O
next	O
sample	B-Algorithm
is	O
taken	O
.	O
</s>
<s>
This	O
procedure	O
is	O
sometimes	O
referred	O
to	O
as	O
a	O
'	O
box-car	O
generator	O
 '	O
,	O
or	O
a	O
zero	O
order	O
sample	B-Algorithm
and	O
hold	O
.	O
</s>
<s>
The	O
result	O
so	O
obtained	O
is	O
the	O
convolution	O
of	O
a	O
rectangular	O
pulse	O
with	O
step	O
size	O
W	O
with	O
the	O
impulses	O
located	O
at	O
the	O
sampling	O
instants	O
with	O
weights	O
equal	O
to	O
the	O
sample	B-Algorithm
values	O
.	O
</s>
<s>
In	O
consequence	O
,	O
the	O
spectrum	O
of	O
interest	O
will	O
have	O
superimposed	O
upon	O
it	O
the	O
frequency	O
response	O
of	O
the	O
sample	B-Algorithm
and	O
hold	O
,	O
and	O
the	O
spectrum	O
of	O
the	O
sampled	O
signal	O
Ss	O
is	O
given	O
by	O
:	O
</s>
<s>
the	O
'	O
sin(x )	O
/x	O
'	O
part	O
,	O
is	O
the	O
frequency	O
response	O
of	O
the	O
sample	B-Algorithm
and	O
hold	O
.	O
</s>
<s>
Its	O
amplitude	O
decreases	O
with	O
frequency	O
and	O
it	O
falls	O
to	O
63%	O
of	O
its	O
peak	O
value	O
at	O
half	O
the	O
sampling	B-Algorithm
frequency	I-Algorithm
and	O
it	O
is	O
zero	O
at	O
multiples	O
of	O
that	O
frequency	O
(	O
since	O
fs	O
=	O
1/W	O
)	O
.	O
</s>
<s>
The	O
second	O
term	O
in	O
the	O
equation	O
is	O
called	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
discrete	O
signal	O
sn	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
an	O
exponential	O
chirp	B-Application
(	O
with	O
its	O
top	O
frequency	O
well	O
below	O
the	O
Nyquist	O
limit	O
)	O
is	O
sampled	O
at	O
256	O
points	O
,	O
as	O
shown	O
.	O
</s>
<s>
Although	O
the	O
characteristic	O
of	O
the	O
zero	O
order	O
hold	O
has	O
a	O
small	O
influence	O
on	O
this	O
result	O
,	O
the	O
slope	O
is	O
mainly	O
due	O
to	O
the	O
properties	O
of	O
the	O
chirp	B-Application
.	O
</s>
<s>
(	O
A	O
linear	O
chirp	B-Application
,	O
on	O
the	O
other	O
hand	O
,	O
has	O
a	O
nominally	O
flat	O
spectrum	O
because	O
its	O
frequencies	O
are	O
swept	O
at	O
the	O
same	O
rate	O
,	O
as	O
shown	O
in	O
some	O
earlier	O
plots	O
)	O
.	O
</s>
<s>
Consequently	O
,	O
more	O
efficient	O
algorithms	O
have	O
been	O
developed	O
,	O
especially	O
Fast	O
Fourier	B-Algorithm
Transforms	I-Algorithm
(	O
FFT	O
)	O
.	O
</s>
<s>
Computer	O
programs	O
which	O
implement	O
the	O
FFT	O
are	O
widely	O
available	O
in	O
the	O
literature	O
and	O
in	O
proprietary	O
CAD	O
programs	O
such	O
as	O
Mathcad	B-Application
,	O
MATLAB	B-Language
,	O
and	O
Mathematica	B-Language
.	O
</s>
<s>
In	O
the	O
following	O
example	O
a	O
linear	O
chirp	B-Application
with	O
time	O
bandwidth	O
product	O
of	O
25	O
is	O
sampled	O
at	O
128	O
points	O
(	O
i.e.	O
</s>
<s>
There	O
is	O
very	O
little	O
spectral	O
content	O
beyond	O
the	O
sweep	O
frequency	O
range	O
of	O
a	O
chirp	B-Application
pulse	O
and	O
this	O
is	O
especially	O
true	O
for	O
waveforms	O
where	O
the	O
time-bandwidth	O
product	O
is	O
large	O
.	O
</s>
<s>
The	O
full	O
line	O
on	O
the	O
graph	O
of	O
the	O
adjacent	O
figure	O
shows	O
results	O
for	O
linear	O
chirps	B-Application
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
non-linear	O
chirp	B-Application
,	O
or	O
a	O
linear	O
chirp	B-Application
shaped	O
by	O
amplitude	O
weighting	O
,	O
the	O
fraction	O
of	O
power	O
outside	O
F	O
is	O
even	O
lower	O
,	O
as	O
is	O
shown	O
on	O
the	O
graph	O
,	O
where	O
the	O
dashed	O
line	O
is	O
for	O
spectra	O
with	O
Hamming	O
profiles	O
.	O
</s>
<s>
This	O
low	O
spectral	O
spread	O
is	O
particularly	O
significant	O
when	O
baseband	O
signals	O
are	O
to	O
be	O
digitized	O
since	O
it	O
permits	O
a	O
sampling	B-Algorithm
frequency	I-Algorithm
to	O
be	O
chosen	O
which	O
is	O
only	O
slightly	O
higher	O
than	O
twice	O
the	O
maximum	O
frequency	O
excursion	O
of	O
the	O
chirp	B-Application
.	O
</s>
<s>
The	O
Fresnel	O
ripples	O
on	O
a	O
chirp	B-Algorithm
spectrum	I-Algorithm
are	O
very	O
obtrusive	O
,	O
especially	O
when	O
time-bandwidth	O
products	O
are	O
low	O
(	O
under	O
50	O
,	O
say	O
)	O
and	O
their	O
presence	O
leads	O
to	O
high	O
time	O
sidelobe	O
levels	O
when	O
chirps	B-Application
are	O
subject	O
to	O
pulse	B-Algorithm
compression	I-Algorithm
as	O
in	O
radar	B-Application
and	O
sonar	B-Application
systems	O
.	O
</s>
<s>
They	O
arise	O
because	O
of	O
the	O
sudden	O
discontinuities	O
in	O
the	O
chirp	B-Application
waveform	O
at	O
the	O
commencement	O
and	O
termination	O
of	O
the	O
pulse	O
.	O
</s>
<s>
Furthermore	O
,	O
some	O
of	O
the	O
methods	O
require	O
amplitude	O
shaping	O
,	O
or	O
amplitude	O
modulation	O
,	O
of	O
the	O
chirp	B-Application
pulse	O
and	O
this	O
makes	O
those	O
methods	O
unsuitable	O
when	O
,	O
for	O
example	O
,	O
the	O
chirp	B-Application
pulses	O
are	O
to	O
be	O
transmitted	O
by	O
a	O
power	O
amplifier	O
operating	O
in	O
a	O
near-limiting	O
condition	O
.	O
</s>
<s>
If	O
the	O
transitions	O
at	O
the	O
start	O
and	O
end	O
of	O
the	O
chirp	B-Application
are	O
made	O
less	O
sudden	O
(	O
or	O
more	O
'	O
rounded	O
 '	O
)	O
,	O
then	O
a	O
reduction	O
in	O
ripple	O
amplitude	O
is	O
achieved	O
.	O
</s>
<s>
Two	O
examples	O
show	O
the	O
spectra	O
of	O
linear	O
chirps	B-Application
with	O
finite	O
rise-times	O
.	O
</s>
<s>
The	O
first	O
is	O
for	O
a	O
chirp	B-Application
with	O
time-bandwidth	O
of	O
250	O
,	O
where	O
the	O
rise	O
and	O
fall	O
times	O
are	O
4%	O
of	O
the	O
total	O
pulse	O
duration	O
and	O
the	O
second	O
is	O
for	O
a	O
chirp	B-Application
with	O
time-bandwidth	O
of	O
25	O
,	O
where	O
the	O
rise	O
and	O
fall	O
times	O
are	O
10%	O
of	O
the	O
total	O
.	O
</s>
<s>
These	O
two	O
spectra	O
show	O
a	O
marked	O
reduction	O
in	O
ripple	O
amplitude	O
compared	O
to	O
the	O
spectra	O
of	O
unmodified	O
linear	O
chirps	B-Application
shown	O
earlier	O
.	O
</s>
<s>
A	O
analogous	O
technique	O
can	O
be	O
applied	O
to	O
the	O
frequency	O
characteristic	O
of	O
the	O
chirp	B-Application
waveform	O
by	O
adding	O
linear	O
FM	O
distortion	O
segments	O
(	O
quadratic	O
phase	O
modulation	O
distortion	O
)	O
to	O
the	O
frequency	O
characteristic	O
of	O
the	O
chirp	B-Application
,	O
as	O
shown	O
.	O
</s>
<s>
This	O
method	O
uses	O
an	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
in	O
order	O
to	O
derive	O
a	O
waveform	O
which	O
has	O
a	O
spectrum	O
with	O
the	O
phase	O
characteristic	O
of	O
a	O
chosen	O
chirp	B-Application
but	O
a	O
new	O
amplitude	O
profile	O
which	O
is	O
rectangular	O
and	O
ripple	O
free	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
a	O
linear	O
chirp	B-Application
waveform	O
with	O
a	O
time	O
bandwidth	O
of	O
25	O
is	O
shown	O
together	O
with	O
its	O
spectrum	O
magnitude	O
(	O
shown	O
by	O
a	O
full	O
line	O
)	O
which	O
,	O
as	O
demonstrated	O
earlier	O
,	O
has	O
a	O
large	O
ripple	O
component	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
find	O
,	O
by	O
means	O
of	O
an	O
inverse	O
FFT	O
,	O
a	O
chirp	B-Application
waveform	O
which	O
,	O
in	O
the	O
frequency	O
domain	O
,	O
has	O
the	O
same	O
phase	O
characteristic	O
as	O
before	O
,	O
but	O
with	O
the	O
rectangular	O
magnitude	O
characteristic	O
shown	O
by	O
the	O
dashed	O
line	O
on	O
the	O
plot	O
.	O
</s>
<s>
The	O
chirp	B-Application
waveform	O
resulting	O
from	O
this	O
process	O
has	O
a	O
very	O
long	O
time	O
duration	O
,	O
but	O
when	O
it	O
is	O
truncated	O
to	O
say	O
,	O
a	O
length	O
2T	O
,	O
then	O
the	O
spectrum	O
acquires	O
some	O
ripple	O
once	O
more	O
,	O
as	O
shown	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
a	O
chirp	B-Application
waveform	O
is	O
compressed	O
by	O
means	O
of	O
its	O
matched	O
filter	O
,	O
then	O
the	O
resultant	O
waveform	O
approximates	O
to	O
the	O
sinc	B-Algorithm
function	I-Algorithm
and	O
,	O
consequently	O
,	O
has	O
annoyingly	O
high	O
sidelobes	O
.	O
</s>
<s>
Similar	O
problems	O
arise	O
in	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
where	O
the	O
spectral	O
shaping	O
is	O
provided	O
by	O
a	O
window	B-Language
function	I-Language
,	O
a	O
process	O
sometimes	O
called	O
apodization	B-Algorithm
.	O
</s>
<s>
Although	O
spectral	O
shaping	O
of	O
a	O
chirp	B-Application
could	O
be	O
applied	O
in	O
the	O
frequency	O
domain	O
,	O
better	O
results	O
are	O
obtained	O
if	O
the	O
shaping	O
is	O
carried	O
out	O
in	O
the	O
time	O
domain	O
.	O
</s>
<s>
Examples	O
of	O
this	O
process	O
are	O
shown	O
for	O
linear	O
chirps	B-Application
with	O
time-bandwidth	O
products	O
of	O
250	O
and	O
25	O
.	O
</s>
<s>
Non-linear	O
chirps	B-Application
can	O
be	O
devised	O
that	O
have	O
a	O
bell	O
shaped	O
spectrum	O
,	O
such	O
as	O
the	O
Blackman-Harris	O
window	O
just	O
discussed	O
,	O
and	O
consequently	O
will	O
exhibit	O
reduced	O
ripple	O
compared	O
to	O
the	O
linear	O
chirp	B-Application
.	O
</s>
<s>
As	O
examples	O
,	O
plots	O
of	O
the	O
spectral	O
magnitudes	O
of	O
non-linear	O
chirps	B-Application
with	O
spectral	O
profiles	O
of	O
Blackman-Harris	O
windows	O
and	O
with	O
time-bandwidth	O
products	O
of	O
250	O
and	O
25	O
are	O
shown	O
below	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
,	O
there	O
is	O
some	O
ripple	O
reduction	O
,	O
but	O
the	O
disappointing	O
performance	O
can	O
be	O
attributed	O
to	O
the	O
fact	O
that	O
these	O
chirps	B-Application
,	O
although	O
they	O
have	O
reduced	O
energy	O
content	O
in	O
their	O
outer	O
frequency	O
regions	O
,	O
they	O
still	O
have	O
amplitude	O
profiles	O
with	O
fast	O
rise	O
and	O
fall	O
times	O
.	O
</s>
