Digital Signal Processing Reference
In-Depth Information
is the shape of the pulse
p
(
t
) that makes the autocorrelation
R
pp
(
τ
) nar-
rowest (in some well defined mathematical sense)? This is called the
pulse
compression
problem because
p
(
t
) is “compressed” into
R
pp
(
τ
). The pulse
compression ratio, defined in the next section, is the extent to which the
duration has been compressed when going from
p
(
t
)to
R
pp
(
τ
)
.
The reader
interested in this important topic should pursue references such as Van
Trees [2001] and Levanon and Mozeson [2004].
2.5.2 The chirp or LFM signal
A commonly used waveform in radar applications is the
chirp
waveform
p
(
t
)=
e
jπKt
2
.
(2
.
57)
Here
K
is a positive constant called the chirp parameter. It has the dimension
(Hz)
2
so that
Kt
2
is dimensionless. The above signal is called the
complex chirp
to distinguish it from the real version cos(
πKt
2
). The finite-duration version
of this waveform provides an excellent compression ratio as we shall see. For
a preview of what the waveform looks like, see Figs. 2.32 and 2.35. We can
regard
p
(
t
) as a signal with “instantaneous frequency” which increases linearly
with time. For the waveform
e
jω
0
t
the fixed frequency
ω
0
is the derivative of
the phase
ω
0
t
. Similarly for the chirp waveform the “instantaneous frequency”
is the derivative
d
(
πKt
2
)
dt
=2
πKt
rad
/
s
.
Dividing by 2
π
we get the instantaneous frequency in hertz:
f
t
=
Kt
Hz.
(2
.
58)
Since this is linearly increasing with time,
p
(
t
) is also called a linear frequency
modulated (or
LFM
) waveform. It can be shown that the Fourier transform of
p
(
t
) is given by [Papoulis, 1968]
11
1
1
√
K
e
jπ/
4
e
−jω
2
/
4
πK
=
√
K
e
jπ/
4
e
−jπf
2
/K
,
P
(
jω
)=
(2
.
59)
=1
/
√
K
for all
ω,
and
where we have used
ω
=2
πf.
Note that
=
1 for all
t.
Thus, both the function
and
its Fourier transform are spread out
uniformly! The real part of the complex chirp is the real-chirp function
|
P
(
jω
)
|
|
p
(
t
)
|
p
re
(
t
) = cos(
πKt
2
)
,
(2
.
60)
and this has the Fourier transform (Problem 2.10)
√
K
cos
ω
2
.
1
π
4
P
re
(
jω
)=
4
πK
−
(2
.
61)
11
See Problem 2.10. Note that
p
(
t
) is not absolutely integrable, nor square integrable. Its
Fourier transform exists only if it is interpreted as a Cauchy principal value [Papoulis, 1968],
that is, as the limit of
a
−a
p
(
t
)
e
−j
2
πft
dt
as
a →∞.
Search WWH ::
Custom Search