Digital Signal Processing Reference
In-Depth Information
2.5.2.A Finite-duration chirp
In practice the chirp is truncated to be a finite duration pulse. Thus the trun-
cated complex chirp is
⎧
⎨
e
jπKt
2
T
2
≤
T
2
−
t
≤
p
T
(
t
)=
(2
.
62)
⎩
0
otherwise.
The subscript
T
is temporary, for added clarity. The instantaneous frequency
f
t
=
Kt
changes from
−
KT/
2to
KT/
2, so the total frequency sweep, denoted
as
F
m
,isgivenby
F
m
=
KT.
(2
.
63)
The product
D
=
TF
m
=
KT
2
(2
.
64)
is dimensionless and is called the
time-bandwidth product
of the chirp. Note
that
p
T
(
t
) is an even function, that is,
p
T
(
t
)=
p
T
(
−
t
). The Fourier transform
therefore satisfies
P
T
(
jω
)=
P
T
(
−jω
)
.
(2
.
65)
|
2
is (real and) even, which shows that the autocorrelation
of the complex chirp
p
T
(
t
) is real and even:
In particular
|
P
T
(
jω
)
R
(
τ
)=
R
(
−τ
)
.
(2
.
66)
2.5.2.B Fourier transform of finite-duration chirp
The Fourier transform
P
T
(
jω
) is a convolution of
P
(
jω
) in Eq. (2.59) with an
appropriate sinc function. There is no simple closed form for this, but it has
been shown [Klauder
et al.,
1960] that for large values of the time-bandwidth
product
D
, most of the energy of the Fourier transform is in the region
F
m
2
≤ f ≤
F
m
2
−
,
(2
.
67)
as demonstrated in Fig. 2.30. (Actually there are ripples in the plot which we
have not shown. See Fig. 2.34 for a preview.) The magnitude at
f
=
F
m
/
2is
about 6 dB below the zero-frequency value. It has been claimed [Klauder
et al.,
1960] that, when
D
= 10, about 95% of the energy is contained in
|
f
|≤
F
m
/
2,
whereas, for
D
= 100, about 99% of the energy is in
F
m
/
2
.
Thus the
time-limited chirp is
approximately bandlimited
with total bandwidth
|
f
|≤
≈
F
m
.
2.5.2.C Autocorrelation of finite-duration chirp
The autocorrelation of
p
T
(
t
) has been shown to be
⎧
⎨
−
|
t
T
sin
πF
m
t
1
−
|
t
T
T
1
πF
m
t
1
−
|
t
T
−
T
≤
t
≤
T
R
(
τ
)=
(2
.
68)
⎩
0
otherwise,
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