Digital Signal Processing Reference
In-Depth Information
ω
−
ω
0
2
W
ω
+
ω
0
2
W
(c)
X
3
(
ω
)
=
rect
+
rect
;
(d)
X
4
(
ω
)
=
u
(
ω − ω
0
)
−
u
(
ω −
2
ω
0
)
.
9.3
The converse of the uncertainty principle, explained in Problem 9.2, is
also true. In other words, a time-limited signal cannot be band-limited. By
calculating the CTFT of the following time-limited signals, show that the
converse of the uncertainty principle is indeed true (assume that
τ,
T
, and
α
are real, positive constants):
(a)
x
1
(
t
)
=
cos(
ω
0
t
)[
u
(
t
+
T
)
−
u
(
t
−
T
)];
t
τ
t
τ
(b)
x
2
(
t
)
=
rect
∗
rect
(
∗
denotes the CT convolution operator);
t
τ
(c)
x
3
(
t
)
=
e
−α
t
rect
;
(d)
x
4
(
t
)
= δ
(
t
−
5)
+ δ
(
t
+
5).
9.4
The CT signal
x
(
t
)
=
v
1
(
t
)
v
2
(
t
) is sampled with an ideal impulse train:
∞
s
(
t
)
=
δ
(
t
−
kT
s
)
.
k
=−∞
(a) Assuming that
v
1
(
t
) and
v
2
(
t
) are two baseband signals band-limited
to 200 Hz and 450 Hz, respectively, compute the minimum value of
the sampling rate
f
s
that does not introduce any aliasing.
(b) Repeat part (a) if the waveforms for
v
1
(
t
) and
v
2
(
t
) are given by the
following expression:
v
1
(
t
)
=
sinc(600
t
)
and
v
2
(
t
)
=
sinc(1000
t
)
.
(c) Assuming that a sampling interval of
T
s
=
2 ms is used to sample
x
(
t
)
=
v
1
(
t
)
v
2
(
t
) specified in part (b), sketch the spectrum of the sam-
pled signal. Can
x
(
t
) be accurately recovered from the sampled signal?
(d) Repeat part (c) for a sampling interval of
T
s
=
0
.
1 ms.
9.5
The CT signal
x
(
t
)
=
sin(400
π
t
)
+
2 cos(150
π
t
) is sampled with an ideal
impulse train. Sketch the CTFT of the sampled signal for the following
values of the sampling rate:
(a)
f
s
=
100 samples/s;
(b)
f
s
=
200 samples/s;
(c)
f
s
=
400 samples/s;
(d)
f
s
=
500 samples/s.
In each case, calculate the reconstructed signal using an ideal LPF with the
transfer function given in Eq. (9.7) and a cut-off frequency of
ω
s
/
2
= π
f
s
.
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