Digital Signal Processing Reference
In-Depth Information
12.18.
Find and sketch
X[k],
the Discrete Fourier Transform (DFT) of
x[n] = e
j
6pn
,
8
where
n = 0, 1, Á , 7.
12.19.
In Figure P12.19, four Discrete Fourier Transforms (DFTs) are with plotted with
labels DFT A, DFT B, DFT C, and DFT D. Match the DFTs to the signals
x
1
[n], x
2
[n], x
3
[n], and x
4
[n].
1
8
e
-
j
2p3n
(a)
x
1
[n] =
, n = 0, 1, Á , 7
8
x
2
[n] = 2, n = 0, 1, Á , 7
(b)
(c)
1
8
e
j
2p3n
x
3
[n] =
, n = 0, 1, Á , 7
8
x
4
[n] = d[n], n = 0, 1, Á , 7
(d)
12.20.
Given the two four-point sequences
x[n] = [1, .75, .5, .25]
and
y[n] = [.75, .5, .25, 1],
express the DFT
Y[k]
in terms of the DFT
X[k].
12.21.
The DFT of the analog signal
f(t) = 7 cos(100 t)cos(40 t)
is to be computed.
(a)
What is the minimum sampling frequency to avoid aliasing?
(b)
If a sampling frequency of is used, how many samples must be
taken to give a frequency resolution of 1 rad/s?
v
s
= 300 rad/s
12.22.
An analog signal is sampled at 1024 equally spaced times within 1 second, and its DFT
is computed.
(a)
What is the separation in rad/s between successive frequency components?
(b)
What is the highest frequency that can be allowed in the analog signal if aliasing is
to be prevented?
12.23
Consider two length-4 discrete time signals
x[n] = e
j
2pn
4
, n = 0, 1, 2, 3
and
h[n] = 2d[n] + d[n - 2], n = 0, 1, 2, 3.
(a)
Find
X[k],
the DFT of
x[n]
.
(b)
Find
H[k],
the DFT of
h[n]
.
