Digital Signal Processing Reference
In-Depth Information
Exercise 5.4: Convolution - I.
Let
x
[
]
n
be a discrete-time sequence de-
fined as
⎧
⎨
⎩
l
g
r
,
y
i
d
.
,
©
,
L
s
M
−
n
0
≤
n
≤
M
[
]=
x
n
M
+
n
−
M
≤
n
≤
0
0
otherwise
for some odd integer
M
.
(a) Show that
x
can be expressed as the convolution of two discrete-
time sequences
x
1
[
n
]
[
n
]
and
x
2
[
n
]
.
(b) Using the previous results, compute the DTFT of
x
[
n
]
.
Exercise 5.5: Convolution - II.
Consider the following discrete-time sig-
nals:
x
[
n
]=
cos
(
1.5
n
)
5
sinc
n
1
y
[
n
]=
5
Compute the convolution:
x
]
2
[
n
∗
y
[
n
]
Exercise 5.6: System properties.
Consider the following input-output
relations and, for each of the underlying systems, determine whether the
system is linear, time invariant, BIBO stable, causal or anti-causal. Charac-
terize the eventual LTI systems by their impulse response.
(a)
y
[
n
]=
x
[
−
n
]
.
(b)
y
[
n
]=
e
−j
ω
n
x
[
n
]
.
n
+
n
0
(c)
y
[
n
]=
x
[
k
]
.
k
=
n
−
n
0
(d)
y
[
n
]=
ny
[
n
−
1
]+
x
[
n
]
,suchthatif
x
[
n
]=
0for
n
<
n
0
,then
y
[
n
]=
0
for
n
<
n
0
.
Exercise 5.7: Ideal filters.
Derive the impulse response of a bandpass
filter with center frequency
ω
0
and passband
ω
b
:
⎧
⎨
⎩
1
ω
−
ω
/
2
≤
ω
≤
ω
+
ω
/
2
0
b
0
b
e
j
ω
)=
(
H
bp
1
−
ω
0
−
ω
b
/
2
≥
ω
≥−
ω
0
+
ω
b
/
2
0 lewh e
