Digital Signal Processing Reference
In-Depth Information
Exercise 7.4: Fourier transforms and filtering.
Consider the following
signal:
l
g
r
,
y
i
d
.
,
©
,
L
s
−
1
(
n
/
2
)+
1
for
n
even
x
[
n
]=
0
for
n
odd
(a) Sketch
x
[
n
]
in time.
(b) Which is the most appropriate Fourier representation for
x
[
n
]
?(DFT,
DFS, DTFT?) Explain your choice and compute the transform.
e
j
ω
)
(c) Compute the DTFT of
x
[
n
]
,
X
(
, and plot its magnitude and phase.
(d) Consider a filter with the impulse response
sin
n
π
h
[
n
]=
n
and compute
y
[
n
]=
x
[
n
]
∗
h
[
n
]
.
Exercise 7.5: FIR filters.
Consider the following set of complex numbers:
e
j
π
(
1
−
2
−k
)
,
=
=
z
k
k
1,2,...,
M
=
For
M
4,
=
(a) Plot
z
k
,
k
1, 2, 3, 4, on the complex plane.
(
)
(b) Consider an FIR whose transfer function
H
z
has the following zeros:
z
1
,
z
2
,
z
∗
1
,
z
∗
2
,
1
−
and write out explicitly the expression for
H
(
z
)
.
(c) Howmany nonzero taps does the impulse response
h
[
n
]
have atmost?
e
j
ω
)
(d) Sketch the magnitude of
H
(
.
(e) What can you say about this filter: What FIR type is it? (I, II, etc.)
Is it lowpass, bandpass, highpass?
Is it equiripple?
Is this a “good” filter? (By “good” we mean a filter which is close to 1 in
the passband, close to zero in the stopband and which has a narrow
transition band.)
