Digital Signal Processing Reference
In-Depth Information
narrowband signal: just like the impulse response of an ideal filter is neces-
sarily an infinite two-sided sequence, so any perfectly narrowband signal
cannot have an identifiable “beginning”. When we think of “applying” the
input to the filter, we are implicitly assuming a one-sided (or, more likely,
a finite-support) signal and this signal has nonzero spectral components at
all
frequencies. The net effect of these is that the overall delay for the signal
will always be nonnegative.
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Further Reading
Discrete-time filters are covered in all signal processing topics, e.g. a good
review is given in
Discrete-Time Signal Processing
,byA.V.Oppenheimand
R. W. Schafer (Prentice-Hall, last edition in 1999).
Exercises
Exercise 5.1: Linearity and time-invariance - I.
Consider the trans-
formation
x
]
=
[
n
nx
[
n
]
.Does
define an LTI system?
Exercise 5.2: Linearity and time-invariance - II.
Consider a discrete-
time system
cos
(
n
,the
{·}
. When the input is the signal
x
[
n
]=
2
π/
5
)
x
]
=
sin
(
π/
n
. Can the system be linear and time-
output is
[
n
2
)
invariant? Explain.
Exercise 5.3: Finite-support convolution.
Consider the finite-support
signal
h
[
n
]
defined as
1 r
M
0 th rwie
|
n
|≤
h
[
n
]=
(a) Compute the signal
x
[
n
]=
h
[
n
]
∗
h
[
n
]
for
M
=
2 and sketch the result.
e
j
ω
)
(b) Compute the DTFT of
x
[
n
]
,
X
(
, and sketch its value in the interval
.
(c) Give a qualitative description of how
X
[
0,
π
]
changes as
M
grows.
(
e
j
ω
)
(d) Compute the signal
y
2 and sketch the result.
For a general
M
, is the behavior of the sequence
y
[
n
]=
x
[
n
]
∗
h
[
n
]
for
M
=
[
n
]
? (E.g. is it linear
in
n
?Isitquadratic?)
e
j
ω
)
(e) Compute
Y
(
and sketch its value.
