Digital Signal Processing Reference
In-Depth Information
time signal
x
[
n
]
with
I
(
t
)
, we obtain a continuous-time signal:
I
t
−
nT
s
T
s
x
(
t
)=
x
[
n
]
n
l
g
r
,
y
i
d
.
,
©
,
L
s
which looks like this:
1.0
0.5
0
1
2
3
4
5
−
0.5
−
1.0
Assume that the spectrum of
x
[
n
]
between
−
π
and
π
is
1 r
|
ω
|≤
2
π/
3
e
j
ω
)=
X
(
0 rwie
(with the obvious 2
π
-periodicity over the entire frequency axis).
(a) Compute and sketch the Fourier transform
I
(
j
Ω)
of the interpolating
function
I
. (Recall that the triangular function can be expressed as
the convolution of rect
(
t
/
2
)
with itself ).
(
t
)
(b) Sketch the Fourier transform
X
(
j
Ω)
of the interpolated signal
x
(
t
)
;in
Ω
=
π/
particular, clearly mark the Nyquist frequency
T
s
.
N
(c) The use of
I
(
t
)
instead of a sinc interpolator introduces two types of
errors: briefly describe them.
[
−
Ω
Ω
]
(d) To eliminate the error
in the baseband
N
,
we can pre-filter the
N
signal
x
[
n
]
with a filter
h
[
n
]
before
interpolating with
I
(
t
)
. Write the
e
j
ω
)
frequency response of the discrete-time filter
H
(
.
Exercise 9.3: Another view of sampling.
One of the standard ways of
describing the sampling operation relies on the concept of “modulation by
a pulse train”. Choose a sampling interval
T
s
and define a continuous-time
pulse train
p
(
t
)
as
∞
p
(
t
)=
δ
(
t
−
kT
s
)
=
−∞
k
