Digital Signal Processing Reference
In-Depth Information
j
ω
H
(
e
)
(a)
ω
−π/3
−π/6
π/3
π/2
2π/3
−π
0
π
j
(
ω −
2
π
/ 3)
H
(
e
)
(b)
ω
0
π/3
2π/3
−π
π
j
(
ω −
4
π
/ 3)
H
(
e
)
(c)
ω
−π
−π/3
π
0
Figure 3.12
. (a) A multiband alias-free(3) decimation filter response
H
(
e
jω
). (b), (c)
The shifted versions
H
(
e
j
(
ω−
2
π/
3)
)and
H
(
e
j
(
ω−
4
π/
3)
) as seen in the region
−π ≤ ω<π.
3.4
Interpolation filters
An interpolation filter is a discrete-time filter used at the output of an expander
as shown in Fig. 3.13(a). This combination of an expander followed by an
interpolation filter is said to be a discrete-time
interpolator
. From its definition
we know that the expander inserts zero-valued samples between adjacent samples
of the input. The purpose of the interpolation filter is to replace these zeros with
weighted averages of the input samples, as we shall explain later.
First consider what happens in the frequency domain. Recall that the ex-
pander has
M
squeezed copies (images) of the input Fourier transform
X
(
e
jω
)
.
The interpolation filter retains one out of these
M
copies of the Fourier trans-
form. This is demonstrated in Fig. 3.13 for the case when the filter is lowpass
and
M
=3
.
Thus the interpolated output
Y
(
e
jω
) is nothing but a squeezed
version of
X
(
e
jω
) with the images removed.
3.4.1 Time-domain view of interpolation filter
Returning to the time domain, Fig. 3.14 shows the various signals involved.
The output of the interpolation filter is the convolution of its input
s
(
n
)with
the impulse response
h
(
n
)
,
that is,
∞
∞
y
(
n
)=
s
(
k
)
h
(
n
−
k
)=
s
(
kM
)
h
(
n
−
kM
)
,
k
=
−∞
k
=
−∞
where the second equality follows because
s
(
k
) = 0 unless
k
is a multiple of
M
.
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