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F I GU R E 14 . 15
Spectrum of the upsampled signal.
filters. Because they decompose the source output into components, they are also called
analysis
filters. The filters after the upsampling operation are used to recompose the original
signal; therefore, they are called
synthesis
filters. We can also view these filters as interpolating
between nonzero values to recover the signal at the point that we have inserted zeros. Therefore,
these filters are also called
interpolation
filters.
Although the use of ideal filters would give us perfect reconstruction of the source output,
in practice we do not have ideal filters available. When we use more realistic filters in place of
the ideal filters, we end up introducing distortion. In the next section we look at this situation
and discuss how we can reduce or remove this distortion.
14.6 Perfect Reconstruction Using Two-Channel
Filter Banks
Suppose we replace the ideal low-pass filter in Figure
14.11
with a more realistic filter with the
magnitude response shown in Figure
14.4
. The spectrum of the output of the low-pass filter is
showninFigure
14.16
. Notice that we now have nonzero values for frequencies above
2
.Ifwe
now subsample by two, we will end up sampling at
less
than twice the highest frequency, or in
other words, we will be sampling at below the Nyquist rate. This will result in the introduction
of aliasing distortion, which will show up in the reconstruction. A similar situation will occur
when we replace the ideal high-pass filter with a realistic high-pass filter.
In order to get perfect reconstruction after synthesis, we need to somehow get rid of the
aliasing and imaging effects. Let us look at the conditions we need to impose upon the filters
H
1
(
in order to accomplish this. These conditions are called
perfect reconstruction
(PR) conditions.
Consider Figure
14.17
. Let's obtain an expression for
X
z
),
H
2
(
z
),
K
1
(
z
)
, and
K
2
(
z
)
(
z
)
in terms of
H
1
(
z
),
H
2
(
z
)
,
K
1
(
z
)
, and
K
2
(
z
)
. We start with the reconstruction:
X
(
z
)
=
U
1
(
z
)
+
U
2
(
z
)
(42)
=
V
1
(
z
)
K
1
(
z
)
+
V
2
(
z
)
K
2
(
z
)
(43)
