Digital Signal Processing Reference
In-Depth Information
3. A discussion on the computational complexity of the filter bank for
scaling
First, we discuss the computation cost of the introduced filter bank drawn in Figure 2 as
well as the sequential structure shown in Figure 1.
The sequential structure decomposes the input signal into sub-band frequency components
first with down-sampling by the factor
D
. Next, the decomposed signals are synthesized
after up-sampling by
D
. Such sub-band processing with the filter bank is followed by the
direct scaling structure, where further up-sampling by
U
is carried out. Thus, the sampling
rate increases with two successive up-sampling processes. After this, the sampling rate de‐
creases with down-sampling by
D
.
On the other hand, the introduced filter bank structure also decomposes the input into mul‐
tiple sub-band signals with down-sampling by
D
first, and then synthesizes the signals with
up-sampling by
U
.
Accordingly, we see that the sequential structure requires two down-sampling and up-sam‐
pling processes, while the introduced filter bank structure is carried out with only one
down-sampling and up-sampling process. Such a structural difference can cause different
computation time for scaling signals.
If we assume that both the systems have the same length of the analysis filters, then the dif‐
ference in length between both the systems will be in their synthesis part. The difference will
also cause different computation time. We also discuss these concerns with the design and
simulation in both of the systems in detail.
4. A discussion on the filter bank structure for scaling
4.1. The frequency decomposition and synthesis characteristics of the filter bank
Next, we discuss the performance of the frequency decomposition and synthesis with the fil‐
ter bank. Figure 3(a) shows an original signal
X
(
z
)
where the bold line represents π [rad] on
the frequency axis. Figure 3(b) shows the frequency decomposition characteristics with the
analysis low-pass filter
H
0
(
z
)
followed by down-sampling by the factor
D
. We also show
the counterpart of the high-pass filter
H
D
-1
(
z
)
and down-samplingin Figure 3(c). The fre‐
quency components colored with dots represent the aliasing components. Please note that
these decomposition characteristics in the analysis part are the same as those of traditional
(perfect reconstruction) filter banks.
