Digital Signal Processing Reference
In-Depth Information
Now we turn our attention to the bottom channel of Figure 9.10. The delay unit
shifts our inputs over one unit, so its rst output is zero, followed by x[0], x[1], x[2];
etc. When this signal reaches the down-sampler, it has the eect of removing every
other value. Since 0 is the rst input, it passes through, while x[0] is discarded.
Thus, the down-sampler's output is 0, x[1], x[3], x[5]; etc. At the end of the forward
transform for the bottom channel, we see that we have the odd values for x. The
synthesis side of the QMF for the bottom channel simply up-samples the signal.
This results in the pattern 0, 0, x[1], 0, x[3], 0, x[5]; etc.
Combining the top channel with the bottom channel with addition results in
0 + 0, x[0] + 0, 0 + x[1], x[2] + 0, 0 + x[3], x[4] + 0, 0 + x[5]; etc.
This demonstration shows that performing down and up-sampling on the chan-
nels of a QMF is not as crazy as it sounds. In eect, the QMF of Figure 9.8 breaks
the input into its even values and its odd values, then adds them back together.
This results in an output that matches the input, except for a delay.
9.5.2
Down-Sampling and Up-Sampling with 2 Coecients
With wavelets, the lter bank structure will include down-samplers and up-samplers
in the following conguration, Figure 9.11. In each channel, the signal undergoes
several changes. To keep them distinct, they are each labeled. In the upper channel,
we have z[n], which is the output from the rst or \analyzing" FIR lter, then z
d
[n]
corresponds to z[n] after it has been down-sampled, then z
u
[n] represents the signal
after up-sampling, and nally z
f
[n] gives us the nal values for that channel.
z [n]
u
z[n]
z [n]
d
z [n]
f
b, −a
2
2
−a, b
Input
Output
x[n]
a, b
2
2
b, a
y[n]
w[n]
w [n]
d
w [n]
u
w [n]
f
Figure 9.11: A two-channel lter bank with down/up-samplers.
Signals w[n] and z[n] are just as they were in section 9.1:
w[n] = ax[n] + bx[n1]
z[n] = bx[n]ax[n1]:
Signals w[n1] and z[n1], are again found by replacing n above with n1:
w[n1] = ax[n1] + bx[n2]
