Digital Signal Processing Reference
In-Depth Information
more) pair of analysis lters. For the wavelet transform, we do this with the lowpass
lter's output. Wavelet packets, however, use an additional lter pair for each chan-
nel. We will examine multiresolution starting with the Daubechies four-coecient
wavelet transform, with down-sampling and up-sampling, for two levels of resolution
(octaves), as shown in Figure 9.22.
z[n]
z [n]
d
2
d, −c, b, −a
Input
z2[n]
z2 [n]
d
d, −c, b, −a
2
x[n]
a, b, c, d
2
w[n]
w [n]
d
a, b, c, d
2
w2[n]
w2 [n]
d
z [n]
d
z [n]
u
z [n]
f
2
−a, b, −c, d
z2 [n]
d
z2 [n]
u
z2 [n]
f
Output
2
−a, b, −c, d
w [n]
d
2
d, c, b, a
y[n]
w2 [n]
d
w [n]
u
w [n]
f
2
d, c, b, a
w2 [n]
u
w2 [n]
f
Figure 9.22: Two levels of resolution.
Signals w[n], w
d
[n], z[n], and z
d
[n] are the same as before.
w[n] = ax[n] + bx[n1] + cx[n2] + dx[n3]
z[n] = dx[n]cx[n1] + bx[n2]ax[n3]
w
d
[n] = w[n]; n is even; 0 otherwise
z
d
[n] = z[n]; n is even; 0 otherwise
To generate w2[n], w2
d
[n], z2[n], and z2
d
[n], we rst notice that they have the
same relationship to w
d
[n] as w
d
[n] and z
d
[n] have to x[n]. In other words, we can
reuse the above equations, and replace x[n] with w
d
[n]. Also, since there is a down-
sampler between w2[n] and w2
d
[n], every other value of w2[n] will be eliminated.
Signal w2[n], by denition, is based upon w
d
[n], a down-sampled version of w[n].
Using the original n, therefore, means that we must say that every other value from
the even values of n will be eliminated. In other words, the only values that will get
