Digital Signal Processing Reference
In-Depth Information
y[n]
= acx[n1] + bdx[n1]
+aax[n3] + bbx[n3] + ccx[n3] + ddx[n3]
+acx[n5] + bdx[n5]; n is even
y[n]
= acx[n1] + bdx[n1]
+aax[n3] + bbx[n3] + ccx[n3] + ddx[n3]
+acx[n5] + bdx[n5]; n is odd:
In this nal step, we notice two things. First, the expressions are the same regardless
of whether n is even or odd, so now we can represent it as a single statement:
y[n]
= aax[n3] + bbx[n3] + ccx[n3] + ddx[n3]
+acx[n1] + bdx[n1] + acx[n5] + bdx[n5]:
Second, we see that we have x[n1] and x[n5] terms again, but that these can
be eliminated if we require ac =bd. Assuming this is the case, we have our nal
expression for y[n]:
y[n] = (aa + bb + cc + dd)x[n3]:
If we compare this result to the earlier case (without down-sampling) we see that the
dierence is a factor of 2. This explains why the sum of the squares of the Daubechies
coecients, (aa + bb + cc + dd) equals 1. When that is the case, y[n] = x[n3], or
y[n] is simply a copy of x[n], delayed by 3 time units.
9.6
Breaking a Signal Into Waves
Just as the discrete Fourier transform breaks a signal into sinusoids, the discrete
wavelet transform generates \little waves" from a signal. These waves can then be
added together to reform the signal. Figure 9.13 shows the wavelet analysis for
three octaves on the left, with the standard synthesis (reconstruction) on the right.
\LPF" stands for lowpass lter, while \HPF" means highpass lter. The \ILPF"
and \IHPF" are the inverse low- and highpass lters, respectively. Although the
down-sampling and up-sampling operations are not explicitly shown, they can be
included in the lters without loss of generality. On the right, there is an implied
addition when two lines meet.
Figure 9.14 shows an alternate way to do the reconstruction. While not as
ecient as Figure 9.13, it serves to demonstrate the contribution of each channel.
