Digital Signal Processing Reference
In-Depth Information
LP
LP
LP
LP
LP
LP
Tree structure
LP
LP
LP
LP
LP
LP
BP
LP
BP
LP
Filter 1
LP
BP
Filter 1
LP
BP
Filter 2
BP
BP
Filter 1
LP
Filter 2
BP
Filter 1
LP
Filter 2
BP
Filter 2
BP
Dyadic
cascade system composed of dual channel systems
f
t
“Uncertainty blocks” arising from narrowing of frequency domain
Δ
t
∗ Δ
f
= konstant
Illustration 240:
Discrete wavelet transformation with a dyadic cascade subband system
This circuit is generally used for DWT. Discrete wavelet transformation is the result of convolution of the
BP pulse responses or wavelets with the relevant input signal. Only the LP- filtered signal is split up again
and taken to a different step. The set of data is reduced by half by downsampling by factor 2 (“dyadic”). In
theory, this process could be continued until only one reading was left. In practice, the last LP-filtered
part is not changed once resolution is high enough.
Yet the overall image at the bottom of Illustration 65 is highly redundant. Everything is
dealt with a dozen times over. This results in the problem of finding the minimal number
of the wavelet scalings whose CWTs contain all the information of the wavelet transfor-
mation without any redundancy.
The problem of this discrete form of wavelet transformation DWT has luckily been
solved. As is so often the case in scientific research, already tried and tested methods in
signal and image processing turned out to be special forms of DWT. This includes the dy-
adic cascade subband system shown in Illustration 240. The pulse responses of the band-
pass filters are the discrete wavelets that are each convoluted with the relevant signal.
