Digital Signal Processing Reference
In-Depth Information
Delta-pulse
FI R
Ti me domai n
FF T
Fr eq. domai n
10,0
7,5
5,0
2,5
0,0
-2,5
10,0
7,5
5,0
2,5
0,0
-2,5
10,0
7,5
5,0
2,5
0,0
-2,5
0,050
0,045
0,040
0,035
0,030
0,025
0,020
0,015
0,010
0,005
0,000
0,0250
0,0225
0,0200
0,0175
0,0150
0,0125
0,0100
0,0075
0,0050
0,0025
0,0000
0,0125
0,0100
0,0075
0,0050
0,0025
0,0000
255
255
260
260
265
265
270
270
275
275
280
280
285
285
290
290
295
295
300
300
305
305
-500
-250
0
250
500
ms
ms
Hz
Illustration 239:
Changing the properties of a filter by subsampling the pulse response
Downsampling by factor 2 results in a reduced duration of the signal or the pulse response. As a result of
the Uncertainty Principle
UP
, the bandwidth of the new FIR filter is doubled. If downsampling is used on
the new FIR filter another FIR filter with the fourfold bandwidth is created).
To sum up: the same HP-LP filter set that was used initially can be used in each step of
the cascade system. Downsampling in each of these steps has the effect of an (additional)
magnifying glass. The first step divides the bandwidth of the source signal into an HP and
a LP range of equal size, the second step, in turn, divides each of these two bands into
an HP and a LP range and so forth.
To use a metaphor: While the original image shows a compact forest, step by step scaling
gradually exposes individual trees, branches and eventually leaves.
Discrete Wavelet Transformation and Multi-Scale Analysis MSA
Wavelet transformation following (windowed) short-term FFT (GABOR Transforma-
tion) was already dealt with at the end of Chapter 3 which demonstrates that wavelet
transformation enables us to extend the uncertainty phenomenon from the time-frequency
problem to other patterns contained in the mother wavelet, such as steps and
discontinuities. Wavelet transformation is specialised in efficiently analysing, filtering
and storing almost any form of change.
Illustration 65 presents these results. In the continuous wavelet transformation CWT used
here, a wavelet is continuously compressed, i.e. its scaling is continuously changed. The
present, momentary CWT is represented for some of the scalings. All these CWTs are
combined in an overall image at the bottom of the Illustration. This image reproduces the
structure of the test signal in much more detail than the FOURIER transformation.
