Digital Signal Processing Reference
In-Depth Information
1,75
Translation
1,50
1,25
1,00
0,75
0,50
0,25
0,00
-0,25
-0,50
-0,75
50
100
150
200
250
300
350
400
450
500
ms
6
Broad wavelet
Scaling
5
t = 0 s
4
3
2
1
0
-1
Narrow wavelet
-2
25
50
75
100
125
150
175
200
225
250
275
300
32
ms
2,5
1,5
0,5
-0,5
-1,5
-2,5
1,75
1,25
0,75
0,25
-0,25
-0,75
1,25
0,50
-0,25
-1,00
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
Illustration 63:
Translation, scaling of a "Mexican hat“ wavelet and sliding averaging
Top Illustration: A wavelet with constant scaling slides on the time scale from left to right “along the
signal”. This “Mexican hat” wavelet has a certain similarity to “a very short sine”.
Middle Illustration: With t = 0 s on the rear time scale the broad (“mother”) wavelet can be seen; in the
next stage the wavelet is slightly compressed but remains unchanged in this translation. Here the
compressed wavelets for a whole series of stages are to be seen each with a translation of c. 95ms.
Bottom Illustration of the CWT. The signal is - from a mathematical point of view - multiplied by the
signal. This produces the bottom signal segment. As the wavelet has negative values at the edges these part
signals appear with the opposite sign to the original signal: the signal segment changes in the “rhythm” of
the wavelet! If the “rhythmic agreement” between the wavelet and the signal is large the following
averaging is also automatically large.
