Digital Signal Processing Reference
In-Depth Information
Time domain
Symmetrical spectrum
1, 20
0,050
0,045
0,040
0,035
0,030
0,025
0,020
0,015
0,010
0,005
0,000
0, 60
0, 00
-0,60
-1,20
-100
-75
-50
-25
0
25
50
75
10
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
Hz
ms
1, 20
0,050
0,045
0,040
0,035
0,030
0,025
0,020
0,015
0,010
0,005
0,000
0, 60
0, 00
-0,60
-1,20
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
-100
-75
-50
-25
0
25
50
75
10
ms
Hz
1, 20
0,050
0,045
0,040
0,035
0,030
0,025
0,020
0,015
0,010
0,005
0,000
0, 60
0, 00
-0,60
-1,20
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
-100
-75
-50
-25
0
25
50
75
10
ms
Hz
Illustration 236:
Subband coding from the perspective of filters, pulse responses and wavelets
Quadrature mirror filters QMF are used to prevent distortions or aliasing by overlapping adjacent fre-
quency bands. At first sight this seems (!) to be best achieved using high-quality filters with a rectangle-
like transmission function H(f). This would result in the fundamental form of all pulse responses. As
already pointed out in Illustration 233, the si-function becomes thus the mother function, also referred to
as the scaling function. All the other pulse responses are the result of multiplying the tops si-function by the
mid-frequency of the bandpass filters. The centre of the Illustration therefore shows the si-function as an
envelope. The si-functions were locally limited with a Blackman window and reduced to zero on the left
and right hand side.
The pulse responses in this Illustration appear as an analog representation to better illustrate the physical
relationships. What might be the connection between this Illustration and the wavelet transformation?
This can be seen particularly clearly in Illustration 236. The mother function or scaling
function is the above Si-function. The pulse responses h(t)
BP
of the four bandpass filters
are the result of multiplying the top si-function by the mid-frequencies 20, 40, 60 and 80
Hz. The Si-function is therefore always the envelope of these pulse responses.
Of particular interest are the two top pulse responses h(t)
TP
(0-255 Hz) and h(t)
BP
(256-512
Hz) in Illustration 234. The multiplication by the mid-frequency described here is
identical with the sign inversion of every other value (see also lists 0 and 1 in Illustration
234, bottom).
