Digital Signal Processing Reference
In-Depth Information
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Frequency dom ain
Time domain
FFT
Freq. domain
delta pulse
Lowpass
Delay
LP and HP
Sum
Multipl. 1/2
Filter pattern of LP and HP or of
sliding averager and differencer
0
50
100
150
200
250
300
350
400
450
500
Hz
4
3
2
1
0
-1
-2
-3
-4
7,5
5,0
2,5
0,0
-2,5
-5,0
-7,5
7,5
5,0
2,5
0,0
-2,5
-5,0
-7,5
-10,0
4
3
2
1
0
-1
-2
-3
-4
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
4
3
2
1
0
-1
-2
-3
-4
4,0
3,5
3,0
2,5
2,0
1,5
1,0
0,5
0,0
Source signal
Sliding mean value equivalent to lowpass signal
Pulse
response (h(t)
of the sliding
mean value ,
i.e. of the low
pass filter
Sliding difference equivalent to highpass signal
Pulse
response h (t)
of the sliding
d
iff
erenc
e , i
.e
.
of the high
pass filter
Sum of LP and HP signal equivalent to source signal
25
50
75
100
20.0
25.0
30.0
ms
ms
Illustration 245:
LP and BP signal splitting with 2-tap FIR filter in the time domain
So far mainly pulse responses or wavelets and scaling functions with “taps” between 256 and 1024 dis-
crete support values were selected. For this reason nothing good is to be expected of a 2-tap FIR filter. The
result here is surprising: although the frequency domains of LP and BP strongly overlap (top right) the
source signal can again be completely reconstructed from the filtered signals! If we had selected a system
with several steps the result would be the same as the same thing happens on each subsequent level as on
the first. This is proved here not only as a formula ( Illustration 243) but also experimentally (see lower
sequence).
So far the choice of windowed Si-functions as a scaling function prevented the frequency
domains of adjacent filters from overlapping. This avoided distortions by aliasing and at
the same time took in the entire frequency domain which guaranteed freedom from errors
and at the same time complete inclusion of all the information present.
In Illustration 245 the frequency bands overlap very strongly as a result of the 2-tap filter
(top right). Nevertheless, the complete reconstruction of the source signal from the
filtered LP and BP signals is possible and this independently of the number of DWT steps.
