Digital Signal Processing Reference
In-Depth Information
Time domain
Symmetrical spectrum
7,5
5,0
2,5
0,0
-2,5
-5,0
-7,5
100
25
-50
-125
100
50
0
-50
-100
1500
1000
50
0
-500
-1000
-1500
1500
1000
50
0
-500
-1000
-1500
25000
15000
5000
-5000
-15000
-25000
20000
5000
-10000
-25000
4E+5
2E+5
0E+5
-2E+5
-4E+5
3,0E+5
2,0E+5
1,0E+5
0,0E+5
-1,0E+5
0,200
0,150
0,100
0,050
0,000
4,0
3,0
2,0
1,0
0,0
4,0
3,0
2,0
1,0
0,0
80
6
40
20
0
70
50
30
10
1500
1250
1000
750
500
25
0
1250
0
25000
20000
15000
10000
500
0
100000
75000
50000
2500
0
0
50
150
250
350
450
550
650
750
850
950
-500
-250
0
250
500
ms
Hz
Illustration 242:
Simulation of a four-step DWT system in the time and frequency domain
This simulation with DASYLab was carried out direct using special filters, and not using a concrete DWT
system, so that all signals in the time and frequency domain can be represented in one Illustration. The
three LP signals, which are each divided into an LP signal (of half the length) and a BP signal, are also
depicted. The time domain reveals whether it is a LP or a BP signal.
The cascade subband systems in Illustration 238 and Illustration 240 therefore each
represent an entire DWT system. It consequently includes the entire information con-
tained in the source signal because the complete frequency range is registered. In addition,
it is indirectly also a limiting case of the sampling principle.
In order to be able to show the signal information as redundancy-free and compact as
possible using DWT, the scaling function and the relevant wavelets require an optimum
adaptation from the start. This sounds as if you need a template which, to a certain degree,
is an optimum representation of the original. In practice, it is necessary to develop -
irrespective of the theoretical knowledge of DWT - a physical sense of form and duration
of the contents of a signal that determines its information. This is the fine art of CWT and
DWT.
What does the optimum solution look like? There is a simple answer: The coefficients of
the DW-transformed signals - i.e. the filtered BP signals and the sequence of digits - must
have as few areas as possible where readings deviate significantly from zero. From the
point of view of physics this means that the energy is concentrated in only a few of these
coefficients. This is the case when the selection of the “basic pattern” is optimal. In this
case it is generally easy to compress a signal strongly or to largely eliminate the noise of
a signal.
