Digital Signal Processing Reference
In-Depth Information
0,08
0,06
0,04
0,02
12,5
10,0
7,5
5,0
2,5
0,0
-2,5
5,0
2,5
0,0
-2,5
-5,0
10,0
7,5
5,0
2,5
0,0
-2,5
-5,0
7,5
5,0
2,5
0,0
-2,5
-5,0
-7,5
7,5
Pulse response h(t)
Amplitude Spectrum
200
100
0
-100
-200
0,08
0,06
0,04
0,02
0,00
200
100
0
-100
-200
0,09
0,06
0,03
0,00
200
100
0
-100
-200
Phase Spectrum
LP -filtered rectangular signal
Pulse response h (t)
Amplitude Spectrum
Phase Spectrum
LP -filtered rectangular signal
Pulse response h(t)
Amplitude Spectrum
-5,0
7,5
5,0
2,5
0,0
-2,5
-5,0
-7,5
Phase Spectrum
LP- filtered rectangular signal
0 25
75
125
175
225
275
325
375
25
50
75
100
125
150
175
ms
Hz
Illustration 142:
Traditional analog lowpass filters of the 10th order
The arrangement of the signals or their amplitude and phase curve corresponds exactly to to Illustration
141. However, all the lowpass filters here are 10th order, roughly the maximum of what can be created in
analog terms with a reasonable expenditure of effort.
In the distortions of the rectangular signal you see the weaknesses and virtues of the different filter types.
Note that the pulse response h(t) lasts longer the steeper the edges of the filter. The trained person can give
very precise information on the curve of the transfer function H(f) from the curve of h(t).
The steps in the phase curve should not confuse you. They always go from
or from 180 degrees to -
180 degrees. Both angles are identical. For this reason it is usual to enter the phase curve only between
these two cutoff values. An angle of 210 degrees is plotted at -180 + 30 = -150 degrees.
π
to -
π
Bottom right you see a phase curve with „random steps“. It results from the numerical calculation and has
nothing to do with real curve. Precisely speaking, the computer has difficulty with the division of values
which are „close to zero“.
The BUTTERWORTH filter represents an important compromise between these two
types. It has a reasonably linear amplitude curve in the conducting state region and toler-
able edge steepness in the transition from the conducting state region to the blocking state
region.
In the case of the traditional analog filters there is therefore only the possibility of obtai-
ning a tolerable filter characteristic at the expense of other filter values. There are two
straightforward reasons for this:
