>"Sam (r鰮 sans -no-sp-am)" <totalsam-no-sp-am@hotmail.com> wrote in
message
>news:3ff754cd$0$726$5402220f@news.sunrise.ch...
>> Hi all !
>>
>> Thanks for reading ! I would like to design a 1st order digital high
pass
>> filter with a 50 us time constant. The analog transfer function is
>> Ha(s)=50E-6*s/(1+50E-6*s).
>>
>> Then I use the bilinear transformation. Sampling freq is 48 kHz. The
>> pre-warped pulsation is 0.4228 rad/sec and the z transfer function is
>> H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
>>
>> When I compute this filter, it seems to be unstable at 24 kHz, and the
FFT
>> of a filtered signal gives a spike at this frequency.
>>
>> Is something wrong in my design ?
>
>Yes, but not in your calculations for the digital fitler.
>
>> How can I avoid having this oscillation ?
>
>You need to plot the results for a 20khz signal for 1 millisecond to see
how
>the sampling is affecting your results.
>
>> Putting a low-pass filter after this one seems not to be a good idea I
>think
>> ...
>>
>> Who could help me ??
>>
>> Thanks a LOT in advance !
>>
>> Sam
>>
>>
>You are right. Try putting the analog low pass filter before the
digital
>filter.
>Check out anti-aliasing.
>
>Peter Nachtwey
>I suggest to introduce a ZERO at this point, this hopefully will cancel
the effect of oscillation.
regards
al
>
>
Reply Start a New Thread
Robin Clark wrote:
> On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:
>
>> Sam (r�p. sans -no-sp-am) wrote:
>>
>>> Hi all !
>>>
>>> Thanks for reading ! I would like to design a 1st order
>>> digital high pass filter with a 50 us time constant. The
>>> analog transfer function is Ha(s)=50E-6*s/(1+50E-6*s).
>>>
>>> Then I use the bilinear transformation. Sampling freq is 48
>>> kHz. The pre-warped pulsation is 0.4228 rad/sec and the z
>>> transfer function is H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
>>
>> Something must have gone wrong with the bilinear transform.
>> It would have been useful to show your calculations. The BLT
>> substitutes s <-- c*(1-1/z)/(1+1/z). To make the responses
>> equal at frequency fc < fs/2, choose c = 2*Pi*fc *
>> cot(PI*fc/fs), where fs is the sampling frequency.
>>
>> Assuming that by "pre-warped pulsation" you meant the
>> quantity 2*Pi*fc, I get the discrete transfer function
>>
>> H(z) = 0.8275862069 * (1-1/z) / (1-0.6551724138/z)
>>
>> for your Ha and fs.
>>
>>
>> Martin
>
>
> I plotted a frequency sweep for the coeffs you calculated above
> and got a nice unity gain high pass with a 3 bd point around 1
> tenth of the nyquist frequency
Robin,
thanks for also making your plots available in a readily human-
readable form. They coincide with mine. The short succession and
identical contents (apart from the plot values) of your first two
responses give a certain sense of urgency, but what did you actually
intend to tell me?
Martin
Reply Start a New Thread
Assuming you dropped a minus the plots for this quite nice high
passs filter. 1000 == nyquist frequency
http://80.3.72.34/~robin/stuff2/
Reply Start a New Thread
On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:
> Sam (r�p. sans -no-sp-am) wrote:
>
>> Hi all !
>>
>> Thanks for reading ! I would like to design a 1st order digital
>> high pass filter with a 50 us time constant. The analog transfer
>> function is Ha(s)=50E-6*s/(1+50E-6*s).
>>
>> Then I use the bilinear transformation. Sampling freq is 48 kHz.
>> The pre-warped pulsation is 0.4228 rad/sec and the z transfer
>> function is H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
>
> Something must have gone wrong with the bilinear transform.
> It would have been useful to show your calculations. The BLT
> substitutes s <-- c*(1-1/z)/(1+1/z). To make the responses
> equal at frequency fc < fs/2, choose c = 2*Pi*fc * cot(PI*fc/fs),
> where fs is the sampling frequency.
>
> Assuming that by "pre-warped pulsation" you meant the
> quantity 2*Pi*fc, I get the discrete transfer function
>
> H(z) = 0.8275862069 * (1-1/z) / (1-0.6551724138/z)
>
> for your Ha and fs.
>
>
> Martin
I plotted a frequency sweep for the coeffs you calculated above
and got a nice unity gain high pass with a 3 bd point around 1 tenth
of the nyquist frequency
Reply Start a New Thread
On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:
>> Thanks for reading ! I would like to design a 1st order digital
>> high pass filter with a 50 us time constant. The analog transfer
>> function is Ha(s)=50E-6*s/(1+50E-6*s).
>>
> Assuming that by "pre-warped pulsation" you meant the
> quantity 2*Pi*fc, I get the discrete transfer function
>
> H(z) = 0.8275862069 * (1-1/z) / (1-0.6551724138/z)
>
Plotting the above co-effs gives a high pass filter
with a gain of 1 and a 3db point around a tenth of the nyquist
frequency.
I plot biquads using a c program and gnuplot in solaris.
Here are the complex number results for a frequency sweep of
this filter configuration from it.
# a0 0.827586 a1 -0.827586 a2 0.000000 b0 1.000000 b1 -0.655170 b2 0.000000
# MAG_NOTCH PHASE_NOTCH MAG_SPEC_INV MAG_SPEC_PHASE RADIANS
1 0.007540 1.563257 0.999972 -0.007540 angle 0.003142
2 0.015078 1.555718 0.999886 -0.015078 angle 0.006283
3 0.022614 1.548181 0.999744 -0.022616 angle 0.009425
4 0.030146 1.540646 0.999546 -0.030150 angle 0.012566
5 0.037673 1.533115 0.999290 -0.037682 angle 0.015708
6 0.045194 1.525587 0.998978 -0.045209 angle 0.018850
7 0.052707 1.518065 0.998610 -0.052732 angle 0.021991
8 0.060212 1.510548 0.998186 -0.060248 angle 0.025133
9 0.067707 1.503038 0.997705 -0.067759 angle 0.028274
10 0.075190 1.495535 0.997169 -0.075261 angle 0.031416
11 0.082662 1.488040 0.996578 -0.082756 angle 0.034558
12 0.090120 1.480554 0.995931 -0.090242 angle 0.037699
13 0.097563 1.473078 0.995229 -0.097718 angle 0.040841
14 0.104990 1.465612 0.994473 -0.105184 angle 0.043982
15 0.112401 1.458158 0.993663 -0.112639 angle 0.047124
16 0.119793 1.450715 0.992799 -0.120081 angle 0.050265
17 0.127166 1.443286 0.991882 -0.127511 angle 0.053407
18 0.134518 1.435869 0.990911 -0.134927 angle 0.056549
19 0.141849 1.428467 0.989888 -0.142329 angle 0.059690
20 0.149158 1.421080 0.988813 -0.149716 angle 0.062832
21 0.156442 1.413709 0.987687 -0.157088 angle 0.065973
22 0.163702 1.406354 0.986510 -0.164442 angle 0.069115
23 0.170937 1.399016 0.985282 -0.171780 angle 0.072257
24 0.178144 1.391696 0.984004 -0.179100 angle 0.075398
25 0.185324 1.384395 0.982677 -0.186402 angle 0.078540
26 0.192476 1.377112 0.981302 -0.193684 angle 0.081681
27 0.199597 1.369850 0.979878 -0.200947 angle 0.084823
28 0.206688 1.362608 0.978407 -0.208189 angle 0.087965
29 0.213748 1.355387 0.976889 -0.215410 angle 0.091106
30 0.220775 1.348187 0.975325 -0.222609 angle 0.094248
31 0.227769 1.341011 0.973715 -0.229786 angle 0.097389
32 0.234729 1.333857 0.972061 -0.236940 angle 0.100531
33 0.241655 1.326726 0.970362 -0.244071 angle 0.103673
34 0.248544 1.319620 0.968621 -0.251177 angle 0.106814
35 0.255398 1.312538 0.966836 -0.258259 angle 0.109956
36 0.262214 1.305481 0.965010 -0.265316 angle 0.113097
37 0.268992 1.298450 0.963142 -0.272346 angle 0.116239
38 0.275732 1.291446 0.961235 -0.279351 angle 0.119381
39 0.282433 1.284468 0.959287 -0.286329 angle 0.122522
40 0.289094 1.277517 0.957301 -0.293280 angle 0.125664
41 0.295714 1.270594 0.955277 -0.300203 angle 0.128805
42 0.302294 1.263699 0.953215 -0.307098 angle 0.131947
43 0.308831 1.256833 0.951117 -0.313964 angle 0.135088
44 0.315327 1.249995 0.948983 -0.320801 angle 0.138230
45 0.321780 1.243187 0.946814 -0.327609 angle 0.141372
46 0.328190 1.236410 0.944612 -0.334387 angle 0.144513
47 0.334557 1.229662 0.942376 -0.341135 angle 0.147655
48 0.340879 1.222945 0.940107 -0.347852 angle 0.150796
49 0.347157 1.216259 0.937807 -0.354538 angle 0.153938
50 0.353391 1.209604 0.935476 -0.361193 angle 0.157080
51 0.359579 1.202981 0.933115 -0.367816 angle 0.160221
52 0.365721 1.196389 0.930725 -0.374407 angle 0.163363
53 0.371818 1.189831 0.928306 -0.380966 angle 0.166504
54 0.377868 1.183304 0.925860 -0.387493 angle 0.169646
55 0.383872 1.176811 0.923386 -0.393986 angle 0.172788
56 0.389829 1.170350 0.920887 -0.400446 angle 0.175929
57 0.395740 1.163923 0.918363 -0.406873 angle 0.179071
58 0.401603 1.157530 0.915814 -0.413267 angle 0.182212
59 0.407419 1.151171 0.913241 -0.419626 angle 0.185354
60 0.413188 1.144845 0.910646 -0.425952 angle 0.188496
61 0.418909 1.138554 0.908029 -0.432243 angle 0.191637
62 0.424582 1.132297 0.905390 -0.438500 angle 0.194779
63 0.430207 1.126075 0.902730 -0.444722 angle 0.197920
64 0.435784 1.119887 0.900051 -0.450910 angle 0.201062
65 0.441314 1.113735 0.897353 -0.457062 angle 0.204204
66 0.446795 1.107617 0.894637 -0.463180 angle 0.207345
67 0.452228 1.101535 0.891902 -0.469262 angle 0.210487
68 0.457613 1.095488 0.889152 -0.475309 angle 0.213628
69 0.462950 1.089476 0.886384 -0.481321 angle 0.216770
70 0.468239 1.083500 0.883602 -0.487297 angle 0.219911
71 0.473480 1.077559 0.880805 -0.493238 angle 0.223053
72 0.478673 1.071654 0.877994 -0.499143 angle 0.226195
73 0.483818 1.065785 0.875169 -0.505012 angle 0.229336
74 0.488915 1.059951 0.872332 -0.510846 angle 0.232478
75 0.493965 1.054153 0.869482 -0.516644 angle 0.235619
76 0.498966 1.048391 0.866622 -0.522406 angle 0.238761
77 0.503920 1.042665 0.863750 -0.528132 angle 0.241903
78 0.508827 1.036975 0.860869 -0.533822 angle 0.245044
79 0.513687 1.031321 0.857978 -0.539477 angle 0.248186
80 0.518499 1.025702 0.855078 -0.545095 angle 0.251327
81 0.523265 1.020119 0.852170 -0.550678 angle 0.254469
82 0.527984 1.014572 0.849255 -0.556225 angle 0.257611
83 0.532657 1.009061 0.846332 -0.561736 angle 0.260752
84 0.537283 1.003585 0.843403 -0.567212 angle 0.263894
85 0.541863 0.998145 0.840467 -0.572652 angle 0.267035
86 0.546397 0.992741 0.837527 -0.578056 angle 0.270177
87 0.550885 0.987372 0.834581 -0.583425 angle 0.273319
88 0.555329 0.982039 0.831631 -0.588758 angle 0.276460
89 0.559727 0.976741 0.828678 -0.594056 angle 0.279602
90 0.564080 0.971479 0.825721 -0.599318 angle 0.282743
91 0.568388 0.966252 0.822761 -0.604546 angle 0.285885
92 0.572652 0.961059 0.819799 -0.609738 angle 0.289027
93 0.576873 0.955902 0.816835 -0.614895 angle 0.292168
94 0.581049 0.950780 0.813869 -0.620017 angle 0.295310
95 0.585182 0.945693 0.810903 -0.625104 angle 0.298451
96 0.589271 0.940640 0.807936 -0.630157 angle 0.301593
97 0.593318 0.935622 0.804968 -0.635175 angle 0.304734
98 0.597323 0.930639 0.802002 -0.640158 angle 0.307876
99 0.601284 0.925690 0.799036 -0.645108 angle 0.311018
100 0.605205 0.920774 0.796071 -0.650023 angle 0.314159
101 0.609083 0.915893 0.793107 -0.654904 angle 0.317301
102 0.612920 0.911046 0.790145 -0.659751 angle 0.320442
103 0.616716 0.906233 0.787186 -0.664564 angle 0.323584
104 0.620472 0.901453 0.784229 -0.669344 angle 0.326726
105 0.624187 0.896707 0.781276 -0.674091 angle 0.329867
106 0.627862 0.891994 0.778325 -0.678804 angle 0.333009
107 0.631498 0.887314 0.775378 -0.683484 angle 0.336150
108 0.635094 0.882667 0.772435 -0.688131 angle 0.339292
109 0.638652 0.878052 0.769496 -0.692745 angle 0.342434
110 0.642171 0.873471 0.766562 -0.697327 angle 0.345575
111 0.645651 0.868921 0.763633 -0.701876 angle 0.348717
112 0.649094 0.864404 0.760709 -0.706393 angle 0.351858
113 0.652499 0.859919 0.757790 -0.710878 angle 0.355000
114 0.655867 0.855466 0.754877 -0.715331 angle 0.358142
115 0.659199 0.851045 0.751970 -0.719753 angle 0.361283
116 0.662493 0.846655 0.749069 -0.724142 angle 0.364425
117 0.665752 0.842296 0.746174 -0.728501 angle 0.367566
118 0.668975 0.837969 0.743286 -0.732828 angle 0.370708
119 0.672162 0.833672 0.740405 -0.737125 angle 0.373850
120 0.675314 0.829407 0.737531 -0.741391 angle 0.376991
121 0.678432 0.825171 0.734664 -0.745626 angle 0.380133
122 0.681515 0.820966 0.731805 -0.749831 angle 0.383274
123 0.684564 0.816792 0.728953 -0.754006 angle 0.386416
124 0.687580 0.812647 0.726110 -0.758151 angle 0.389557
125 0.690562 0.808532 0.723274 -0.762266 angle 0.392699
126 0.693511 0.804446 0.720447 -0.766351 angle 0.395841
127 0.696428 0.800390 0.717628 -0.770408 angle 0.398982
128 0.699312 0.796363 0.714817 -0.774435 angle 0.402124
129 0.702164 0.792365 0.712016 -0.778433 angle 0.405265
130 0.704985 0.788395 0.709223 -0.782402 angle 0.408407
131 0.707775 0.784454 0.706439 -0.786343 angle 0.411549
132 0.710533 0.780542 0.703664 -0.790256 angle 0.414690
133 0.713261 0.776657 0.700899 -0.794140 angle 0.417832
134 0.715959 0.772801 0.698143 -0.797997 angle 0.420973
135 0.718627 0.768972 0.695397 -0.801826 angle 0.424115
136 0.721265 0.765171 0.692660 -0.805627 angle 0.427257
137 0.723874 0.761397 0.689933 -0.809401 angle 0.430398
138 0.726454 0.757650 0.687216 -0.813148 angle 0.433540
139 0.729006 0.753930 0.684509 -0.816868 angle 0.436681
140 0.731529 0.750236 0.681812 -0.820561 angle 0.439823
141 0.734024 0.746569 0.679125 -0.824228 angle 0.442965
142 0.736492 0.742929 0.676448 -0.827869 angle 0.446106
143 0.738932 0.739314 0.673781 -0.831484 angle 0.449248
144 0.741345 0.735725 0.671125 -0.835072 angle 0.452389
145 0.743732 0.732162 0.668479 -0.838635 angle 0.455531
146 0.746092 0.728625 0.665844 -0.842173 angle 0.458673
147 0.748426 0.725112 0.663220 -0.845685 angle 0.461814
148 0.750734 0.721625 0.660606 -0.849173 angle 0.464956
149 0.753017 0.718163 0.658002 -0.852635 angle 0.468097
150 0.755274 0.714725 0.655410 -0.856073 angle 0.471239
151 0.757507 0.711312 0.652828 -0.859486 angle 0.474380
152 0.759715 0.707923 0.650257 -0.862875 angle 0.477522
153 0.761899 0.704558 0.647697 -0.866240 angle 0.480664
154 0.764059 0.701217 0.645148 -0.869581 angle 0.483805
155 0.766195 0.697899 0.642610 -0.872898 angle 0.486947
156 0.768307 0.694605 0.640082 -0.876192 angle 0.490088
157 0.770397 0.691335 0.637566 -0.879463 angle 0.493230
158 0.772463 0.688087 0.635061 -0.882711 angle 0.496372
159 0.774507 0.684863 0.632567 -0.885935 angle 0.499513
160 0.776528 0.681661 0.630083 -0.889137 angle 0.502655
161 0.778528 0.678481 0.627611 -0.892317 angle 0.505796
162 0.780505 0.675324 0.625150 -0.895474 angle 0.508938
163 0.782461 0.672189 0.622700 -0.898609 angle 0.512080
164 0.784396 0.669076 0.620262 -0.901722 angle 0.515221
165 0.786310 0.665985 0.617834 -0.904813 angle 0.518363
166 0.788202 0.662915 0.615417 -0.907883 angle 0.521504
167 0.790075 0.659867 0.613012 -0.910931 angle 0.524646
168 0.791927 0.656840 0.610617 -0.913958 angle 0.527788
169 0.793759 0.653834 0.608234 -0.916964 angle 0.530929
170 0.795571 0.650849 0.605862 -0.919949 angle 0.534071
171 0.797363 0.647885 0.603501 -0.922913 angle 0.537212
172 0.799136 0.644941 0.601151 -0.925857 angle 0.540354
173 0.800891 0.642017 0.598812 -0.928781 angle 0.543496
174 0.802626 0.639114 0.596484 -0.931684 angle 0.546637
175 0.804342 0.636230 0.594167 -0.934568 angle 0.549779
176 0.806041 0.633366 0.591862 -0.937431 angle 0.552920
177 0.807721 0.630522 0.589567 -0.940275 angle 0.556062
178 0.809383 0.627698 0.587283 -0.943100 angle 0.559203
179 0.811027 0.624893 0.585010 -0.945905 angle 0.562345
180 0.812654 0.622107 0.582748 -0.948691 angle 0.565487
181 0.814263 0.619339 0.580497 -0.951459 angle 0.568628
182 0.815855 0.616591 0.578257 -0.954207 angle 0.571770
183 0.817431 0.613861 0.576028 -0.956937 angle 0.574911
184 0.818990 0.611150 0.573810 -0.959648 angle 0.578053
185 0.820532 0.608457 0.571602 -0.962341 angle 0.581195
186 0.822058 0.605783 0.569406 -0.965015 angle 0.584336
187 0.823568 0.603126 0.567220 -0.967672 angle 0.587478
188 0.825062 0.600487 0.565044 -0.970311 angle 0.590619
189 0.826540 0.597866 0.562880 -0.972932 angle 0.593761
190 0.828003 0.595262 0.560726 -0.975536 angle 0.596903
191 0.829450 0.592676 0.558582 -0.978122 angle 0.600044
192 0.830882 0.590107 0.556450 -0.980691 angle 0.603186
193 0.832300 0.587555 0.554328 -0.983243 angle 0.606327
194 0.833702 0.585020 0.552216 -0.985778 angle 0.609469
195 0.835090 0.582501 0.550115 -0.988297 angle 0.612611
196 0.836464 0.580000 0.548024 -0.990798 angle 0.615752
197 0.837823 0.577515 0.545943 -0.993284 angle 0.618894
198 0.839168 0.575046 0.543873 -0.995752 angle 0.622035
199 0.840500 0.572593 0.541813 -0.998205 angle 0.625177
200 0.841817 0.570157 0.539764 -1.000642 angle 0.628319
201 0.843122 0.567736 0.537725 -1.003062 angle 0.631460
202 0.844412 0.565331 0.535695 -1.005467 angle 0.634602
203 0.845690 0.562942 0.533676 -1.007856 angle 0.637743
204 0.846954 0.560568 0.531667 -1.010230 angle 0.640885
205 0.848206 0.558210 0.529669 -1.012588 angle 0.644026
206 0.849444 0.555867 0.527680 -1.014932 angle 0.647168
207 0.850671 0.553539 0.525701 -1.017260 angle 0.650310
208 0.851884 0.551226 0.523732 -1.019573 angle 0.653451
209 0.853086 0.548927 0.521773 -1.021871 angle 0.656593
210 0.854275 0.546644 0.519823 -1.024154 angle 0.659734
211 0.855452 0.544375 0.517884 -1.026423 angle 0.662876
212 0.856618 0.542121 0.515954 -1.028678 angle 0.666018
213 0.857771 0.539880 0.514033 -1.030918 angle 0.669159
214 0.858913 0.537654 0.512123 -1.033144 angle 0.672301
215 0.860044 0.535443 0.510222 -1.035356 angle 0.675442
216 0.861163 0.533245 0.508330 -1.037554 angle 0.678584
217 0.862271 0.531061 0.506448 -1.039738 angle 0.681726
218 0.863369 0.528890 0.504576 -1.041908 angle 0.684867
219 0.864455 0.526734 0.502712 -1.044065 angle 0.688009
220 0.865530 0.524590 0.500858 -1.046208 angle 0.691150
221 0.866595 0.522460 0.499014 -1.048338 angle 0.694292
222 0.867649 0.520344 0.497178 -1.050455 angle 0.697434
223 0.868693 0.518240 0.495352 -1.052558 angle 0.700575
224 0.869727 0.516150 0.493535 -1.054649 angle 0.703717
225 0.870750 0.514072 0.491727 -1.056726 angle 0.706858
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804 0.997814 0.066151 0.066103 -1.504663 angle 2.525840
805 0.997838 0.065793 0.065745 -1.505022 angle 2.528982
806 0.997861 0.065434 0.065388 -1.505380 angle 2.532124
807 0.997884 0.065076 0.065031 -1.505738 angle 2.535265
808 0.997908 0.064719 0.064674 -1.506096 angle 2.538407
809 0.997931 0.064362 0.064317 -1.506453 angle 2.541548
810 0.997954 0.064005 0.063961 -1.506810 angle 2.544690
811 0.997976 0.063648 0.063605 -1.507167 angle 2.547832
812 0.997999 0.063292 0.063250 -1.507524 angle 2.550973
813 0.998021 0.062936 0.062894 -1.507880 angle 2.554115
814 0.998044 0.062580 0.062539 -1.508235 angle 2.557256
815 0.998066 0.062225 0.062185 -1.508591 angle 2.560398
816 0.998088 0.061870 0.061831 -1.508946 angle 2.563540
817 0.998110 0.061515 0.061477 -1.509301 angle 2.566681
818 0.998131 0.061161 0.061123 -1.509655 angle 2.569823
819 0.998153 0.060807 0.060769 -1.510009 angle 2.572964
820 0.998174 0.060453 0.060416 -1.510363 angle 2.576106
821 0.998196 0.060100 0.060064 -1.510717 angle 2.579248
822 0.998217 0.059747 0.059711 -1.511070 angle 2.582389
823 0.998238 0.059394 0.059359 -1.511423 angle 2.585531
824 0.998259 0.059041 0.059007 -1.511775 angle 2.588672
825 0.998279 0.058689 0.058655 -1.512128 angle 2.591814
826 0.998300 0.058337 0.058304 -1.512480 angle 2.594956
827 0.998321 0.057986 0.057953 -1.512832 angle 2.598097
828 0.998341 0.057634 0.057602 -1.513183 angle 2.601239
829 0.998361 0.057283 0.057252 -1.513534 angle 2.604380
830 0.998381 0.056932 0.056902 -1.513885 angle 2.607522
831 0.998401 0.056582 0.056552 -1.514236 angle 2.610663
832 0.998421 0.056232 0.056202 -1.514586 angle 2.613805
833 0.998440 0.055882 0.055853 -1.514936 angle 2.616947
834 0.998460 0.055532 0.055504 -1.515286 angle 2.620088
835 0.998479 0.055183 0.055155 -1.515635 angle 2.623230
836 0.998498 0.054834 0.054806 -1.515984 angle 2.626371
837 0.998517 0.054485 0.054458 -1.516333 angle 2.629513
838 0.998536 0.054137 0.054110 -1.516682 angle 2.632655
839 0.998555 0.053788 0.053762 -1.517030 angle 2.635796
840 0.998574 0.053440 0.053415 -1.517378 angle 2.638938
841 0.998592 0.053093 0.053068 -1.517726 angle 2.642079
842 0.998610 0.052745 0.052721 -1.518074 angle 2.645221
843 0.998629 0.052398 0.052374 -1.518421 angle 2.648363
844 0.998647 0.052051 0.052028 -1.518768 angle 2.651504
845 0.998665 0.051705 0.051682 -1.519115 angle 2.654646
846 0.998683 0.051358 0.051336 -1.519462 angle 2.657787
847 0.998700 0.051012 0.050990 -1.519808 angle 2.660929
848 0.998718 0.050666 0.050645 -1.520154 angle 2.664071
849 0.998735 0.050321 0.050299 -1.520500 angle 2.667212
850 0.998753 0.049975 0.049954 -1.520845 angle 2.670354
851 0.998770 0.049630 0.049610 -1.521191 angle 2.673495
852 0.998787 0.049285 0.049265 -1.521536 angle 2.676637
853 0.998804 0.048941 0.048921 -1.521880 angle 2.679779
854 0.998821 0.048596 0.048577 -1.522225 angle 2.682920
855 0.998837 0.048252 0.048233 -1.522569 angle 2.686062
856 0.998854 0.047908 0.047890 -1.522913 angle 2.689203
857 0.998870 0.047564 0.047546 -1.523257 angle 2.692345
858 0.998887 0.047221 0.047203 -1.523601 angle 2.695486
859 0.998903 0.046878 0.046861 -1.523944 angle 2.698628
860 0.998919 0.046535 0.046518 -1.524287 angle 2.701770
861 0.998935 0.046192 0.046176 -1.524630 angle 2.704911
862 0.998950 0.045850 0.045833 -1.524973 angle 2.708053
863 0.998966 0.045507 0.045492 -1.525316 angle 2.711194
864 0.998981 0.045165 0.045150 -1.525658 angle 2.714336
865 0.998997 0.044823 0.044808 -1.526000 angle 2.717478
866 0.999012 0.044482 0.044467 -1.526342 angle 2.720619
867 0.999027 0.044140 0.044126 -1.526683 angle 2.723761
868 0.999042 0.043799 0.043785 -1.527025 angle 2.726902
869 0.999057 0.043458 0.043444 -1.527366 angle 2.730044
870 0.999072 0.043117 0.043104 -1.527707 angle 2.733186
871 0.999086 0.042777 0.042764 -1.528048 angle 2.736327
872 0.999101 0.042437 0.042424 -1.528388 angle 2.739469
873 0.999115 0.042096 0.042084 -1.528729 angle 2.742610
874 0.999130 0.041756 0.041744 -1.529069 angle 2.745752
875 0.999144 0.041417 0.041405 -1.529409 angle 2.748894
876 0.999158 0.041077 0.041066 -1.529749 angle 2.752035
877 0.999172 0.040738 0.040727 -1.530088 angle 2.755177
878 0.999185 0.040399 0.040388 -1.530427 angle 2.758318
879 0.999199 0.040060 0.040049 -1.530767 angle 2.761460
880 0.999212 0.039721 0.039711 -1.531106 angle 2.764602
881 0.999226 0.039383 0.039372 -1.531444 angle 2.767743
882 0.999239 0.039044 0.039034 -1.531783 angle 2.770885
883 0.999252 0.038706 0.038696 -1.532121 angle 2.774026
884 0.999265 0.038368 0.038359 -1.532460 angle 2.777168
885 0.999278 0.038030 0.038021 -1.532798 angle 2.780309
886 0.999291 0.037693 0.037684 -1.533136 angle 2.783451
887 0.999304 0.037355 0.037347 -1.533473 angle 2.786593
888 0.999316 0.037018 0.037010 -1.533811 angle 2.789734
889 0.999329 0.036681 0.036673 -1.534148 angle 2.792876
890 0.999341 0.036344 0.036336 -1.534485 angle 2.796017
891 0.999353 0.036008 0.036000 -1.534822 angle 2.799159
892 0.999365 0.035671 0.035664 -1.535159 angle 2.802301
893 0.999377 0.035335 0.035327 -1.535496 angle 2.805442
894 0.999389 0.034999 0.034991 -1.535832 angle 2.808584
895 0.999401 0.034663 0.034656 -1.536169 angle 2.811725
896 0.999412 0.034327 0.034320 -1.536505 angle 2.814867
897 0.999424 0.033991 0.033985 -1.536841 angle 2.818009
898 0.999435 0.033656 0.033649 -1.537177 angle 2.821150
899 0.999446 0.033320 0.033314 -1.537512 angle 2.824292
900 0.999457 0.032985 0.032979 -1.537848 angle 2.827433
901 0.999468 0.032650 0.032644 -1.538183 angle 2.830575
902 0.999479 0.032315 0.032310 -1.538519 angle 2.833717
903 0.999490 0.031980 0.031975 -1.538854 angle 2.836858
904 0.999501 0.031646 0.031641 -1.539189 angle 2.840000
905 0.999511 0.031312 0.031306 -1.539523 angle 2.843141
906 0.999521 0.030977 0.030972 -1.539858 angle 2.846283
907 0.999532 0.030643 0.030638 -1.540193 angle 2.849425
908 0.999542 0.030309 0.030305 -1.540527 angle 2.852566
909 0.999552 0.029975 0.029971 -1.540861 angle 2.855708
910 0.999562 0.029642 0.029637 -1.541195 angle 2.858849
911 0.999572 0.029308 0.029304 -1.541529 angle 2.861991
912 0.999581 0.028975 0.028971 -1.541863 angle 2.865133
913 0.999591 0.028642 0.028638 -1.542197 angle 2.868274
914 0.999601 0.028309 0.028305 -1.542530 angle 2.871416
915 0.999610 0.027976 0.027972 -1.542864 angle 2.874557
916 0.999619 0.027643 0.027639 -1.543197 angle 2.877699
917 0.999628 0.027310 0.027307 -1.543530 angle 2.880840
918 0.999637 0.026978 0.026974 -1.543864 angle 2.883982
919 0.999646 0.026645 0.026642 -1.544197 angle 2.887124
920 0.999655 0.026313 0.026310 -1.544529 angle 2.890265
921 0.999664 0.025981 0.025978 -1.544862 angle 2.893407
922 0.999672 0.025649 0.025646 -1.545195 angle 2.896548
923 0.999681 0.025317 0.025314 -1.545527 angle 2.899690
924 0.999689 0.024985 0.024982 -1.545860 angle 2.902832
925 0.999697 0.024653 0.024651 -1.546192 angle 2.905973
926 0.999705 0.024322 0.024319 -1.546524 angle 2.909115
927 0.999713 0.023990 0.023988 -1.546857 angle 2.912256
928 0.999721 0.023659 0.023656 -1.547189 angle 2.915398
929 0.999729 0.023327 0.023325 -1.547521 angle 2.918540
930 0.999737 0.022996 0.022994 -1.547853 angle 2.921681
931 0.999744 0.022665 0.022663 -1.548184 angle 2.924823
932 0.999752 0.022334 0.022333 -1.548516 angle 2.927964
933 0.999759 0.022004 0.022002 -1.548848 angle 2.931106
934 0.999766 0.021673 0.021671 -1.549179 angle 2.934248
935 0.999773 0.021342 0.021341 -1.549511 angle 2.937389
936 0.999780 0.021012 0.021010 -1.549842 angle 2.940531
937 0.999787 0.020681 0.020680 -1.550173 angle 2.943672
938 0.999794 0.020351 0.020350 -1.550504 angle 2.946814
939 0.999801 0.020021 0.020020 -1.550836 angle 2.949956
940 0.999807 0.019691 0.019690 -1.551167 angle 2.953097
941 0.999814 0.019361 0.019360 -1.551498 angle 2.956239
942 0.999820 0.019031 0.019030 -1.551829 angle 2.959380
943 0.999826 0.018701 0.018700 -1.552160 angle 2.962522
944 0.999832 0.018371 0.018370 -1.552491 angle 2.965663
945 0.999838 0.018042 0.018041 -1.552821 angle 2.968805
946 0.999844 0.017712 0.017711 -1.553152 angle 2.971947
947 0.999850 0.017383 0.017382 -1.553483 angle 2.975088
948 0.999856 0.017053 0.017053 -1.553814 angle 2.978230
949 0.999861 0.016724 0.016723 -1.554144 angle 2.981371
950 0.999867 0.016395 0.016394 -1.554475 angle 2.984513
951 0.999872 0.016066 0.016065 -1.554806 angle 2.987655
952 0.999877 0.015737 0.015736 -1.555136 angle 2.990796
953 0.999883 0.015408 0.015407 -1.555467 angle 2.993938
954 0.999888 0.015079 0.015078 -1.555798 angle 2.997079
955 0.999892 0.014750 0.014749 -1.556128 angle 3.000221
956 0.999897 0.014421 0.014421 -1.556459 angle 3.003363
957 0.999902 0.014092 0.014092 -1.556790 angle 3.006504
958 0.999906 0.013764 0.013763 -1.557120 angle 3.009646
959 0.999911 0.013435 0.013435 -1.557451 angle 3.012787
960 0.999915 0.013107 0.013106 -1.557782 angle 3.015929
961 0.999920 0.012778 0.012778 -1.558113 angle 3.019071
962 0.999924 0.012450 0.012449 -1.558444 angle 3.022212
963 0.999928 0.012121 0.012121 -1.558775 angle 3.025354
964 0.999932 0.011793 0.011793 -1.559106 angle 3.028495
965 0.999935 0.011465 0.011465 -1.559437 angle 3.031637
966 0.999939 0.011137 0.011136 -1.559768 angle 3.034779
967 0.999943 0.010809 0.010808 -1.560100 angle 3.037920
968 0.999946 0.010481 0.010480 -1.560431 angle 3.041062
969 0.999950 0.010152 0.010152 -1.560763 angle 3.044203
970 0.999953 0.009825 0.009824 -1.561095 angle 3.047345
971 0.999956 0.009497 0.009496 -1.561427 angle 3.050486
972 0.999959 0.009169 0.009169 -1.561759 angle 3.053628
973 0.999962 0.008841 0.008841 -1.562092 angle 3.056770
974 0.999965 0.008513 0.008513 -1.562425 angle 3.059911
975 0.999968 0.008185 0.008185 -1.562759 angle 3.063053
976 0.999970 0.007858 0.007858 -1.563092 angle 3.066194
977 0.999973 0.007530 0.007530 -1.563427 angle 3.069336
978 0.999975 0.007202 0.007202 -1.563762 angle 3.072478
979 0.999978 0.006875 0.006875 -1.564097 angle 3.075619
980 0.999980 0.006547 0.006547 -1.564434 angle 3.078761
981 0.999982 0.006220 0.006220 -1.564771 angle 3.081902
982 0.999984 0.005892 0.005892 -1.565109 angle 3.085044
983 0.999986 0.005565 0.005565 -1.565449 angle 3.088186
984 0.999987 0.005237 0.005237 -1.565790 angle 3.091327
985 0.999989 0.004910 0.004910 -1.566133 angle 3.094469
986 0.999991 0.004582 0.004582 -1.566478 angle 3.097610
987 0.999992 0.004255 0.004255 -1.566825 angle 3.100752
988 0.999993 0.003927 0.003927 -1.567177 angle 3.103894
989 0.999995 0.003600 0.003600 -1.567532 angle 3.107035
990 0.999996 0.003273 0.003273 -1.567893 angle 3.110177
991 0.999997 0.002945 0.002945 -1.568261 angle 3.113318
992 0.999998 0.002618 0.002618 -1.568640 angle 3.116460
993 0.999999 0.002291 0.002291 -1.569033 angle 3.119602
994 0.999999 0.001964 0.001964 -1.569448 angle 3.122743
995 1.000000 0.001636 0.001636 -1.569898 angle 3.125885
996 1.000000 0.001309 0.001309 -1.570410 angle 3.129026
997 1.000001 0.000982 0.000982 -1.571045 angle 3.132168
998 1.000001 0.000655 0.000655 -1.571988 angle 3.135309
999 1.000001 0.000327 0.000327 -1.574161 angle 3.138451
Reply Start a New Thread
Sam (r�p. sans -no-sp-am) wrote:
> The error was that I substituted s with
> 2*cot(Pi*fc/fs)*(1-z^-1)/(1+z^-1), as I read it from a paper.
>
> But the real transformation is 2/T * (1-z^-1)/(1+z^-1), so are
> you sure that c = 2*Pi*fc*cot(...) and not c=2*Pi*fs*cot(...)
> ????
Yes, I'm sure. Have you compared some filters made with
each? Also consider how the two c's behave as functions of fc.
Martin
Reply Start a New Thread
"Sam (r�p. sans -no-sp-am)" <totalsam-no-sp-am@hotmail.com> wrote in message
news:3ff754cd$0$726$5402220f@news.sunrise.ch...
> Hi all !
>
> Thanks for reading ! I would like to design a 1st order digital high pass
> filter with a 50 us time constant. The analog transfer function is
> Ha(s)=50E-6*s/(1+50E-6*s).
>
> Then I use the bilinear transformation. Sampling freq is 48 kHz. The
> pre-warped pulsation is 0.4228 rad/sec and the z transfer function is
> H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
>
> When I compute this filter, it seems to be unstable at 24 kHz, and the FFT
> of a filtered signal gives a spike at this frequency.
>
> Is something wrong in my design ? How can I avoid having this oscillation
?
> Putting a low-pass filter after this one seems not to be a good idea I
think
> ...
>
Well, your design has a zero at z^-1=1=z
And, it has a pole at z^-1=-1.00473224 or z= -0.99527 which is *extremely*
close to the unit circle in the z-plane.
On the other hand, Ha(s) has a pole at -20,000 which would yield a -3dB
break point at 20kHz I believe... which is 20/24=.833 of fs/2 which
corresponds to 0.833*pi or 0.4165Hz if you're normalizing to a sample
interval of T=1 second.
[that is if T=1 then fs=2*pi radians per second = 1Hz.
To get a -3dB break point at 0.4165Hz, or equivalently 0.833*24kHz= 20kHz,
you need to have the pole located just where it ended up! So, it has to be
that the bilinear transformation for this high pass filter isn't going to
work. I don't have much experience with this but here are a couple of
thoughts:
You want to have a single-order high-pass filter with -3dB down point at
20kHz and you want to have a sample rate of 48kHz.
1) A single-pole filter isn't very sharp. The transition from stop to pass
band is quite wide. So, in order to get -3dB at 20kHz, you have to push the
location of the pole very close to the unit circle if the sample rate is
48kHz. Having the pole real and close to the unit circle pretty much
guarantees a high degree of ringing at fs/2. Real world arithmetic could
push the pole to the outside of the circle and yield instability.
2) Neglect sampling in the time domain for a moment but keep sampling in the
frequency domain. The high pass response that you want should go to 1.0 at
some high frequency. Unless you include those frequencies that don't
essentially have gain of 1.0, then the filter you want hasn't been
"captured" - there is going to be frequency aliasing. So, you have to
increase the sample rate until this is no longer the case or use another
method.
Somewhere the point of high-pass filters and the suitability of using the
bilinear transformation must be written up as well as tips for how to use it
or where to not use it.
We can avoid all that this way:
How about starting with a nice FIR filter with coefficients 1/2,-1/2 so that
the gain at dc is zero and the gain at fs/2 is 1.0 (which has a pole at the
orgin of the unit circle in the z-plane).
H(z)=(z-1)/(z)
Then, you can move the pole along the negative real axis until you get
something acceptable - turning it into a recursive filter. For example:
H(z)=0.25*(z-1)/(z+0.5)
or
H(z)=[(k-1)/2]*(z-1)/(z+k) with the scaling (k-1)/2 so that gain at fs/2=1
In the case you have, k is so close to 1 that the thing blows up. Notice no
bilinear transformation here, simply moving a first order pole around.
It's all a matter of moving the pole to a location that's acceptable.
Either that or go to a filter of higher order.
Fred
Reply Start a New Thread
"Sam (r�p. sans -no-sp-am)" <totalsam-no-sp-am@hotmail.com> wrote in message
news:3ff754cd$0$726$5402220f@news.sunrise.ch...
> Hi all !
>
> Thanks for reading ! I would like to design a 1st order digital high pass
> filter with a 50 us time constant. The analog transfer function is
> Ha(s)=50E-6*s/(1+50E-6*s).
>
> Then I use the bilinear transformation. Sampling freq is 48 kHz. The
> pre-warped pulsation is 0.4228 rad/sec and the z transfer function is
> H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
>
> When I compute this filter, it seems to be unstable at 24 kHz, and the FFT
> of a filtered signal gives a spike at this frequency.
>
> Is something wrong in my design ?
Yes, but not in your calculations for the digital fitler.
> How can I avoid having this oscillation ?
You need to plot the results for a 20khz signal for 1 millisecond to see how
the sampling is affecting your results.
> Putting a low-pass filter after this one seems not to be a good idea I
think
> ...
>
> Who could help me ??
>
> Thanks a LOT in advance !
>
> Sam
>
>
You are right. Try putting the analog low pass filter before the digital
filter.
Check out anti-aliasing.
Peter Nachtwey
Reply Start a New Thread
The error was that I substituted s with 2*cot(Pi*fc/fs)*(1-z^-1)/(1+z^-1),
as I read it from a paper.
But the real transformation is 2/T * (1-z^-1)/(1+z^-1), so are you sure that
c = 2*Pi*fc*cot(...) and not c=2*Pi*fs*cot(...) ????
Sam
Reply Start a New Thread
Sam (r�p. sans -no-sp-am) wrote:
> Hi all !
>
> Thanks for reading ! I would like to design a 1st order digital
> high pass filter with a 50 us time constant. The analog transfer
> function is Ha(s)=50E-6*s/(1+50E-6*s).
>
> Then I use the bilinear transformation. Sampling freq is 48 kHz.
> The pre-warped pulsation is 0.4228 rad/sec and the z transfer
> function is H(z)=236.4E-6*(1-z^-1)/(1+0.999527*z^-1).
Something must have gone wrong with the bilinear transform.
It would have been useful to show your calculations. The BLT
substitutes s <-- c*(1-1/z)/(1+1/z). To make the responses
equal at frequency fc < fs/2, choose c = 2*Pi*fc * cot(PI*fc/fs),
where fs is the sampling frequency.
Assuming that by "pre-warped pulsation" you meant the
quantity 2*Pi*fc, I get the discrete transfer function
H(z) = 0.8275862069 * (1-1/z) / (1-0.6551724138/z)
for your Ha and fs.
Martin
Reply Start a New Thread