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IIR design and stability

Started by Unknown January 3, 2004
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
...

Who could help me ??

Thanks a LOT in advance !

Sam


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
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


"Sam (r&#2013265929;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
"Sam (r&#2013265929;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
Sam (r&#2013265929;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
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 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On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:

> Sam (r&#2013265929;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
 

Assuming you dropped a minus the plots for this quite nice high
passs filter. 1000 == nyquist frequency

http://80.3.72.34/~robin/stuff2/
Robin Clark wrote:

> On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote: > >> Sam (r&#2013265929;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