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)PE-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)#6.4E-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 (rép. 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)PE-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)#6.4E-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 (rép. sans -no-sp-am)" <t...@hotmail.com> wrote in
message
news:3ff754cd$0$726$5...@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)PE-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)#6.4E-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 (rép. sans -no-sp-am)" <t...@hotmail.com> wrote in message
news:3ff754cd$0$726$5...@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)PE-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)#6.4E-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 (rép. 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)PE-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 
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 655  0.992227  0.124776  0.124453  -1.446030  angle 2.057743 
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 657  0.992335  0.123900  0.123583  -1.446906  angle 2.064026 
 658  0.992389  0.123463  0.123150  -1.447343  angle 2.067168 
 659  0.992443  0.123027  0.122717  -1.447779  angle 2.070310 
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 661  0.992549  0.122156  0.121853  -1.448650  angle 2.076593 
 662  0.992602  0.121723  0.121422  -1.449084  angle 2.079734 
 663  0.992655  0.121289  0.120992  -1.449517  angle 2.082876 
 664  0.992707  0.120857  0.120563  -1.449949  angle 2.086018 
 665  0.992759  0.120425  0.120134  -1.450381  angle 2.089159 
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 669  0.992964  0.118706  0.118427  -1.452101  angle 2.101725 
 670  0.993015  0.118278  0.118002  -1.452529  angle 2.104867 
 671  0.993065  0.117850  0.117578  -1.452956  angle 2.108009 
 672  0.993115  0.117424  0.117154  -1.453383  angle 2.111150 
 673  0.993165  0.116998  0.116731  -1.453809  angle 2.114292 
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 676  0.993313  0.115724  0.115466  -1.455082  angle 2.123717 
 677  0.993361  0.115301  0.115046  -1.455505  angle 2.126858 
 678  0.993410  0.114879  0.114626  -1.455928  angle 2.130000 
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 682  0.993601  0.113196  0.112955  -1.457611  angle 2.142566 
 683  0.993649  0.112777  0.112539  -1.458030  angle 2.145708 
 684  0.993696  0.112359  0.112123  -1.458448  angle 2.148849 
 685  0.993742  0.111942  0.111708  -1.458865  angle 2.151991 
 686  0.993789  0.111525  0.111294  -1.459282  angle 2.155133 
 687  0.993835  0.111108  0.110880  -1.459699  angle 2.158274 
 688  0.993881  0.110693  0.110467  -1.460114  angle 2.161416 
 689  0.993927  0.110278  0.110054  -1.460529  angle 2.164557 
 690  0.993972  0.109864  0.109643  -1.460944  angle 2.167699 
 691  0.994018  0.109450  0.109232  -1.461357  angle 2.170841 
 692  0.994063  0.109037  0.108821  -1.461770  angle 2.173982 
 693  0.994107  0.108625  0.108411  -1.462183  angle 2.177124 
 694  0.994152  0.108213  0.108002  -1.462595  angle 2.180265 
 695  0.994196  0.107802  0.107593  -1.463006  angle 2.183407 
 696  0.994240  0.107391  0.107185  -1.463416  angle 2.186548 
 697  0.994284  0.106982  0.106778  -1.463826  angle 2.189690 
 698  0.994328  0.106572  0.106371  -1.464235  angle 2.192832 
 699  0.994371  0.106164  0.105965  -1.464644  angle 2.195973 
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 701  0.994457  0.105349  0.105154  -1.465459  angle 2.202256 
 702  0.994500  0.104942  0.104749  -1.465866  angle 2.205398 
 703  0.994542  0.104536  0.104346  -1.466272  angle 2.208540 
 704  0.994585  0.104130  0.103942  -1.466677  angle 2.211681 
 705  0.994627  0.103726  0.103540  -1.467082  angle 2.214823 
 706  0.994668  0.103321  0.103138  -1.467487  angle 2.217964 
 707  0.994710  0.102918  0.102736  -1.467890  angle 2.221106 
 708  0.994751  0.102515  0.102335  -1.468294  angle 2.224248 
 709  0.994792  0.102112  0.101935  -1.468696  angle 2.227389 
 710  0.994833  0.101710  0.101535  -1.469098  angle 2.230531 
 711  0.994874  0.101309  0.101136  -1.469499  angle 2.233672 
 712  0.994914  0.100908  0.100737  -1.469900  angle 2.236814 
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 714  0.994995  0.100108  0.099941  -1.470700  angle 2.243097 
 715  0.995034  0.099709  0.099544  -1.471099  angle 2.246239 
 716  0.995074  0.099311  0.099148  -1.471498  angle 2.249380 
 717  0.995113  0.098913  0.098752  -1.471895  angle 2.252522 
 718  0.995152  0.098516  0.098356  -1.472293  angle 2.255664 
 719  0.995191  0.098119  0.097962  -1.472690  angle 2.258805 
 720  0.995230  0.097723  0.097567  -1.473086  angle 2.261947 
 721  0.995269  0.097327  0.097174  -1.473482  angle 2.265088 
 722  0.995307  0.096932  0.096780  -1.473877  angle 2.268230 
 723  0.995345  0.096538  0.096388  -1.474271  angle 2.271371 
 724  0.995383  0.096144  0.095995  -1.474665  angle 2.274513 
 725  0.995421  0.095750  0.095604  -1.475059  angle 2.277655 
 726  0.995458  0.095357  0.095213  -1.475452  angle 2.280796 
 727  0.995495  0.094965  0.094822  -1.475844  angle 2.283938 
 728  0.995533  0.094573  0.094432  -1.476236  angle 2.287079 
 729  0.995569  0.094182  0.094043  -1.476627  angle 2.290221 
 730  0.995606  0.093791  0.093653  -1.477018  angle 2.293363 
 731  0.995643  0.093401  0.093265  -1.477409  angle 2.296504 
 732  0.995679  0.093011  0.092877  -1.477798  angle 2.299646 
 733  0.995715  0.092622  0.092489  -1.478188  angle 2.302787 
 734  0.995751  0.092233  0.092102  -1.478576  angle 2.305929 
 735  0.995786  0.091845  0.091716  -1.478965  angle 2.309071 
 736  0.995822  0.091457  0.091330  -1.479352  angle 2.312212 
 737  0.995857  0.091070  0.090944  -1.479740  angle 2.315354 
 738  0.995892  0.090683  0.090559  -1.480126  angle 2.318495 
 739  0.995927  0.090297  0.090175  -1.480512  angle 2.321637 
 740  0.995962  0.089912  0.089790  -1.480898  angle 2.324779 
 741  0.995996  0.089526  0.089407  -1.481283  angle 2.327920 
 742  0.996031  0.089142  0.089024  -1.481668  angle 2.331062 
 743  0.996065  0.088758  0.088641  -1.482052  angle 2.334203 
 744  0.996099  0.088374  0.088259  -1.482436  angle 2.337345 
 745  0.996133  0.087991  0.087877  -1.482819  angle 2.340487 
 746  0.996166  0.087608  0.087496  -1.483202  angle 2.343628 
 747  0.996199  0.087226  0.087115  -1.483584  angle 2.346770 
 748  0.996233  0.086844  0.086735  -1.483966  angle 2.349911 
 749  0.996266  0.086463  0.086355  -1.484348  angle 2.353053 
 750  0.996298  0.086082  0.085976  -1.484728  angle 2.356194 
 751  0.996331  0.085702  0.085597  -1.485109  angle 2.359336 
 752  0.996364  0.085322  0.085218  -1.485489  angle 2.362478 
 753  0.996396  0.084942  0.084840  -1.485868  angle 2.365619 
 754  0.996428  0.084563  0.084463  -1.486247  angle 2.368761 
 755  0.996460  0.084185  0.084086  -1.486626  angle 2.371902 
 756  0.996491  0.083807  0.083709  -1.487004  angle 2.375044 
 757  0.996523  0.083429  0.083333  -1.487381  angle 2.378186 
 758  0.996554  0.083052  0.082957  -1.487759  angle 2.381327 
 759  0.996586  0.082676  0.082581  -1.488135  angle 2.384469 
 760  0.996617  0.082299  0.082207  -1.488512  angle 2.387610 
 761  0.996647  0.081924  0.081832  -1.488887  angle 2.390752 
 762  0.996678  0.081548  0.081458  -1.489263  angle 2.393894 
 763  0.996708  0.081173  0.081084  -1.489638  angle 2.397035 
 764  0.996739  0.080799  0.080711  -1.490012  angle 2.400177 
 765  0.996769  0.080425  0.080338  -1.490386  angle 2.403318 
 766  0.996799  0.080051  0.079966  -1.490760  angle 2.406460 
 767  0.996829  0.079678  0.079594  -1.491133  angle 2.409602 
 768  0.996858  0.079305  0.079222  -1.491506  angle 2.412743 
 769  0.996888  0.078933  0.078851  -1.491878  angle 2.415885 
 770  0.996917  0.078561  0.078480  -1.492250  angle 2.419026 
 771  0.996946  0.078190  0.078110  -1.492622  angle 2.422168 
 772  0.996975  0.077819  0.077740  -1.492993  angle 2.425310 
 773  0.997004  0.077448  0.077371  -1.493364  angle 2.428451 
 774  0.997032  0.077078  0.077002  -1.493734  angle 2.431593 
 775  0.997061  0.076708  0.076633  -1.494104  angle 2.434734 
 776  0.997089  0.076339  0.076265  -1.494473  angle 2.437876 
 777  0.997117  0.075970  0.075897  -1.494843  angle 2.441017 
 778  0.997145  0.075601  0.075529  -1.495211  angle 2.444159 
 779  0.997173  0.075233  0.075162  -1.495579  angle 2.447301 
 780  0.997200  0.074865  0.074795  -1.495947  angle 2.450442 
 781  0.997228  0.074498  0.074429  -1.496315  angle 2.453584 
 782  0.997255  0.074131  0.074063  -1.496682  angle 2.456725 
 783  0.997282  0.073764  0.073697  -1.497049  angle 2.459867 
 784  0.997309  0.073398  0.073332  -1.497415  angle 2.463009 
 785  0.997336  0.073032  0.072967  -1.497781  angle 2.466150 
 786  0.997362  0.072666  0.072603  -1.498146  angle 2.469292 
 787  0.997389  0.072301  0.072238  -1.498512  angle 2.472433 
 788  0.997415  0.071937  0.071875  -1.498876  angle 2.475575 
 789  0.997441  0.071572  0.071511  -1.499241  angle 2.478717 
 790  0.997467  0.071208  0.071148  -1.499605  angle 2.481858 
 791  0.997493  0.070845  0.070786  -1.499968  angle 2.485000 
 792  0.997518  0.070482  0.070423  -1.500332  angle 2.488141 
 793  0.997544  0.070119  0.070061  -1.500695  angle 2.491283 
 794  0.997569  0.069756  0.069700  -1.501057  angle 2.494425 
 795  0.997594  0.069394  0.069339  -1.501419  angle 2.497566 
 796  0.997619  0.069032  0.068978  -1.501781  angle 2.500708 
 797  0.997644  0.068671  0.068617  -1.502143  angle 2.503849 
 798  0.997669  0.068310  0.068257  -1.502504  angle 2.506991 
 799  0.997694  0.067949  0.067897  -1.502865  angle 2.510133 
 800  0.997718  0.067589  0.067538  -1.503225  angle 2.513274 
 801  0.997742  0.067229  0.067178  -1.503585  angle 2.516416 
 802  0.997766  0.066869  0.066820  -1.503945  angle 2.519557 
 803  0.997790  0.066510  0.066461  -1.504304  angle 2.522699 
 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 
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 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 

On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:

> Sam (rép. 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)PE-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)#6.4E-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 (rép. 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)PE-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)#6.4E-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