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# Discussion Groups | Comp.DSP | IIR design and stability

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# IIR design and stability - Sam (rép. sans -no-sp-am) - 2004-01-03 18:51:00

```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

```
______________________________

# Re: IIR design and stability - Martin Eisenberg - 2004-01-04 00:40:00

```Sam (rÃ©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)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)

Martin
```
______________________________

# Re: IIR design and stability - Sam (rép. sans -no-sp-am) - 2004-01-04 05:14:00

```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

```
______________________________

# Re: IIR design and stability - Peter Nachtwey - 2004-01-04 06:09:00

```"Sam (rÃ©p. 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

```
______________________________

# Re: IIR design and stability - Fred Marshall - 2004-01-04 18:24:00

```"Sam (rÃ©p. 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

```
______________________________

# Re: IIR design and stability - Martin Eisenberg - 2004-01-04 19:43:00

```Sam (rÃ©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
```
______________________________

# Re: IIR design and stability - Robin Clark - 2004-01-11 04:35:00

```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
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718  0.995152  0.098516  0.098356  -1.472293  angle 2.255664
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721  0.995269  0.097327  0.097174  -1.473482  angle 2.265088
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723  0.995345  0.096538  0.096388  -1.474271  angle 2.271371
724  0.995383  0.096144  0.095995  -1.474665  angle 2.274513
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726  0.995458  0.095357  0.095213  -1.475452  angle 2.280796
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730  0.995606  0.093791  0.093653  -1.477018  angle 2.293363
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733  0.995715  0.092622  0.092489  -1.478188  angle 2.302787
734  0.995751  0.092233  0.092102  -1.478576  angle 2.305929
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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
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741  0.995996  0.089526  0.089407  -1.481283  angle 2.327920
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744  0.996099  0.088374  0.088259  -1.482436  angle 2.337345
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747  0.996199  0.087226  0.087115  -1.483584  angle 2.346770
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787  0.997389  0.072301  0.072238  -1.498512  angle 2.472433
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789  0.997441  0.071572  0.071511  -1.499241  angle 2.478717
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795  0.997594  0.069394  0.069339  -1.501419  angle 2.497566
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811  0.997976  0.063648  0.063605  -1.507167  angle 2.547832
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815  0.998066  0.062225  0.062185  -1.508591  angle 2.560398
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817  0.998110  0.061515  0.061477  -1.509301  angle 2.566681
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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
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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
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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

```
______________________________

# Re: IIR design and stability - Robin Clark - 2004-01-11 04:45:00

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

> Sam (rÃ©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)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
```
______________________________

# Re: IIR design and stability - Robin Clark - 2004-01-11 08:43:00

```

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

http://80.3.72.34/~robin/stuff2/
```
______________________________

# Re: IIR design and stability - Martin Eisenberg - 2004-01-12 17:02:00

```Robin Clark wrote:

> On Sun, 04 Jan 2004 05:40:32 +0000, Martin Eisenberg wrote:
>
>> Sam (rÃ©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)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,