Thanks for your update Cyrille. This seems a useful approach to find approximations. I should really learn how to do that with libre office. Do you have tutorial links? So far I found this: https://newtonexcelbach.wordpress.com/2015/06/28/using-linest-for-non-linear....
For the filter recipe the curve must go through coordinates [0 1] and [pi -1], to get the expected behavior when cutoff frequency is set to DC or Nyquist. [hip~ 0] should not block DC, which is only possible when the coefficient is exactly 1 like you get it with (1-sin(0))/cos(0). I tuned my factors 'by hand' to do so. That may be possible with your libre office result as well. However, when done in double precision the calculation will give slightly different result. Above or below 1, I don't know yet. In any case this has to be considered when writing the C.
Katja
On Wed, Oct 19, 2016 at 4:18 PM, cyrille henry ch@chnry.net wrote:
since you manually adjust the coefficient, i wanted to see the difference with coefficient adjusted/optimised by a computer. I update the calc using X from -0.5 to 0.5 with X = (x-0.5)*Pi The coefficient i get are really different from yours (when considering the Pi factor between them).
I update you patch with this coefs. I don't understand why, but your coefs works better for low frequency. so, nothing better here, but I still think it's worth sharing, in case i'm not the only one wondering...
also, there was a typo in my original mail, it's not a linear regression, but a polynomial regression (since the result was obviously polynomial)
cheers c
Le 19/10/2016 à 15:06, katja a écrit :
Changing the thread title to reflect the new approach. Extract of the original thread;
- I suggested using iemlib's hi pass filter recipe to improve
frequency response of [hip~]
- Christof Ressi pointed to formula in
http://www.arpchord.com/pdf/coeffs_first_order_filters_0p1.pdf
- this formula calculates feedback coefficient k = (1 - sin(a)) /
cos(a) where a = 2 * pi * fc / SR
- the filter implementation is y[n] = (x[n] - x[n-1]) * (1 + k) / 2
- k * y[n-1]
- following convention in d_filter.c (and pd tilde classes in
general), trig functions should best be approximated
- Cyrille provided libre office linear regression result for
(1-sin(x))/cos(x)
Thanks for the useful infos and discussion. My 'math coach' suggested using odd powers of -(x-pi/2) in an approximation polynomial for (1-sin(x))/cos(x). The best accuracy/performance balance I could get is with this 5th degree polynomial:
(-(x-pi/2))*0.4908 - (x-pi/2)^3*0.04575 - (x-pi/2)^5*0.00541
Using this approximation in the filter formula, response at cutoff frequency is -3 dB with +/-0.06 dB accuracy in the required range 0 < x < pi. It can be efficiently implemented in C, analogous to an approximation Miller uses in [bp~]. So that is what I'll try next.
Attached patch hip~-models.pd illustrates and compares filter recipes using vanilla objects:
- current implementation, most efficient, accuracy +/- 3 dB
- implementation with trig functions, least efficient, accuracy +/- 0.01
dB
- implementation with approximation for trig functions, efficient,
accuracy +/- 0.06 dB
A note on efficiency: coefficients in [hip~] are only recalculated when cutoff frequency is changed. How important is performance for a function rarely called? I'm much aware of the motto 'never optimize early', yet I spent much time on finding a fast approximation, for several reasons: it's a nice math challenge, instructive for cases where performance matters more, and I want to respect Miller's code efficiency when proposing a change. Today pd is even deployed on embedded devices so the frugal coding approach is still relevant. After 20 years.
Katja
On Tue, Oct 18, 2016 at 10:28 AM, cyrille henry ch@chnry.net wrote:
Le 18/10/2016 à 00:47, katja a écrit :
The filter recipe that Christof pointed to was easy to plug into the C code of [hip~] and works perfectly. But when looking further in d_filter.c I came across an approximation function 'sigbp_qcos()' used in the bandpass filter. It made me realize once more how passionate Miller is about efficiency. I'm not going to make a fool of myself by submitting a 'fix' using two trig functions to calculate a filter coefficient when a simple approximation could do the job. So that is what I'm now looking into, with help of a math friend: an efficient polynomial approximation for (1-sin(x))/cos(x).
according to libre office linear regression, for x between 0 and Pi, (1-sin(x))/cos(x) is about : -0.057255x³ + 0.27018x² - 0.9157x + 0.99344
the calc is in attachment, if you want to tune the input source or precision. cheers c
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