Matched Lowpass Filter

[lalalandsynth]

Sweet , testing these.
Thanks martin

[juha_tp]

I have made these approximation formulas to calculate coefficients for Butterworth 1st order LPF (fixed fs=44100Hz):

Code: Select all

if x<1000
  b0 = -2.8877914930158800E-17*x^4 + 5.1505099601836300E-13*x^3 - 8.3042766124760100E-09*x^2 + 1.1658923888554000E-04*x + 4.7538137317132600E-09;
  b1 = -8.5798500741765000E-18*x^4 + 1.5075985241148900E-13*x^3 - 1.8436891078309800E-09*x^2 + 2.5886403730853200E-05*x + 1.4813928303349300E-09;
  a1 = -1.0000210773722100E+00 * exp(-1.4259268000113900E-04*x);
else
  b0 = -1.4099442035756000E-30 * x^7 + 1.0017110602452500E-25 * x^6 - 2.5601690530276300E-21 * x^5 + 2.3521737834624400E-17 * x^4 + 1.5572294695099200E-13 * x^3 - 7.1802325915484600E-09 * x^2 + 1.1519383331259800E-04 * x + 4.0299292935725700E-04;
  b1 = -4.7895187177706000E-30 * x^7 + 3.6547584697795600E-25 * x^6 - 1.0692798201684300E-20 * x^5 + 1.4808481293564600E-16 * x^4 - 9.5370681840518000E-13 * x^3 + 1.8314251162411300E-09 * x^2 + 2.1044667135830600E-05 * x + 1.4837308218941300E-03;
  a1 = -6.1994629213463400E-30 * x^7 + 4.6564695300249300E-25 * x^6 - 1.3252967254712400E-20 * x^5 + 1.7160655077027700E-16 * x^4 - 7.9798387145426300E-13 * x^3 - 5.3488074753069200E-09 * x^2 + 1.3623850044842700E-04 * x - 9.9811327624874200E-01;
end if


where x = cut-off frequency in Hz.

Approximation process wasn't simple because of you need to first calculate enough coefficients for the approximation so, not a real time process. In this 1st order example (coefficient calculation is based on Massberg's method), I calculated coefficients for every (whole) Hz in range 0.01...N (using Octave) and then approximated each coefficient column separately (using LibreOffice Calc). Maybe less samples could have been enough. Approximation formulas are taken from Calc's trend line equations and the R^2 values were around 0.999.
I had to split the range into two parts to get polynomial degrees lowered. I don't know what approximation method LibreOffice Calc implements).

Could this type of implementation give any advantages in real time applications ... (at least in case of (Butterworth) LP and HP filters with fixed Q)?

[martinvicanek]

Could this type of implementation give any advantages in real time applications ... (at least in case of (Butterworth) LP and HP filters with fixed Q)?

In principle, yes. Polynomial approximatios may be very efficient and have a smaller footprint than lookup tables. Personally I wonder what application would require such an extraordinarily accurate match to the analog magnitude response of a first order filter? Polynomials of seventh(!) degree, hmmm. Then, as you note, higher order filters have more independent parameters e.g. Q in addition to the cutoff frequency, which makes a polynomial fit a lot messier.

[juha_tp]

In principle, yes. Polynomial approximatios may be very efficient and have a smaller footprint than lookup tables. Personally I wonder what application would require such an extraordinarily accurate match to the analog magnitude response of a first order filter? Polynomials of seventh(!) degree, hmmm. Then, as you note, higher order filters have more independent parameters e.g. Q in addition to the cutoff frequency, which makes a polynomial fit a lot messier.

Yes, the Q and gain change would mean another sets of approximations ... (dunno yet how linear are the changes in coefficients when these parameters are changed .... Another issue would be that sample rate is fixed... .

This (base) 1st order filter was just an easy example (you know the math needed to get it that accurate... ) but, I have tried similar approximations for 2nd and 4th order Butterworth HPF and LPF and it looks like those works well. Polynomial degree can be dropped by splitting the frequency range and also, its possible to mix various implementations (Massberg, Orfandis, MZTi, MIM, BLT, MZT, IIM etc...) when carefully select those ranges... .