Once upon a time I wrote a nonlinear program to compute the unit sample response (i.e. to "design") FIR filters with the following characteristics: 1) Minimax solution to all pass. 2) The unit sample response could not be the trivial single unit sample - rather, it had to have: a) "length" b) the samples had to be in a magnitude range something like from 1 to 2.5 with arbitrary signs. c) no attempt to get linear phase - in fact that would be undesirable. I found that I could get relatively short filters with a "sinusoidal" magnitude response within an acceptable 3dB ripple .. say length 10 for example. It seemed that the filters had unit sample responses that looked like Bessel functions or sampled Bessel functions or ..... something like that. I've forgotten. Later, Robert Israel helped me figure out that one can analytically define an "all pass" filter with magnitude response that's sinusoidal. Using a trick from Hermann and Schussler, I took the "square roots" of this filter to get a number of filter candidates. Then, I sorted their individual unit sample responses to find the one with the least variance in the coefficient magnitudes (or something like that) to get them closest to the 1 to 2.5 range I mention above. Because the amplitude response is a constant plus a sinusoid, I rather envision the frequency response being a spiral trajectory of a rotating vector with major axis in frequency - which traverses a cylinder. And, that, in some way suggests a Bessel function I think.... I'm not good enough at this stuff to relate the Bessel function that I observed from the rather empirical results to the more analytical approach. It would be interesting to know how to tie them together. So: why should an equiripple all pass have a unit sample response that approximates or *is* a sampled Bessel function of some order? Fred

# FIR all pass and Bessel Function unit sample response

Started by ●June 23, 2009

Reply by ●June 23, 20092009-06-23

Fred Marshall wrote:> So: why should an equiripple all pass have a unit sample > response that approximates or *is* a sampled Bessel function of > some order?Not sure if it leads anywhere, but the Bessel Fourier transform is F{J_n}(w) = { 0, |w| > 1; 2*j^n*T_n(w)/sqrt(1-w^2), |w| < 1 }. The singularity probably means thst your IR is not directly a Bessel function, but it could be a linear combination of such if you managed to approximate some unit-magnitude spectrum with a Chebyshev series that vanishes at |w| = 1. Martin -- Quidquid latine scriptum est, altum videtur.

Reply by ●June 23, 20092009-06-23

Martin Eisenberg wrote:> Fred Marshall wrote: > >> So: why should an equiripple all pass have a unit sample >> response that approximates or *is* a sampled Bessel function of >> some order? > > Not sure if it leads anywhere, but the Bessel Fourier transform is > > F{J_n}(w) = { 0, |w| > 1; 2*j^n*T_n(w)/sqrt(1-w^2), |w| < 1 }. > > The singularity probably means thst your IR is not directly a Bessel > function, but it could be a linear combination of such if you managed > to approximate some unit-magnitude spectrum with a Chebyshev series > that vanishes at |w| = 1. > > > MartinMartin, Thanks. Well, all the frequency magnitude functions are 1.0 +/-e where e could be 3dB. There aren't any zeros so it wouldn't vanish at |w| = 1. The reference thread with Robert Israel is: http://groups.google.com/group/sci.math/browse_thread/thread/119795948f67e8c/1f381f8676a380f7?q=sci.math+israel+fmarshall So, I believe the closed form function I used was: Let b = sqrt(2 k^N/(1+k^(2N))) and C = sqrt(1+k^(2N)). Thus z = C sqrt(1 + b cos(N theta)). Fred

Reply by ●June 23, 20092009-06-23

MeAmI.org wrote: Fred Marshall wrote:> Martin Eisenberg wrote: > > Fred Marshall wrote: > > > >> So: why should an equiripple all pass have a unit sample > >> response that approximates or *is* a sampled Bessel function of > >> some order? > > > > Not sure if it leads anywhere, but the Bessel Fourier transform is > > > > F{J_n}(w) = { 0, |w| > 1; 2*j^n*T_n(w)/sqrt(1-w^2), |w| < 1 }. > > > > The singularity probably means thst your IR is not directly a Bessel > > function, but it could be a linear combination of such if you managed > > to approximate some unit-magnitude spectrum with a Chebyshev series > > that vanishes at |w| = 1. > > > > > > Martin > > Martin, > > Thanks. Well, all the frequency magnitude functions are 1.0 +/-e where e > could be 3dB. There aren't any zeros so it wouldn't vanish at |w| = 1. > > The reference thread with Robert Israel is: > http://groups.google.com/group/sci.math/browse_thread/thread/119795948f67e8c/1f381f8676a380f7?q=sci.math+israel+fmarshall > > So, I believe the closed form function I used was: > > Let b = sqrt(2 k^N/(1+k^(2N))) and C = sqrt(1+k^(2N)). Thus > z = C sqrt(1 + b cos(N theta)). > > FredYou're welcome: Rev. Mod. Phys. 54, 437 (1982): Ando et al. - Electronic ...Korotkikh, V. L., A. L. Musatov, and V. D. Shadrin, 1978, "Influence of ...... Gornik, and Logan (1980) from inversion layers excited by pass- ing an electric ...... Photo-induced response as a function of the gate vol- tage VG for the ...... where Ko and K1 are the modified Bessel functions and we have multiplied ...http://link.aps.org/doi/10.1103/ RevModPhys.54.437 Full text of "Russian Music From The Beginning Of The Nineteenth ...To pass on to new generations 84 RUSSIAN MUSIC the mastery of the school of ...... The bass part begins to acquire all the typical functional traits of the ...... images of Borisov-Musatov have found here their faded reincarnation. ..... This preface appears in the edition of Rus- sia published by Bessel in 1889. ...http:// www.archive.org/stream/russianmusicfrom010506mbp/russianmusicfrom010506mbp_djvu.txt aa aaa aaaa aaaaa aaaaaaaaaooooooowwww aaaaaahhhhh ...... alloy alloyed alloy alloys alloy allpass allport allred allreds allright ...... bessarabia bessarabian besse bessel bessell bessemer bessen bessenbacher ...... fiorucci fip fipp fippin fipple fiqui fir fire fireanti firearm firearms ...... musar musard musashi musashibo musatov musavian musawar musawi musberger ...http:// www.csse.monash.edu.au/~ingrid/Publications/QA/LemmaDictUnique%3Fref%3DSex%25C5%259Ehop.Com Martin