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FIR all pass and Bessel Function unit sample response

Started by Fred Marshall June 23, 2009
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 


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.
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)). Fred
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)). > > Fred
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