> 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
You'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
Reply by Fred Marshall●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.
>
>
> Martin
> 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 Fred Marshall●June 23, 20092009-06-23
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