>
>> I'm just wondering why I had such a hard time to find reference to a Bessel
>> filter in the digital domain.
>
> Google for Thiran-Filter instead.
>
> See: Jean-Pierre Thiran: Recursive Digital Filters with Maximally Flat Group Delay. IEEE Transactions On Circuit Theory, Vol. CT-18, No. 6, November 1971.

You do not usually need them. A Bessel filter is an approximation
to a constant-delay filter, but in the digital domain, it is
possible to construct FIR (finite impulse response) filters with
constant delay.
(For completeness, it is possible to contruct a FIR filter in
the analog domain, if suitable delay elements are available).
--
-TV

Reply by ●March 5, 20182018-03-05

> I'm just wondering why I had such a hard time to find reference to a Bessel
> filter in the digital domain.

Google for Thiran-Filter instead.
See: Jean-Pierre Thiran: Recursive Digital Filters with Maximally Flat Group Delay. IEEE Transactions On Circuit Theory, Vol. CT-18, No. 6, November 1971.

Reply by ger_lough●August 13, 20142014-08-13

>.. but if I use an elliptic (Cauer) or Chebyshev filter, it won't be that
>much different from the above, and I get it with standard design methods.
>
>These filters have significant group delay ripple (clearly asymmetric
>impulse response) but are fast overall (the IR "kicks in" rather

quickly).

>I understood that is the requirement, otherwise this would be a little

out

>of place under the "Bessel" title...
>
>_____________________________
>Posted through www.DSPRelated.com
>

mnentwig, thanks for your assistance here. I haven't looked through your
script yet as I had parked this for a while, but will definitely have a
look through it as soon as this issue comes back to bite me in the ass in
the not so distant future!
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Reply by mnentwig●July 11, 20142014-07-11

.. but if I use an elliptic (Cauer) or Chebyshev filter, it won't be that
much different from the above, and I get it with standard design methods.
These filters have significant group delay ripple (clearly asymmetric
impulse response) but are fast overall (the IR "kicks in" rather quickly).
I understood that is the requirement, otherwise this would be a little out
of place under the "Bessel" title...
_____________________________
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Reply by mnentwig●July 11, 20142014-07-11

Hi,
I put an old IIR filter design script here:
https://drive.google.com/file/d/0B1gLUU8hXL7vN0ZQZHFjUlo3VG8/edit?usp=sharing
I make no promises on this algorithm...
The intention is to design a minimum-phase (=>low group delay) filter for a
given magnitude response. As it doesn't know the phase, it designs for some
starting value, then takes the phase of the result as input to the
least-squares solver and tries again several times.
You can control the result by
- specifying the frequency response you want (i.e. "1" for passband, "0"
for stopband
- setting a weight for different regions, trading passband ripple against
stopband rejection. Leaving "don't-care" regions with "0" weight will
improve performance in other areas.
You could also have a look here:
http://www.dsprelated.com/showcode/20.php
Slightly different approach - it creates the phase from the amplitude
spectrum via Hilbert transform.
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Reply by Dave●July 11, 20142014-07-11

Robert - did you get the copy of the paper I sent you?
The email address tied to my Google login isn't valid anymore, so I wouldn't have gotten any replies.
Best,
Dave

Reply by ger_lough●July 3, 20142014-07-03

Rob, thanks for your response. I think I may have been in for a
disappointment had I splashed out on Schlichtharle. I am only interested in
minimizing the group delay; not too concerned about overshoot. Most people
say to just use an FIR, but CPU will probably be an issue for my
application, so may not be feasible.
I will look into FDLS. How exactly would one approach digital Bessel design
using least squares? How is least squares used for this problem?
Thanks
Ger
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Reply by glen herrmannsfeldt●July 2, 20142014-07-02

> On Tue, 01 Jul 2014 14:41:38 -0400, robert bristow-johnson
> <rbj@audioimagination.com> wrote:

>>so Greg, in this statement about gaussian having the smallest
>>time-bandwidth product depend on a definition of time-width and
>>bandwidth like that above?

> From my ancient 1999 AES paper: "... duration is defined as the
> normalized second moment of the square of the time signal. Bandwidth
> is similarly defined as the second moment of the squared magnitude of
> the signal's Fourier Transform. Some familiar quantities that are
> also based upon the second moment include variance, standard
> deviation, mean-squared duration, and RMS duration."

In theory, you could use any even moment, or linear combination
of them. Or, even more generally, appropriately weighted even
functions around the (mean, median, mode, etc.) of the signal.
-- glen

Reply by Greg Berchin●July 2, 20142014-07-02

On Tue, 01 Jul 2014 14:41:38 -0400, robert bristow-johnson
<rbj@audioimagination.com> wrote:

>so Greg, in this statement about gaussian having the smallest
>time-bandwidth product depend on a definition of time-width and
>bandwidth like that above?

From my ancient 1999 AES paper: "... duration is defined as the
normalized second moment of the square of the time signal. Bandwidth is
similarly defined as the second moment of the squared magnitude of the
signal's Fourier Transform. Some familiar quantities that are also
based upon the second moment include variance, standard deviation,
mean-squared duration, and RMS duration."
Greg

Reply by glen herrmannsfeldt●July 1, 20142014-07-01

robert bristow-johnson <rbj@audioimagination.com> wrote:

> On 7/1/14 11:54 AM, Greg Berchin wrote:
>> The Gaussian filter
>> exhibits the smallest time-bandwidth product of any filter.

> i didn't know that, for some reason.

A favorite subject in quantum mechanics, where it minimizes the
product of position uncertainty and momentum uncertainty, or the
prodcut of time uncertainty and energy uncertainty.

> doesn't this depend on how "time width" and "bandwidth" are defined.

I suppse so, but there aren't so many choices that otherwise
make sense.

> one cool thing about gaussian filters is that they map back to
> themselves through the Fourier transform (FT). specifically:

> FT{ e^(-pi*t^2) } = e^(-pi*f^2)

> using the unitary FT (so that the inverse FT is essentially
> identical to the forward FT and all definitions are consistent).

You can put some 2*pi in there, too.

> of course, with scaling:

> FT{ e^(-pi*(t/T)^2) } = T * e^(-pi*(T*f)^2)

> for any T>0. the time width is T and the bandwidth is 1/T. some
> product equal to 1.

> we can define the time width and bandwidth of both of those functions to
> be 1. so their "time edges" and "band edges" are both at -1/2 and +1/2.
> and the time-bandwidth product for this family of curves is 1.

> then we can define the energy that is under the curve of

> +1/2
> integral |e^(-pi*t^2)|^2 dt} = 0.92368*sqrt(1/2)
> -1/2

> and divide that by the energy under the whole curve

> +inf
> integral |e^(-pi*t^2)|^2 dt} = sqrt(1/2)
> -inf

> and say that the time width or band width is the smallest possible width
> under the curve which has the same portion of the total energy, which
> appears to be 92.368 % of the total energy.

> so Greg, in this statement about gaussian having the smallest
> time-bandwidth product depend on a definition of time-width and
> bandwidth like that above? like no other family of functions other than
> gaussian can have a time-bandwidth product equal to or less than 1?

How many peak-shaped (decrease monotonically to zero on either size
of the maximum) functions are there that also have a peak-shaped
Fourier transform?

> just curious.
>

>> If that is what you need to optimize, then there is no other filter,
>> digital or analog or otherwise, that will do better than the Gaussian,

> what about "prolate spheroidal filters"? i confess i have never,
> myself, looked into these or the math behind them, but i remember being
> told that these are the filters that are *virtually* both bandlimited
> and timelimited which normally we say cannot be the case.