Bessel Filter (Digital)

Hi,
I want to implement a bessel filter (Digital) (in c++, well but that i can
think of my own how to implement it in c++, right now i just want to know
how to implement it!!). Can you please guide me? (I'll fetch you further
details abt. the filter viz. 4-pole etc. , once i get started). I want
basically the following info to get started:
    1.> What all consists of a digital bessel filter? I mean what all
should the final output GENERALLY show? (given an input signal).

    2.> What are the design equations?

PS: My input signal will be sampled in time-domain. And yes, i have already
developed fourier and bessel functions , if required , for any operations.

Thanks & Regards,
Raj!

>Hi,
>I want to implement a bessel filter (Digital) (in c++, well but that i
can
>think of my own how to implement it in c++, right now i just want to know
>how to implement it!!). Can you please guide me? (I'll fetch you further
>details abt. the filter viz. 4-pole etc. , once i get started). I want
>basically the following info to get started:
>    1.> What all consists of a digital bessel filter? I mean what all
>should the final output GENERALLY show? (given an input signal).
>
>    2.> What are the design equations?
>
>PS: My input signal will be sampled in time-domain. And yes, i have
already
>developed fourier and bessel functions , if required , for any
operations.
>
>Thanks & Regards,
>Raj!
>

You can probably find everything you need along with a working C++ program
here:
http://www-users.cs.york.ac.uk/~fisher/mkfilter/

raj malhotra wrote:
> Hi,
> I want to implement a bessel filter (Digital) (in c++, well but that i can
> think of my own how to implement it in c++, right now i just want to know
> how to implement it!!). Can you please guide me? (I'll fetch you further
> details abt. the filter viz. 4-pole etc. , once i get started). I want
> basically the following info to get started:
>     1.> What all consists of a digital bessel filter? I mean what all
> should the final output GENERALLY show? (given an input signal).
>     2.> What are the design equations?
> PS: My input signal will be sampled in time-domain. And yes, i have already
> developed fourier and bessel functions , if required , for any operations.


1. There is no direct digital counterpart to the Bessel filter function 
defined in analog domain. You can only approximate Bessel filter one way 
or another. This is neither simple nor elegant.

2. There is generally no need for digital Bessel filters as you can get 
perfectly linear phase with FIR.

3. Book: D. Schlichtcharlie "Digital Filters". He derives approximations 
for Bessel filter functions, too.


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

On 11/16/2010 6:07 AM, Vladimir Vassilevsky wrote:
>
>
> raj malhotra wrote:
>> Hi,
>> I want to implement a bessel filter (Digital) (in c++, well but that i
>> can
>> think of my own how to implement it in c++, right now i just want to know
>> how to implement it!!). Can you please guide me? (I'll fetch you further
>> details abt. the filter viz. 4-pole etc. , once i get started). I want
>> basically the following info to get started:
>> 1.> What all consists of a digital bessel filter? I mean what all
>> should the final output GENERALLY show? (given an input signal).
>> 2.> What are the design equations?
>> PS: My input signal will be sampled in time-domain. And yes, i have
>> already
>> developed fourier and bessel functions , if required , for any
>> operations.
>
>
> 1. There is no direct digital counterpart to the Bessel filter function
> defined in analog domain. You can only approximate Bessel filter one way
> or another. This is neither simple nor elegant.
>
> 2. There is generally no need for digital Bessel filters as you can get
> perfectly linear phase with FIR.
>
> 3. Book: D. Schlichtcharlie "Digital Filters". He derives approximations
> for Bessel filter functions, too.
>
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Design Consultant
> http://www.abvolt.com

3) Yes, and they're horrible to work with.  I'm not necessarily sure 
there's blame to be ascribed there; at some point horrible math requires 
doing horrible math.  But even those approximations are really only any 
good for a cutoff frequency WAY below Nyquist.

Unless you've got both a) a very low cutoff frequency ( < 0.1 Fs ) and 
b) a serious aversion to doing polyphase filtering, the aforementioned 
FIR filter will make you a much happier person.

-- 
Rob Gaddi, Highland Technology
Email address is currently out of order

On 11/16/2010 06:07 AM, Vladimir Vassilevsky wrote:
>
>
> raj malhotra wrote:
>> Hi,
>> I want to implement a bessel filter (Digital) (in c++, well but that i
>> can
>> think of my own how to implement it in c++, right now i just want to know
>> how to implement it!!). Can you please guide me? (I'll fetch you further
>> details abt. the filter viz. 4-pole etc. , once i get started). I want
>> basically the following info to get started:
>> 1.> What all consists of a digital bessel filter? I mean what all
>> should the final output GENERALLY show? (given an input signal).
>> 2.> What are the design equations?
>> PS: My input signal will be sampled in time-domain. And yes, i have
>> already
>> developed fourier and bessel functions , if required , for any
>> operations.
>
>
> 1. There is no direct digital counterpart to the Bessel filter function
> defined in analog domain. You can only approximate Bessel filter one way
> or another. This is neither simple nor elegant.
>
> 2. There is generally no need for digital Bessel filters as you can get
> perfectly linear phase with FIR.
>
> 3. Book: D. Schlichtcharlie "Digital Filters". He derives approximations
> for Bessel filter functions, too.

Conclusion: if you're implementing a Bessel filter because you're 
replicating some creaky old analog thing in digital then you want to 
re-think what you're doing, and consider a FIR filter.  If you know you 
have good reasons to implement an IIR filter with an approximately* 
linear phase delay** then feel free to proceed -- but you may have a 
long row to hoe.

* 'cause it'll never be exact.

** I.e., severe processing resource constraints.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

On Nov 16, 12:54&#4294967295;pm, Tim Wescott <t...@seemywebsite.com> wrote:
>
> Conclusion: if you're implementing a Bessel filter because you're
> replicating some creaky old analog thing in digital then you want to
> re-think what you're doing, and consider a FIR filter. &#4294967295;If you know you
> have good reasons to implement an IIR filter with an approximately*
> linear phase delay** then feel free to proceed -- but you may have a
> long row to hoe.
>
> * 'cause it'll never be exact.
>
> ** I.e., severe processing resource constraints.
>

could we be more clear as to why Bessel seems okay for analog and not
with digital?

i have never implemented a Bessel filter in either domain, i don't
remember even doing it in school.  i do remember seeing it and the
design equations and such in sections besides that of other filters
(Butterworth, Tchebyshev 1 and 2, Elliptical, Gaussian).  i remember
seeing plots of their group delay were pretty damn flat until well
higher than the cutoff frequency (which was pretty sloppy compared to
a Butterworth).  the phase delay would also be pretty flat.

now, i can imagine using the bilinear transform on it, the magnitude
response will simply be warped with that tan() function, and the phase
response will be similarly warped, but group delay or phase delay is
calculated with the warped phase response and the *unwarped*
frequency.  if it was phase-linear in the analog domain, i wouldn't
expect it to be in the digital domain, unless the cutoff frequency was
much smaller than Nyquist.

is that the essential reason that the discrete-time Bessel is a
bitch?  or is there something else i hadn't thought of?

r b-j

On 11/16/2010 2:27 PM, robert bristow-johnson wrote:
> On Nov 16, 12:54 pm, Tim Wescott<t...@seemywebsite.com>  wrote:
>>
>> Conclusion: if you're implementing a Bessel filter because you're
>> replicating some creaky old analog thing in digital then you want to
>> re-think what you're doing, and consider a FIR filter.  If you know you
>> have good reasons to implement an IIR filter with an approximately*
>> linear phase delay** then feel free to proceed -- but you may have a
>> long row to hoe.
>>
>> * 'cause it'll never be exact.
>>
>> ** I.e., severe processing resource constraints.
>>
>
> could we be more clear as to why Bessel seems okay for analog and not
> with digital?
>
> i have never implemented a Bessel filter in either domain, i don't
> remember even doing it in school.  i do remember seeing it and the
> design equations and such in sections besides that of other filters
> (Butterworth, Tchebyshev 1 and 2, Elliptical, Gaussian).  i remember
> seeing plots of their group delay were pretty damn flat until well
> higher than the cutoff frequency (which was pretty sloppy compared to
> a Butterworth).  the phase delay would also be pretty flat.
>
> now, i can imagine using the bilinear transform on it, the magnitude
> response will simply be warped with that tan() function, and the phase
> response will be similarly warped, but group delay or phase delay is
> calculated with the warped phase response and the *unwarped*
> frequency.  if it was phase-linear in the analog domain, i wouldn't
> expect it to be in the digital domain, unless the cutoff frequency was
> much smaller than Nyquist.
>
> is that the essential reason that the discrete-time Bessel is a
> bitch?  or is there something else i hadn't thought of?
>
> r b-j
>
>

The Schlichtcharlie book that VV mentioned has discrete-time Bessel 
equations that basically prewarp the phase response so as to give an IIR 
filter with the same sort of mostly linearish phase though the pass 
band.  My results with his equations have been pretty mediocre.

What I think is going on is that, since the roll-off on a Bessel is so 
sloppy, as you start pushing Fc up towards Nyquist, the aliases of the 
filter response come back into play.  Since the Bessel approximation is 
only linear through the passband, the out-of band alias of the filter 
response comes back around and delinearizes the higher frequency 
components until the attenuation finally wins.

Either that or my transcription between Pascal and Octave was sloppy.

-- 
Rob Gaddi, Highland Technology
Email address is currently out of order

On Nov 16, 11:54&#4294967295;am, Tim Wescott <t...@seemywebsite.com> wrote:

> Conclusion: if you're implementing a Bessel filter because you're
> replicating some creaky old analog thing in digital then you want to
> re-think what you're doing, and consider a FIR filter. &#4294967295;

Alas, the poor Bessel Filter gets no respect.

It's not always about (approximately) linear phase, and it's not
always about rolloff characteristics.  The Bessel Filter is an IIR
approximation to a Gaussian Filter.  At filter orders greater than six
or eight, it's a rather good approximation.

A Gaussian Filter has the sometimes very attractive characteristics of
zero overshoot and zero ringing in the time domain, as well as
exhibiting the smallest duration-bandwidth product of any filter.
Most of the time people don't much care about these things, but
sometimes they do.  When they do care, the Gaussian is the best.

A Bessel Filter is very close to a Gaussian Filter in both domains.
In the time domain, there's a tiny bit of overshoot and a tiny bit of
ringing, but it's a lot better than other classical filter
approximations and it's orders of magnitude better than linear-phase
filters designed strictly in the frequency domain.

Bilinear transform approximations are pretty good when the cutoff
frequency is far below the folding frequency.  FDLS does a rather nice
job of approximation at higher cutoff frequencies, within reason.

Greg

Greg Berchin wrote:

> Alas, the poor Bessel Filter gets no respect.
> 
> It's not always about (approximately) linear phase, and it's not
> always about rolloff characteristics.  The Bessel Filter is an IIR
> approximation to a Gaussian Filter.  At filter orders greater than six
> or eight, it's a rather good approximation.
> 
> A Gaussian Filter has the sometimes very attractive characteristics of
> zero overshoot and zero ringing in the time domain, as well as
> exhibiting the smallest duration-bandwidth product of any filter.
> Most of the time people don't much care about these things, but
> sometimes they do.  When they do care, the Gaussian is the best.

I was surprised to learn that if the task is calculating moving average 
over the noisy data (using IIR filter), the Bessel filter of the 2nd 
order is the function to use. Bessel 2nd has the best settling time vs 
noise bandwidth. Although a simple real pole filter of the 2nd order is 
not very different from that. In the other words, Bessel response is 
good for measurement purposes, if the data is corrupt with white noise.
Somehow this сommon fact isn't mentioned in EE textbooks.

> A Bessel Filter is very close to a Gaussian Filter in both domains.
> In the time domain, there's a tiny bit of overshoot and a tiny bit of
> ringing, but it's a lot better than other classical filter
> approximations and it's orders of magnitude better than linear-phase
> filters designed strictly in the frequency domain.
> Bilinear transform approximations are pretty good when the cutoff
> frequency is far below the folding frequency.  FDLS does a rather nice
> job of approximation at higher cutoff frequencies, within reason.
> 
> Greg


Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

>>Hi,
>>I want to implement a bessel filter (Digital) (in c++, well but that i
>can
>>think of my own how to implement it in c++, right now i just want to
know
>>how to implement it!!). Can you please guide me? (I'll fetch you further
>>details abt. the filter viz. 4-pole etc. , once i get started). I want
>>basically the following info to get started:
>>    1.> What all consists of a digital bessel filter? I mean what all
>>should the final output GENERALLY show? (given an input signal).
>>
>>    2.> What are the design equations?
>>
>>PS: My input signal will be sampled in time-domain. And yes, i have
>already
>>developed fourier and bessel functions , if required , for any
>operations.
>>
>>Thanks & Regards,
>>Raj!
>>
>
>You can probably find everything you need along with a working C++
program
>here:
>http://www-users.cs.york.ac.uk/~fisher/mkfilter/



Hi,
Thanks for replying! I have checked out the link : 
http://www-users.cs.york.ac.uk/~fisher/mkfilter/
The link contains everything what i want, it contains the c++ source code
for bessel filter which i can download. 

But a little problem which i am facing is that , there are 4 directories
listed on the link: 
http://www-users.cs.york.ac.uk/~fisher/software/mkfilter/ 
And many files contain the c++ source code for bessel filter (along with
butterworth and Chebyshev ). I am pretty ignorant abt. this topic, and thus
pretty confused which code i can use out of the many listed there. Could
you please help me out in streaming down the c++ code for ONLY bessel
filter? Thanks in advance :)

Raj!