IIR Filter Co-efficients Formula

Hello All,

I am in the process of creating a front end for a butterworth low pass
filter of order 4.This will allow the user to enter the cut off
frequency and the software will calculate the co-efficients for the
user.
I used the information(formula for co-efficients) from this website
http://www.apicsllc.com/apics/Sr_3/Sr_3.htm

The formula is correct for the band stop filter but it doesn't work
correctly for the low pass filter.I have verified my results with
winfilter and MATLAB.Can anyone help me with a generic formula for the
co-efficients for low pass 4th order filter??

Thanks a bunch
Vj

On 16 Okt, 16:56, right05 <vidu...@gmail.com> wrote:
> Hello All,
>
> I am in the process of creating a front end for a butterworth low pass
> filter of order 4.This will allow the user to enter the cut off
> frequency and the software will calculate the co-efficients for the
> user.
> I used the information(formula for co-efficients) from this websitehttp://www.apicsllc.com/apics/Sr_3/Sr_3.htm
>
> The formula is correct for the band stop filter but it doesn't work
> correctly for the low pass filter.I have verified my results with
> winfilter and MATLAB.Can anyone help me with a generic formula for the
> co-efficients for low pass 4th order filter??

Well, if the formulaf or the BP filter works, there
is no reason why the LP formulas shouldn't work.

What's left, then, is the question whether a 4th order
LP filter can meet the spec. If you Restrict the order
to exactly 4, you have done away with one very imprtant
degree of freedom in filter design.

Rune

Thanks Rune.But the filter is part of our front end and that  suffice
our application.If you look at the website ,http://www.apicsllc.com/
apics/Sr_3/Sr_3.htm

the formula for the co-efficients for low pass and high pass looks
exactly the same( Just the denominator co-efficients for order 4)
LPF (Order 4)
n0=1,
n1=4,
n2=6
n3=4
n4=1
d0= (0.07*C^2 +0.13*C+0.07)*(0.03*C^2+0.002*C+0.03)
d1= -0.032*(C^2-1)*(125*C^2+163.168*C+125)
d2=6*C^4-6.83*C^2+6
d3=-0.032*(C^2-1)*(125*C^2-163.168*C+125)
d4=(0.07*C^2 -0.13*C+0.07)*(0.03*C^2-0.002*C+0.03)
HPF (order 4)
n0=C^4
n1=4* C^4
n2=6* C^4
n3=4* C^4
n4= C^4
d0= (0.07*C^2 +0.13*C+0.07)*(0.03*C^2+0.002*C+0.03)
d1= -0.032*(C^2-1)*(125*C^2+163.168*C+125)
d2=6*C^4-6.83*C^2+6
d3=-0.032*(C^2-1)*(125*C^2-163.168*C+125)
d4=(0.07*C^2 -0.13*C+0.07)*(0.03*C^2-0.002*C+0.03)

I would greatly appreciate if someone can give me a generic formula
for fourth order low pass filter.

Thanks
Vj

On Tue, 16 Oct 2007 14:56:04 +0000, right05 wrote:

> Hello All,
> 
> I am in the process of creating a front end for a butterworth low pass
> filter of order 4.This will allow the user to enter the cut off
> frequency and the software will calculate the co-efficients for the
> user.
> I used the information(formula for co-efficients) from this website
> http://www.apicsllc.com/apics/Sr_3/Sr_3.htm
> 
> The formula is correct for the band stop filter but it doesn't work
> correctly for the low pass filter.I have verified my results with
> winfilter and MATLAB.Can anyone help me with a generic formula for the
> co-efficients for low pass 4th order filter??
> 
> Thanks a bunch
> Vj

Somewhere in that discussion the cutoff frequency and the sampling rate
get dropped -- also if you look at the "any DSP text" that he references
you'll find that pole warping is far more involved (but still doable) than
he states.

I'd look for a better source.  I'm away from my reference shelf right now,
but I'll bet that Rick Lyons's book goes into this.  Trolling your local
university library for "any DSP text" may be good, too, if it doesn't take
a day to get there.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Tim Wescott wrote:
> On Tue, 16 Oct 2007 14:56:04 +0000, right05 wrote:
> 
>> Hello All,
>>
>> I am in the process of creating a front end for a butterworth low pass
>> filter of order 4.This will allow the user to enter the cut off
>> frequency and the software will calculate the co-efficients for the
>> user.
>> I used the information(formula for co-efficients) from this website
>> http://www.apicsllc.com/apics/Sr_3/Sr_3.htm
>>
>> The formula is correct for the band stop filter but it doesn't work
>> correctly for the low pass filter.I have verified my results with
>> winfilter and MATLAB.Can anyone help me with a generic formula for the
>> co-efficients for low pass 4th order filter??
>>
>> Thanks a bunch
>> Vj
> 
> Somewhere in that discussion the cutoff frequency and the sampling rate
> get dropped -- also if you look at the "any DSP text" that he references
> you'll find that pole warping is far more involved (but still doable) than
> he states.
> 
> I'd look for a better source.  I'm away from my reference shelf right now,
> but I'll bet that Rick Lyons's book goes into this.  Trolling your local
> university library for "any DSP text" may be good, too, if it doesn't take
> a day to get there.
> 
Rick Lyons, "Understanding Digital Signal Processing", Prentice-Hall, 
2004, chapter 6.

Or Oppenheim & Schafer, "Digital Signal Processing", Prentice-Hall, 
1975.  It's definitely a less accessible book than the Lyons book, both 
in getting your hands on the old thing and in understanding what the 
heck is being said once you have it -- but it's what was sitting on my 
shelf...

-- 

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

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

On Oct 16, 12:35 pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Tue, 16 Oct 2007 14:56:04 +0000, right05 wrote:
> > Hello All,
>
> > I am in the process of creating a front end for a butterworth low pass
> > filter of order 4.This will allow the user to enter the cut off
> > frequency and the software will calculate the co-efficients for the
> > user.
> > I used the information(formula for co-efficients) from this website
> >http://www.apicsllc.com/apics/Sr_3/Sr_3.htm
>
> > The formula is correct for the band stop filter but it doesn't work
> > correctly for the low pass filter.I have verified my results with
> > winfilter and MATLAB.Can anyone help me with a generic formula for the
> > co-efficients for low pass 4th order filter??
>
> > Thanks a bunch
> > Vj
>
> Somewhere in that discussion the cutoff frequency and the sampling rate
> get dropped -- also if you look at the "any DSP text" that he references
> you'll find that pole warping is far more involved (but still doable) than
> he states.

actually, at least as far as the bilinear transform (BLT) is
concerned, the "warping" and unwarping issues apply to signficant
frequencies, not directly to the complex values of the poles (or
zeros) themselves.  if it's a Butterworth, there is exactly one
significant frequency, the cutoff frequency, that gets warped (and
should be compensated for that).

if the s-plane transfer function is normalized (the significant
frequency in rad/unit_time is 1), then the substitution is (stolen
from the audio eq cookbook):

                                  1         1 - z^-1
      (normalized)   s  <--  ----------- * ----------
                              tan(w0/2)     1 + z^-1

   where  w0 = 2*pi*f0/Fs

   and f0 is the unnormalized significant (or cutoff) frequency.

   one can then make use of these trig identities:

                     sin(w0)                               1 - cos(w0)
      tan(w0/2) = -------------           (tan(w0/2))^2 =
-------------
                   1 + cos(w0)                             1 + cos(w0)


anyway, as far as i can tell, the analog Q for each Butterworth biquad
stage in the LPF prototype is not changed by the warping effects of
the BLT. if the order of the Butterworth, N, is even, then the Q is

    1/Q = 2*cos(pi/N*(n+1/2))   where  0 <= n < N/2

if it's a simple Butterworth LPF or HPF, the biquad formula for
coefficients is worked out (in the cookbook) and the frequency warping
is taken care of.  for BPF or BRF (notch) filters, i am not sure if
it's as straightforward, but i think it is.  maybe a little messier.

for LPF, HPF, BPF, and BRF using Tchebyshev Type I or II (of which
Butterworth is a limiting case), someone should work out the closed
form solutions to the filters (assuming BLT and prewarping the single
significant frequency) and publish it once and for all (maybe somebody
did).  they might also have to fudge the bandwidth of the BPF and BRF
due to BLT frequency warping because the bandwidth gets compressed as
the center frequency gets closer to Nyquist.  i had a simple first-
order fudging factor that i used to map the bandwidth in octaves of
the target digital filter to bandwidth in octaves of the prototype
analog filter:

     BW = w0/sin(w0) * bw

BW is for the analog prototype, bw is the bandwidth (in units of log
frequency) of the design target.  it's not perfect, but it's not too
bad.

r b-j

boy, that didn't wrap well.  i wish Google Groups would allow us to
turn off word wrapping.

this is better:

On Oct 16, 2:06 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
>
> if the s-plane transfer function is normalized (the significant
> frequency in rad/unit_time is 1), then the substitution is (stolen
> from the audio eq cookbook):
>
>                                   1         1 - z^-1
>       (normalized)   s  <--  ----------- * ----------
>                               tan(w0/2)     1 + z^-1
>
>    where  w0 = 2*pi*f0/Fs
>
>    and f0 is the unnormalized significant (or cutoff) frequency.
>
>    one can then make use of these trig identities:
>
>                 sin(w0)                          1 - cos(w0)
>   tan(w0/2) = -------------     (tan(w0/2))^2 = -------------
>                1 + cos(w0)                       1 + cos(w0)

r b-j

Thanks,Tim. I got hold of a Oppenheim & Schafer textbook and working
out a formula for the co-efficients for the fourth order low pass
filter.
But if I have 2 second order low pass filters in cascade, can I use
the second order co-efficients for both the  filters..I mean using the
second order formula.

Thanks Robert. Thats a complex math for a fourth order LPF.Wish there
is a generic formula for the coefficients.This website does
http://www.apicsllc.com/apics/Sr_3/Sr_3.htm but its not right.

On Oct 16, 2:17 pm, right05 <vidu...@gmail.com> wrote:
> Thanks Robert. Thats a complex math for a fourth order LPF.Wish there
> is a generic formula for the coefficients.This website doeshttp://www.apicsllc.com/apics/Sr_3/Sr_3.htmbut its not right.

Hello,

See if my paper helps ou out.

http://www.claysturner.com/dsp/Butterworth%20Filter%20Formulae.pdf

IHTH,

Clay