Jon Harris wrote:

   ...

> When the OP mentioned sample rate ("0.05"), since no units were given, I assumed
> this was normalized to the sample rate.  If that is indeed the case, then the
> order and normalized cut-off would be enough to calculate the phase shift vs.
> frequency.

I assumed that too, and .05 makes agreement between the digital filter
and its analog prototype very good. The OP seemed to want a single
number, though, not a function of frequency. If that's the case, the
rest of any discussion is beside the point. Let's wait to hear from him.

   ...

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

"Jerry Avins" <jya@ieee.org> wrote in message
news:406d8bf0$0$3057$61fed72c@news.rcn.com...
> Stewart wrote:
>
> > Hi All,
> >
> > Hoping someone can help me out.  I'm working with Butterworth Filters
> > and have some great equations for calculating the coefficients as a
> > function of cut-off frequency:
> >
> > http://www.planetanalog.com/showArticle.jhtml?articleID=12802683&pgno=3
> >
> > However I need a function which gives me the phase delay of the filter
> > as a function of cut-off frequencies.  For example, a second order
> > Butterworth Filter (Low Pass) with a cut-off frequency of 0.05 has a
> > lag (phase shift) of ??? radians.
> >
> > Hope someone has a formula for this.  If it was genereric for N-Order
> > that would be best.
> >
> > Regards,
> >
> > Stewart
>
> The phase performance of a digital Butterworth filter varies not only
> with the particular frequency in the passband, but also with the ratio
> of cut-off to sampling frequencies. It's not just one number.

When the OP mentioned sample rate ("0.05"), since no units were given, I assumed
this was normalized to the sample rate.  If that is indeed the case, then the
order and normalized cut-off would be enough to calculate the phase shift vs.
frequency.

However, it is not a simple calculation, especially for higher orders.  In the
past, when I've needed this, I've started with the s-domain transfer function,
and just worked it out with brute-force algebra (substitute s = jw, separate
real/imaginary parts, take the arctan of imaginary/real).  I also found I had to
do some pre-warping correction for improved accuracy near the Nyquist frequency.
It was messy and ugly and what I came up with wasn't that general--only did 1st
and 2nd order filters.  However, if you have that, you can work out the higher
order filters by first decomposing them to 1st/2nd order sections (factoring)
and then simply summing the phase responses.

Probably there is a better way to do this directly in the z-domain, but I don't
know how.  That's the way Matlab does it.  If you don't have Matlab, then you'd
probably at least want to use Excel or some other tool for crunching the
numbers.

Stewart wrote:

> Hi All,
> 
> Hoping someone can help me out.  I'm working with Butterworth Filters
> and have some great equations for calculating the coefficients as a
> function of cut-off frequency:
> 
> http://www.planetanalog.com/showArticle.jhtml?articleID=12802683&pgno=3
> 
> However I need a function which gives me the phase delay of the filter
> as a function of cut-off frequencies.  For example, a second order
> Butterworth Filter (Low Pass) with a cut-off frequency of 0.05 has a
> lag (phase shift) of ??? radians.
> 
> Hope someone has a formula for this.  If it was genereric for N-Order
> that would be best.
> 
> Regards,
> 
> Stewart

The phase performance of a digital Butterworth filter varies not only 
with the particular frequency in the passband, but also with the ratio 
of cut-off to sampling frequencies. It's not just one number.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Hi All,

Hoping someone can help me out.  I'm working with Butterworth Filters
and have some great equations for calculating the coefficients as a
function of cut-off frequency:

http://www.planetanalog.com/showArticle.jhtml?articleID=12802683&pgno=3

However I need a function which gives me the phase delay of the filter
as a function of cut-off frequencies.  For example, a second order
Butterworth Filter (Low Pass) with a cut-off frequency of 0.05 has a
lag (phase shift) of ??? radians.

Hope someone has a formula for this.  If it was genereric for N-Order
that would be best.

Regards,

Stewart