Determing whether or not a filter is minimum phase

Hello all
I have a filter I inherited from someone else (long gone) that is supposed
to be minimum phase but I would like to make sure.  Could someone please
tell me the best way to do this?  The filter is rather long (~500).  Is the
only method to use a generic polynomial root-finding algorithm, see where
the zeros lie, and pray for stability/convergence?  Thanks for your time
Alex 

On 11 Jun, 23:54, "alexryu" <ryu.a...@gmail.com> wrote:
> Hello all
> I have a filter I inherited from someone else (long gone) that is supposed
> to be minimum phase but I would like to make sure. &#4294967295;Could someone please
> tell me the best way to do this? &#4294967295;The filter is rather long (~500). &#4294967295;Is the
> only method to use a generic polynomial root-finding algorithm, see where
> the zeros lie, and pray for stability/convergence? &#4294967295;

Yes.

Rune

On Jun 11, 10:54 pm, "alexryu" <ryu.a...@gmail.com> wrote:
> Hello all
> I have a filter I inherited from someone else (long gone) that is supposed
> to be minimum phase but I would like to make sure.  Could someone please
> tell me the best way to do this?  The filter is rather long (~500).  Is the
> only method to use a generic polynomial root-finding algorithm, see where
> the zeros lie, and pray for stability/convergence?  Thanks for your time
> Alex

Perhaps this would be of help:

http://en.wikibooks.org/wiki/Control_Systems/Jurys_Test

(Obviously, use your filter polynomial, rather than the non-existent
denominator polynomial!)

--
Oli

On Jun 11, 5:54 pm, "alexryu" <ryu.a...@gmail.com> wrote:
> Hello all
> I have a filter I inherited from someone else (long gone) that is supposed
> to be minimum phase but I would like to make sure.  Could someone please
> tell me the best way to do this?  The filter is rather long (~500).

sounds like an FIR.

>  Is the
> only method to use a generic polynomial root-finding algorithm, see where
> the zeros lie, and pray for stability/convergence?

and yet, one more answer that is different than either of the previous
ones:

for a minimum-phase system, the natural logarithm of the magnitude
frequency response (the even-symmetry double-sided magnitude response
including negative frequencies) is the Hilbert Transform of the phase
response (phase measured in radians).

that is, if

   H(z) = SUM{ h[n] * z^(-n) }

is minimum-phase, then


   log|H(e^(jw))| = Re{ log(H(e^(jw)) } = Hilbert{ arg{H(e^(jw))} }

where

   arg{H(e^(jw))} = Im{ log(H(e^(jw)) } = phi(w)

where

   H(e^(jw)) = |H(e^(jw))| * e^(j*phi(w))


also, for a causal h[n] (that is h[n] = 0 for every n<0), the real and
imaginary parts of H(e^(jw)) (no logarithm) are a Hilbert transform
pair.

so,

1. take your FIR coefficients, the impulse response h[n].
2. put them into the first half of an FFT buffer (zero pad as
necessary and zero the second half corresponding to negative time).
3. FFT
4. complex log every bin in the result of the FFT
5. inverse FFT

the result should be "causal", the negative-time half (or the second
half) should be strictly zero for every bin (within a very small
amount because of numericial issues).

try that out.

r b-j

alexryu wrote:
> Hello all
> I have a filter I inherited from someone else (long gone) that is supposed
> to be minimum phase but I would like to make sure.  Could someone please
> tell me the best way to do this?  The filter is rather long (~500).  Is the
> only method to use a generic polynomial root-finding algorithm, see where
> the zeros lie, and pray for stability/convergence?  Thanks for your time
> Alex 
> 
> 
Here's one more answer to add to your pile:

If the filter (I assume it's an FIR) is symmetrical around the center 
then it's not minimum phase.

-- 

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 12 Jun, 02:01, robert bristow-johnson <r...@audioimagination.com>
wrote:

> for a minimum-phase system, the natural logarithm of the magnitude
> frequency response (the even-symmetry double-sided magnitude response
> including negative frequencies) is the Hilbert Transform of the phase
> response (phase measured in radians).

That's a very interesting approach, which might be worth a try.
I would assume there would be some issues wrt the unwrapping of
the phase response?

Rune

alexryu wrote:
> Hello all
> I have a filter I inherited from someone else (long gone) that is supposed
> to be minimum phase but I would like to make sure.  Could someone please
> tell me the best way to do this?  The filter is rather long (~500).  Is the
> only method to use a generic polynomial root-finding algorithm, see where
> the zeros lie, and pray for stability/convergence?
>  Thanks for your time

The best way is redesign the filter from scratch.

1. The filter with 500 taps just can't be the right thing. Especially if 
it is supposed to be a minimum phase. There should be more efficient 
approach to the problem.

2. With non-trivial polynomial of that high order, you are very likely 
to run into the numeric problems which would not allow you to prove 
anything. Especially with the FFT approach suggested by RBJ.


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

On Jun 11, 10:43 pm, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 12 Jun, 02:01, robert bristow-johnson <r...@audioimagination.com>
> wrote:
>
> > for a minimum-phase system, the natural logarithm of the magnitude
> > frequency response (the even-symmetry double-sided magnitude response
> > including negative frequencies) is the Hilbert Transform of the phase
> > response (phase measured in radians).
>
> That's a very interesting approach, which might be worth a try.
> I would assume there would be some issues wrt the unwrapping of
> the phase response?

yes, there are (and i forgot about that).  phase needs to be odd-
symmetry and decently unwrapped.

sometimes, like for chirp signals, the phase wraps so much that to
unwrap it properly (with uniformly-spaced discrete points from the
FFT), i applied the standard unwrapping alg to the *differences* of
adjacent phase points and then discretely "integrate" the unwrapped
differences to get the properly unwrapped phase.

r b-j

On Jun 11, 11:56 pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:
> alexryu wrote:
> > Hello all
> > I have a filter I inherited from someone else (long gone) that is supposed
> > to be minimum phase but I would like to make sure.  Could someone please
> > tell me the best way to do this?  The filter is rather long (~500).  Is the
> > only method to use a generic polynomial root-finding algorithm, see where
> > the zeros lie, and pray for stability/convergence?
> >  Thanks for your time
>
> The best way is redesign the filter from scratch.
>
> 1. The filter with 500 taps just can't be the right thing. Especially if
> it is supposed to be a minimum phase. There should be more efficient
> approach to the problem.

you can have minimum-phase, maximum-phase, or linear-phase FIRs.
nowadays 500 taps ain't so uncommon.  especially for non-realtime FIR
filtering.   MATLAB's "remez" or "firls" work pretty good for FIRs as
long as 512 taps.

> 2. With non-trivial polynomial of that high order, you are very likely
> to run into the numeric problems which would not allow you to prove
> anything. Especially with the FFT approach suggested by RBJ.

i thought that the numerical problems that the OP would get have to do
with factoring a 499th order polynomial into hundreds of irreducible
quadratics and 1st-order terms (and then checking to see where if the
real or complex roots are inside the unit circle).  i think that's
what the Jury criterion (which is a discrete-time counterpart to the
Routh-Hurwitz stability criterion) is there to deal with.

not much of a numerical problem with a 1024-point FFT, is there Vlad?

r b-j

"robert bristow-johnson" <rbj@audioimagination.com> wrote in message 
news:d525a000-3e2e-45d4-9166-c4b0ee7569af@d1g2000hsg.googlegroups.com...
> On Jun 11, 11:56 pm, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
> wrote:
>> alexryu wrote:
>> > Hello all
>> > I have a filter I inherited from someone else (long gone) that is 
>> > supposed
>> > to be minimum phase but I would like to make sure.  Could someone 
>> > please
>> > tell me the best way to do this?  The filter is rather long (~500).  Is 
>> > the
>> > only method to use a generic polynomial root-finding algorithm, see 
>> > where
>> > the zeros lie, and pray for stability/convergence?
>> >  Thanks for your time
>>
>> The best way is redesign the filter from scratch.
>>
>> 1. The filter with 500 taps just can't be the right thing. Especially if
>> it is supposed to be a minimum phase. There should be more efficient
>> approach to the problem.
>
> you can have minimum-phase, maximum-phase, or linear-phase FIRs.
> nowadays 500 taps ain't so uncommon.  especially for non-realtime FIR
> filtering.   MATLAB's "remez" or "firls" work pretty good for FIRs as
> long as 512 taps.

Well, it is not a problem to do the design however a FIR with 500 taps is 
ridiculous. It obviously indicates the attempt to solve something by brute 
force. There must be more efficient approach.


>> 2. With non-trivial polynomial of that high order, you are very likely
>> to run into the numeric problems which would not allow you to prove
>> anything. Especially with the FFT approach suggested by RBJ.
>
> i thought that the numerical problems that the OP would get have to do
> with factoring a 499th order polynomial into hundreds of irreducible
> quadratics and 1st-order terms (and then checking to see where if the
> real or complex roots are inside the unit circle).  i think that's
> what the Jury criterion (which is a discrete-time counterpart to the
> Routh-Hurwitz stability criterion) is there to deal with.
>
> not much of a numerical problem with a 1024-point FFT, is there Vlad?

If you are OK with the fuzzy results of FFT, then why can't we just make the 
1/H(z) filter in the direct form and run a pulse through it? If it diverges, 
then the filter is not minimal phase. The numerical problems apply, of 
course.

VLV