ringing: minimum vs linear phase

In my experience, if a minimum phase FIR filter does (doesn't) ring
then the linear phase equivalent of that filter also will (won't) ring.

Anybody have any counterexamples or else a proof that the above is
always true?

John

On 27 Mar 2005 04:31:09 -0800, "toobs" <johndhancock@gmail.com> wrote:

>In my experience, if a minimum phase FIR filter does (doesn't) ring
>then the linear phase equivalent of that filter also will (won't) ring.
>
>Anybody have any counterexamples or else a proof that the above is
>always true?
>
>John

If the frequency selectivity of the filters are the same then the
ringing behavior should be expected to be very similar.   The deletion
of the high-frequency terms results in the ringing (as in, "Gibb's
Phenomena"), so it won't matter much whether it's minimum phase or
not.


Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

"toobs" <johndhancock@gmail.com> wrote in message 
news:1111926669.968049.298090@o13g2000cwo.googlegroups.com...
> In my experience, if a minimum phase FIR filter does (doesn't) ring
> then the linear phase equivalent of that filter also will (won't) ring.
>
> Anybody have any counterexamples or else a proof that the above is
> always true?
>
> John

Well, certainly in linear phase filters the ringing is a function of the 
shape of the transition region.  Sharp transitions lead to ringing.  Smooth 
transitions less.  Of course this is a great simplification of all the 
factors that go into it.

One might also observe that the position of zeros (and poles if present) 
affects ringing for really the same reasons.

I've not thought much about how changing from linear phase to minimum phase 
or maximum phase might affect the ringing.  I guess I've always just assumed 
or understood that it wouldn't matter and that the transient response is a 
reflection of the "Q" of the system - which tends to be large around the 
transitions.  High Q - long time.  I suppose there could be a 
counter-example but I kinda doubt it.

I guess you should define exactly what you mean by "the linear phase 
equivalent"

Fred 

in article 1111926669.968049.298090@o13g2000cwo.googlegroups.com, toobs at
johndhancock@gmail.com wrote on 03/27/2005 07:31:

> In my experience, if a minimum phase FIR filter does (doesn't) ring
> then the linear phase equivalent of that filter also will (won't) ring.

can you define exactly what you mean by "ringing"?

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Fred Marshall wrote:
> "toobs" <johndhancock@gmail.com> wrote in message
> news:1111926669.968049.298090@o13g2000cwo.googlegroups.com...
> > In my experience, if a minimum phase FIR filter does (doesn't) ring
> > then the linear phase equivalent of that filter also will (won't)
ring.
> >
> > Anybody have any counterexamples or else a proof that the above is
> > always true?
> >
> > John
>
> Well, certainly in linear phase filters the ringing is a function of
the
> shape of the transition region.  Sharp transitions lead to ringing.
Smooth
> transitions less.

Agreed.

> Of course this is a great simplification of all the
> factors that go into it.

Uh... is it? I can't really see what more than the width of transition
regions (and possibly the width of the pass/stop bands of the filters,
if they are very narrow) go into the question of ringing. I can't see
how the phase response would have anything to do with ringing.

But then, I'm more than groggy after 10 days with the flu(?), so I
might 
very well be wrong...

Rune

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:1111949034.814482.201270@o13g2000cwo.googlegroups.com...
>
> Fred Marshall wrote:
>> "toobs" <johndhancock@gmail.com> wrote in message
>> news:1111926669.968049.298090@o13g2000cwo.googlegroups.com...
>> > In my experience, if a minimum phase FIR filter does (doesn't) ring
>> > then the linear phase equivalent of that filter also will (won't)
> ring.
>> >
>> > Anybody have any counterexamples or else a proof that the above is
>> > always true?
>> >
>> > John
>>
>> Well, certainly in linear phase filters the ringing is a function of
> the
>> shape of the transition region.  Sharp transitions lead to ringing.
> Smooth
>> transitions less.
>
> Agreed.
>
>> Of course this is a great simplification of all the
>> factors that go into it.
>
> Uh... is it? I can't really see what more than the width of transition
> regions (and possibly the width of the pass/stop bands of the filters,
> if they are very narrow) go into the question of ringing. I can't see
> how the phase response would have anything to do with ringing.

Thanks Rune.  Maybe the simplification wasn't so "great".

The factors I was thinking of have to do with the 2nd order things like the 
derivatives of the transition.  One can find a lowpass filter with a 
temporal transition to a step input such that the temporal transition has 
minimum width while being monotonic (no ringing).  That's the sort of thing 
that I had in mind.  But these are really details!

Fred 

Eric,

I agree with your statement and I have a question of more detail.  I
agree that a linear phase filter  and a min phase filter both can
suffer from ringing due to Gibbs Phenomena (assumming they bith are
similar otherwise, i.e. magnitude response)  .  My question is this....
  Is it true that the difference will be that  in the linear phase
filter case, that the ringing will equally come both before and after
the "step" in time, while in the min phase filter case, the ringing
will all be after the step in time.  To be specific, I'm thinking of
low pass filters with step functions passing through them.

thanks

Mark

On 27 Mar 2005 18:00:59 -0800, "Mark" <makolber@yahoo.com> wrote:

>Eric,
>
>I agree with your statement and I have a question of more detail.  I
>agree that a linear phase filter  and a min phase filter both can
>suffer from ringing due to Gibbs Phenomena (assumming they bith are
>similar otherwise, i.e. magnitude response)  .  My question is this....
>  Is it true that the difference will be that  in the linear phase
>filter case, that the ringing will equally come both before and after
>the "step" in time, while in the min phase filter case, the ringing
>will all be after the step in time.  To be specific, I'm thinking of
>low pass filters with step functions passing through them.
>
>thanks
>
>Mark

Yup, and you only need to look at the impulse response to understand
why.

A symmetric LPF with a sinx/x impulse response exhibits ringing before
and after and edge since the convolution will encounter the tails of
the sinx/x before and after the main lobe.   Just imagine the
convolution integrating as the edge of the square wave passes throught
he impulse response:  first it encounters the leading tails of the
sinx/x and the output (the dot product of coefficient with the input
as the edge passes through) will wiggle up and down as the undulation
in the tails affects the output sum.   When the edge passes the main
lobe the output edge will transition at a rate (i.e., slope)
determined by the width of the main lobe.  (A wise mentor in my early
DSP days said that the edge rate, and therefore the frequency
response, of a FIR is controlled by the width of the main lobe.  This
is why.)  As the edge passes the trailing lobes of the sinx/x the
ringing continues as they affect the output sum accordingly.

So the magnitude of the ringing will be proportional to the amplitudes
of the sinx/x sidelobes, and the slope of the edge is a function of
the width of the main lobe.

For a minimum phase filter there are typically few or no leading
sidelobes in the impulse response, so, just as you described, there
won't be any ringing preceding the edge in the output.  The rate of
the edge is still controlled by the width of the main lob and the
amplitude of the ringing is still going to be proportional to the
amplitudes of the trailing sidelobes in the impulse response.   The
difference, just as you indicated, is that the lack of leading
sidelobes in the impulse response eliminates the leading ringing in
the step response output.




Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

Eric,

OK thanks.

Now lets put this into an audio context.

It would seem (and this is subjective now)  that the ringing due to min
phase filter would be less objectionable because it is all post edge
and would sound similar to reverberation.  The linear phase filter has
pre ringing and I suspect that would sound very objectionable.

In any case they both probably sound pretty bad so the key to good
transient response (as I have believed for a while) is not linear phase
filters  or min phase filters, but rather using a more gradual rolloff
to avoid  the Gibbs phenomenon ringing in the first place.

In more general terms, any rapid change in the frequency response
(magnitude response) causes ringing problems and thus gradual filtering
may often be preferable to "brickwall" filtering.  This applies to both
low pass and high pass.  (Again this discussion is limited to an audio
context)

To add one more level to the discussion.... digital audio systems that
operate at  44.1 ksps typically employ  sharp anti-aliasing and
reconstruction low pass filters at about 20 kHz.  It seems from the
previous discussion that this filter (because it is typically designed
to have a sharp cutoff) typically will  create ringing no matter if it
is designed as min phase or linear phase.    Now this may sound like I
think these system sound bad due to this, but I also believe that the
saving grace is the fact that the rest of the system (mics, tweeters,
ears etc) is seldom flat to 20 kHz.  I know my ears aren't.  So even in
the worst case where the mics and tweeters and everything else were
flat to 20 kHz, the anti alias and reconstruction filters may create
ringing at 20 kHz, I'm not going to hear it anyway.    And in the more
typical case, the mics and speakers are not flat to 20 kHz but rather
will create a more gradual rolloff and since the Gibbs phenomenon
depends on the entire system having a sharp roll off, in this typical
case the ringing won't even occur due to the gradual rolloff created in
other parts of the system.   Comments?

thanks
Mark

Eric Jacobsen wrote:

   ...

> For a minimum phase filter there are typically few or no leading
> sidelobes in the impulse response, so, just as you described, there
> won't be any ringing preceding the edge in the output.  The rate of
> the edge is still controlled by the width of the main lob and the
> amplitude of the ringing is still going to be proportional to the
> amplitudes of the trailing sidelobes in the impulse response.   The
> difference, just as you indicated, is that the lack of leading
> sidelobes in the impulse response eliminates the leading ringing in
> the step response output.

This doesn't directly bear on Mark's question, but I think it ties in.

With the minimum-phase filter, some frequencies will be delayed more 
than others. Consider a signal of finite length that resides in a file 
both before and after being filtered. If the samples in the file are 
rear backwards in time -- last first, and so on -- the delayed 
frequencies will be seen as advanced, leading edges will be seen as 
trailing edges, and their associated ringing will happen before them. If 
the time-reversed signal is filtered again with the same filter, the 
frequencies most advanced will be most retarded, and ringing will follow 
the trailing edges, matching that which preceded them.

There are two major effects: the impulse response of the combined 
forward/backward filter will have symmetry, and the time delay of all 
frequencies will be the same. In fact, either condition implies the 
other. I'm content to let this scenario be the demonstration of that.

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