On Wed, 06 Dec 2006 18:33:45 -0500, Jerry Avins <jya@ieee.org> wrote:

>Vladimir Vassilevsky wrote:
>> 
>> 
>> Jerry Avins wrote:
>> 
>> 
>>> Basically, it's not phase error that causes the ringing, but the sharp 
>>> cutoff.
>> 
>> To be more exact, the ringing is caused by the cutoff sharpness vs delay 
>> in the filter. One can build a filter as sharp as he like and without 
>> any ringing. That can be either FIR or IIR structure, it does not matter.
>> 
>>> Sharp corners in frequency make for ringing in time. It's harder to 
>>> show the reverse.
>> 
>> 
>> Imagine a cascade of moving average filters for FIR or a cascade of 1st 
>> order filters for IIR. The cutoff can be as sharp as you like depending 
>> on the number of stages, the phase will be nonlinear for IIR, however 
>> there will be no ringing at all.
>
>Interesting. A cascade of moving average filters is a binomial filter, a 
>digital approximation to a Gaussian. Do Gaussians not ring? As 2-D 
>filters, they are separable. What other remarkable properties do they have?
>
>Jerry

Hi Jer,
  well, if we have a Gaussian time-domain sequence 
whose duration is finite, the Fourier transform of 
that sequence certainly does have ringing in 
the form of "sidelobes" in the freq domain.

The Fourier transform of an infinitely-long 
Gaussian function is itself a Gaussian function.
However we can't have infinitely-long functions 
(sequences) in our computers. 

See Ya,
[-Rick-]

On Wed, 06 Dec 2006 15:30:41 -0500, Jerry Avins <jya@ieee.org> wrote:

>Sharp corners in frequency make for ringing in time. It's harder to show 
>the reverse.
>
>Jerry

It's not that bad, even for a single-stage FIR.   This was pointed out
by a really insightful DSP guru early in my career:  The steepness of
the step response depends on the width of the main lobe in the filter.

It isn't too hard to imagine a simple FIR convolving with a step
function and that the steepness of the output step is dependent almost
completely on the width of the main lobe.   The "ringing" is then just
an artifact of the sidelobes.   That was a really useful insight for
me as it gives yet another meaningful connection between the impulse
response and other filter characteristics.   It's not a big leap to
realize that a narrow main lobe is also a sharp filter in frequency,
so there's an intuitive connection between ringing in time and
sharpness in frequency (as well as the steepness of the step
response).   They're all related.


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

>Because a Linkwitz-Riley pair does not sum to unity, it sums to allpass.
>In other words, when you sum the transfer functions of the lowpass
>section and the highpass section, you don't get "1", you get the
>transfer function of an allpass filter.  Allpass filters have unity
>magnitude response but nonlinear phase response.  That nonlinear phase
>response is what causes the poor summed transient response that you
>observe.


Greg thank you for turning on the light!!  Now I understand!

Thanks everyone for your comments.  Sometimes it takes me a while to "get
it".

On Wed, 06 Dec 2006 14:04:54 -0600, "jeff227" <rocksonics@earthlink.net>
wrote:

>So if the phase BETWEEN the LP and HP outputs is always constant why
>doesn't summing the LP and HP outputs reproduce a square wave?  What am I
>not understanding here?

Because a Linkwitz-Riley pair does not sum to unity, it sums to allpass.
In other words, when you sum the transfer functions of the lowpass
section and the highpass section, you don't get "1", you get the
transfer function of an allpass filter.  Allpass filters have unity
magnitude response but nonlinear phase response.  That nonlinear phase
response is what causes the poor summed transient response that you
observe.

-- Greg

jeff227 wrote:

> I tried cascading damped 2nd order sections (Bessel, I believe) with the
> assumption you stated above (aka, 2nd order Linkwitz-Riley "Transient
> perfect" crossover).

LR arrangement consists of 4 Butterworth filters of the same order: two 
cascaded in the lowpass branch, two cascadded in the highpass branch. In 
this case the frequency response will be perfect.


  No ringing on the LP/HP individual outputs but the
> SUMMED outputs still had very ugly square wave response even though the
> summed frequency response was flat.  Why now?

In the crossover frequency area, the group delay is at maximum, whereas 
it is zero at high and low frequencies. What you got by combining the 
HPF and LPF is the allpass filter which rings a lot. So, LR crossover is 
ringing.

   Ringing, in this case, is
> not the reason.

What is your goal? Yes, it is possible to design the crossover without 
ringing and with near perfect combined response, however it is not going 
to be LR.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

Vladimir Vassilevsky wrote:
> 
> 
> Jerry Avins wrote:
> 
> 
>> Basically, it's not phase error that causes the ringing, but the sharp 
>> cutoff.
> 
> To be more exact, the ringing is caused by the cutoff sharpness vs delay 
> in the filter. One can build a filter as sharp as he like and without 
> any ringing. That can be either FIR or IIR structure, it does not matter.
> 
>> Sharp corners in frequency make for ringing in time. It's harder to 
>> show the reverse.
> 
> 
> Imagine a cascade of moving average filters for FIR or a cascade of 1st 
> order filters for IIR. The cutoff can be as sharp as you like depending 
> on the number of stages, the phase will be nonlinear for IIR, however 
> there will be no ringing at all.

Interesting. A cascade of moving average filters is a binomial filter, a 
digital approximation to a Gaussian. Do Gaussians not ring? As 2-D 
filters, they are separable. What other remarkable properties do they have?

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

jeff227 skrev:
> >Imagine a cascade of moving average filters for FIR or a cascade of 1st
> >order filters for IIR. The cutoff can be as sharp as you like depending
> >on the number of stages, the phase will be nonlinear for IIR, however
> >there will be no ringing at all.
>
>
> I tried cascading damped 2nd order sections (Bessel, I believe) with the
> assumption you stated above (aka, 2nd order Linkwitz-Riley "Transient
> perfect" crossover).  No ringing on the LP/HP individual outputs but the
> SUMMED outputs still had very ugly square wave response even though the
> summed frequency response was flat.

Why do you expect the sum of a LP'd and HP'd signal to be OK?

Rune

>Imagine a cascade of moving average filters for FIR or a cascade of 1st 
>order filters for IIR. The cutoff can be as sharp as you like depending 
>on the number of stages, the phase will be nonlinear for IIR, however 
>there will be no ringing at all.


I tried cascading damped 2nd order sections (Bessel, I believe) with the
assumption you stated above (aka, 2nd order Linkwitz-Riley "Transient
perfect" crossover).  No ringing on the LP/HP individual outputs but the
SUMMED outputs still had very ugly square wave response even though the
summed frequency response was flat.  Why now?  Ringing, in this case, is
not the reason.

Jerry Avins wrote:


> Basically, it's not phase error that causes the ringing, but the sharp 
> cutoff.

To be more exact, the ringing is caused by the cutoff sharpness vs delay 
in the filter. One can build a filter as sharp as he like and without 
any ringing. That can be either FIR or IIR structure, it does not matter.

> Sharp corners in frequency make for ringing in time. It's harder to show 
> the reverse.


Imagine a cascade of moving average filters for FIR or a cascade of 1st 
order filters for IIR. The cutoff can be as sharp as you like depending 
on the number of stages, the phase will be nonlinear for IIR, however 
there will be no ringing at all.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com

jeff227 wrote:

> I coded a very nice crossover using 4th order Linkwitz-Riley filters
> whereby their sine wave outputs do indeed sum to a flat frequency
> response.

And this is not particularly useful since you need to take the whole 
audio system into the account.

> 
> However, the square wave response of the summation is terrible.  I am
> getting over 4dB of overshoot and ringing.

Doesn't matter. Listen with the ears, not with the scope.

> I understand Butterworth IIR filters have non-linear phase shift - with
> respect to their inputs.

The linearity of phase and the amount of ringing are the two unrelated 
subjects.

   But in the 4th order Linkwitz-Riley alignment
> the HP and LP filters are always 360 degrees out of phase with respect to
> EACH OTHER.

So what?

> So if the phase BETWEEN the LP and HP outputs is always constant why
> doesn't summing the LP and HP outputs reproduce a square wave?  What am I
> not understanding here?

The LR filter is basically a Butterworth filter which has the known 
property to "ring". The ringing is more the higher the filter order is.

> 
> (This is the reason I am pursuing linear phase FIR filters in my other
> post)

This is a popular misconception about linear/nonlinear phases and ringing.

Vladimir Vassilevsky

DSP and Mixed Signal Design Consultant

http://www.abvolt.com