# questions raised by reading and thinking with possibly missing background

Started by December 10, 2005
```Randy Yates wrote:
> Richard Owlett <rowlett@atlascomm.net> writes:
>
> > Richard Owlett wrote:
> >
> >> ...
> >> That got me thinking ;<
> >
> > What are the *NECESSARY* conditions for a FIR filter of an arbitrary
> > shape in the frequency domain to be "linear phase".
> >
> > One of the references I was reading stated that "a FIR filter would be
> > 'linear phase' if its coefficients were symmetric about the middle
> > coefficient."
> >
> > Is that a "sufficient" condition or a "necessary" condition?
> > What implication does it have for the passband response?
...
> It is a sufficient condition. A trivial example of an FIR filter
> that does not meet this condition but is still linear phase is
> the FIR given by h[0] = 0, h[1] = 0, and h[2] = 1.

That, to me, is just an obfuscation which can be remedied by
a more thorough definition of "middle coefficient", and the
addition/removal of up to an infinite number of zero terms.

IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M

```
```Randy Yates wrote:
> Randy Yates <yates@ieee.org> writes:
> > [...]
> > I've heard that a linear-phase filter has magnitude and phase
> > responses that are Hilbert transforms of each other, but I've
> > never been interested enough to investigate.
>
> Sorry - correction!: Those are *minimum-phase* filters.

Minimum-phase FIR filters are interesting if speed of response
is more important than the phase linearity.  For low pass filters,
minimum-phase filters would seem to me to be far more "natural"
than linear-phase filters, given that linear-phase low pass filters
have a "pre-ringing" response that sounds extremely unnatural
compared to any natural or analog filtering process.  And
minimum-phase filters have the fastest mean response or
delay for a given pile of poles and zeros.

But the advantage of linear-phase filters for the OP is that,
given matched delays, they can be summed without worrying
about any phase cancellations of some frequency bands.

IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M

```
```"Ron N." <rhnlogic@yahoo.com> writes:

> Randy Yates wrote:
>> Randy Yates <yates@ieee.org> writes:
>> > [...]
>> > I've heard that a linear-phase filter has magnitude and phase
>> > responses that are Hilbert transforms of each other, but I've
>> > never been interested enough to investigate.
>>
>> Sorry - correction!: Those are *minimum-phase* filters.
>
> Minimum-phase FIR filters are interesting if speed of response
> is more important than the phase linearity.  For low pass filters,
> minimum-phase filters would seem to me to be far more "natural"
> than linear-phase filters, given that linear-phase low pass filters
> have a "pre-ringing" response that sounds extremely unnatural
> compared to any natural or analog filtering process.  And
> minimum-phase filters have the fastest mean response or
> delay for a given pile of poles and zeros.

Hey Ron, how do you know so much about minimum-phase filters? This
is still, at my ripe-old-age, one of the topics I have yet to broach
in my career.

Say, do you have an example of a linear-phase filter and corresponding
minimum-phase filter in which the linear-phase version exhibits the
"pre-ringing" phenomenom? I'd love to try this out for myself.
--
%  Randy Yates                  % "Midnight, on the water...
%% Fuquay-Varina, NC            %  I saw...  the ocean's daughter."
%%% 919-577-9882                % 'Can't Get It Out Of My Head'
%%%% <yates@ieee.org>           % *El Dorado*, Electric Light Orchestra
```
```Randy Yates wrote:
> "Ron N." <rhnlogic@yahoo.com> writes:
...
> > Minimum-phase FIR filters are interesting if speed of response
> > is more important than the phase linearity.  For low pass filters,
> > minimum-phase filters would seem to me to be far more "natural"
> > than linear-phase filters, given that linear-phase low pass filters
> > have a "pre-ringing" response that sounds extremely unnatural
> > compared to any natural or analog filtering process.  And
> > minimum-phase filters have the fastest mean response or
> > delay for a given pile of poles and zeros.
>
> Hey Ron, how do you know so much about minimum-phase filters?

I don't.  That I have anything at all to say about the topic falls
into the category of "random walk" continuing ed.

> This is still, at my ripe-old-age, one of the topics I have yet to
> broach in my career.

I was experimenting with cepstral methods for pitch recognition,
and found in my reading that a cepstrum calculation could also
be used to construct minimum phase FIR filters.  There's a long
comp.dsp thread on the subject that I started somewhere around
mid-March of 2004.

> Say, do you have an example of a linear-phase filter and corresponding
> minimum-phase filter in which the linear-phase version exhibits the
> "pre-ringing" phenomenom? I'd love to try this out for myself.

I don't have the c code handy, but I started with this description
found on the net:

>> wn = [ones(1,m); 2*ones((n+odd)/2-1,m) ; ones(1-rem(n,2),m);
>>   zeros((n+od d)/2-1,m)];
>> y = real(ifft(exp(fft(wn.*real(ifft(log(abs(fft(x)))))))));

rewrote it, and fed it some very low pass (relative to the sample
rate) linear-phase FIR filters.

Minimum-phase FIR filters converted from symmetric windowed
Sincs can also be used for upsampling interpolation (and, if an
analog or physical filter was used for the original Nyquist
limiting process before sampling, a non-linear-phase reconstruction
might perhaps be closer to the original signal than a symmetric
windowed-Sinc reconstruction).

IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M

```
```Richard Owlett wrote:
> Richard Owlett wrote:
>
> > ...
> > That got me thinking ;<
>
> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
> shape in the frequency domain to be "linear phase".
>
> One of the references I was reading stated that "a FIR filter would be
> 'linear phase' if its coefficients were symmetric about the middle
> coefficient."
>
> Is that a "sufficient" condition or a "necessary" condition?

We discussed this last June:

> What implication does it have for the passband response?

None that I could think of, apart from the linear phase.

Regards,
Andor

```
```Ron N. wrote:
...
> Minimum-phase FIR filters are interesting if speed of response
> is more important than the phase linearity.

Also, if number of coefficients is important - a given magnitude
response can usually be met with less coefficients if the
phase-linearity condition is dropped.

> For low pass filters,
> minimum-phase filters would seem to me to be far more "natural"
> than linear-phase filters, given that linear-phase low pass filters
> have a "pre-ringing" response that sounds extremely unnatural
> compared to any natural or analog filtering process.

Perhaps that is the reason why digital audio sounds so extremely
unnatural - it's them damn linear-phase reconstruction filters!

Regards,
Andor

```
```abariska@student.ethz.ch writes:

> Richard Owlett wrote:
>> Richard Owlett wrote:
>>
>> > ...
>> > That got me thinking ;<
>>
>> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
>> shape in the frequency domain to be "linear phase".
>>
>> One of the references I was reading stated that "a FIR filter would be
>> 'linear phase' if its coefficients were symmetric about the middle
>> coefficient."
>>
>> Is that a "sufficient" condition or a "necessary" condition?
>
> We discussed this last June:
>

Consider the filter coefficients determined as

function y = test(x)
%function y = test(x)
n = [-25 : 25];
Fs = 1;
Ts = 1/Fs;
t = n*Ts;
plot(sinc(t+1/7));

These are neither symmetric nor antisymmetric in the sense you defined,
and yet this is a linear phase filter, is it not?
--
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface,
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO
```
```Randy Yates wrote:
> abariska@student.ethz.ch writes:
>
> > Richard Owlett wrote:
> >> Richard Owlett wrote:
> >>
> >> > ...
> >> > That got me thinking ;<
> >>
> >> What are the *NECESSARY* conditions for a FIR filter of an arbitrary
> >> shape in the frequency domain to be "linear phase".
> >>
> >> One of the references I was reading stated that "a FIR filter would be
> >> 'linear phase' if its coefficients were symmetric about the middle
> >> coefficient."
> >>
> >> Is that a "sufficient" condition or a "necessary" condition?
> >
> > We discussed this last June:
> >
>
> Consider the filter coefficients determined as
>
> function y = test(x)
> %function y = test(x)
> n = [-25 : 25];
> Fs = 1;
> Ts = 1/Fs;
> t = n*Ts;
> plot(sinc(t+1/7));
>
> These are neither symmetric nor antisymmetric in the sense you defined,
> and yet this is a linear phase filter, is it not?

No, it's not.

Regards,
Andor

```
```Randy Yates wrote:

> Consider the filter coefficients determined as
>
> function y = test(x)
> %function y = test(x)
> n = [-25 : 25];
> Fs = 1;
> Ts = 1/Fs;
> t = n*Ts;
> plot(sinc(t+1/7));
>
> These are neither symmetric nor antisymmetric in the sense you defined,
> and yet this is a linear phase filter, is it not?

I'm with Andor:

>> X = grpdelay(sinc(t+1/7),1,20);
>> X

X =

24.7190
24.8790
24.9947
24.7873
24.7215
24.9915
24.9908
24.5986
24.7294
25.6582
32.6615
25.6582
24.7294
24.5986
24.9908
24.9915
24.7215
24.7873
24.9947
24.8790

Compare that with:

>> X2 = grpdelay(sinc(t),1,20);
>> X2

X2 =

25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000
25.0000

Ciao,

Peter K.

```
```Richard Owlett wrote:
> Jerry Avins wrote:

...

>> The the differences between the shapes of filters is subtle. If those
>> filters without steps at the ends, I find it difficult to distinguish
>> a Blackman from Nuttall, Blackman-Harris, von Hann, and others. What
>> distinguishing feature of Blackman attracts you?
>
>
> I have a pdf of unknown title ( got saved as Windows.pdf ) written by
> Craig Stuart Sapp <craig@ccrma.stanford.edu> 25 Feb 1997.
>
> I has a collection of various windows and their transforms. The
> particular Blackman window illustrated had a "nice" central lobe and all
> the residual lobes were of "uniform" shape and at least 60 dB down.

Those plots are not the shapes of the windows. Rather, they are the
shapes of the frequency responses obtained by applying the windows to a
filter, not at all what you wrote. Better shapes than any of them (but
not by much) are filters optimized by Parks-McClellan and such. Look up
"windowed sinc".

> *DARN YOU MR. AVINS*
>  You just made me read rather than just look at pretty pictures ;{

:-)

> The plot of the particular Blackman-Harris window had max side lobes
> another 20 dB down, but scale of drawing emphasized the side lobes near
> the central one.
>
> Transform of illustrated Hann window -- too much slop
> Transform of illustrated Hann-Poisson window has a "pleasing shape" with
> less "rejection" off central peak.
>
> I've been "hit over head with 2x4" on another issue.
> What a implications of all these being symmetric about some point.
> Obviously if I'm going to have
> "passband 1 of width a centered at freq b"
> and
> "passband 2 of width y centered at freq z"
> what strange effects will asymmetry have?

Try it and see. Won't ScopeDSP do it for you?

...

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