Importance of phase in Audio

Hello all,

    i am designing a low pass filter and high pass filter to pass my
audio samples. in matlab i checked out both butterworth and
chebyshev(cheby2) 3rd order filter.
   I have observed there is a lot of differance in the phase plot of
both the filters.
   Now my query went to what actuall effect of phase in audio? please
anybody help me in solving this.

  Looking for the reply,

  With warm regards,
  -Prasad

mailprassi@gmail.com wrote:
...
>     i am designing a low pass filter and high pass filter to pass my
> audio samples. in matlab i checked out both butterworth and
> chebyshev(cheby2) 3rd order filter.
>    I have observed there is a lot of differance in the phase plot of
> both the filters.
>    Now my query went to what actual effect of phase in audio? please
> anybody help me in solving this.

there is some basic stuff that i think everybody agrees with and then
some other stuff that becomes more controversial.  a few years ago, the
JAES published a letter i wrote where i was a bit critical of Andrew
Horner, who was writing a lot of papers on Wavetable Synthesis (not to
be confused with the PCM sample playback that the marketing dept. of
some soundcard chip manufacturers like to call "wavetable synthesis")
with many good parts in some the papers, who, without any real
justification, was just setting the phase components of all of the
harmonics of his wavetable tone to 0.  i thought (and still think) that
it was both unnecessary to zero the phase of the harmonics in
wavetable, changed the waveshape and potentially made for discernable
differences.

here is the more uncontroversial (for single mono-channel sounds, for
stereo if there is phase shift applied to one channel that is different
than what is applied to the other, you will notice a difference):

1.  a delay line is a sorta "all-pass filter" (APF) that causes a
linear phase shift (with respect to frequency) which gets translated to
constant "phase delay" and constant "group delay" (w.r.t. frequency)
and the two are equal to each other.  the transfer function in the
z-plane is:

    H(z)  =  z^(-N)

where N is the number of samples delay.  the continuous-time (s-plane)
transfer function is:

    H(s)  =  e^(-s*N*T)

where 1/T is the sampling frequency.

if the delay is short enough that you cannot detect the delay, the
phase shift is also undetectable.  if the delay is long enough for you
to detect it (that is somehow you noticed that the audio is delayed)
then the sole effect of the phase shift is nothing but delaying the
sound.  accounting for the delay, there would be no additional effect.
i think this is obvious.

2. there is another kind of APF which is not phase linear (nor, as a
consequence, constant phase delay or group delay w.r.t. frequency).
these APFs have a z-plane transfer function of:

    H(z)  =  (1- p*z^N) / (z^N - p)  =  (z^(-N) - p) / (1 - p*z^(-N))

which can be made out of a single delay line (of length N samples) and
some feedback, scaling (p can be anything so that |p| < 1)), and
summers.  if N=1, it's a simple 1st-order APF and p is equal to the
single pole while q=1/p is the reflected zero.  if N>1, then there are
more poles and zeros (ya gotta factor the S.O.B.) but all the poles
will have zeros reflected about the unit circle with the mapping q=1/p.
 if you were to represent this as a continuous-time system (s-plane),
the transfer function would be

    H(s)  =  (e^(-s*N*T) - p) / (1 - p*e^(-s*N*T)) .

so the question might be, what effect of this APF on phase will be
detectable?

if |p| is close to 1 and the length of the delay is perceptually long
(you will hear *both* the recursively delayed input signal and the
first undelayed input scaled by -p) say N*T = 1/2 second, there will
clearly be a perceived effect.  it will sound like some echo box or, as
the delay is shortened, a cheap reverberator.  but there will be values
of p and N*T that the effect of the APF becomes less noticable or not
noticeable.  if |p| << 1 or if N*T <<< 1 second (notice the 3 "<<<"),
you might not notice the effect on any kind of signal.

even with these APFs, the discernability of phase shift will also
depend on the nature of the input signal.  in the case of the a solid
tone that has been on forever (no transients), i don't think any of us
will detect an effect of any APF, if there are no amplitude
non-linearities in the signal chain (make sure speakers/amps are not
overdriven).  an example to try with MATLAB (or whatever) is comparing
this bandlimited square wave:

     x(t) = cos(w0*t) - (1/3)*cos(3*w0*t) + (1/5)*cos(5*w0*t) -
(1/7)*cos(7*w0*t) ...

(stop at some finite term less than Nyquist) to this waveform

     y(t) = cos(w0*t) + (1/3)*cos(3*w0*t) + (1/5)*cos(5*w0*t) +
(1/7)*cos(7*w0*t) ...

clearly x(t) and y(t) do not have similar waveshape.  y(t) is not a
simple delayed version of x(t).  x(t) approaches a square wave (with
finite number of terms) and y(t) becomes very spikey (which is why i
put in the non-linearity caveat).  if your speakers or amp or anything
else in the signal chain do not clip those spikes in y(t), and if you
make sure that the number of terms is low enough that the highest
frequency term is well below Nyquist, i doubt any of us will hear a
difference between x(t) and y(t).  i might have posted a MATLAB program
demonstrating this in the past (can't find it now).

Now that being said, even in cases where absolute phase cannot be
heard, the *change* in phase (with respect to time) causes a frequency
detuning (equal to that time rate of change) and if that detuning is
sufficiently large (that's another whole perceptual thing that needs to
be explored in a decent scholarly sense), you will hear a difference.

also, that APF thought experiment above (with sufficiently large |p|
and N*T) will show an effect of phase shift on *transient* and changing
signals (not constant tones that were on forever).  but my complaint
with Horner in the AES letter is that these transient (quasi-periodic)
signals are the interesting ones we are trying to synthesize with
Wavetable Synthesis.

so, if you apply identical APFs to both channels of your stereo system,
you will probably not hear much difference unless both the feedback
coefficient, p, and delay length, N*T, get sufficiently large and that
these APFs are applied to dynamic music/audio.  in some special cases,
you can find that the APFs have no discernable effect.

r b-j

robert bristow-johnson wrote:

> 
> mailprassi@gmail.com wrote:
> ...
>>     i am designing a low pass filter and high pass filter to pass my
>> audio samples. in matlab i checked out both butterworth and
>> chebyshev(cheby2) 3rd order filter.
>>    I have observed there is a lot of differance in the phase plot of
>> both the filters.
>>    Now my query went to what actual effect of phase in audio? please
>> anybody help me in solving this.
> 
* snip *
> 
> so, if you apply identical APFs to both channels of your stereo system,
> you will probably not hear much difference unless both the feedback
> coefficient, p, and delay length, N*T, get sufficiently large and that
> these APFs are applied to dynamic music/audio.  in some special cases,
> you can find that the APFs have no discernable effect.
> 

It seems that some are able to hear a pre-echo in transient sounds (e.g.,
percussions) filtered with certain linear-phase FIR filters. Thus, not only
amplitude response and phase response affect the sound, but also the shape
of the impulse response. Of course impulse response is defined by amplitude
and phase response (and vice versa), however, pre-echo is explained more
intuitively by the shape of the impulse response.

Then again, I have never heard this phenomenom (or even tried to hear) and
some claim that the pre-echo shouldn't be audible (in LP-FIR EQ
applications). Audibility obviously depends on the impulse response.

What I'm trying to say, is that when it comes to high-end audio, listener is
the judge of good sound. Linear phase isn't everything, some even consider
it bad. To answer the OPs question, there is likely no audible effect on
sound caused by the non-linear phase. Nonlinear phase causes different
parts of the signal to delay unequally. When the phase is sufficiently
close to linear, this effect isn't audible. However, in extreme cases, two
consecutive bandlimited pulses can even change places when filtered (due to
delay of one pulse w.r.t. other).

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Prasad wrote:

> Hello all,
>
>     i am designing a low pass filter and high pass filter to pass my
> audio samples. in matlab i checked out both butterworth and
> chebyshev(cheby2) 3rd order filter.
>    I have observed there is a lot of differance in the phase plot of
> both the filters.
>    Now my query went to what actuall effect of phase in audio? please
> anybody help me in solving this.

I got some interesting articles by googling for "is phase audible".

Andor wrote:

>
> I got some interesting articles by googling for "is phase audible".

"article_s_" (plural)?  i got one hit.  some guy's powerpoint in pdf
about watermarking and hiding bits:

    When is phase audible?
      =B7 Sometimes ... in certain compound tones with frequency
        components lying within a critical band

    s(t) =3D A*[ (1/2)*cos((w1-w2 )*t)
                              + cos(w1*t + phi)
                                    + (1/2)*cos((w1+w2 )*t) ]


that's not the only case.  but it's another good special "pathological"
case.

i invite anyone with MATLAB to run this program (or translate it to
Mathcad or Octave or Python or Mathematica or whatever).  the waveform
transistions in such a way that it looks pathologically different, but
sounds the same.  especially if max_harmonic is about 10 or 15.

___________________

%
%   square_phase.m
%
%   a test to see if we can really hear phase changes
%   in the harmonics of a Nyquist limited square wave.
%
%   (c) 2004 rbj@audioimagination.com
%

if ~exist('Fs')
    Fs =3D 44100                      % sample rate, Hz
end

if ~exist('f0')
    f0 =3D 110.25                     % fundamental freq, Hz
end

if ~exist('tone_duration')
    tone_duration =3D 2.0             % seconds
end

if ~exist('change_rate')
    change_rate =3D 1.0               % Hz
end

if ~exist('max_harmonic')
    max_harmonic =3D floor((Fs/2)/f0) - 1
end

if ~exist('amplitude_factor')
    amplitude_factor =3D 0.25   % this just keeps things from clipping
end

if ~exist('outFile')
    outFile =3D 'square_phase.wav'
end

                            % make sure we don't uber-Nyquist anything
max_harmonic =3D min(max_harmonic, floor((Fs/2)/f0)-1);

t =3D linspace((-1/4)/f0, tone_duration-(1/4)/f0, Fs*tone_duration+1);

detune =3D change_rate;

x =3D cos(2*pi*f0*t);                  % start with 1st harmonic

n =3D 3;                               % continue with 3rd harmonic
while (n <=3D max_harmonic)
                     % lessee if it's an "even" or "odd" term
    if ((n-1) =3D=3D 4*floor((n-1)/4))
        x =3D x + (1/n)*cos(2*pi*n*f0*t);
     else
        x =3D x - (1/n)*cos(2*pi*(n*f0+detune)*t);
        detune =3D -detune;     % comment this line in an see some
    end                        % funky intermediate waveforms
    n =3D n + 2;                   % continue with next odd harmonic
end

x =3D amplitude_factor*x;

% x =3D sin(pi/2*x);               % toss in a little soft clipping

plot(t, x);                      % see
sound(x, Fs);                    % hear
wavwrite(x, Fs, outFile);        % remember

___________________

if you run this under ideal linear playback conditions, it's unlikely
you'll be able to hear the change in waveshape.  if there is a slight
non-linearity in between, you may very well.  i have an ongoing
disagreement with Andrew Horner who has just published, in the JAES,
his umpteenth paper about Wavetable Synthesis where he just assumes, at
the very beginning, that phase
is inaudible and he can set phase to whatever he wants.=20

r b-j

ooops.

robert bristow-johnson wrote:

...
> if you run this under ideal linear playback conditions, it's unlikely
> you'll be able to hear the change in waveshape.  if there is a slight
> non-linearity in between, you may very well.  i have an ongoing
> disagreement with Andrew Horner who has just published, in the JAES,
> his umpteenth paper about Wavetable Synthesis

this was copied from an earlier post of mine and i didn't read it
enough to see it contained dated information.  Andrew published that
paper a couple of years ago.  he may have published continuations to it
since, but i can't remember.

> where he just assumes, at
> the very beginning, that phase is inaudible and he can set phase to whatever he wants.
> 
> r b-j

robert bristow-johnson wrote:
> Andor wrote:
>
> >
> > I got some interesting articles by googling for "is phase audible".
>
> "article_s_" (plural)?  i got one hit.  some guy's powerpoint in pdf
> about watermarking and hiding bits:

Don't know which Google you used, mine revealed this:

http://www.google.ch/search?hl=de&q=is+phase+audible&meta=

The second hit seemed quite relevant. There's more further down.

Regards,
Andor

Andor wrote:
> robert bristow-johnson wrote:
> > Andor wrote:
> >
> > >
> > > I got some interesting articles by googling for "is phase audible".
> >
> > "article_s_" (plural)?  i got one hit.  some guy's powerpoint in pdf
> > about watermarking and hiding bits:
>
> Don't know which Google you used,

i used quotes (which i mistakenly thought you did).

> mine revealed this:
>
> http://www.google.ch/search?hl=de&q=is+phase+audible&meta=
>
> The second hit seemed quite relevant.

ya.

> There's more further down.

i'm wondering - did anyone run my MATLAB program?  you can use the
axis() command to zoom in and look at the waveform.

r b-j

robert bristow-johnson wrote:
...
> i'm wondering - did anyone run my MATLAB program?  you can use the
> axis() command to zoom in and look at the waveform.

Yes. A normal square wave I get with this code:

%        x = x - (1/n)*cos(2*pi*(n*f0+detune)*t);
        x = x - (1/n)*cos(2*pi*(n*f0)*t);

The wave that your original code

        x = x - (1/n)*cos(2*pi*(n*f0+detune)*t);
%        x = x - (1/n)*cos(2*pi*(n*f0)*t);

generates sounds like a flanged version of the first. Very audible
difference.

Regards,
Andor

Andor wrote:
> robert bristow-johnson wrote:
> ...
> > i'm wondering - did anyone run my MATLAB program?  you can use the
> > axis() command to zoom in and look at the waveform.
>
> Yes. A normal square wave I get with this code:
>
> %        x = x - (1/n)*cos(2*pi*(n*f0+detune)*t);
>         x = x - (1/n)*cos(2*pi*(n*f0)*t);
>
> The wave that your original code
>
>         x = x - (1/n)*cos(2*pi*(n*f0+detune)*t);
> %        x = x - (1/n)*cos(2*pi*(n*f0)*t);
>
> generates sounds like a flanged version of the first. Very audible
> difference.

try it with a reduced number of harmonics.  maybe with max_harmonic =
10.

r b-j