| Author |
Message |
Ron N.
Guest
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 Posted:
Wed Dec 14, 2005 1:16 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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Richard Owlett wrote:
| Quote: | Picking numbers *AT RANDOM*
Are you saying that a "low pass" filter with an overall passband of 10
kHz with "peaks" at 2 kHz, 3.14 kHz, and 7.9631 kHz with relative
amplitudes of 1, 2.91234, and 1.167 *could not* be linear phase?
|
Three peaks implies a multiple number of poles and zeros,
which can be reflected around (left or right half plane) for
multiple phase relationships, none of which might be linear,
even though for identical frequency response filters.
Linear phase implies that if you know the delay of one
frequency (the time it takes a zero crossing at the input
to appear at the output), it's the same for all frequencies
(realistically, only within the passband of the filter).
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
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Jerry Avins
Guest
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| Posted:
Wed Dec 14, 2005 7:54 am Post subject:
Crossover networks. Can someone recall the name? |
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Vladimir Vassilevsky wrote:
| Quote: |
Jerry Avins wrote:
A linear-phase phono equalizer completely louses up the transient
response. A "perfect" linear-phase speaker crossover often sounds much
worse that the minimum-phase analog approximation that it replaced.
Don't look at the transient response and linear phase will sound just as
good as the minimal phase :) We are entering the area of the holy wars
of the blunt-pointed vs sharp pointed. From my experience the only
observable difference results from the implementation issues like
overflows, loss of accuracy, group delay or frequency response mismatch
and such.
BTW, what do you think about Bessel filters, which are the minimum phase
approximations of the linear phase?
|
Bessel filters are good for many applications, but not for crossovers.
Their approximation to linear phase breaks down at the crossover
frequency, where it matters most. As far as I know, Bessel filters are
low-pass. The filters produced by the usual low- to high-pass
transformation approximate linear phase very poorly. It's a mess.
There is an excellent crossover -- Linkwitz-Riley -- that consists of
cascaded second-order Butterworth (Butterworth^2) sections. When used
with appropriate offset, it allows a very wide angle of good
performance. Most crossovers are evaluated only on axis, hardly a
reasonable basis, especially in a theater.
http://www.rane.com/note147.html http://www.rane.com/note160.html
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Randy Yates
Guest
|
| Posted:
Wed Dec 14, 2005 7:58 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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Richard Owlett <rowlett@atlascomm.net> writes:
| Quote: | 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:
http://groups.google.com/group/comp.dsp/msg/9be6c8f2861d1d3a
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?
je ne comprend pas ;]
Can you give me code understood by Scilab so I can know what point you
wish to make ;[
|
Sorry, Richard, I don't know Scilab. You can download Octave (a Matlab
clone) and try the code there.
You should be able to construct the basic response in your head
anyway. It's just a since function, shifted left by 1/7 of a second
(but still sampled at n*Ts intervals). Since I set my sample rate to 1
Hz (Ts = 1 second), this means the 1/7 second is a fraction of a
sample delay, so the sinc() function, which is usually nice and
symmetric about 0, is no longer symmetric.
--
% Randy Yates % "I met someone who looks alot like you,
%% Fuquay-Varina, NC % she does the things you do,
%%% 919-577-9882 % but she is an IBM."
%%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO
http://home.earthlink.net/~yatescr |
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Randy Yates
Guest
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| Posted:
Wed Dec 14, 2005 8:00 am Post subject:
Re: questions raised by reading and thinking with possibly m |
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Jerry Avins <jya@ieee.org> writes:
| Quote: | Randy Yates wrote:
"Peter K." <p.kootsookos@iolfree.ie> writes:
[...]
I'm with Andor:
X = grpdelay(sinc(t+1/7),1,20);
X
X =
24.7190
24.8790
[...]
Those ripples are apparently from truncation. I haven't
proved it, but the longer you extend the sequence, the
smaller they become. As I stated to Andor, I think b[n] = sinc(n +
1/7)
is linear phase, and doesn't match the symmetric/antisymmetric
requirement. Think about it: If you start with a linear phase
impulse response,
then delay it by a fractional sample amount, it's still linear phase,
but it ain't necessarily symmetric anymore.
The symmetry condition is sufficient, not necessary.
I think I stated earlier, in an oblique way tailored to Richard, that
pure delay added to a linear-phase transfer function won't impair the
phase linearity.
|
Oh? I missed that, Jerry. Then I follow in your footsteps.
--
% 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
http://home.earthlink.net/~yatescr |
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Jerry Avins
Guest
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| Posted:
Wed Dec 14, 2005 9:15 am Post subject:
Re: Crossover networks. Can someone recall the name? |
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Greg Berchin wrote:
...
| Quote: | Linkwitz-Riley crossovers are particular cases of sum-to-allpass
crossovers, where the responses of the lowpass and highpass sections sum
to an allpass characteristic (constant amplitude, nonlinear phase).
Second-order Linkwitz-Riley crossovers sum to a first-order allpass
filter; fourth-order Linkwitz-Riley crossovers sum to a second-order
allpass filter, and so on. An additional attribute is that the lowpass
section and the highpass section are exactly in-phase with each other at
all frequencies.
|
On axis. Their outstanding property is superior (but hardly perfect)
performance off axis.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Greg Berchin
Guest
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| Posted:
Wed Dec 14, 2005 9:15 am Post subject:
Re: Crossover networks. Can someone recall the name? |
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On Tue, 13 Dec 2005 22:29:32 -0500, Jerry Avins <jya@ieee.org> wrote:
| Quote: | On axis. Their outstanding property is superior (but hardly perfect)
performance off axis.
|
Correct. For perfect off-axis response, the low frequency driver and
the high frequency driver would have to be coincident. I think of that
not as a limitation of the crossover, but of the loudspeaker.
Greg |
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Jerry Avins
Guest
|
| Posted:
Wed Dec 14, 2005 9:16 am Post subject:
Re: Crossover networks. Can someone recall the name? |
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Greg Berchin wrote:
| Quote: | On Tue, 13 Dec 2005 22:29:32 -0500, Jerry Avins <jya@ieee.org> wrote:
On axis. Their outstanding property is superior (but hardly perfect)
performance off axis.
Correct. For perfect off-axis response, the low frequency driver and
the high frequency driver would have to be coincident. I think of that
not as a limitation of the crossover, but of the loudspeaker.
|
Right. Love my Tannoys! By rights, I should delay the woofer because the
tweeter horn driver is behind it, but on;y by the length of the pole piece.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Greg Berchin
Guest
|
| Posted:
Wed Dec 14, 2005 9:16 am Post subject:
Re: Crossover networks. Can someone recall the name? |
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|
On Tue, 13 Dec 2005 20:54:01 -0500, Jerry Avins <jya@ieee.org> wrote:
| Quote: | Bessel filters are good for many applications, but not for crossovers.
|
When used in the "traditional" fashion, with a Bessel LPF and a Bessel
HPF, this is true. But there's more than one way to skin a cat.
| Quote: | Their approximation to linear phase breaks down at the crossover
frequency, where it matters most. As far as I know, Bessel filters are
low-pass. The filters produced by the usual low- to high-pass
transformation approximate linear phase very poorly. It's a mess.
|
What happens is that the Bessel LPF maximally-flat group-delay property,
referenced to 0 Hz, becomes referenced to infinity Hz as a result of the
lowpass-to-highpass transformation. This creates a HPF with really
nasty transient response.
But when a Bessel LPF is combined with a matched-delay subtractive
highpass filter, the results are superb: "Perfect Reconstruction
Digital Crossover Exhibiting Optimum Time Domain Transient Response in
All Bands", AES 107th Convention, September 1999, Preprint 5010.
| Quote: | There is an excellent crossover -- Linkwitz-Riley -- that consists of
cascaded second-order Butterworth (Butterworth^2) sections. When used
with appropriate offset, it allows a very wide angle of good
performance. Most crossovers are evaluated only on axis, hardly a
reasonable basis, especially in a theater.
|
Linkwitz-Riley crossovers are particular cases of sum-to-allpass
crossovers, where the responses of the lowpass and highpass sections sum
to an allpass characteristic (constant amplitude, nonlinear phase).
Second-order Linkwitz-Riley crossovers sum to a first-order allpass
filter; fourth-order Linkwitz-Riley crossovers sum to a second-order
allpass filter, and so on. An additional attribute is that the lowpass
section and the highpass section are exactly in-phase with each other at
all frequencies.
Greg |
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Vladimir Vassilevsky
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
Greg Berchin wrote:
| Quote: | Let's take a typical subwoofer LPF with -3dB/100Hz and -24dB/200Hz.
The FIR version will look like exp(-x^2) function without any ripples.
... assuming that it was designed as an approximation to a Gaussian
response.
|
No. Assumming the reasonable rolloff speed and passband flatness.
| Quote: | However, consider how many taps a 100Hz FIR Gaussian filter
will require -- for 48kHz sampling with 24-bit coefficients, it's on the
order of 3000.
|
Brute force approach :(
First, for FIR you don't need 24 bit coefficients - a mere 10 bits is
sufficient.
Second, I would do such filter as a part of multirate filterbank. If
your goal is a subwoofer, there is no need to burn at the full speed of
48kHz.
| Quote: | Design for a flatter passband, and the number of taps
(and, consequently, the duration of the impulse response) increases.
|
If the design spec is unreasonable, then the results will be
unreasonable. It does not matter if it is FIR or IIR.
| Quote: | The IIR version could be 4th order Butterworth.
Or 4th order Bessel, which would be a pretty good approximation for the
Gaussian, up to the cutoff frequency. And it could be implemented with
a couple of biquads.
|
To have reasonable audio performance, those biquads should use 32x32 =
64 bit multiplications at least, as well as the noise shaping at
truncation of 64 to 32 bits. This relates to the straightforward 48kHz
implementation.
Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com |
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Greg Berchin
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
On Wed, 14 Dec 2005 16:44:44 GMT, Vladimir Vassilevsky
<antispam_bogus@hotmail.com> wrote:
| Quote: | ... assuming that it was designed as an approximation to a Gaussian
response.
No. Assumming the reasonable rolloff speed and passband flatness.
|
You mentioned exp(-x^2) specifically. That's a Gaussian.
| Quote: | However, consider how many taps a 100Hz FIR Gaussian filter
will require -- for 48kHz sampling with 24-bit coefficients, it's on the
order of 3000.
Brute force approach :(
First, for FIR you don't need 24 bit coefficients - a mere 10 bits is
sufficient.
|
If you use 10 bit coefficients and eliminate all coefficients whose
magnitude falls below that 10 bit limit, then you have effectively
applied a rectangular window to your time domain convolution. You can
do some error control on the coefficient quantizations, but 14 bits of
quantization error is a lot to deal with.
| Quote: | Second, I would do such filter as a part of multirate filterbank. If
your goal is a subwoofer, there is no need to burn at the full speed of
48kHz.
|
Ah! Now I see what you're saying. Your "mere 10 bits is sufficient"
comment comes in the context of a multirate filterbank approach.
The multirate filterbank approach has some pitfalls. Adjacent banks
overlap, and when the sample rate is reduced in each bank, aliasing
occurs. The structure of the filterbank guarantees that the aliasing
products mathematically cancel in the reconstruction process. However,
mathematical reconstruction in a processor and acoustical reconstruction
in a loudspeaker system can be two very different things.
| Quote: | To have reasonable audio performance, those biquads should use 32x32 =
64 bit multiplications at least, as well as the noise shaping at
truncation of 64 to 32 bits. This relates to the straightforward 48kHz
implementation.
|
I respectfully disagree. I have implemented this in the DSP56K with 24
bit data, 24 bit coefficients, 56 bit accumulator, and no noise shaping.
The performance is quite "reasonable". Admittedly, though, a 100Hz
filter in a 48kHz context with 24 bit coefficients and 24 bit data is
approaching the limits of precision for a direct-form biquad
implementation.
Greg |
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Greg Berchin
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
On Wed, 14 Dec 2005 15:19:23 GMT, Al Clark <dsp@danvillesignal.com>
wrote:
| Quote: | Imagine someone hits a drum. You can hear the drum before he hits it.
If the filters have a fairly regular passband ripple (kind of
sinusoidal), such as might be expected from a PM filter, then the pre-
echo will be concentrated. You can reduce this effect, by using a filter
with a more "random" like passband ripple. BTW: (I learned all this from
a discussion with rbj).
|
Peter Craven published an interesting article about this (in the context
of antialias filters) last year: "Antialias Filters and System
Transient Response at High Sample Rates"; Journal of the Audio
Engineering Society, Volume 52, Number 3, March 2004.
Greg |
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Greg Berchin
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
On Wed, 14 Dec 2005 15:53:45 GMT, Vladimir Vassilevsky
<antispam_bogus@hotmail.com> wrote:
| Quote: | I think this is a misconception.
Let's take a typical subwoofer LPF with -3dB/100Hz and -24dB/200Hz.
The FIR version will look like exp(-x^2) function without any ripples.
|
.... assuming that it was designed as an approximation to a Gaussian
response. However, consider how many taps a 100Hz FIR Gaussian filter
will require -- for 48kHz sampling with 24-bit coefficients, it's on the
order of 3000. Design for a flatter passband, and the number of taps
(and, consequently, the duration of the impulse response) increases.
| Quote: | The IIR version could be 4th order Butterworth.
|
Or 4th order Bessel, which would be a pretty good approximation for the
Gaussian, up to the cutoff frequency. And it could be implemented with
a couple of biquads.
Greg |
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Vladimir Vassilevsky
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
Al Clark wrote:
| Quote: | I think the main criticism with Linear Phase FIR filters is that you can
often hear pre-echos caused by the fact that the impulse response is
symmetrical around the center.
|
I think this is a misconception.
Let's take a typical subwoofer LPF with -3dB/100Hz and -24dB/200Hz.
The FIR version will look like exp(-x^2) function without any ripples.
The IIR version could be 4th order Butterworth.
Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com |
|
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Vladimir Vassilevsky
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
Jerry Avins wrote:
| Quote: | A linear-phase phono equalizer completely louses up the transient
response.
Don't look at the transient response and linear phase will sound just
as good as the minimal phase :)
|
| Quote: |
BTW, what do you think about Bessel filters, which are the minimum
phase approximations of the linear phase?
Bessel filters are good for many applications, but not for crossovers.
Their approximation to linear phase breaks down at the crossover
frequency, where it matters most. As far as I know, Bessel filters are
low-pass. The filters produced by the usual low- to high-pass
transformation approximate linear phase very poorly. It's a mess.
|
Of course the linear phase will not remain after the nonlinear 1/s
transformation. There is another issue with Bessels: the design should
be done with the impulse invariant methods, not with the usual BLT.
However I know some people who are saying that Bessel is the best for
Xovers.
| Quote: | There is an excellent crossover -- Linkwitz-Riley -- that consists of
cascaded second-order Butterworth (Butterworth^2) sections. When used
with appropriate offset, it allows a very wide angle of good
performance.
|
Yes, LR is a pretty elegant idea. BTW, did they patented it?
| Quote: | Most crossovers are evaluated only on axis, hardly a
reasonable basis, especially in a theater.
|
In the real world, the difference in the speaker locations and responses
is much more significant then the relatively small difference due to
the Xover filter type. The typical Butterworth filter works just as good
as any other filter.
Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com |
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Al Clark
Guest
|
| Posted:
Wed Dec 14, 2005 5:16 pm Post subject:
Re: Crossover networks. Can someone recall the name? |
|
|
Jerry Avins <jya@ieee.org> wrote in
news:M8GdnUu2qJ2i5gLenZ2dnUVZ_sSdnZ2d@rcn.net:
| Quote: | Vladimir Vassilevsky wrote:
Jerry Avins wrote:
A linear-phase phono equalizer completely louses up the transient
response. A "perfect" linear-phase speaker crossover often sounds
much worse that the minimum-phase analog approximation that it
replaced.
|
I think the main criticism with Linear Phase FIR filters is that you can
often hear pre-echos caused by the fact that the impulse response is
symmetrical around the center.
Imagine someone hits a drum. You can hear the drum before he hits it.
If the filters have a fairly regular passband ripple (kind of
sinusoidal), such as might be expected from a PM filter, then the pre-
echo will be concentrated. You can reduce this effect, by using a filter
with a more "random" like passband ripple. BTW: (I learned all this from
a discussion with rbj).
Linkwitz-Riley filters are popular because it is important for the phase
response to be continuous. You still need to correct for the differences
in the acoustical centers of each driver (time alignment)
| Quote: |
Don't look at the transient response and linear phase will sound just
as good as the minimal phase :) We are entering the area of the holy
wars of the blunt-pointed vs sharp pointed. From my experience the
only observable difference results from the implementation issues
like overflows, loss of accuracy, group delay or frequency response
mismatch and such.
BTW, what do you think about Bessel filters, which are the minimum
phase approximations of the linear phase?
Bessel filters are good for many applications, but not for crossovers.
Their approximation to linear phase breaks down at the crossover
frequency, where it matters most. As far as I know, Bessel filters are
low-pass. The filters produced by the usual low- to high-pass
transformation approximate linear phase very poorly. It's a mess.
There is an excellent crossover -- Linkwitz-Riley -- that consists of
cascaded second-order Butterworth (Butterworth^2) sections. When used
with appropriate offset, it allows a very wide angle of good
performance. Most crossovers are evaluated only on axis, hardly a
reasonable basis, especially in a theater.
http://www.rane.com/note147.html http://www.rane.com/note160.html
Jerry
|
--
Al Clark
Danville Signal Processing, Inc.
--------------------------------------------------------------------
Purveyors of Fine DSP Hardware and other Cool Stuff
Available at http://www.danvillesignal.com |
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