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questions raised by reading and thinking with possibly missing background

Started by Richard Owlett December 10, 2005
Jerry Avins <jya@ieee.org> writes:

> 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
On Tue, 13 Dec 2005 20:54:01 -0500, Jerry Avins <jya@ieee.org> wrote:

>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.
>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.
>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
Greg Berchin wrote:

   ...

> 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. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
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. Greg
Greg Berchin wrote:
> 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. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins <jya@ieee.org> wrote in
news:M8GdnUu2qJ2i5gLenZ2dnUVZ_sSdnZ2d@rcn.net: 

> 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)
>> >> 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

Jerry Avins wrote:

>>> 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 :)
>> >> 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.
> 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?
> 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.
> > http://www.rane.com/note147.html http://www.rane.com/note160.html > > Jerry
Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com

Al Clark wrote:

> 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
On Wed, 14 Dec 2005 15:53:45 GMT, Vladimir Vassilevsky
<antispam_bogus@hotmail.com> wrote:

>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.
>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
On Wed, 14 Dec 2005 15:19:23 GMT, Al Clark <dsp@danvillesignal.com>
wrote:

>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