The importance of crossover steepness
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gberchin
diyAudio Member

Join Date: Feb 2008
Quote:
 Originally Posted by CopperTop If the Gaussian filter's shape is, well, Gaussian, in both the frequency and time domains, I presume that means you can't have a pure Gaussian low-, or high-, pass filter with an arbitrarily long flat frequency response, then a fall-off..? So how is it useful for an audio crossover?
You can vary the width of the Gaussian. The lowpass Gaussian is centered upon 0 Hz (DC), and extends into both positive and negative frequencies (as do all lowpass filters). By varying the width of the Gaussian, you vary the cutoff frequency of the lowpass filter. As the width of the frequency-domain Gaussian increases, the duration of the time-domain Gaussian decreases, and vice-versa.

Quote:
 And why do we have Linkwitz Riley, Bessel, Butterworth etc. (plus your own filters)? Are they, in essence, Gaussian-derived filters?
Each optimizes a different parameter. Linkwitz-Riley crossovers are in-phase at all frequencies, and sum to an allpass filter. Bessel filters have the most linear phase possible for an IIR filter. Butterworth filters have the flattest possible passband for an IIR filter. Elliptic filters have the narrowest possible transition band for a given filter order and ripple. Of all of these, only the Bessel has any relationship to the Gaussian shape.

Quote:
 Are there other super-performing filters that might be named after someone, if only they were to happen upon the right combination of FIR coefficients, or is the whole theory pretty well known by now?
It's pretty well understood. The classical filters -- Butterworth, Bessel, Elliptic, etc., were formulated in the days of analog signal processing. Digital filters can exploit the rules in ways that analog filters cannot, so alternate designs are possible. (Don't conclude from that that digital filters are inherently superior to analog filters -- there are things that analog filters can do that digital filters cannot.)

Quote:
 (Just trying to work out whether digital crossover design is about managing the practical deviation from an 'obvious' perfect theory, or whether there is no obvious perfect crossover filter to start with).
You have just described the difference between mathematics and engineering.

CopperTop
diyAudio Member

Join Date: Feb 2009
Location: UK
Quote:
 Originally Posted by gberchin You can vary the width of the Gaussian. The lowpass Gaussian is centered upon 0 Hz (DC), and extends into both positive and negative frequencies (as do all lowpass filters). By varying the width of the Gaussian, you vary the cutoff frequency of the lowpass filter. As the width of the frequency-domain Gaussian increases, the duration of the time-domain Gaussian decreases, and vice-versa.
Could you elaborate on this a little? As I understand it, there is no (finite) part of the Gaussian shape ('bell curve') that is horizontal and flat, and if I decrease the width to increase the steepness of the fall-off, I must also reduce the cutoff frequency..? How can I have a Gaussian response that is flat up to 2 kHz say, and then falls off steeply?

And what about the corresponding high pass? I could subtract the output of the lowpass filter from the signal, to give me the high pass response, presumably. Would the equivalent filter still have some sort of Gaussian classification, or would it be classified as something else?

gberchin
diyAudio Member

Join Date: Feb 2008
Quote:
 Originally Posted by CopperTop Could you elaborate on this a little? As I understand it, there is no (finite) part of the Gaussian shape ('bell curve') that is horizontal and flat, and if I decrease the width to increase the steepness of the fall-off, I must also reduce the cutoff frequency..? How can I have a Gaussian response that is flat up to 2 kHz say, and then falls off steeply?
You can't. "Flat" is relative, when you are talking about filters. In the case of a Gaussian, "flat" means that it is monotonically decreasing from 0 dB at 0 Hz to -6 dB at a cutoff frequency that you decide. Same definition applies to Linkwitz-Riley. With Bessel or Butterworth, the definition is the same except that you use -3 dB instead of -6 dB. For an elliptic filter, "flat" means "+/- some tolerance".

You can vary the width of your Gaussian lowpass filter, and thus change its cutoff frequency, but you cannot change the steepness of the cutoff. If you attempt to do so, what you end up with will no longer be a Gaussian.

Quote:
 And what about the corresponding high pass? I could subtract the output of the lowpass filter from the signal, to give me the high pass response, presumably. Would the equivalent filter still have some sort of Gaussian classification, or would it be classified as something else?
It would be "one minus Gaussian" in frequency response, or "delta minus Gaussian" in time response. See my paper, referenced in an earlier response. If you cannot find a copy, send me a PM and I'll get you a copy.

 8th July 2012, 08:45 PM #34 steph_tsf   diyAudio Member     Join Date: Mar 2008 Consider the following: S: Sinc (Sinus cardinal) delivers the most selective lowpass when used as FIR coefficients, unfortunately with a lot of preshoot and ringing in time domain. Dirac minus Sinc delivers the most selective highpass when used as FIR coefficient, unfortunately with a lot of preshoot and ringing in time domain. G: Gaussian delivers a quite selective lowpass when used as FIR coefficient, with the advantage of no preshoot and no ringing in time domain. Dirac minus Gaussian FIR only delivers a 2nd-order (24db/octave slope) highpass, which is insufficient. F: We have the so-called "Flat-Top" window. What if we use the "Flat-Top" shape, not as window, but as FIR lowpass coefficients? What are the properties of a Dirac minus "Flat-Top", as complementary highpass? A new lowpass FIR could be computed, combining the three above approaches. What kind of combination? I would suggest: (S at the power of s) * (G at the power of g) * (F at the power of f) With s, g and f as real numbers. Start with s=g=f=1 maybe. Let us define "F6db" the desired -6 dB crossover frequency. S would feature a -3dB cutoff frequency equal to "F6db" * sf G would feature a -3dB cutoff frequency equal to "G3db" * gf F would feature a -3dB cutoff frequency equal to "F3dB" * ff Start with sf=gf=ff=1 maybe. How to iterate the s, g, f, sf, gf, ff values? You would check for a 0.2 dB corridor in the lowpass passband, a 0.2 dB corridor in the highpass passband, a 36dB/octave slope for the lowpass, and a 36dB/octave slope for the highpass. Other constraints? - the FIR would feature 128 taps maximum - quite welcome would be the FIR to feature lots of zero coefficients - the desired FIR "F6dB" could not be arbitrary, but an integer fraction of the sampling frequency, say Fs/12 (4 kHz for Fs=48 kHz) Some relaxes? A "don't care" scheme at frequencies where both the targeted amplitude and the realized amplitude are less than -40 dB.
gberchin
diyAudio Member

Join Date: Feb 2008
Quote:
 Originally Posted by steph_tsf Consider the following:
You appear to be reinventing the wheel. Look up "window-based filter design". It can be applied in two ways: prototype frequency response is convolved with a window for improved impulse response; or, by duality, prototype impulse response is convolved with a window for improved frequency response.

 8th July 2012, 11:10 PM #36 steph_tsf   diyAudio Member     Join Date: Mar 2008 Say Fs = 48 kHz. Say you want the -6dB cutoff to be 4 kHz. Say you want a 36dB/octave slope. You want the FIR lowpass to feature a monotonic amplitude response. You want the FIR lowpass to feature a linear phase. Those design parameters are especially appropriated when designing the FIR in frequency domain. It is like operating a graphic equalizer with equally spaced frequencies from DC to Fs/2. You draw the amplitude response at will, following the targeted amplitude response, and for each frequency point, you set the phase in order to remain phase linear. However, we get an infinity of FIRs implementing the above constraints, as we have said nothing about the transition region: is it Bessel, Butterworth, or anything else? We will now select one and only one FIR in the infinity of available FIRs. We will now focus on the transition band. We will start from a maximally-flat design emulating a phase-linear 36dB/octave Butterworth. We will then assess the complementary highpass, equal to Dirac less the lowpass. If the complementary highpass doesn't exhibit the targeted 36dB/octave slope and a symmetrical transition band (compared to the lowpass), we will soften the FIR lowpass transition band. Then reassess the complementary highpass. And so on, iteratively. There should be rules enabling great success if asking for particular feature sets. Looking at the Philips DSS-930 design, I have the impression that the Fs/12 -6dB point combined to the 36dB slope is one particular, high efficiency feature set. I would like to know more about this. By designing the FIR in frequency domain, you have the temptation of relying on a power of two FIR lenght. This may not be optimal. There is a great probability that the high efficiency feature sets get mathematically coupled to particular FIR lenghts, not fitting into the power of two numbers. The whole thing gets thus complicated, because the exact FIR lenght may be of great importance. Physically, there is major problem with phase-linear FIRs having a power of two lenghts. Physically, you would always design a phase-linear FIR as having an odd amount of coefficients. In a 3 tap FIR, the central coefficient would describe the main energy locus, while the two side coefficients would describe the energy leak over time, before and after the main energy locus. You would add precision to the energy leak description using a 5 tap FIR, then 7 tap FIR, and so on. This makes me say that the Fs/12 -6dB point combined to the 36dB slope in the Philips DSS-930, is a particular feature set leading to a great computing efficiency, but at this stage, we don't know the exact FIR lenght. Philips says a 79 tap FIR lenght for the lowpass, and a 30 tap FIR lenght for the highpass. That's quite bizarre. With different FIR lengths, how could they exactly derive the complementary highpass, from the lowpass? Anyway, we are dealing with medium length FIRs. We don't want to execute them in the frequency domain. We thus need to translate them back in time domain, in an exact way, using the appropriate math. This was about the crossover. Now, if we want to add the driver compensation feature (a Bode Plot inversion), nothing prevents embedding it into the final FIR. If we need a better frequency resolution for compensating the driver, nothing prevents from basing the whole design on a longer FIR, featuring a length that's compatible with the efficiency set we are starting from. I guess that if the high efficiency FIR was N taps, that a (2N+1) FIR lenght or (3N+1) FIR lenght won't harm the efficiency pattern. Possibly, in the Philips DSS-930 design, the 30 tap tweeter FIR managed to do both the highpass crossover, and the tweeter Bode plot inversion. Possibly, what Philips calls a 30 tap FIR, is in reality a 31 tap FIR, as a 31 tap FIR is only requiring 30 memory cells. Possibly, in the Philips DSS-930 design, a 30 tap woofer FIR was too short for embedding the woofer Bode plot inversion. When looking at the woofer unfiltered frequency response featuring a steep extinction above 5 kHz, for sure such woofer was exhibiting a quite long ringing at 4.5 KHz or so, even if such natural ringing was shortened using a dedicated, IIR-based notch. Possibly Philips needed to extend the woofer FIR length to 79 tap (1.65 ms), for encompassing the 4.5 kHz residual ringing untill a decent extinction. The 1.65 ms FIR duration equal 7.4 periods at 4.5 kHz.
steph_tsf
diyAudio Member

Join Date: Mar 2008
Quote:
 Originally Posted by gberchin You appear to be reinventing the wheel. Look up "window-based filter design". It can be applied in two ways: prototype frequency response is convolved with a window for improved impulse response; or, by duality, prototype impulse response is convolved with a window for improved frequency response.
I'm afraid that using this you will never find the required lowpass and complementary highpass pair featuring same 36db/octave slopes. Aren't you?

 8th July 2012, 11:36 PM #38 CopperTop   diyAudio Member   Join Date: Feb 2009 Location: UK @steph and berchin (Still trying to catch up to a fraction of what you know) Something I missed totally earlier was the possibility of the high pass being an exact complement of the low pass in the crossover region, so that the ringing of the woofer is 'neutralised' by the opposite ringing of the tweeter (give or take the discrepancies due to off-axis listening). If this is the case (or can be made the case), is the issue of overshoot one of a waste of power and/or needless displacement of the driver?
gberchin
diyAudio Member

Join Date: Feb 2008
Quote:
 Originally Posted by steph_tsf I'm afraid that using this you will never find the required lowpass and complementary highpass pair featuring same 36db/octave slopes. Aren't you?
It's just another design constraint.

gberchin
diyAudio Member

Join Date: Feb 2008
Quote:
 Originally Posted by CopperTop @steph and berchin Something I missed totally earlier was the possibility of the high pass being an exact complement of the low pass in the crossover region, so that the ringing of the woofer is 'neutralised' by the opposite ringing of the tweeter (give or take the discrepancies due to off-axis listening). If this is the case (or can be made the case), is the issue of overshoot one of a waste of power and/or needless displacement of the driver?
For a given "squared area under the curve" in the frequency domain, the "squared area under the curve" in the time domain is always the same [1]. Thus, overshoot and ringing are really just moving that constant squared area around in time.

The real issue is the fact that, even though the lowpass and highpass time-domain responses may cancel mathematically, they do not necessarily cancel acoustically. So your first thought was correct.

[1] Parseval's Relation

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