Self's book on crossovers has a brief reference to Gaussian filters but he doesn't seem (or claim) to know much about them.
This has piqued my interest, it's very educational to try to understand the difficult cases.
What I can find is mostly related to optical filters and not easily applicable to analog speaker crossovers.
I can see how it would work as a FIR filter in DSP, but not how to translate this.
The Gaussian is not a rational function and I don't see how to determine the poles to make an approximation.
I suspect the filter coefficients should be symmetrical for any order, as also true for L/R.
I think that in the second order case, the Gaussian filter reduces to a Linkwitz/Riley.
I don't know about the fourth order.
Does anyone here understand this or have a reference?
David
This has piqued my interest, it's very educational to try to understand the difficult cases.
What I can find is mostly related to optical filters and not easily applicable to analog speaker crossovers.
I can see how it would work as a FIR filter in DSP, but not how to translate this.
The Gaussian is not a rational function and I don't see how to determine the poles to make an approximation.
I suspect the filter coefficients should be symmetrical for any order, as also true for L/R.
I think that in the second order case, the Gaussian filter reduces to a Linkwitz/Riley.
I don't know about the fourth order.
Does anyone here understand this or have a reference?
David
Are you using a Gaussian filter as your crossover or for testing your audio circuit? Usually the Gaussian filter will not have a very wide bandwidth, even across the audio frequencies. I imagine Self was trying to explain how a Gaussian filter could be used to test an impulse response, as the sharper or more narrow the curve, the more "harsh" the sound will be.
I haven't seen Gaussian used as a frequency filter. Gaussian is usually used in the context of a time window or, as you say, is popular in imaging as a smooth blurring function.
Certainly an FIR filter following the Gaussian shape can be defined. An IIR filter could be made (as with any FIR filter) by using the FIR coefficients on the feed forward side and nothing on the feedback side. Otherwise you are into the usual curve fitting approaches to get an approximation in, say, second order sections.
The typical Gaussian shape would represent a low pass filter. I'm not sure what the high pass counterpart would be.
Viewed as a time window, the wiki page was pretty good and gave a lot of frequency response equivalents of the standard time windows.
Window function - Wikipedia, the free encyclopedia
The Fourier Transform of a Gaussian window is another Gaussian, but they show the log magnitude FFT so you can see the side lobes.
It all sounds a bit like Bessel filters which people get excited about from time to time.
Good Luck,
David S.
Certainly an FIR filter following the Gaussian shape can be defined. An IIR filter could be made (as with any FIR filter) by using the FIR coefficients on the feed forward side and nothing on the feedback side. Otherwise you are into the usual curve fitting approaches to get an approximation in, say, second order sections.
The typical Gaussian shape would represent a low pass filter. I'm not sure what the high pass counterpart would be.
Viewed as a time window, the wiki page was pretty good and gave a lot of frequency response equivalents of the standard time windows.
Window function - Wikipedia, the free encyclopedia
The Fourier Transform of a Gaussian window is another Gaussian, but they show the log magnitude FFT so you can see the side lobes.
It all sounds a bit like Bessel filters which people get excited about from time to time.
Good Luck,
David S.
Just curious, you don't seem to be too excited of Bessel? Would you care to explain🙂 I have the option to try L/R, Butterworth and Bessel in my DSP. I get different results of course, but can't say the other one is better?! It all depends on where and how it's implemented, or?
Peter
Nothing wrong with Bessel filters. The low pass is inherently linear phase or constant group delay. The high pass is not, so the combination isn't much better than other options.
Rane does a nice write up here and they also reference Lipshitz and Vanderkooy
A Bessel Filter Crossover, and Its Relation to Others
"Bessels are historically low-pass or all-pass. A crossover however requires a separate high-pass, and this needs to be derived from the low-pass. There are different ways to derive a high-pass from a low-pass, but here we discuss a natural and traditional one that maximizes the cutoff slope in the high-pass. Deriving this high-pass Bessel, we find that it no longer has linear phase. Other derivations of the high-pass can improve the combined phase response, but with tradeoffs."
Regards,
David
Rane does a nice write up here and they also reference Lipshitz and Vanderkooy
A Bessel Filter Crossover, and Its Relation to Others
"Bessels are historically low-pass or all-pass. A crossover however requires a separate high-pass, and this needs to be derived from the low-pass. There are different ways to derive a high-pass from a low-pass, but here we discuss a natural and traditional one that maximizes the cutoff slope in the high-pass. Deriving this high-pass Bessel, we find that it no longer has linear phase. Other derivations of the high-pass can improve the combined phase response, but with tradeoffs."
Regards,
David
Yes.
That would depend entirely on the crossover frequency
Nor I, but Gaussian function is often the limit case, so of interest.
I have hazy recollection from Mathematical Physics 301 that a Gaussian impulse has the lowest possible combined uncertainty of time/frequency, expressed in some appropriate metric.
Not necessarily minimizes what we are psycho-acoustically sensitive to, but worth a look.
Yes. This makes it a natural for FIR convolution but not so easy for analog.
I have seen plenty of "implement your analog filter in DSP" information but little in the other direction.
I have some feel about how to convert back and forward between frequency response/poles and zeros/transfer polynomials but not frequency response/time domain convolution.
So this is kind of an educational test.
If the filter coefficients are symmetrical then the frequency inverted filter will also be a Gaussian.
This is what I expect but haven't proved yet, it would eliminate the minor problem that Bessel filters have in this respect.
Thank you for the well considered reply
Best wishes
David
I have had a closer look at Self, I assume he has copied Gaussian filter parameters from Filtershop but they don't look strictly correct.
Filtershop probably has some approximation parameter, this would explain the otherwise mysterious Gaussian 12db and 6dB suffixes.
Anyone here with Filtershop?
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Hi Dave Zan,
Yes, I have LinearX's Filtershop and it does synthesize Gaussian Filters as does their Crossovershop as well. They describe this filter as follows;
"Gaussian
The Gaussian family is very similar to the Bessel, but has slightly slower cutoff rate and a more shallow group delay knee. It is commonly used in applications where transient response or linear time delay is important. Bessel and Linear Phase are also similar equivalents."
I have not attempted to use this type of filter because the HP4+LP4 responses do not appear to sum flat at any overlapping point.
Peter
Yes, I have LinearX's Filtershop and it does synthesize Gaussian Filters as does their Crossovershop as well. They describe this filter as follows;
"Gaussian
The Gaussian family is very similar to the Bessel, but has slightly slower cutoff rate and a more shallow group delay knee. It is commonly used in applications where transient response or linear time delay is important. Bessel and Linear Phase are also similar equivalents."
I have not attempted to use this type of filter because the HP4+LP4 responses do not appear to sum flat at any overlapping point.
Peter
Hi David,
I haven't been able to find my original data, so I put together a 2nd and 4th order crossover at 1KHz assuming -3dB and -6dB respectively at the crossover points.
These are attached. If you know of a better overlap in frequency/magnitude, let
me know and I'll plug them in and let you know results. The only other thing I can think of is for you to contact Chris Strahm at LinearX.
Peter
I haven't been able to find my original data, so I put together a 2nd and 4th order crossover at 1KHz assuming -3dB and -6dB respectively at the crossover points.
These are attached. If you know of a better overlap in frequency/magnitude, let
me know and I'll plug them in and let you know results. The only other thing I can think of is for you to contact Chris Strahm at LinearX.
Peter
Hi Peter,
You can iteratively fine tune the overlap yourself with a little trial and error. Your second order case was about 3dB high, so pulling both sections away from the 1000Hz point will drop the mid point level. Either guess and try, or look at the curves and see how far you need to go to be another 3dB down the slope on the stop band side. Best to shift both sides an equal amount so that you can still call it a 1kHz crossover (even if it requires 950 on one side and 1060 on the other).
The combined sections phase response shows that there is a lump of delay at crossover, so these aren't any magical flat phase solution (plus this is an electrical sim and, of course, ignores driver delay).
Regards,
David
You can iteratively fine tune the overlap yourself with a little trial and error. Your second order case was about 3dB high, so pulling both sections away from the 1000Hz point will drop the mid point level. Either guess and try, or look at the curves and see how far you need to go to be another 3dB down the slope on the stop band side. Best to shift both sides an equal amount so that you can still call it a 1kHz crossover (even if it requires 950 on one side and 1060 on the other).
The combined sections phase response shows that there is a lump of delay at crossover, so these aren't any magical flat phase solution (plus this is an electrical sim and, of course, ignores driver delay).
Regards,
David
Hi Dave,
I summed the Gaussian HP4 and LP4 at -4dB through to -7dB crossover points as shown attached. As you can see, there is no point where they sum flat. From these curves it seems that a -6dB crossover point provides the closest to a flat response, with the worse point being -1dB down at the crossover frequency of 1KHz.
Since David Zan has doubts about the accuracy of the LinearX Filtershop Gaussian filter parameters, all of this may or may not be correct.
Peter
I summed the Gaussian HP4 and LP4 at -4dB through to -7dB crossover points as shown attached. As you can see, there is no point where they sum flat. From these curves it seems that a -6dB crossover point provides the closest to a flat response, with the worse point being -1dB down at the crossover frequency of 1KHz.
Since David Zan has doubts about the accuracy of the LinearX Filtershop Gaussian filter parameters, all of this may or may not be correct.
Peter
I am fairly sure that values are inaccurate in the Gaussian Filter table, 7.12 of Self's Active Crossover book, p.181.
I don't know the source of the error, he references Filtershop so I assume that is what he has used, but perhaps the mistake is his.
The Q's and filter frequencies do not produce correct results for the 2nd order filter.
The 3rd order filter does not look consistent with the step response printed immediately before it, or the amplitude response on p.180.
I haven't checked the 4th order values yet but they don't look correct either, I expect they should be much closer to the Bessel filter values.
Thanks for the results, some of them are unusual too.
Best wishes
David
Hi Peter,
You have a phase issue there. The green curve should add highest yet it actually adds lowest. That means that the phase curves of the individual sections are not close enough , they overlap and are partially cancelling. When you went to the lower amounts at crossover,by pulling the sections apart you got an increase in the sum because the phase curves were getting closer. Improving the phase gave more than enough to offset reduced level at crossover. You should see this if you plot the individual rather than the combined phase curves.
Its all a bit academic as doing anything with drivers included will contribute some arbitrary time delay between sections. Matters might improve or they might get worse.
Can your simulater add delay to one section or the other?
I would still stick with LR4 or similar since it is clear that there are no magical phase properties from Gaussian filters (see the 3 slope phase curves.)
Good stuff though,
David
Hi David
It was difficulties with time delay that made me look at this in the first place.
I have a compression driver mid that is a bit too far behind the woofer.
If practicable I would like to delay the woofer in the crossover and avoid DSP.
So I started to look at Bessel and similar (Gaussian) responses to see how they compared to LR in this application.
Any advice, references or hints?
Best wishes
David
When we go from theoretical, delay free crossover examples to real crossovers with drivers with arbitrary physical depth, we will always have to deal with the phase shift due to acoustic center location.
Most of the time we look at band pass sections and how they add. Frequently you will see a novice that gets excited with first order Butterworth or some other scheme that has linear phase or some ideal phase blend. This is only for the ideal case of no delay or the adding of purely electrical filters.
Typically the low pass (woofer) section will have a phase that falls above the corner frequency. The high pass (tweeter) section will have a phase that rises. In both cases the phase shift is relative to frequencies in the center of their respective passbands which hang around zero degrees.
As such they will have phases that are inherently moving away from each other. Still, we can get a good phase overlap if we remember that two phase curves that are 180 degrees apart add well if polarity of one section is flipped. Likewise, if the phase shift between units becomes 360 degrees then they are actually back in phase again.
In the case of real drivers we just need to find any scenario where the phase shift of the networks, plus the inherent phase shift of the drivers, plus a possible phase flip of a section, gets us to good enough phase alignment over the range of frequencies of crossover. There will always be some scenario, including asymmetrical slopes, that gets us close enough. It may just take some trial and error to get there.
Usually we have more woofer depth than tweeter depth and the right phase blend is easy to find. Horns have greater physical depth and that makes things a little more tricky, but there should still be a solutions.
Good Luck,
David
I think that the physics of the horn situation is actually more favourable.
The roll-off of the woofer adds some delay inherently, so we need to move the mid back to compensate but the typical direct radiator speaker flat baffle layout typically moves the mid forward, the exact opposite.
Whereas a horn naturally moves the mid driver in the desired direction, which is what allowed me to think an analog solution is practicable.
Why would it be more tricky?
Asymmetrical hi/low orders looks like an excellent idea, I found some other helpful discussion too.
Best wishes
David
I need to think about Richard Heyser's paper that showed Group Delay is NOT necessarily time delay.
Any comments?
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With a long horn then the high pass inherent physical delay is greater. I haven't worked with that case much and was assuming it was trickier. If you have found it easier then that's great.
Filter asymmetry will often be helpful. We typically talk about bending the phase to get the phase curves to cross at the crossover point (or be exactly 180 out). This is the bare bones requirement but even better we would wish that the slope of the 2 phase curves are the same. That means that phase closeness occurs over a range above and below the crossover point and we get the best possible blend. (This also represents equal group delay of both units in the crossover region.)
In practice we can only add phase shift in the stop band of a section. We can add orders to the low pass to bend the phase downwards or add orders to the high pass to bend phase upwards. Whichever section has the lesser slope (lesser delay) would need crossover orders added to try and get the slopes to align.
This is the art of crossover design, at least for those that still use non DSP methods.
Regards,
David
Filter asymmetry will often be helpful. We typically talk about bending the phase to get the phase curves to cross at the crossover point (or be exactly 180 out). This is the bare bones requirement but even better we would wish that the slope of the 2 phase curves are the same. That means that phase closeness occurs over a range above and below the crossover point and we get the best possible blend. (This also represents equal group delay of both units in the crossover region.)
In practice we can only add phase shift in the stop band of a section. We can add orders to the low pass to bend the phase downwards or add orders to the high pass to bend phase upwards. Whichever section has the lesser slope (lesser delay) would need crossover orders added to try and get the slopes to align.
This is the art of crossover design, at least for those that still use non DSP methods.
Regards,
David
Well, I haven't found the solution yet😉 But it looks doable.
I have played with amplifier Return Ratio optimisation with similar techniques so this is very much how I conceptualize it too.
But there's some differences I need to think over, thanks as always for your ideas.
Best wishes
David
For those that are new to the concepts of crossover networks and real phase curves and phase overlap, here is a tutorial in the context of a crossover redesign of an old AR4x.
Crossover mods for the AR4x - Mods, Tweaks, and Upgrades to the Classics - The Classic Speaker Pages Discussion Forums
Regards,
David
Crossover mods for the AR4x - Mods, Tweaks, and Upgrades to the Classics - The Classic Speaker Pages Discussion Forums
Regards,
David
The highly regarded Accuphase F25 analog crossover used Gaussian filters in a GIC configuration.
I'm trying to change the crossover points of some Accuphase frequency boards but having trouble working out what the component values should be.
I've tried using an online LC component value generator for a Gaussian filter and then used the transformation described in the Burr-Brown paper referenced above to work out the GIC resistor and capacitor values but I'm not getting anything like the values on the existing Accuphase boards.
The problem may be in my use of the online LC filter tool.
Any suggestions please?
I'm trying to change the crossover points of some Accuphase frequency boards but having trouble working out what the component values should be.
I've tried using an online LC component value generator for a Gaussian filter and then used the transformation described in the Burr-Brown paper referenced above to work out the GIC resistor and capacitor values but I'm not getting anything like the values on the existing Accuphase boards.
The problem may be in my use of the online LC filter tool.
Any suggestions please?
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