The importance of crossover steepness

[gberchin]

Originally Posted by CopperTop

The 'standard' filter is based solely on the formula given in post #71 ... I'm using Q = 0.5.

Ah, thanks. I keep getting thrown by the reference to "standard". What you have there appears to be an approximation of the magnitude response of low-Q analog filters, such as Bessel or Butterworth. While those filters are standard, that approximation is not one that I have seen in common use, at least not for higher-order filters.

[gberchin]

Originally Posted by CopperTop

Unfortunately, the system has no longer sounded as thrilling in recent days; a bit 'muddled' perhaps. Now, I'm the arch-sceptic when it comes to trusting my own hearing with subtle changes - experimenter bias and all that - but... I decided to go back to the original linear crossover slope. And to me, it really does sound better.

CopperTop, as you read the following please understand that I in no way intend to question or belittle your perceptions -- your preferences are your own and neither I nor anybody else has the right to judge them. But your statement appears to be contrary to the conventional wisdom in the audiophile industry. As I understand it, extremely steep, linear phase crossovers have been tried and rejected due to poor sound quality -- generally for softening of transients. (I have never tried them myself.) I do not know what specific factors contribute to your perception of better sound despite the increased filter ringing, but if you like the sound, use the crossovers.

[CopperTop]

Originally Posted by gberchin

As I understand it, extremely steep, linear phase crossovers have been tried and rejected due to poor sound quality -- generally for softening of transients. (I have never tried them myself.) I do not know what specific factors contribute to your perception of better sound despite the increased filter ringing, but if you like the sound, use the crossovers.

I don't quite follow you. What I am saying is that in order to get the same level of attenuation at the frequency extremes as I get with the 'linear' crossover, I would need a 'standard' (sorry!) crossover which is far steeper in the middle of the crossover region, giving rise to heavy ringing and overshoot. The linear crossover has the benefit of being not particularly steep in the middle (contributing to not very bad ringing or overshoot as shown in the screenshot) but gives total attenuation of the frequencies that might cause the drivers to misbehave. I'm suggesting that the 'fun with numbers' aspect of all this might have something to it after all. Can we create a filter that sharpens up the corners without steepening the middle, and yet performs well in the ringing and overshoot stakes? The linear crossover is half way there, it seems to me.

Edit: maybe what I've just said means no sense at all in dBs and log frequency scales, but it looks quite clear to me on the linear/linear scale.

[gberchin]

Originally Posted by CopperTop

What I am saying is that in order to get the same level of attenuation at the frequency extremes as I get with the 'linear' crossover, I would need a 'standard' (sorry!) crossover which is far steeper in the middle of the crossover region, giving rise to heavy ringing and overshoot. The linear crossover has the benefit of being not particularly steep in the middle (contributing to not very bad ringing or overshoot as shown in the screenshot) but gives total attenuation of the frequencies that might cause the drivers to misbehave.

Hmm ... I understand what you're saying, and it makes sense. I wonder how some of what I called the "Gaussian-like" higher-order filter pairs would compare.

Can we create a filter that sharpens up the corners without steepening the middle, and yet performs well in the ringing and overshoot stakes? The linear crossover is half way there, it seems to me.

Well, in general you want to avoid sharp corners. I suggest that you research "window-based filter design", then start with your piecewise-linear frequency response and convolve that with a nice window to round the corners; something symmetrical, like a Hann window.

[CopperTop]

Originally Posted by gberchin

Well, in general you want to avoid sharp corners. I suggest that you research "window-based filter design", then start with your piecewise-linear frequency response and convolve that with a nice window to round the corners; something symmetrical, like a Hann window.

I was reasonably convinced that I was doing something like window-based design at the moment:

My procedure is to calculate (or draw, or whatever) the desired frequency response and load up the FFT with it (real only, imaginary set to zero for a linear phase filter, symmetrical about the Nyquist point, DC and Nyquist elements set to zero), then I calculate the inverse FFT to get the impulse response. I window that with a raised cosine whose width is proportional to the FFT size. Then I calculate the forward FFT again and use the result as my filter in the real time crossover program (which is implemented using overlap-add, incidentally).

I also plot the frequency response only after the windowing which, for a large FFT is usually pretty much identical to the input - when viewed on a linear scale at least. Until now, I've been using the window purely to allow me to specify any frequency response I like without worrying about discontinuities at the edges, rather than attempting to use it the other way round i.e. to influence the frequency response.

Steph also mentioned the idea of using the window as the tool for reducing ringing and overshoot some time ago. Thanks, I will look into it.

[gberchin]

Originally Posted by CopperTop

I was reasonably convinced that I was doing something like window-based design at the moment

Well, you can either multiply by a window or convolve with a window. I am suggesting that you convolve your frequency response with a raised cosine (Hann) window, to round-off those sharp corners. That is equivalent to multiplying the impulse response by the inverse transform of the raised cosine. The remainder of your procedure will still be necessary.

Ultimately you will find yourself trying to find a compromise between rounded corners, steep transition band, and attenuation above 4 kHz. (Again I suggest that you also examine your frequency responses on a dB scale.)

(real only, imaginary set to zero for a linear phase filter, symmetrical about the Nyquist point, DC and Nyquist elements set to zero),

Was that a typographical error? You want the DC element of your lowpass filter to equal 1.0, and the highpass filter to be 1.0 at half the sampling frequency.

[CopperTop]

Originally Posted by gberchin

Was that a typographical error? You want the DC element of your lowpass filter to equal 1.0, and the highpass filter to be 1.0 at half the sampling frequency.

Ooh, you're right. I never thought about that. Can't see any difference on the impulse response plots, however..?

Well, you can either multiply by a window or convolve with a window. I am suggesting that you convolve your frequency response with a raised cosine (Hann) window, to round-off those sharp corners. That is equivalent to multiplying the impulse response by the inverse transform of the raised cosine. The remainder of your procedure will still be necessary.

I like it. I will do it.

[CopperTop]

@gberchin

I haven't got round to anything more advanced yet, but I did try simply reducing the width of the raised cosine window that I multiply the impulse response with. This does, in fact, result in the rounding off of the corners of the linear crossover. Presumably, the ultra-sharpness of the corners was possible because of the low amplitude, but extended, ringing on either side of the central impulse which is now gone. I enclose a screenshot of the new rounded linear crossover, plus a 'standard' filter with a similar level of suppression of the frequencies at the edge of the crossover region. I'm thinking that the rounded linear crossover is a pretty good compromise.

A bit light on details, I know, but I'll endeavour to pull it all together with better on-screen stats and those log scales!

[CopperTop]

@gberchin

Finally some plots with log/log scales. (More prettifying to do yet, though)

The 'standard' crossovers are as the formula mentioned previously (Q = 0.5).

The Gaussian is as per the formula you supplied a few pages ago.

The experimental linear crossover is shown with a wide window i.e. sharp corners on the lin/lin plot.

I can switch/sweep between shapes, cutoff frequency, order and window width in real time while listening to the result.

[steph_tsf]

compare with iDFT_Lab
see attached .jpg and .zip