Mixer Math

[cebec]

I'm continually impressed by Rob's workshops, patches, and contributions. I often dissect and cannibalize his patches for the terrific 'building blocks' included within them.

However, there're at least two areas of many of these building blocks that I have difficulty understanding; the math performed with the mixer modules and the feedback paths.

For example, Rob's latest Matrix Synth has a modified NM Classic filter which uses a few mixers to add a DIY resonance control and probably to tune the filter differently. It sounds terrific!

If I could understand the basic rules and principles for using such feedback paths and performing math with mixers I could learn to design or at least tweak some DIY circuits.

Can Rob, Tim, et al. provide any insight into this? Where do I begin?

Thanks!

[Tim Kleinert]

Cebec,

I don't have the patch you are referring to so I can't give specific answers. Maybe if you could post a link or a specific question...

[cebec]

Here's the patch --

Take a look at the lefthand column and the Filter and Noise blocks. There's feedback and mixer trickery going on here. In fact, a lot of Rob's designs use mixers in this way, and I've built many of the DIY filters from his workshops but still don't understand what's being accomplished with these routings.

And Rob's workshops are here:

http://www.xs4all.nl/~rhordijk/G2Pages/index.htm

I should read the section on resonant filters several more times, at least.

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matrix synth.pch2
Description:		Download
Filename:	matrix synth.pch2
Filesize:	9.67 KB
Downloaded:	1839 Time(s)

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[Tim Kleinert]

The one thing I can already say about mixers and feedback paths is thus:

Digital feedback paths can be prone to unwanted HF oscillations, mostly at half of the sampling frequency. This can and must be taken care of by filtering. Here's were the mixers come in. If you feedback a mixer with itself, it acts as a cheap 6dB lowpass filter. (Just be careful that the sum of the inputs of the signal and the feedback is never more than 127. (Another technique is to use a Xfade module with linear response and feedbacking the output to one of the inputs --my preferred solution))

You'll find these kind of self-feedbacked mixers in many of Robs DIY filter designs.

Just for fun, attached is a 24dB filter made of four cascaded Xfade modules as 6dB poles, plus feedback path.

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No Filter Module.pch2
Description:		Download
Filename:	No Filter Module.pch2
Filesize:	1.51 KB
Downloaded:	1522 Time(s)

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[Tim Kleinert]

OK, I checked it out. As far as I can see:

The first mixer in the blue filter section serves to bandlimit and boost the signal at the same time. (Boosting happens because the sum of the input levels is bigger than unity gain. In this case this boosting is desired because the signal has to be sufficiently boosted anyway within a filter feedback path i n order for resonance to occur. This mixer serves both purposes neatly.)

The second mixer inverts the signal and adds some of the original signal before sending it back into the loop. The inversion is needed for the resonance to work properly with a 24dB lowpass filter, and the "original signal injection" serves to combat the passband loss that usually occurs at high resonance settings. This makes the filter sound "bassier" even with alot of resonance.

[cebec]

Thanks, Tim. I'm just processing your analysis right now.
Are these design choices based in the convergence of both DSP and subtractive synthesis? Or is it something else?

[Tim Kleinert]

[Rob]

Allow me to explain the how and why. I was not really satisfied with the amplitude drop of the Classic filter when the resonance is raised. The Classic filter has a little bit of compression built in, which is absent on the other filters, but the drop in amplitude does not really make use of this. I wanted to improve on these matters.

The old trick to compensate for a drop in amplitude in an analog ladder filter (or digital emulation) is to add some of the input signal just before the internal mixer knob that controls the resonance. But as this signal path is a 'module internal' path, the only solution is to create an extra external signal path for resonance, and not use the resonance knob on the filter module anymore. As the amplitude is dropped by about -12dB for a high resonance setting it means that quite a lot of the input signal must be added. Plus 9 dB seems just about right to both compensate for the overall signal drop and make use of the compressive effect in the filter to get a slightly saturated resonance peak. In the resonance feedback path the signal must also be amplified and slightly filtered to prevent high frequency oscillation if the filter is fully opened.



All in all everything can be done with just three extra mixer modules. The first mixer module acts as both the new signal input (on the chain input) and the resonance control knob. Then comes the filter module itself, and this module also has the filtered output signal on its output. The filtered output signal is then amplified and slightly filtered in the next mixer to create a properly boosted resonance feedback signal. The last mixer receives the input signal on two inputs to add the extra 9 dB for a high resonance setting, and inverts the boosted resonance feedback signal. The output of the last mixer is then routed back to the first mixer.

So, if the resonance control is closed the input signal will just fall through the first mixer into the filter module and there is the standard non-resonant filtering effect.
If the new resonance knob is opened it will not only feed the resonance signal back into the filter module input, but also add a 9 dB of extra input signal. Now you can clearly hear the effect of the compressive saturation in the Classic filter module, which produces extra odd harmonics.

It is possible to add even more saturation to get a distortive filter sound. This can be done by e.g. adding an extra non-linear gain cell in the resonance feedback path. An example is in the second attachment where some extra even harmonic distortion is added. Note that as this is an asymmetrical type of distortion (a positive signal is boosted while a negative signal is compressed) only a little bit must be used to avoid the resonance feedback signal to clip to the positive headroom limit.

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Robs ladder filter.pch2
Description:	Classic ladder filter with compensation for the amplitude drop at higher resonance settings	Download
Filename:	Robs ladder filter.pch2
Filesize:	1.52 KB
Downloaded:	1603 Time(s)

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Robs ladder filter with even harmonic distortion.pch2
Description:	Classic ladder filter with compensation for the amplitude drop at higher resonance settings and some extra even harmonic distortion	Download
Filename:	Robs ladder filter with even harmonic distortion.pch2
Filesize:	1.5 KB
Downloaded:	1660 Time(s)

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[Rob]

And here is the test to show the saturation in the Classic filter module. Four types of filters get a maximum amplitude sinewave while their cutoffs are fully opened. With the four way switch you can select one of the filters.

Only on the Classic filter you can hear how a third harmonic is added to the sinewave. If you reduce the sinewave amplitude to the normal value and open the resonance knob you will hear that the odd harmonic distortion is hardly present at all.

So, some boost of the input signal in relation to the resonance setting is necessary to intensify the odd harmonics effect. Additionally, if an external resonance feedback path is created the odd harmonic distortion will build up properly from the third to the fifth, the seventh, etc.

This type of distortion can sound quite nice on a single oscillator waveform, but when filtering a multi-oscillator unison sound you must take care not to overdo it.

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SaturationInClassicFilter.pch2
Description:	Test to show the third harmonic saturation distortion in the Classic filter module.	Download
Filename:	SaturationInClassicFilter.pch2
Filesize:	1.32 KB
Downloaded:	1500 Time(s)

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[cebec]

Thank you, Rob. This is all very helpful!

[Afro88]

Thanks very much Rob!