Hi all,

Been using PD for a little while for making MobMuPlat patches.
While the filters found in pd extended are certainly good, I am looking for a higher order filter (LP and HP) this time. Preferably, an 8th order filter.

It is a requirement to use pd-extended only, for keeping MobMuPlat support.
The filters are required to have a cutoff input and a resonance input (neither needs to be audio signal in, but that would be a nice bonus).

While I did manage to get a patch working using rifft~ and basically graphing the desired frequency spectrum, the result isn't very great and probably adds unnecessary overhead.

Does anyone know of a freely available patch for these filters?
Or perhaps pointers on how to use the pole~ objects, so that I can create it myself?

Thanks in advance.

If the intent is to get a 48 dB/octave rolloff, you could chain 4 2-pole filters in series. (Often, a 4-pole aka 24 dB/oct filter is implemented by chaining 2 of them, because a true recursive 4th-order filter requires too much numeric precision or else it blows up. I'd imagine an 8th-order filter would be subject to the same concern.)

If the intent is for very specific filter characteristics, that's outside of my expertise.

hjh

@elwinbran as @ddw_music says you can just cascade filters, when you cascade filters you use [delwrite~] and [delread] to implement a feedback loop to get the resonance. You need to something to limit the feedback so it does not blow up and in this case I used [clip~] which is not the greatest solution but it works and is in pd-extended. You can do this same thing with any filters in pd-extended, so find a replacement for [lop~] with signal rate frequency input and your filter will have signal rate inputs for everything. Probably would want to stick this in an abstraction for ease of use/reuse but that is simple enough. The vast majority of classic synth filters are just cascaded 6db filters with a common feedback loop.
lop8.pd

Thanks a ton so far, I will work on this

Is there a cookbook for (or gentle introduction to) making filters with the pole/zero objects? I always found this one buried in J07.oversampling.pd to be interesting.

Im still working on this project, yet have to implement it on mobile devices. However, testing the 'cascade filter' design came with some weird drawbacks. Perhaps this was already due to the lop~ and hip~ objects, but i'll accept the weird resonances across the spectrum. What isn't really acceptable is the wonky response to resonance and both the filters seemingly not functioning above half nyquist range.

I have used a arctangent to make a smoother clip for the resonance, further design is unchanged.

@elwinbran Yeah, clip is fairly raw. Not being able to go above half of nyquist must be a quirk of Extended, I don't have that issue in vanilla. To get a better resonance response I often use something like [pow 8] on the resonance input but you will want to stick a [clip 0 1] before it and a [* n} after if you need to go higher than 1. Generally I can get the response good enough by playing with the value to [pow]. Adding a bit of compression in the feedback loop can help as well and can also help smooth out some of the harshness.

Edit: actually [pow] may not work for you, not sure if the issue with the response is the touchiness or just not being able to go above half nyquist.

@elwinbran I can't tell you how to build a proper 8th order resonant filter ... (it's certainly not an easy task)
But here's how you could maybe improve the cascaded filter from above:

First, you should use the smallest possible delay for the feedback, so use a subpatch with [block~ 1], maybe even try oversampling.

A zero at the nyquist [rzero~ -1] may help to suppress unwanted resonances. With a 0.5 gain it's a simple averaging filter. (The three zeros in the Butterworth example do the exact same thing, by the way.) Maybe you could even use one zero for each pole, but I'm not sure about that.

The result would then be something like this: lop8-block1.pd

@oid
i wonder what you think what this feedbackloop should be good for.

@Obineg ? I will guess you thought I meant that the [pow] should be in the audio path? Was referring to the control input, will give more control of the resonance in the region around self oscillation. Compression was the only thing I meant to be in the audio path.

Non-linear elements in the feedback path can be fun and do some interesting things but far from a general purpose filter, no clue if pd will play well with that sort of abuse.

@oid said:

Non-linear elements in the feedback path can be fun and do some interesting things but far from a general purpose filter, no clue if pd will play well with that sort of abuse.

In my humble experience: very well. I've been working on an "acoustic feedback" patch for a while now. With allpass/diffusion, peak filter and basic distortion in the loop, it was quite lively and convincing.

@oid I was assuming that you thought of analogue ladder filter design a la Moog when you suggested the feedback path to get the resonance. And since analog feedback is basically delayless (which in digital land is of course impossibe), I suggested to make the delay as short as possible.

But there's one thing, that I forgot to consider: The whole magic of the Moog filter only works with four one-pole filters in series! I can't remeber how exactly, but it has to do with the phase response of each one-pole filter and the sum of the phase shifts corresponding to the resonant frequency.

So if you care about controlling the frequency of the resonance, it seem to be necessary to implement the 8-pole filter as two resonant 4-pole filters in series. Of course, this is still a very crude approximation ...

lop8_2x4.pd

@kyro By non-linear I was referring to things like [pow~] getting stuck in the feedback loop, not a correct terminology but one I fall back on with synths because it was a common term in the synth diy community 20 years ago to refer to anything that is not the normal distortion/filtering type stuff in a filters feedback loop.

@manuels The magic of the moog (as in minimoog) is primarily in the output amp which affects the entire filter and is why the ARP version sounds so different, they 'fixed' it and lost the characteristic flavor of the mini's filter. The output amp on the mini's filter does not have anywhere high enough of an impedance to deal with the ladder which drags down frequency response and distorts things, feedback loop comes off output amp so nothing in the filter is untouched by the output amps flaws and it is more the heart of the filter than the ladder. Later Moog ladders are closer to the ARP version, output amp has a high enough impedance and we loose most of the mini's character but we get a decent response in the upper registers in trade.

Most of the classic VCFs are just cascaded 6db filters with a feedback loop for resonance and is my standard, the various 'proper' filters created in the digital world are lacking in many ways to my ears. The cascaded nature of many circuits in the analog world is part of what the digital world is missing, in a 4 pole filter that means we at least have input amp, 4 filter stages, output stage and feedback each giving their own distrotion and frequency characteristic. The filter stages we try and balance and get them to cancel out any quirks to work together but they still have an individual character and input, output and feedback have a very strong character which gives us a rather dynamic result when everything is combined.

And since analog feedback is basically delayless (which in digital land is of course impossibe)

Check my thread on feedback which actually came about because of trying to get a decent filter executed in pd, getting closer.

@oid Thanks for the insights! Fascinating stuff ...

I'm still wondering how the resonant frequency relates to the cut-off frequency when you use this filter topology with other than four cascaded one-pole fiters. It kind of seems to work but I can't figure out how to control the resonant frequency.

To be more precise about the phase shifts I mentioned earlier: The one-pole filters have a phase response of 45 degrees at the cut-off frequency, and so the total phase shift of the four one-pole filters is 180 degrees, which corresponds to inversion of the signal and therefore resonates with the inverted feedback. Well, at least that's what I read and how I understand it.

But how does this resonance work for more (or less) than four cascaded filters?

@manuels The moog is nothing special here just a standard cascaded filter. What do you think about this?

Which also should solve @elwinbran's issue and I had completely spaced out until you got my neglected filter theory working again, higher order VCFs generally put the feedback around only part of the filter and even a few lower VCFs did this. As you steepen the filter you change the frequency which has the proper cumulative phase alignment.

Edit: Out of curiosity I pulled out the Blofeld to see how they deal with this in hardware, the displayed filter cutoff is actually the frequency which the filter oscillates at and not the usual cutoff point. I wanted to also test this on the Micron which has an 8pole filter but it has some quirks regarding getting its filters to self oscillate and I don't seem to quite recall what they are at this moment or it might just be that not all of its filters can self oscillate, scrolling through my patches it seems I never use oscillating filters on that synth.

Another Edit: The Blofeld could actually be using a 2-pole resonance loop on the 4-pole filter, I tested by seeing if the frequency was the same between the two which I now realize is flawed. How the sound changed makes me suspect this is not the case and I was correct, I will do a more proper test later and maybe figure out the Micron, this got me curious.

@oid That's kind of what I would have expected. In the 4-pole version you shouldn't have a delay at all (or at least a much smaller one), and in the lower order filters the 64 sample delay is either too big or too small to produce the required phase shift ...

Which led me to the following idea: Couldn't we calculate the exact cumulative phase shift of the [lop~] filters and add a variable delay to get the required 180 degrees (or pi radians)? I found the formula for the frequency response in Julius O. Smith's book on digital filters and just tried:

resonance-fix.pd

To my surprise it really seems to work!

Interestingly the delay of the 4-pole filter shouldn't be as small as possible (as I thought) but depend on the actual cut-off frequency. The reason for that (if I understand correctly) is that the phase shift of the one-pole filter at the cut-off frequency actually seems to be somewhat less than 45 degrees.

Likewise, the Q value for self-oscillation (> 4) also seems to be an approximation. I scaled it to 1 by using the calculated amplitude response.

Edit FYI: Smith's one-pole filter analysis

Another edit: The deviation from the expected 45 degree phase shift depends on the sample rate, and in fact seems to be proportional to the sampling interval.

@manuels The 2x4 pole 'moog filter approximation' does seem to provide pretty good results. Had to fix the patch a bit, cause of compatability with extended, but it does have a good resonance.

As for your more recent message, it will take me a bit more time to fix the resonance fix patch, but I assume then that I can take that design and scale it up to more poles?

Also @oid thanks for the contribution, I think Im learning a lot here.

@elwinbran said:

As for your more recent message, it will take me a bit more time to fix the resonance fix patch, but I assume then that I can take that design and scale it up to more poles?

I don't think that's possible, four poles is the maximum here. But considering what @oid said about higher order VCFs, you could put the feedback path around only part of the cascade. I didn't think about this possibility earlier. So maybe it's better to have only one feedback path in the 2x4 pole filter? I don't have any experience with this kind of digital filter design and you should keep in mind that it's highly unconventional! Better you don't tell anybody that it's supposed to sound like a Moog filter or something.

Here's another way to fix the resonance frequency of the 4-pole filter:

resonance-fix-rzero.pd

In my previous attempt I calculated the required delay for the 180 degree phase shift, and it turned out to be around 3 samples for a wide range of cut-off frequencies. So introducing 4 averaging filters (or replacing each one-pole filter by a one-pole one-zero lowpass filter) reduces the optimal delay to around 1 sample, which is the smallest possible dalay anyway. Not tested yet, but it might be an alternative.

@manuels The compensation is interesting and I have been playing with it for the past couple days. Sort of a double edged sword, with the compensation I find it much more difficult to get a dynamic sound full of character and nuance that I seek but the resonance is much easier to deal with. Need to read more of Smith's filter book and understand things better so I can find the balance. One of these days I will learn to speak digital above the level of logic gates but the failures of my analogies seem to largely be useful and productive at least.

@oid Not sure where the "character" in your filter with extra delay comes from ... It's probably some kind of distortion. To recreate this effect in a more controllable way you could set the resonance frequency to some percentage of the cut-off frequency, like this:

resonance-fix-extended.pd

For values smaller than 100 the resonance is somehow distorted. Try feeding the filter with impulses at high resonance to hear the effect more clearly.

I removed the arctangent, so there isn't any nonlinearity in the feedback loop, just the [clip~ -1 1].

Edit: Would it be more reasonable to set the cut-off frequency to a percentage of the resonant frequency instead? I'm not sure ...