Hey,
I just wonder if there are any differences between filter implementations..
For example a biquad can be made of 4 "1sample delays" and one "adder" or just two delay but more "adders".. or one can use a pole-zero structure (+ gain)...
So does anyone know something about this?!?

It came to my mind since there is no signal-controlled biquad is there?? Only the pole-zero things are signal-controlled right?!

...and is there no sig-controlled delay??? (leaving out the [vd~] cause of the interpolation-stuff it does)

In general I wonder why some very basic objects are not included in (at least) pd-vanilla.. (like a samplewise-delay etc.. On the other hand there is something like [lrshift~] which I never used due to its limitations, maybe I didn't find a proper job for it at yet..)
(that must be some kind of conspiracy, there are rumors (or rumours; whatever you like most) that 23 important objects are missing )

Seriously, sometimes I really regret not being able to write externals! A simple delay doesn't sound that hard to create, does it?!

Anyways, filter-design rocks!

@Flipp said:

So does anyone know something about this?!?

Yeah...but I can't really tell what you want to know about it.

...and is there no sig-controlled delay??? (leaving out the [vd~] cause of the interpolation-stuff it does)

If you want to leave out the interpolation, this should work:

[pd ms_to_samples]
|
[expr~ int($v1)]
|
[pd samples_to_ms]
|
[vd~]

The ms-to-samples conversion is, of course, dependent on the sample rate, which is why I just stuck them in subpatches there.

In general I wonder why some very basic objects are not included in (at least) pd-vanilla..

Part of it is that Pd-vanilla seems to be more utilitarian, in that it has very few objects, but you can really make a lot with those objects. And the other part is just that Miller seems REALLY hesitant about adding new objects and functionality that he didn't write himself. My guess is because since it's really his baby, he doesn't want to maintain code he doesn't understand or use himself.

But if you really want to know the answer to that question, you should hang out on the list. Discussions about how it takes the devs a long time to implement simple changes that others have already found solutions for happen fairly often there.

Seriously, sometimes I really regret not being able to write externals! A simple delay doesn't sound that hard to create, does it?!

If you have any programming skills, you should look into it. I've only dabbled in a few basic externals, but it's actually not all that hard once you figure it out.

@Maelstorm said:

Yeah...but I can't really tell what you want to know about it.

I just mean some general things. Eg. I guess the "2 delays some adders"-structure is the most cpu-friendly, whereas the pole-zero-structure is better for signal-controlled (constantly changing) filters.. ..that's just a guess, but if you know which structure is best for lets say quality or cpu-friendliness, please let me know as well..

If you want to leave out the interpolation, this should work: ...

Hmm I'll give it a try. But I did this once for the [delread~] and it was just a little messy since I don't really know about its interiors.. It seems it rounds to the closest sample but not really, it rather does round at 0.49999... instead of 0.5, and I don't know what it does at different samplerates. Soooo, a simple samplewise delay would just keep averything "tight"...

@Maelstorm said:

...My guess is because since it's really his baby, he doesn't want to maintain code he doesn't understand or use himself.

That's quite comprehensible I think.
Well if changes don't happen too often, I guess learning how to write externals myself might be still faster even without any (non-graphical) programming-skills..
Could you tell me where to start?!? I already googled for this toppic but I did not really find something usefull since I don't know what to look for...
Is there anything like a pd-template for a certain compiler out there?? Or what?!

Do you know the "faust-online-compiler"?? I tried that one, but it didn't work... I must be doing something wrong there (and I don't like the "external-structure" it creates: something like an external inside an abstraction (with only one sig-inlet??!))

The most efficient implementation is the Direct Form II (the one with only two delays), because it uses half the memory. This is what [biquad~] is doing internally. The structure itself, however, doesn't care whether or not it is controlled by an audio signal.

The "pole-zero" structure is computationally less efficient because it is implemented as four separate first-order filters, and it's made even more complicated by the fact that these filters use complex coefficients, which is not nearly as straight-forward as math using real numbers.

However, in Pd it is more efficient to use the first-order series structure simply because of the one-sample feedback in the filter. You could do the whole thing in [fexpr~], but it is cpu intensive. If you tried to do the feedback with a delay, you would have to use a [block~ 1], which is also cpu intensive. The only other option is to do the feeback delay inside an external, which is what [cpole~] and family do.

It has nothing to do with the signal control, it's just what Pd has to offer. [biquad~] could be made with signal inlets for each coefficient pretty easily, and there really should be one in Pd.

By the way, if you're not aware, there is [z~] for making delays in samples.

First of all thanks.

The most efficient implementation is the Direct Form II (the one with only two delays), because it uses half the memory.

Okay, that's rather obvious but what's about the behavior or quality?! Do all those structures behave the same when eg. the Q or F_cutoff changes?? (no clicks or different transients or what-so-ever). Or whats about a numerical error? I'd say the more math the more potential errors can occur.

However, in Pd it is more efficient to use the first-order series structure simply because of the one-sample feedback in the filter.

Even faster than the data-controlled [biquad~] ?? Oh no! Im on windows overhere so I can't directly measure this but with external programs. ...
...just did it. It seems to be slower ..fortunately..

...the whole [expr]-family is cpu-intensive it seems...

It has nothing to do with the signal control, it's just what Pd has to offer. [biquad~] could be made with signal inlets for each coefficient pretty easily, and there really should be one in Pd.

That's what I thought. But since there is none, only the poles and zeros, I assumed that there might be a reason for that...

By the way, if you're not aware, there is [z~] for making delays in samples.

I know I know... I'd rather use [delay~]. I don't like zexy cause of all the console-messages, although it seems to be coded solidly. Anyways I like cyclone most and I want to use as few libraries as possible.

@Flipp said:

First of all thanks.

No problem.

Do all those structures behave the same when eg. the Q or F_cutoff changes?? (no clicks or different transients or what-so-ever). Or whats about a numerical error? I'd say the more math the more potential errors can occur.

Theoretically, from a purely mathematical perspective, they are exactly the same. They are just more than one way of expressing the same thing. You could take the equation of one form and do a bit of algebra to get the other.

The only potential issue is, as you said, numerical errors due to the quantization limits of digital numbers. But there's not a whole lot you can do about that. I believe the Direct Form I and Direct Form II versions of a biquad filter have the same number of additions and multiplications, so there's probably no difference there. The issue with the first-order series version is that you have to take the biquad coefficients and extrapolate the coefficients for the first-order filters from them, which involves more complicated computation. Take a look at my [biquad.mmb~] abstraction to get an idea of how ugly it can get.

Even faster than the data-controlled [biquad~] ??

Never tested it myself, but I kind of doubt it. I should clarify, I only meant it's more efficient to use the first-order filters when you want audio-rate control of the coefficients because there is no external version of [biquad~] in Pd that does that, so you have to make it yourself somehow. The other options of making it in Pd are less efficient. For a message-rate version, [biquad~] is already there doing the feedback internally, and from what I can see in the source, it's pretty simple. It's just not practical for time-varying coefficients since it will only update them on block boundaries, which can cause clicks.

...the whole [expr]-family is cpu-intensive it seems...

I've never found [expr] or [expr~] to be all that cpu-intensive. Yeah, they're slightly more than using the basic math objects, but not enough to try avoiding them. [fexpr~], on the other hand, is pretty intense.

Theoretically, from a purely mathematical perspective, they are exactly the same.

Should be because math comes before the "physical" implementation. But it is "spectral-math" not "time-domain-math". So the build up of a sine may be different from case to case although they behave the same after the transient oscillations..

I didn't really search but I found this: https://ccrma.stanford.edu/~jos/fp/Direct_Form_II.html
..or just wikipedia..

For a message-rate version, [biquad~] is already there doing the feedback internally, and from what I can see in the source, it's pretty simple.

Can't you modify the source then?! Like just replace "inlet" with "inlet~" and there we have the sig-biquad.. ...I'll gonna have a look at the faust-compiler again...

@Flipp said:

Should be because math comes before the "physical" implementation. But it is "spectral-math" not "time-domain-math". So the build up of a sine may be different from case to case although they behave the same after the transient oscillations..

The implementation is purely time-domain. It can be represented in the frequency domain, and in some cases it may be easier to manipulate in the z-domain (which is very much related to the frequency domain), but the implementation uses time-domain difference equations. It's much easier that way.

I feel like we've had a similar discussion before...;-)

I didn't really search but I found this: https://ccrma.stanford.edu/~jos/fp/Direct_Form_II.html
..or just wikipedia..

Yeah, and this part:

is the time-domain representation. And it's really simple to implement. If you look at the C code for [biquad~], it's pretty much exactly this.

Can't you modify the source then?! Like just replace "inlet" with "inlet~" and there we have the sig-biquad.. ...I'll gonna have a look at the faust-compiler again...

It's a little more complicated than that, but, yeah, it wouldn't be too hard. iirc, there's a signal-controlled version of [biquad~] called [bq~] out there, but I think it doesn't conform to [biquad~] (i.e., the feedback coefficients are reversed). I've had thoughts of making external versions of some of my abstractions, and that would probably be at the top of my list. I just haven't gotten around to it.

The implementation is purely time-domain.

...as every real implementation... (even fft, since its not 'normal' fourier..)
Furthermore Laplace deals with frequencies, the Z-domain rather deals with angles.

I feel like we've had a similar discussion before...;-)

Yeah, I know what you mean.. we happen to celebrate long discussions...

Well I just simply tried it out. I've created biquads with [fexpr~], and the direct form I and II in a subpatch with [block~ 1 1 1] and the pole-zero-structure with cpole~s and czero~s.

And I can confirm that [biquad~] contains the direct form II.
-This is (...obviously...) the direct form I: [fexpr~ $f4*$x1[0]+$f5*$x1[-1]+$f6*$x1[-2]+$f2*$y1[-1]+$f3*$y1[-2]].

However all implementations behave differently on f_cutoff-changes!!!
The pole-zero-structure and direct-form-II can produce large overshoot.
To cut it short:
Direct Form I: f-up: ok, f-down: ok
Direct Form II: f-up: overshoot, f-down: ok
Pole-Zero: f-up: ok, f-down: overshoot

I don't know if no-overshoot stands for quality, but if so I'd say the "direct form I" might be best for a signal-version of a biquad.

iirc, there's a signal-controlled version of [biquad~] called [bq~] out there...

I can't find it. ..maybe cause google just ignores the "~" and "[]"... I just found smooth_biquad~ but that's for max/msp... maybe one can sort of port it..

I think direct form I is best for fixed point math. Direct form II or transposed direct form I are preferred for floating point.

There's a very thorough PhD thesis on the topic: www.tech.plym.ac.uk/spmc/pdf/audio/RobClarkPhD.pdf

There are lots of different possible topologies. The direct form biquads are pretty good all around, but there are quantization problems if fc/fs is very low. In this case it's better to use 64 bit floats for both coefficients and internal signals.

And I don't think you should be discontinuously changing the coefficients in any topology. If they're signal controlled, it's no big deal to change the coefficients using linear ramps over a few ms.

@Flipp said:

...as every real implementation... (even fft, since its not 'normal' fourier..)

Well, the Fourier transform isn't real, it's complex. And FFT is just an efficient algorithm for implementing the DFT and/or the STFT. But once you're in the frequency domain, there is no representation of time. So it isn't really time-domain at that point. The STFT (which is what [fft~] does) is an attempt to make a frequency-domain representation using DFTs over short snippets of time, so you get a two-dimensional signal over time. It actually falls under the category of "time-frequency representation". But there's a trade-off between frequency resolution and time resolution. I wouldn't call it time-domain.

Furthermore Laplace deals with frequencies, the Z-domain rather deals with angles.

I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".

However all implementations behave differently on f_cutoff-changes!!!
The pole-zero-structure and direct-form-II can produce large overshoot.
To cut it short:
Direct Form I: f-up: ok, f-down: ok
Direct Form II: f-up: overshoot, f-down: ok
Pole-Zero: f-up: ok, f-down: overshoot

I'd be very interested in seeing a patch that illustrates this.

@acreil: Thanks! I know very little about the issues in dealing with fixed-point vs. floating-point. Would you happen to know of any good resources that cover the issues in general (i.e. not necessarily dealing with filters)?

Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!

@acreil said:

And I don't think you should be discontinuously changing the coefficients in any topology. ...

I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best signal at a certain jumpsize.

@Maelstorm said:

Well, the Fourier transform isn't real, it's complex...

Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality" NOnono! There is no complex variable in real life. Now you might say "But what's about the cpole~ or czero~?!" -It's just an implementation of complex math but there is nothing like a variable of the form "a+i*b". ...there is a reason why "a" is called real- and "(i*) b" is called imaginary... (..one just feeds in and gets out the real part at the end anyways..)
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.

I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".

Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!

I'd be very interested in seeing a patch that illustrates this.

It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..

I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)

Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)

Btw. one can't upload everything at once, right?!

http://www.pdpatchrepo.info/hurleur/ap2-f_c-down_ac.jpg

Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!

@acreil said:

And I don't think you should be discontinuously changing the coefficients in any topology. ...

I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best signal at a certain jumpsize.

@Maelstorm said:

Well, the Fourier transform isn't real, it's complex...

Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality" NOnono! There is no complex variable in real life. Now you might say "But what's about the cpole~ or czero~?!" -It's just an implementation of complex math but there is nothing like a variable of the form "a+i*b". ...there is a reason why "a" is called real- and "(i*) b" is called imaginary... (..one just feeds in and gets out the real part at the end anyways..)
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.

I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".

Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!

I'd be very interested in seeing a patch that illustrates this.

It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..

I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)

Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)

Btw. one can't upload everything at once, right?!

http://www.pdpatchrepo.info/hurleur/ap2-f_c-down_ac.jpg

Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!

@acreil said:

And I don't think you should be discontinuously changing the coefficients in any topology. ...

I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best signal at a certain jumpsize.

@Maelstorm said:

Well, the Fourier transform isn't real, it's complex...

Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality" NOnono! There is no complex variable in real life. Now you might say "But what's about the cpole~ or czero~?!" -It's just an implementation of complex math but there is nothing like a variable of the form "a+i*b". ...there is a reason why "a" is called real- and "(i*) b" is called imaginary... (..one just feeds in and gets out the real part at the end anyways..)
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.

I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".

Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!

I'd be very interested in seeing a patch that illustrates this.

It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..

I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)

Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)

Btw. one can't upload everything at once, right?!

http://www.pdpatchrepo.info/hurleur/ap2-f_c-down_ac.jpg

Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!

@acreil said:

And I don't think you should be discontinuously changing the coefficients in any topology. ...

I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best signal at a certain jumpsize.

@Maelstorm said:

Well, the Fourier transform isn't real, it's complex...

Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality" NOnono! There is no complex variable in real life. Now you might say "But what's about the cpole~ or czero~?!" -It's just an implementation of complex math but there is nothing like a variable of the form "a+i*b". ...there is a reason why "a" is called real- and "(i*) b" is called imaginary... (..one just feeds in and gets out the real part at the end anyways..)
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.

I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".

Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!

I'd be very interested in seeing a patch that illustrates this.

It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..

I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)

Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)

Btw. one can't upload everything at once, right?!

http://www.pdpatchrepo.info/hurleur/ap2-f_c-down_ac.jpg

@Maelstorm said:

@acreil: Thanks! I know very little about the issues in dealing with fixed-point vs. floating-point. Would you happen to know of any good resources that cover the issues in general (i.e. not necessarily dealing with filters)?

I think it's mostly just general rule of thumb issues: you need to use fixed point for embedded processors without hardware FPUs, and you can more easily determine the effects of quantization. Floating point won't (practically) clip, so it's better for "modular" type systems where you can't guarantee any sort of fixed topology or signal level. But you have to worry about denormals and NaNs and whatnot.

In this case, if I remember correctly, you don't want to use fixed point for direct form II because it needs more dynamic range to avoid clipping internally.