Eric -
Wow! That was fast coding for the interpolation. This is
exactly what I was thinking.
Eric Scheirer wrote:
Excellent point.
I wonder how many applications truly need filter re-design to happen
at the k-rate. At least in the one I had in mind, I'm happy to have
the filters designed at the i-rate, or even at orchestra startup.
People have been looking at this for decades, I have discovered. Apparently this has been considered a hard problem forever.
If you have the luxury of infinite calculation speed, then it is nice to have an arbitrary mapping of user parameters to physical synth parameters. This is a nice general idea in block diagram form:
UserParameters --> Mapping -> PhysicalParameters
Of course, this is a many-to-many transformation, and it is usually non-linear.
In the case of filter design, we have dozens (actually, zillions) of choices for this mapping. In my opinion, Julius Smith has written the most comprehensive study of all of the math behind the mappings for filter design in his thesis. He really needs to be given a year or so to turn this into a book. It is better than any of the other filter design books that I have, except that it is a bit hard. He could add a few sections for implementation details, perhaps.
Short of that, Matlab is my second choice.
Clearly, in practice, it is overkill to have a complete filter design
calculation at K-rate, or even maybe I-rate, as Interpolate Table lookup
works so well in so many cases.
Past the issue of how often the filter is changing, there's the issue
of zipper noise that comes from changing parameters on the fly.
Re: Zippering
Here I would argue that the case is clear. Filter coefficients must be ramped at the sample rate. OK - i have seen people ramp filter coefficients at some lower rate, and in many cases you cannot hear the zippering. The filter itself can provide some filtering of the coefficients as they are ramped.
But the only certain way to avoid zipper noise is to ramp the coefficients at the sample rate. The same goes for amplitude multipliers - actually, they are even more important to ramp.
If you cannot afford to ramp filter coefficients at the sample rate,
then you really have to make sure that the coefficients are not changing
very fast, and you need to test them with suitable program material.
I have found that when you try to cheat here, you can get burned when a
composer tries something you did not test. They come back and complain
that there is something wrong.
How stable are traditional (ie, bilinear transform) methods of
filter design as you change the input parameters? For example,
if I smoothly vary the cutoff frequency of a bandpass designed
with ellip() in Matlab, will the pole/zero locations smoothly
vary?
There are two questions here:
1. Stability when the filter coefficients are changing.
There are at least two definitions of stability.
A. the most traditional definition of filter stability is to constrain the filter poles to remain inside the unit circle. This is necessary and sufficient for time-invariant filters, I think. For this definition of filter stability, you need to consider each filter topology separately, when you are interpolating coefficients.
One of the most popular topologies is the Direct Form II biquad. With this filter, if you start with a coefficient set that is stable, and you end with a coefficient set that is stable, all linear interpolated values in between are also stable. This comes from a wonderful simple proof that I learned about from Andy Moorer. There is a nice drawing (see Jackson's text) that shows the 'triangle of stability', which is a plot of the recursive coefficients a1 versus a2. All stable filters fall within a triangle in this plane, and it is trivial to see that linear interpolations between different a1's or a2's all fall on lines within this triangle, therefore, all intermediate values are stable.
For various ladder filters, interpolated filter coefficients can also be shown to be stable, as long as the coefficients are on the unit interval.
For Direct Form filters of order greater than 2, I am not sure what the case is.
B. Energy Stability definition
This definition is that the energy output of the filter is bounded when the input is bounded. I believe that this is a bit different than requiring that the poles remain inside the unit circle. Actually, I think that the constraint that the poles remain inside the unit circle comes from this initial constraint, but in the time varying case, there are more considerations.
Here I frankly do not understand the math. Sadly, I fell behind in my studies. However, I know that Jean Laroche independently discovered a fundamental proof that can show whether a filter is stable under time varying conditions using the state variable representation of the filter, and some amazing eigenvalue / linear algebra tricks that I could only marvel at from a distance. He then found that his proof was published in a European signal processing journal some years ago, but I think that Jean should re-publish this result in more understandable engineering terms.
Basically, he found that there are several filter topologies that are unconditionally stable under time varying conditions, and that the direct form is not one of them.
This does not mean that you cannot use the direct form for time varying filters. In fact, it behaves very well. You have to get very extreme cases to make the output 'blow up'.
I believe that the normal form for filters is stable under time varying conditions.
Still, I would advise using the direct form if you are comfortable with this structure, unless you plan on some extreme sweeps.
2. Interpolate Filters - do they satisfy the basis filter design constraints?
OK, there is a second major issue in whether the "poles smoothly vary".
What is the intermediate filter shape? Do the interpolated filters still satisfy the desired design constraints of the initial filter prototypes (or 'basis' filters - a nice concept).
For example, a low pass filter design constraints are pretty easy to summarize in text: the filter should be unity gain up to the cutoff frequency, and the zero gain above that in the stop band. Of course, real world filters have pass band ripple, transition band width, and stop band ripple, but the prototype constraints are as above.
Here, I have not seen much published, but I have discovered a couple of things empirically.
First, the parametric eq, when implemented with a 5-multiply direct form biquad, interpolates perfectly. All of the intermediate filters still satisfy the design constraints of a parametric eq, which is nice. You apparently only need 8 basis filters to generate all intermediate parametric eq filters. Note that a 4 multiply bi quad does *NOT* interpolate properly. Again, I can show why they are different with simple algebra, but I cannot show in closed form why the 5-multiply form works so well.
Second, I have found, also empirically, that other filter designs will interpolate very nicely, using linear interpolation of the filter coefficients. What do I mean by 'very nicely'? I plotted the intermediate shapes using Matlab, and the shapes all looked like I wanted them to... Not a precise definition, of course. Still, I was amazed that so many filters worked so well.
Here, I would suggest just taking a couple of basis filters, and then interpolating the coefficients and looking at 16 intermediate filters. It is usually easy to inspect the intermediate filters to determine if they satisfy the design constraints. It is also pretty easy to create a matlab script to test the interpolated filters to see if they still satisfy the constraints.
I can show some cases where this technique fails. In fact, the technique does not work 'properly' with FIR filters. F. Richard Moore shows an excellent example in his Intro to Computer Music book. If you take an FIR filter for a lowpass at say 100 Hz, and linear interpolate the coefficients with the same order filter that has a cutoff of 100 Hz, the intermediate value looks like a sum of the outputs of two filters in parallel - not a low pass filter at all, but a low pass with a funny step in the response. What you probably want is actually a lowpass with a cutoff at an intermediate frequency, which does not happen with an FIR.
What is happening is probably just that the high order FIR filter represents a high order polynomial, and the coefficients do not map simply onto filter parameters.
With a second order direct form filter, the coefficients have simple
relationships to the filter parameters.
Is this why the interpolate ok?
I tried looking empirically at what happens to the poles and zeros of filters whose parameters are being interpolated, and they follow arcs on the z-plane. I did not try to study analytically what was happening. In the case of the parametric eq, amazing weird things happen. I found a number of filters where the poles move smoothly from complex poles to become real poles. Then the real poles would move along the real axis. The movements in the z-plane can be rather complex, actually.
>3. Table lookup with InterpolationParticularly in the complex plane, it's very straightforward, as
the poles and zeros are being interpolated, I think. You're right
that this is a good trick.
And a further generalization of this is
some kind of "basis filter" method, where you have a set of
prototypes and use linear combinations to get the desired
filter as a reconstruction from the set of prototypes, which
acts as a basis for "filter space".3-D audio people use this trick for data reduction on sets of HRTF
filters, I believe. They get the prototype functions from a Karhunen-
Loeve expansion (principal components analysis) of the set of HRTF
vectors itself. Bill Gardner also showed in his WASPAA paper this
year that you can do nearly as well more efficiently by simply choosing
a good subset of the original HRTF set as your basis set.
I wonder if the perspective of filter basis sets could help in understanding
how linear interpolation of direct form coefficients works. See above
regarding FIR coefficient interpolation.
This is exactly the way to test interpolation inside of Matlab, I believe, if I understand the SAOL syntax correctly. I would argue that the best way to try this out is to take a couple of prototype filter parameter sets, for a 2nd order direct form filter, for example, and then interpolate the coefficients, and then plot the results to see what they look like compared to the original filter.kopcode interpfilter(table a, table b, ksig p[2], ksig order) {
By the way, I would not advise using a higher order direct form filter. I do not believe that this will interpolate correctly.
However, lattice filters behave well under interpolation. The all pole case has been used extensively in speech synthesis, with a lot of different filter topologies, and these results are widely published.
In general, outside of the lattice/ladder filters, I would suggest using cascaded second order sections, whether they are direct form or otherwise, for interpolation.
Another filter type that interpolates very badly is the parallel form. (See Jackson, Rabiner Gold, etc.). The parallel form decomposes a higher order transfer function into a parallel combination of second order sections. In this form, a variation in any coefficient causes *all* of the zeros to shift in unpredictable ways. This filter may have nice noise properties, but it is useless for any time varying applications, as far as I can tell. There is no way to get independent parameter variation.
Actually, time varying filters are a huge topic. I wish that I had the time (and the skills) to really study the behavior of various classes of time varying filters analytically. i actually suspect that the math may be intractable for many general cases.
The goal is to design a filter structure that satisfies some design constraint for all values of the parameters under interpolation. Is this tractable?
Anyway, thanks for the engaging discussion.
Cheers -
-dana
---
Dana Massie
dana@beadgame.com
102 Bradley Drive
Santa Cruz, CA 95060
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