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This Sound is just a stereo input split into the Left and Right channels and fed to two identical Kyma Allpass Filters. These filters are order 12, meaning that we should get 1,080 degrees of phase shift centered around the filter frequency. You can narrow the region over which this phase shift occurs by increasing the filter Q.

What I find interesting, harking back to the discussion about dispersion, is that the dispersion of an Allpass filter leads to a group delay for frequencies near the filter tuning frequency. This group delay has been eyeball measured using CoolEdit? spectral displays, and it corresponds approximately to

Tgrp = N*Q/(4*Fc)

where N is the filter order, Q is the filter Q, and Fc is the filter frequency.

With these filters tuned to around 40-50 Hz, and Q's ranging from 0.5 to 2 you hear a very pronounced effect on the bass response of kick drums and the like. Plugging numbers Fc = 40 Hz, and Q = 1 into the above formula shows that the group delay ought to be around 75 ms!! That seems like an enormous delay to me, but the sonic effects are quite attractive.

Group delay is somewhat different from phase delay, although the two are related. Phase delay refers to a progressive displacement of a sinewave after going through the delaying element. All filters produce some amount of phase delay. However, with a pure sinewave it is impossible to tell where the beginning and end of it is, since it lasts for an indefinite (infinite) amount of time. All we can tell with pure sinewaves is that two waves are out of phase synchrony after one is phase delayed relative to the other.

But group delay refers to the physical delay of a small packet of sound with a narrow bandwidth. This delay is a physical delay in that it actually retards the delivery of sound energy. The group delay of a network is defined as the negative derivative of the phase shift with respect to frequency. As such, unless you unwrap the measured or computed phase shift of the filter it is difficult to guage. But using first order reasoning, namely that the bandwidth over which the phase shift of the Allpass filter applies is approximately Fc/Q, and that the overall phase shift from low frequencies to high is about -N*90 deg, you can arrive at the approximate equation shown above.

When you examine a recording of a stream of impulses fed through the Allpass filter the sonogram shows that frequencies in the vicinity of Fc are indeed delayed by approximately the amount shown by that equation. An impulse ordinarily contains an equal mix of all frequencies arriving all at the same instant of time. After sending the impulse through the Allpass filter you see all the frequencies arriving together as before, except that those frequencies in the vicinity of Fc are displaced in time, arriving later.

Hence, by delaying the bass energy in the vicinity of 40-50 Hz by as much as 50-100 ms the bass takes on a deep rich character that can only be appreciated by listening to it. I find this quite interesting. An allpass filter, by design, imparts nothing more than phase shifts to the input signal. The amplitude response of an allpass filter is completely flat with frequency, so the bass enhancement is not a product of boosting the amplitude of the bass energy. Rather the effect is solely the result of selective delaying of the bass energy to the speakers or headphones.

Is this interesting? or what!

[When I first began this investigation, it was unclear to me whether bass response would be enhanced by delaying or advancing the bass frequencies. An allpass filter is only capable of delaying frequencies, not advancing...

It is a common misconception that you can achieve phase linear behavior with IIR filter, as used in most parametric EQ's, by first recording the processed signal through the filter, then time reversing the recording and sending it through the filter again, finally time reversing to restore the original sound. This is simply false, except in the rare cases of completely time symmetric sounds.

So there is no way to use an IIR filter to advance a signal, producing a negative group delay. Oh sure, you could retard all but the frequencies of interest, producing the equivalent effect of group advancement. But such a filter would be very complicated to design. [...and yes, now that I think about it, it would be possible to achieve group advancement by simply applying group delay to a time reversed recording, then reversing the result... not phase linear, but defintely time reversed! I'll give that a try!]

However! You can produce whatever you want by means of convolution. A simple chirp can serve as a convolution filter (FIR filter) to produce either retarded or advanced signal outputs. I tried this very thing using the Kyma FIR filter.

But alas, the Kyma FIR filter is limited in its duration capability, and practical filters, with about 256 taps, don't have enough length to affect the lowest frequencies very well, and the amount of dispersion in time is limited to about half that number of samples. At 44.1 KHz, 128 taps represents only about 3 ms, not nearly the amount of dispersion obtainable with the Kyma 12th order Allpass Filter.

So, in the end, I hear a pleasant bass enhancement by means of group delay with the Allpass Filter. I have no idea what it would sound like with a similar amount of group advancement in the bass.

[... well, I just did the experiment of sending the time reversed recording through the APF and then reversing the result. It sounds horrible. A definite chirp to the upside in all the bass drums. So group delay it is...]


-- DavidMcClain - 16 Dec 2003

i've read that lexicon reverbs are/were built around allpass filters fed into delay lines with some sort of feedback loop (figure-8 feedback loop was the statement i seem to remember) and delay modulation to smooth out the sound.

-- BenPhenix - 17 Dec 2003

Yes! Almost all synthetic reverbs are built with Allpass filters in them. See the Euverb Sounds in Kyma. Up to now, that seemed like about the only application for them. But with typical filter orders and higher frequencies the group delays obtained are minimal, and so it doesn't seem like that is their purpose in these reverbs. But I could be wrong about that...

Any time you have flat amplitude response coupled with some delay you have an allpass system. Reverbs use extremely high order allpass systems to give the impression of echoes. A delay of N samples is an allpass system with a linear phase response. The Allpass filters in Kyma are designed to give nonlinear phase response - which means that instead of a uniform delay for all frequency components, some are delayed more than others.

Anyone have some insight into why Allpass filters are used in synthetic reverbs?

[ I guess one approach would be to dissect the Euverb and remove the Allpass filters in there to see what happens to the sound without them...

Well, I just did that. I took a look. What the Euverb has are Allpass Delays not Allpass Filters. An allpass delay is just about useless without some feedback. I suspect that is their sole purpose.

So does anyone have any other ideas where Allpass Filters are useful? About the only place I can imagine is for use in correcting phase alignment in dispersive systems -- such as periodic insertion along a long telephone cable to correct for the dispersion due to the accumulated inductance of the cable. I have never heard of studio engineers using allpass filters, but I admit to extreme ignorance in this area. Anybody? ]

[ ahh... there is one other place I have used them... A first order Allpass filter can be used to impart a nearly constant group delay at lower frequencies. This is most useful for tuning a delay line with feedback to fractional sample delays. A delay element incorporates the integer number of samples delay, while the first order allpass filter imparts a fractional sample delay in series with the delay element. You'd do this to tune a waveguide synthesis engine. As you go higher in pitch the coarseness offered by integral numbers of samples delay is too great. So the first order allpass allows you to tune the instrument to the needed fractional samples to remain in tune as you go up in pitch.

But other than this, where have people used these higher order Allpass filters? ]

-- DavidMcClain - 18 Dec 2003

Hi Ben and David I didn't know the lexicon used all pass filters and this is an interesting fact.

Good reverbs needs a lots of delays with feed back. In general the more delays the less colouring. If you used just one delay with feed back you would get a comb filter with all the teeth in tune. This would make a reverb sound with one very pronounced ringing note and would therefore colour the sound by a large amount. Maybe by putting Allpass filters in the delays (within the feedback path) the teeth of the combs would no longer be in tune and a lot less delays would be needed to mask the colouring. Also controllable damping is needed by putting filters in the feedback paths, and maybe some clever use of the settings of these allpass filters will do this second job as well. Note: the damping on the Euverb is fixed because the Brightness control in the Resonator module is not hot. When I can get back to doing more DSP stuff I hope to make a module with a bank of resonators with hot parameter for this damping control.

One use of a Hi order all pass filters was the old Phaser. This was a totally analogue device which used a string of about 10 controlled allpass filter and then its out put was added to the input, right at the end, and made a sweeping comb filter with about 5 teeth.

Another thought I had was to use allpass filters in chorus units. They used to use chorus units in string machines to emulate multiple violins, but the chances of points in time where total cancellation or total in phaseness could happen were high. But two real violins playing together at the same time would never cancel out. This is because the real violins would not produce the same harmonic phase relationship as each other(i.e. one harmonic will cancel out from time to time but not all harmonics at the same time). Therefore if an allpass filter were used to feed one of the chorus delay legs , total cancellation would not be possible and should sound more real.

-- PeteJohnston - 18 Dec 2003

The non-linear aspect of Kyma's Allpass filters attracts me and makes me wonder about all sort of applications including reverb, distortion, and oscillator synthesis. I'll try to go back and dig up some info on the Lexicon reverb theory, but Pete's comment about using Allpass to help mask colouring seems right. I have seen some reverbs that were nothing but a series of Allpass filters, but they tend to be metallic, same with comb-filters into allpass filters.

On the Reverb topic, I know FIR filters are becoming more used for the early reflections with All-pass and delay networks filling out the rest. I've heard mention of about wave-guide techniques as well, perhaps David could explain that more. Actually, I don't know what waveguide is to begin with (about time search google).

-- BenPhenix - 18 Dec 200

Ahh, Interesting about using phase shifts in chorusing units to avoid cancellation. Thanks for that Pete!

A waveguide synthesizer is nothing more than a delay line with feedback. Sometimes the feedback incorporates 1-pole Allpass filters to enable the creation of fractional sample delays in addition to the whole number of samples delay from the delay line.

These are called "waveguide synthesizers" because of the analogy to a mechanical resonator. Think of a tube as a waveguide. Boundary conditions on the ends of the tube, e.g., open to the air, or sealed, cause the waveguide to become a resonator. The resonant frequency of the tube depends on its length and those boundary conditions at the ends of the tube. An open tube, or one sealed off at both ends, resonates at frequencies twice as high in pitch as a tube with one end sealed off. Our delay line mimmicks the behavior of the tube, and feedback, positive or negative, provides the equivalent of boundary conditions on the ends of the tube. Positive feedback corresponds to the open tube, or one sealed at both ends, while negative feedback corresponds to a tube sealed at only one end - a so-called "quarter wave resonator". The open tube is a "half-wave resonator".

[gee, I hope I got those two correct... being dislexic means that anytime I have a choice of two alternatives, I'm almost certain to get it wrong the first time. Give me more than two choices and I have no problem...]

A tube is referred to in esoteric circles as a "waveguide" because, using it to pipe sound, you can guide the sound waves to wherever the tube leads. But that's all it really is. Radio systems often use electromagnetic waveguides to do a similar job. You can "pipe" the radio energy from the transmitter to wherever you need the energy using a waveguide. Electromagnetic waveguides take advantage of Maxwell's Equations describing radio and lightwave energy to guide the energy along the inside of a conductive tube. Sometimes these tubes are circular in cross section, but frequenty they are designed with rectangular cross sections to enhance the guiding of particular kinds of waves and attenuate the propagation of other kinds of waves. Optical fibers are another kind of "waveguide" used to pipe light from one place to another. Tubes used for sound piping make use of mechanical properties of the air reflecting off the inside boundary of the tube.

You can find great discussions of waveguide synthesis, and other kinds of "physical modeling synthesis" in the CSound sites and books. A web search for CSound will surely point you there. IRCAM in France also does a lot of work in physical modeling synthesis: search for IRCAM.

The Tassman synthesizer from Montreal is based largely on a variety of "physical modeling" synthesis techniques, including waveguide synthesis. These guys all have Ph.D.'s in Acoustics from IRCAM in France. They moved to Montreal because it is easier to do business from Canada than it is from France.

-- DavidMcClain - 19 Dec 2003

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