David LowenfelsCCRMA Stanford

(MA/MST Graduate)

Virtual AnalogSynthesis

with the Time Varying Comb Filter

Waveform morphing, PWM, detuning, FM.

You’ve probably seen and heard it before... I’ll show you how they are connected.

One simple structure can generate all these classic effects!

Introduction

Sawtooth:

Square:

“Parabola”:

Triangle:

Waveform Definitions

∞∑n=1

1n2

sinnθ

∞∑n=1

1n

sinnθ

∞∑n=1,3,5,...

1n

sinnθ

∞∑n=1,3,5,...

1n2

sinnθ

(Pulse/Rect.)

(Ramp)

BLIT Synthesis

IntegratorBP-BLIT

IntegratorBLIT

Stilson and Smith (ICMC ‘96)

BLIT:

Sawtooth:

Sqr/Pulse:

Tri/Ramp: IntegratorPulse Rescale

N∑k=0

sin (kθ) =sin ((2N + 1)θ)

sin θ

Deriving the Comb Filter

H(z) = 1 − z−T/2

BLIT:

BP-BLIT:

Notice: BP-BLIT(k) = BLIT(k) - BLIT(k - T/2)

Equiv. Comb Filter:

(All harmonics)

(Odd harmonics)

H(z) = α + βz−MGeneral Form:

Period (T)

Comb filter equivalency applies to steady state only.

Glitch in first period while delay is empty.

Delay could be precalculated and “stuffed”.

But it’s perceptually insignificant.

A minor issue...

“Echo” coefficient [0,1]; parametrically controls level of even harmonics.

Morph between sawtooth and pulse waves.

Equivalent to linear crossfading.

Waveform Morphing

β

H(z) = 1 − βz−M

Various effects, depending on modulator waveform and frequency.

Robust modulation needs interpolated delay.

Comb Filter ModulationH(z) = α + βz−M

PWM is commonly done with a feed-foward “null” flanger.

Destructive interference creates spectral notches.

Delay period is inversely proportional to notch spacing.

Works with any sawtooth synthesis method.

Same idea works for tri/ramp “skew” modulation, use parabola wave instead.

Pulse Width Modulation

Rapid phase modulation causes frequency shifts, proportional to derivative of modulator.

Useful for vibrato, Leslie, FM timbres.

Change coefficients for delay-only signal ( ).

“Frequency” Modulation

α = 0

Phase modulate the oscillator with a constant slope modulator (sawtooth).

Narrow or wide detuning (harmonization)

No Crossfade needed!

Vintage “analog” sound.

Can introduce aliasing in some cases.

Unison / Detuning

Sync effects are also possible with this delay modulation method.

The secret is in the modulator waveform!

Hard Sync?!

Simple architecture, many effects.

Generally works with any periodic waveform synthesis technique.

To reduce aliasing, needs high-order interpolation, possibly oversampling.

Summary

Julius O. Smith III

Chris Chafe

music-dsp mailing list at calarts

Hiroko Terasawa and the MST ’03 gang

Acknowledgements