Sound synthesis with Periodically Linear Time Varying Filters - - PowerPoint PPT Presentation

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sound synthesis with periodically linear time varying
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Sound synthesis with Periodically Linear Time Varying Filters - - PowerPoint PPT Presentation

Aug 22, 2022 12 likes •263 views

Context Review Dynamic PD FBAM Conclusions Sound synthesis with Periodically Linear Time Varying Filters Antonio Goulart, Marcelo Queiroz Joseph Timoney, Victor Lazzarini Computer Music Research Group - IME/USP - Brazil Sound and Digital

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Context Review Dynamic PD FBAM Conclusions

Sound synthesis with Periodically Linear Time Varying Filters

Antonio Goulart, Marcelo Queiroz Joseph Timoney, Victor Lazzarini

Computer Music Research Group - IME/USP - Brazil Sound and Digital Music Technology Group - NUIM - Ireland antonio.goulart@usp.br

Linux Audio Conference - 2015/04/10

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Context Review Dynamic PD FBAM Conclusions

Motivations

New synth sounds ⊲ Low computational cost Virtual Analog Oscillators Usage as audio effect

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Context Review Dynamic PD FBAM Conclusions

Motivations

New synth sounds ⊲ Low computational cost Virtual Analog Oscillators Usage as audio effect The challenge: ⊲ “When I first got some - I won’t call it music - sounds out of a computer in 1957, they were pretty horrible. (...) Almost all the sequence of samples - the sounds that you produce with a digital process - are either uninteresting, or disagreeable, or downright painful and dangerous. It’s very hard to find beautiful timbres.” Max Mathews, 2010.

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Context Review Dynamic PD FBAM Conclusions

Contribution

LTV theory approach to distortion techniques h(p, n) H(z, n) H(ω, n)

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Context Review Dynamic PD FBAM Conclusions Phase Distortion

Phaseshaping - US patent 4658691 Casio - CZ

Add a phase distortion function to the regular phase generator Sawtooth: Inflection point on the regular (dashed) index t + g(t) =

  • 0.5 t

d ,

0 ≤ t ≤ d 0.5 t−d

1−d + 0.5,

d < t < 1 For d = 0.05

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Context Review Dynamic PD FBAM Conclusions

The allpass filter

H(z) = −a + z−1 1 − az−1 Flat magnitude response Frequency dependent phase shift φ(ω) = −ω + 2 tan−1 −a sin (ω) 1 − a cos (ω)

  • Reverb, chorus, flanger, phaser, spectral delay

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Context Review Dynamic PD FBAM Conclusions

Amplitude modulation

cos (2πfcn) cos (2πfmn) = 1 2 cos (2πfcn + 2πfmn)+1 2 cos (2πfcn − 2πfmn)

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

Jussi Pekonen, 2008

Coefficient-modulated first-order allpass filter as distortion effect Suggests the method for sound synthesis and audio effects Recall that classic PD is restricted to cyclic tables Derives stability condition |m(n)| ≤ 1 ∀n Recommends appropriate values for m(n) Allpass Dispersion on low frequencies φDC(n) = 1 − m(n) 1 + m(n)

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki, 2009

Spectrally rich phase distortion sound synthesis using allpass filter Time-varying allpass transfer function H(z, n) = −m(n) + z−1 1 − m(n)z−1 Phase distortion φ(ω, n) = −ω + 2 tan−1 −m(n) sin (ω) 1 − m(n) cos (ω)

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

φ(ω, n) = −ω + 2 tan−1 −m(n) sin (ω) 1 − m(n) cos (ω)

  • Knowing φ(ω, n), use tan(x) ≈ x,

m(n) = −(φ(ω, n) + ω) 2 sin (ω) − (φ(ω, n) + ω) cos (ω)

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

Emulate the classic phase distortion technique t + g(t) =

  • 0.5 t

d ,

0 ≤ t ≤ d 0.5 t−d

1−d + 0.5,

d < t < 1 Subtract linear phase from the phase distortion function g(t) =

  • ( 1

2 − d) t d ,

0 ≤ t ≤ d ( 1

2 − d) 1−t 1−d + 0.5,

d < t < 1

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

Range for the allpass modulation should be [−ω, −π] φ(ω, t) = g(t)((1 − 2d)π − ω) (1 − 2d)π − (1 − 2d)π − ω Get the modulation function m(n) = −(φ(ω, n) + ω) 2 sin (ω) − (φ(ω, n) + ω) cos (ω) Implementation with difference equations y(n) = x(n − 1) − m(n)(x(n) − y(n − 1)) ⊲

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

Phase distortion and coefficient modulation functions

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

Outputs with classic PD (solid) and modulated allpass (dashed)

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

J.Timoney, V.Lazzarini, J.Pekonen, V.Valimaki

Classic PD and Modulated allpass spectra

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

Arbitrary distortion function

y(n) = 0.4 cos (f0) + 0.4 cos

  • 2f0 − π

3

  • +

0.35 cos

  • 3f0 + π

7

  • + 0.3 cos
  • 4f0 + 4π

3

  • Shift it to the appropriate range

ys(n) = −π 2 (y(n) + 1) 2 Create your own (: ⊲

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

Arbitrary distortion function

Phase distortion and derived modulation functions

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Context Review Dynamic PD FBAM Conclusions Allpass filters coefficient modulation

Arbitrary distortion function

Waveform and spectrum

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Context Review Dynamic PD FBAM Conclusions

J.Kleimola, V.Lazzarini, J.Timoney, V.Valimaki, 2009

FeedBack Amplitude Modulation (FBAM) Revisiting of an old idea by A.Layzer tested by Risset in the catalogue ⊲ Modulate oscillator amplitude using its output y(n) = cos (ω0n)[1 + βy(n − 1)] with ω0 = 2πf0 and y[0] = 0

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Context Review Dynamic PD FBAM Conclusions

FeedBack Amplitude Modulation

y(n) = cos(ω0n)[1 + y(n − 1)] y(n) = cos(ω0n) + cos(ω0n) cos(ω0[n − 1]) + cos(ω0n) cos(ω0[n − 1]) cos(ω0[n − 2]) + ... = ∞

k=0

k

m=0 cos[ω0(n − m)]

cos2(p) = 1

2(1 + cos(2p))

cos3(p) = 1

4(3 cos(p) + cos(3p))

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Context Review Dynamic PD FBAM Conclusions

FeedBack Amplitude Modulation

LPTV interpretation y(n) = x(n) + βa(n)y(n − 1) x(n) = a(n) = cos (ω0n) in this case (but could be =) 1 pole coefficient modulated IIR → Dynamic PD ⊲

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Context Review Dynamic PD FBAM Conclusions

Feedback Amplitude Modulation

β similar to FM’s modulation index

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Context Review Dynamic PD FBAM Conclusions

Feedback Amplitude Modulation

Stability condition

  • β

N

  • m=1

cos (ω0m)

  • < 1

Aliasing before instability

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Context Review Dynamic PD FBAM Conclusions

2nd order FBAM

Two previous outputs with individual βs y(n) = cos (ω0n)[1 + β1y(n − 1) + β2y(n − 2)] Narrower pulse and wider band ⊲

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Context Review Dynamic PD FBAM Conclusions

Conclusions

Reissue of a classic technique Different kind of implementation Input and modulation can be arbitrary signals Deeper investigation of LTV Studying 2nd and higher order systems stability Thanks a lot! antonio.goulart@usp.br

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