Real-time simulation of a family of

fractional-order low-pass filters

Thomas Hélie Equipe analyse et synthèse des sonsIRCAM-CNRS UMR 9912-UPMC

1, place Igor Stravinsky75004 Paris, France

135th Convention of the

Audio Engineering Society

17 October 2013, New York, USA

A1. Motivation1. First-order low-pass filter (cutoff frequency fc)

1. Below fc: unit gain2. Above fc:

attentuation of -6dB/octave, Dephasing of -180°

2. Zero-order filter = constant gain [0dB/oct,0°]

Question: Is there something in between ?Can we go from [0dB/oct,0°] to [-6dB/oct,-180°]

in a continuous way ? Fractional order 0≤α≤1

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A2. DefinitionFamily of stable causal low-pass filters of

orderα Cutoff freq.: fc in [20Hz,20kHz]

Order: αin [0,1]

Laplace domain Re(s)>0 : Transfer function:

Fα,fc(s) = Hα( s/(2π.fc) )

with

choosing the principal value of the power to α in the complex plane.

Bode diagram: Hα(s=iω)

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OutineA. Motivation and definition of filtersB. Exact representation (complex analysis)C. Finite dimensional approximations (2 methods)

1. Interpolation of the state2. Optimization w.r.t. an audio objective function

D. Simulation and sound examplesE. Conclusion

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B1. Exact representation:What is the difficulty? An example to start

A. The causal time integrator of order ½. What is it ?

Property: 1. Long memory (~1/√t)2. |H| ~ -3dB/oct ~

-10dB/decade (Fourier domain: s=2iπ f )

B. Integral (also called, diffusive) representation. What is it?

C. In summary: One aggregates an infinite continuous set of one-pole filters over C !

1. LT-1t>0:

2. H is analytic over(here, the cut

is ) 3. Residue theorem:

Result & definition:a)

b)

c) Well-posed if

Difficult to simulate

Result (a-c) can be applied to our family of filters.

B2. Exact representation of Hα: cut C, weight μ ? (0<α<1)

1. Analysis Transfer function:

Cut: is cut on

Weight:

Well-posed? Yes!

2. Result Exact formula:

Interpretation: Hαis an infinite continuous combination of one pole-filters where: Poles σ=-1-ξdescribes C Associated gains at f=0

are μ(σ)

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C. Finite dimensional approximations (2 methods)

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?

C1. Finite dim. approximation: Method 1: interpolation of the state1. Method 1

(details in the paper)

1. Pole placement on the cut C:

for a large geometrical sequence

2. Interpolation of the dynamic state associated with poles σ on C by those of the finite set σn

3. Closed-form formula

2. Result (N poles between l0=-10 and

lN+1=+10)

N=20 poles

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C2. Finite dim. approximation: Method 1: Results

N=40 poles

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N=20 poles

C3. Finite dim. approximation: Method 1: conclusion

In practice, this method requires:

A large range for the poles with l0=-10 and lN+1=+10

A large number N of poles (about 40)

Question (especially for real-time issues):Can we reduce N while preserving accuracy ?

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C4. Finite dim. approximation: Method 2: optimization

1. Principle: 1. Pole placement: similar to

method 12. Optimization of weights3. Objective function based on

audio features:a) The frequency range for Hα

covers the audible range.b) Frequencies are perceived

according to a log-scale.c) Errors are perceived

relatively to the exact values

2. Objective function: In theory:

with ωmin=10-3 and ωmax= 10+3 (dimensionless)

In practice (see paper): integralfinite sum Add a Tikhonov penalty

term (condition number)

Matrix formulation & closed-form solution

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C4. Finite dim. Approximation: Method 2: Results (in the paper)

(N poles between l0=-5 and lN+1=+5)

N=20 poles

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N=10 poles

C5. Finite dim. approximation: Method 2: conclusion

A good accuracy is obtained for all:Orders α,Cutoff frequenciesThe audible range

Approximations are obtained for: In the paper: N=20 poles (l0=-5, lN+1=+5)More recently: N=13 poles (& no need of Tikhonov

penalty)

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D. Simulation and sound examples

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?Stable time-domain simulations

1. State-space representation (continuous time)a)

b) With a tunable cutoff frequency: Fα,fc(s) = Hα( s/(2π.fc) ) Replace by

2. Numerical scheme (discrete time) Exact exponential kernels for a sample-and-hold

inputor Bilinear tranform, etc

D1. Simulation: 1. Time domain & 2. Numerical scheme

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1. Simulation (Matlab, sampling frequency=96kHz)a) White noise, fc=440Hz, α goes from 0 to 1

(step=0.1)

b) Square wave, same fiters

2. Real-time simulation :a) FAUST codeb) for N=13 and the bilinear transform

D2. Simulation: Sound examples

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1. Fractional order low-pass filters an infinite continuous combination of one pole-

filters

2. Approximations Finite combinations: N=13 poles with optimized

weights

3. Simulation Numerical schemes applied to one-pole filters Guaranteed stability (even for time-varying parameters)

4. Real-time program (in FAUST language)ALSO AVAILABLE: IEEE-TASLP paper

E. Conclusion

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Thank you for your attention !