Real-time simulation of a family of fractional-order low-pass filters TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA Thomas Hélie Equipe analyse et synthèse des sons IRCAM-CNRS UMR 9912-UPMC 1, place Igor Stravinsky Paris, France 135 th Convention of the Audio Engineering Society 17 October 2013, New York, USA

A1. Motivation 1. First-order low-pass filter (cutoff frequency f c ) 1. Below f c : unit gain 2. Above f c :  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 2

A2. Definition Family of stable causal low-pass filters of orderα Cutoff freq.: f c in [20Hz,20kHz] Order: α in [0,1] Laplace domain Re(s)>0 : Transfer function: F α,fc (s) = H α ( s/(2π.f c ) ) with choosing the principal value of the power to α in the complex plane. Bode diagram: H α (s=iω) 3

Outine  Motivation and definition of filters  Exact representation (complex analysis)  Finite dimensional approximations (2 methods) 1. Interpolation of the state 2. Optimization w.r.t. an audio objective function  Simulation and sound examples  Conclusion 4

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA 5 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| ~ -3 dB/oct ~ -10 dB/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 -1 t>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 H α : cut C, weight μ ? (0<α<1) 1. Analysis Transfer function: Cut: is cut on Weight: Well-posed? Yes! 2. Result Exact formula: Interpretation: H α is an infinite continuous combination of one pole- filters where: Poles σ=-1-ξdescribes C Associated gains at f=0 are μ(σ) 6

C. Finite dimensional approximations (2 methods) 7 ?

C1. Finite dim. approximation: Method 1: interpolation of the state  Method 1 (details in the paper) 1. Pole placement on the cut C: for a large geometrical sequence 1. Interpolation of the dynamic state associated with poles σ on C by those of the finite set σ n 2. Closed-form formula 2. Result (N poles between l 0 =-10 and l N+1 =+10) N=20 poles 8

C2. Finite dim. approximation: Method 1: Results N=40 poles 9 N=20 poles

C3. Finite dim. approximation: Method 1: conclusion In practice, this method requires: A large range for the poles with l 0 =-10 and l N+1 =+10 A large number N of poles (about 40) Question (especially for real-time issues) : Can we reduce N while preserving accuracy ? 10

C4. Finite dim. approximation: Method 2: optimization  Principle: 1. Pole placement: similar to method 1 2. Optimization of weights 3. Objective function based on audio features: a) The frequency range for H α 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 ω min =10 -3 and ω max = (dimensionless) In practice (see paper) : integral  finite sum Add a Tikhonov penalty term (condition number) Matrix formulation & closed- form solution 11

C4. Finite dim. Approximation: Method 2: Results (in the paper) (N poles between l 0 =-5 and l N+1 =+5) N=20 poles 12 N=10 poles

C5. Finite dim. approximation: Method 2: conclusion A good accuracy is obtained for all: Orders α, Cutoff frequencies The audible range Approximations are obtained for: In the paper: N=20 poles (l 0 =-5, l N+1 =+5) More recently: N=13 poles (& no need of Tikhonov penalty) 13

D. Simulation and sound examples 14 ? Stable time-domain simulations

1. State-space representation (continuous time) a) b) With a tunable cutoff frequency: F α,fc (s) = H α ( s/(2π.f c ) )  Replace by 2. Numerical scheme (discrete time) Exact exponential kernels for a sample-and-hold input or Bilinear tranform, etc D1. Simulation: 1. Time domain & 2. Numerical scheme 15

1. Simulation (Matlab, sampling frequency=96kHz) a) White noise, f c =440Hz, α goes from 0 to 1 (step=0.1) b) Square wave, same fiters 2. Real-time simulation : a) FAUST code b) for N=13 and the bilinear transform D2. Simulation: Sound examples 16

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 17

18 Thank you for your attention !