Emulate an Analog Nonlinear Audio Device, is it Possible? ABAV, Belgium, 2016 Schmitz Thomas Department of Electrical Engineering and Computer Science, University of Liège 24 janvier 2016 1 / 18
Communication Outline 1 Why nonlinear models are important? 2 Non-linear modelling IR measurement 3 Real audio devices emulation Objective evaluation of the model Model limitations 4 Conclusion 2 / 18
Why nonlinear models are important? Section 1 Why nonlinear models are important? 3 / 18
Why nonlinear models are important? Nonlinear Devices A system is nonlinear if : it does not respect the principles of superposition (additivity) and homogeneity (scaling) Almost all audio devices exhibit a nonlinear behavior Loudspeakers. Amplifiers. Compressors. Guitar and Keybord sound effects. Microphones. Acoustic field : More generally, the system equations governing fluid dynamics (for sound waves in liquid or gazes) and elasticity (for sound waves in solid) are nonlinear. 4 / 18
Why nonlinear models are important? Why Try to Emulate Audio Devices Figure: Home studio! Figure: Home studio? Advantages of the simulation : Wide variety of sounds and timbres. Weight and overcrowding. Cheaper. 5 / 18
Non-linear modelling Section 2 Non-linear modelling 6 / 18
Non-linear modelling IR measurement Nonlinear Systems Identification Techniques Autonomous models Physical model (non black box). NARMAX model (difficult to identify). MISO model (difficult to identify). Volterra model (difficult to identify). Neural network (too many parameters).... Non autonomous models Extended Kalman filter. Particle filtering.... 7 / 18
Non-linear modelling IR measurement Nonlinear Emulation : Volterra Series Figure: Volterra decomposition in subclass models 8 / 18
Non-linear modelling IR measurement Miso to Hammerstein model Power series model (.) 1 h 1 [n] x[n] (.) 2 h 2 [n] y[n] (.) M h M [n] Figure: Polynomial Hammerstein model y[n] = x[n] h 1 [n] + x[n] 2 h 2 [n] +... + x[n] M h M [n] (1) 9 / 18
Non-linear modelling IR measurement OHKISS method From Identification to Emulation Exponential Sine Sweep ss[n] Device Under Test Input Guitar Signal y 1 [n] x[n] Inverse Sine Sweep ss[n] Computation Hammerstein Kernels h m[n] Hammerstein Model Emulation y 2 [n] Output Emulated Device OHKISS Emulation 10 / 18
Non-linear modelling IR measurement Deconvolved ESS Through a NL Device z[n] = y[n] x 1 [n] (2) 0.1 0.08 0.06 h 5 h 4 h 3 h 2 h 1 0.04 Amplitude 0.02 0 0.02 2 =2.01s 0.04 0.06 3 =3.18s 0.08 0.1 15 16 17 18 19 20 21 Time(s) Figure: z[n], the deconvolution of the ESS through a Nonlinear device From measured kernels to Hammerstein kernels (see [5]) h m[n] compute = h m[n] (3) 11 / 18
Real audio devices emulation Section 3 Real audio devices emulation 12 / 18
Real audio devices emulation Objective evaluation of the model Real Output Device Signal vs Emulated Signal ss[n] NL device under test y 1 ss[n] (.) M Compute Volterra kernels z[n] h m[n] ss m [n] Nonlinear Convolution y 2 0.8 0.6 y 1 y 2 0.4 Amplitude 0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 Sample [n] Figure: Output signal from TSA15 (y 1)and emulated signal (y 2) comparison. 13 / 18
Real audio devices emulation Model limitations So what? 1 y 1 y 2 0.5 Amplitude 0 0.5 Figure: The Grail of emulation techniques? 1 0 50 100 150 200 Sample [n] Figure: Comparison between the real output signal with an input amplitude of 0.5 (y 1) and the simulated signal where Hammerstein kernel where calculated for a input amplitude of 1 (y 2) Principal limitation : Input level dependency 14 / 18
Conclusion Section 4 Conclusion 15 / 18
Conclusion Questions? Hammerstein model do not fits for all input amplitude but works well for a given level. Can we measure several models at different levels and switching between them? Do we have to try others models and identification methods? 16 / 18
Conclusion Thank you for your attention! 17 / 18
Conclusion Bibliography M.Schetzen, The Volterra & Wiener Theory of Nonlinear Systems (John Wiley & Sons, 1980). T.Ogunfunmi, Adaptive Nonlinear System Identification : The Volterra and Wiener Model Approaches (Springer, 2007). F. Kadlec, P. Lotton, A. Novak, & L. Simon, A new method for identification of nonlinear systems using miso model with swept-sine technique : Application to loudspeaker analysis, presented at 124th Convention of the Audio Engineering Society (2008 May), convention paper 7441. A. Farina, A. Bellini, & E. Armelloni, Nonlinear Convolution : A New Approach for the Auralization of Distorting Systems, presented at the 110th Convention of the Audio Engineering Society (2001 May), convention paper 5359. T. Schmitz, J.J. Embrechts, Nonlinear Guitar Loudspeaker Simulation, presented at the 134th Convention of the Audio Engineering Society (2013 May),convention e-brief 96. 18 / 18