Physically-informed Prediction - Tractable Nonlinear Signal Processing
The traditional signal processing approach to such problems as speech, radar, acoustic and seismic processing, involves the use of classical techniques such as the discrete Fourier transform and linear prediction analysis. At the core of these techniques is the elegant theory of time-invariant linear systems, and the interaction with random signals that can be wholely described using only mean and variance [1]. These techniques are powerful but limited to physical situations where the linear, time-invariant approximation is valid. Although this is true in many circumstances, for complex biological or social systems the linear approximation often fails. In these circumstances, a nonlinear signal approach is required.
The description nonlinear signal processing, as defined in the negative, is generally too broad to be useful:
of the many approaches that could be taken, only some have been shown to be valuable. These include methods
based upon higher-order statistics such as the bispectrum and bicoherence [2], Bayesian statistical approaches
such as generalised variational autoregression [3] and general nonlinear predictors
[4].
We have produced new nonlinear signal processing methods based around the use of discrete variational integrators. Variational integration names a class of numerical integration techniques that respect the energy properties of particular models [5]. We have produced a model of nonlinear vocal fold oscillation (model diagram shown above) and, using the variational integration method, constructed a nonlinear predictor for speech signals [6]. This allows effective nonlinear model parameter estimation which could be used in speech analysis tasks such as biometrics and speech compression.
[1] J.G. Proakis, D.G. Manolakis (1996), Digital signal processing : principles, algorithms, and applications. 3rd ed, Upper Saddle River, N.J.: Prentice Hall. 1 v. (various pagings).
[2] C.L. Nikias, A.P. Petropulu (1993), Higher-order spectra analysis : a nonlinear signal processing framework. Prentice Hall signal processing series, Englewood Cliffs, N.J.: PTR Prentice Hall. xxii, 537 p.
[3] S.J. Roberts, W.D. Penny (2002), Variational Bayes for generalized autoregressive models. Ieee Transactions on Signal Processing, 50(9): pp. 2245-2257.
[4] H. Kantz, T. Schreiber (1997), Nonlinear time series analysis. Cambridge nonlinear science series ; 7, Cambridge ; New York: Cambridge University Press. xvi, 304 p.
[5] E. Hairer, C. Lubich, G. Wanner (2002), Geometric numerical integration : structure-preserving algorithms for ordinary differential equations. Springer series in computational mathematics, 31, Berlin ; New York: Springer. xiii, 515 p.
[6] M. Little, P. McSharry, I. Moroz, S. Roberts (2005), A simple nonlinear model of vocal fold dynamics in Proceedings of 3rd International Conference on Non-Linear Speech Processing, NOLISP'05: Barcelona, Spain. pp. 188-203.