The goal of this paper is to describe some applications of fractional order calculus to biomedical signal processing with emphasis on the ability of this mathematical tool to remove noise, enhance useful information, and generate fractal signals. Three types of digital filters are considered, namely, lowpass differentiation filter, smoothing filter, and 1/fβ-noise generation filter. The filter impulse responses are functions of the fractional order and the sampling period only, and thus can be computed easily. Application examples are presented for illustrations.

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The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Padé, Prony and Shanks' methods have been considered in this paper. Using Al-Alaoui's rule as s-to-z transform, computation examples show that both Prony and Shanks' method can achieve more accurate fractional differentiation and integration than Padé method which is equivalent to continued fraction expansion technique.

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