Signal prediction using fractional derivative models

In this chapter the linear prediction (LP) and its generalisation to fractional linear prediction (FLP) is described with the possible applications to one-dimensional (1D) and two-dimensional (2D) signals. Standard test signals, such as the sine wave, the square wave, and the sawtooth wave, as well as the real-data signals, such as speech, electrocardiogram and electroencephalogram are used for the numerical experiments for the 1D case, and greyscale images for the 2D case. The 1D FLP model is proposed to have a similar construction as the LP model, i. e. it uses a linear combination of fractional derivatives with different values of the fractional order. The 2D FLP model uses a linear combination of the fractional derivatives in two directions, horizontal and vertical. The scheme for the computation of the optimal predictor coefficients for both 1D and 2D FLP models is also provided. The performance of the proposed FLP models is compared to the performance of the LP models, confirming that the proposed FLP can be successfully applied in processing of 1D and 2D signals, giving comparable or better performance using the same or even a smaller number of parameters.