Fractional Signal Synthesis And Fractional-Order Filters With Applications

In this dissertation, we will introduce some complex random signals which are characterized by the presence of heavy-tailed distribution or non-negligible dependence between distant observations, from the'fractional'point of view. Furthermore, the analysis techniques for these fractional signals are investigated using the'fractional thinking'. The term'fractional signals'in this dissertation refers to some random signals which manifest themselves by heavy-tailed distribution, long range dependence (LRD)/long memory, or local memory. Fractional processes are widely found in science, technology and engineering systems. Typical heavy-tailed distributed signals include underwater acoustic signals, low-frequency atmospheric noises, many types of man-made noises, and so on[1-3]. Typical LRD/long memory processes and local memory processes can be observed in financial data, communications networks data and biological data[4-6]. These properties, i.e., heavy-tailed distribution, LRD/long memory, and local memory always lead to certain trouble in correctly obtaining the statistical characteristics and extracting desired information from these fractional processes. These properties cannot be neglected in time series analysis, because the tail thickness of the distribution, LRD, or local memory properties of the time series are critical in characterizing the essence of the resulting natural or man-made phenomena of the signals. Therefore, some valuable fractional-order signal processing (FOSP) techniques were provided to analyze these fractional processes. FOSP techniques are based on the fractional calculus, FLOM and FrFT basic theories. This dissertation concentrates on the synthesis of variable-order fractional signals, the realization of variable-order fractional systems, distributed-order fractional filters, and optimal fractional-order damping systems.Simulation of random signals is a valuable tool in random signal processing. Most random processes can be generated by performing time domain integer-order filtering on a white Gaussian process[7,8]. Similarly, the fractional random processes can be simulated by performing the time domain fractional-order filtering on a white Gaussian process or a whiteα-stable process[9,10]. In his dessertation, a synthesis method, which is based on variable-order fractional operators, for multifractional Gaussian noises (mGn) is proposed by studying the relationship of white Gaussian noise (wGn), mGn, and multifractional Brownian motion (mBm). Furthermore, a synthesis method for multifractionalα-stable processes, the generalization of mGn, is proposed in order to more accurately characterize the processes with local scaling characteristics and heavy-tailed distributions.The discussion of the distributed-order fractiona filters is presented in the dessertation. The distributed-order fractionla filter is motivated by the advantages of the classical fractional-order filter which is a generalized case of the integer-order filters. It can be proved that both integer and fractional-order systems are special cases of distributed-order fractional systems. Particularly when the complexity, networks, nonhomogeneous, multi-scale and multispectral are considered, the distributed-order fractional filter becomes a more precise tool to describe the above phenomena[11-15]. Therefore, motivated by the applications of the distributed-order fractional filters to control, filtering and signal processing, the distributed-order fractional filters are derived step by step in the dessertation.This dessertation also explores the potential benefits of three types of fractional-order damping systems using the performance criteria which are based on the step response error in frequency domain and time domain. Three types of fractional-order damping systems are: fractional-order damping without time delay; fractional-order damping with time delay, and distributedorder fractional damping. In frequency domain, the time-delayed fractional-order and distributed-order fractional damping systems were optimized by minimizing the integral of the squared step response error (ISE) performance criterion. In time domain, these three types of fractional-order damping systems were numerically optimized by finding the minimum of ISE, the integral of the time-weighted error squared (ITSE), the integral of absolute error (IAE), and the integral of time-weighted absolute error (ITAE) performance measures. The optimum coefficients and minimum performance indexes for ISE, ITSE, IAE and ITAE criteria of these three types of fractional-order damping systems are provided.Three application examples of FOSP techniques in biomedical signals are presented in the end of this dessertation. These application examples provide the instructions on how to generalize the conventional signal processing methods to FOSP techniques, and how to obtain more valuable information by using FOSP techniques.