Optimal Fractional Signal Analysis And Its Applications

In order to extract more useful information from the complex signal environments,representing and analyzing the complex information through effective signal analysis methods have become the crucial issue.For the complex information,the traditional signal analysis methods are always not effective.However,some fractional order signal analysis methods may solve these problems.The fractional order signal analysis has gradually become a research hot spot in the modern signal analysis and processing.In particular,the fractional Fourier analysis and the fractional order calculus for signal processing have attracted the most attention.In this thesis,some key issues of the fractional Fourier analysis and the extended researches about the fractional order filter design problem that contains fractional order derivative are investigated.Since the fractional order signal processing includes freedom fractional order variables,the optimization methods are employed to optimize these variables for achieving the best performance.The research includes the following aspects:1.Compressed sensing(CS)has shown great potential in accelerating data acquisition procedure for magnetic resonance imaging(MRI).For compressed sensing magnetic resonance imaging(CS-MRI),the incoherence between the sensing matrices and the sparse matrices is a key role of the performance.However,in conventional CS-MRI,the sensing matrix is Fourier matrix which is not optimally incoherent with the sparse transform matrices.Moreover,Fourier encoding weakly spreads out energy and concentrates the energy in the center of the k-space.This will further reduce the randomness of the under-sampling pattern.Therefore,for the Fourier based CS-MRI,incoherence between the sensing matrices and the sparse matrices will be weak and lead to a degradation of images reconstruction quality for highly under-sampling factors.In this thesis,we investigate spread spectrum incoherent sampling compressed sensing MRI using fractional Fourier transform.Simulation results shown that the fractional Fourier transform encoding can spread out the energy more uniformly than the conventional Fourier encoding.Then it is beneficial for designing the incoherent sampling pattern,such as Gaussian random under-sampling matrix,to satisfy the incoherent requirements of the CS-MRI.Furthermore,for the fractional Fourier based CS-MRI,we can select the appropriate fractional Fourier transform orders and reconstruct the images in the optimal fractional Fourier domains.2.The concept of mask operations in fractional Fourier domains is a generalization of the conventional Fourier-based filtering in the frequency domain.The benefits of applying two mask operations in two different rotated fractional Fourier domains are well known especially for signal restoration applications.Compared to just applying a single mask operation in a single rotated fractional Fourier domain,it has been shown that applying two mask operations in two different rotated fractional Fourier domains carefully can improve the restoration performances.However,there is no systematic approach for the globally optimal joint design of these two sets of mask coefficients in two different predefined rotated fractional Fourier domains.In this thesis,this optimal joint design problem is,formulated as a nonconvex optimization problem.Then,a modified filled function method is employed for finding the globally optimal solution of the optimization problem.Computer numerical simulation results show that the obtained restoration system outperforms existing ones.3.It is known that simultaneously employing multiple mask operations in multiple different fractional Fourier domains can lead to significant performance advantages when compared with just employing a single mask operation in a single fractional Fourier domain.However,there is no systematic scheme for optimal joint design of the discrete fractional Fourier transform(DFrFT)matrices and the corresponding sets of mask coefficients.In this thesis,we consider this design problem and construct a formulation which does not depend on the knowledge of noise statistics.We then develop an iterative algorithm,which is a hybrid descent(HD)approach,to solve the formulated optimization problem.For this HD approach,a gradient descent(GD)method is supplemented by a simulated annealing(SA)algorithm.It is employed to find the optimal rotation angles of the DFrFT matrices.During the iterative process,the corresponding sets of mask coefficients can be constructed analytically.Simulation results demonstrate that the proposed scheme is highly effective.4.With the development of the fractional Fourier analysis,other researches on the fractional order signal analysis have been investigated.They are based on the relations with the fractional Fourier transform or the idea of fractional order.In particular,the applications of fractional calculus in signal processing have attracted the most attention.In this thesis,we propose a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components.The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem.The nonsmooth function is approximated by a smooth operator.The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem.To find a near globally optimal solution,the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem.The current or the previous locally optimal solutions are kicked out by adding the random vectors to them.The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded.By iterating the above procedures,a near globally optimal solution is found.The designed filter is applied to perform the denoising.It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method,while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method.Also,the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.In summary,we combine the optimization methods to study some problems of the fractional Fourier analysis and its extensions.This is a meaningful research on the cross innovation of the information science and the mathematical science.The content of this thesis is clear and specific,and it is of great significance to discover the characteristics and rules as well as solve the practical application problems.