Sampling And Multirate Filter Banks Theory Associated With Arbitrary Lattices In The Fractional Fourier Domain

Multi-rate filter banks theory has been developed rapidly in recent years.Its rapid development has been promoted by its wide applications,such as image compression and coding,adaptive filtering,noise elimination,and signal processing in communication.At the same time,with the rapid development of multidimensional signal processing,multidimensional multirate filter banks theory has attracted wide attention.It has been gradually applied to image and video subband coding,multimedia communication and sampling format conversion of different video standards.However,the multidimensional multirate filter banks theory is limited to Fourier domain,which seems powerless to deal with nonstationary signals.Fractional Fourier transform is a very useful time-frequency analysis tool emerging recently.As the generalization of Fourier transform,it has become a powerful tool in dealing with nonstationary signals because of its unique time-frequency characteristics,and has been widely used in image processing,radar,signal separation and time-frequency analysis.At present,multirate filter banks theory with one-dimensional fractional Fourier transform has been successfully proposed.However,due to the increase of freedom in processing multidimensional signals,multidimensional non-separable operation has better advantages than one-dimensional method,which can be fully reflected in image subband coding and multi-resolution analysis.In this paper,the uniform sampling theorem associated with arbitrary lattice sampling is proposed in the fractional Fourier domain.Based on this theorem,multidimensional multirate filter banks method is established,which overcomes the limitation in dealing with nonstationary signals by Fourier transform.The specific work is summarized as follows:The first part aims to propose the uniform sampling theorem for multidimensional bandlimited signals in the fractional Fourier domain.Firstly,Fourier series formula with arbitrary nonsingular matrix as period is defined.Then spectrum formula of sampled signal in fractional domain is deduced based on this formula.Secondly,based on the spectrum relationship a simple alias-free sampling method for arbitrary bandlimited regions is introduced.Furthermore,we derive the reconstruction formula of original bandlimited signal under the premise of alias-free sampling.This reconstruction method brings two advantages: the one is that it can be applied to any bandlimited region,the other is that it overcomes the problem that some integrals with irregular integral regions are calculated difficultly.Finally,an improved alias-free sampling method is proposed based on the idea of segmentation of bandlimited region to help us find the optimal sampling as far as possible.In the second part,multirate conversion method associated with arbitrary lattice is introduced in the fractional Fourier domain.Firstly,we define multidimensional fractional chirp period according to the spectrum formula of sampled signals.Then the definition of multidimensional discrete-time fractional Fourier transform is obtained based on chirp period.At the same time,the corresponding convolution theorem about new transform is proved,which paves the way for multidimensional multirate conversion theory.Secondly,the sampling rate conversion method via integer matrix is carefully studied,including time-frequency analyses of interpolation and decimation,the selections of de-mirror or anti-aliasing filter,interpolation and decimation identical structures.In addition,based on interpolation and decimation via integer matrix,we present the sampling rate conversion method using rational matrix.Finally,according to the advantage that multidimensional interpolation increases the number of sampling points while maintaining the characteristics of the original signal,we propose a new image scaling algorithm and provide simulations to verify the validity of the new algorithm.In the third part,the study of multidimensional multirate filter banks about fractional Fourier transform is presented in this paper.Firstly,the multi-phase representation of multidimensional signals is deduced in the fractional Fourier domain.Based on multi-phase representation,we obtain the efficient multi-phase implementation of multidimensional decimation and interpolation.Secondly,we carefully study the multidimensional m-channel filter banks in the fractional Fourier domain,such as input-output relationship of filter banks,conditions of alias-free reconstruction and complete reconstruction,the multi-phase structure of filter banks.Besides,an efficient design method of alias-free and completely reconstructed filter banks is deduced.This method establish relationship between filter banks in Fourier domain and in fractional Fourier domain.Finally,we propose a method to design multidimensional orthogonal mirror filter banks through the prototype filters of orthogonal mirror filter banks in Fourier domain.And this multidimensional orthogonal mirror filter bank is applied to image subband decomposition.The validity of theoretical application is proved by simulation.