Design Algorithms Of DFT And Interleaved DFT Modulated Filter Banks

Over the past two decades, the theory and design of multirate filter banks have obtained more and more attentions. The important reason to push the development of multirate filter bank is that multirate filter banks are widely used to many fields such as subband coding of speech and images, subband adaptive filtering, interference injection, denoising and communications and radar signal processing. In all kinds of multirate filter banks, the research on modulated filter banks has become popular. Due to its simple structure, great ease in design and high computational efficiency, modulated filter banks include two subclasses: cosine modulated filter banks and DFT (discrete Fourier transform) modulated filter banks. At present, cosine modulated filter banks used to real-value signal processing have been intensively researched. DFT modulated filter banks used to complex-value signal processing is still an important research issue. This dissertation researches the theory and design algorithms of DFT modulated filter bank and interleaved DFT modulated filter bank. This dissertation mainly includes the following four aspects:(1) Design of oversampled near perfect reconstruction DFT modulated filter banks. Firstly,the property of transfer function and aliasing transfer function are analyzed. The relation of the system delay of causal FIR DFT modulated filter bank and the orders of linear phase analysis and syntheis prototype filters is presented and proven. Secondly, design of oversampled DFT modulated filter banks is formulized as a non-convex optimization that minimizes the maximal amplitude of the transfer function distortion and the aliasing transfer functions subject to stopband and transition-band energy as well as the passband flatness of the prototype filters. A bi-iterative second-order cone programming is proposed to solve the optimization by iteratively optimize the synthesis and analysis prototype filter. In each step of iteration, the objective is a linear function on optimized variable and the constraints are quadratic and linear constraints on optimized variable, which is correspond to solve a second order cone programming and the global optimal solution can be obtained. The new method allows component cancellation of each aliasing transfer function. The component cancellation and stopband attenuation of the prototype filters can cooperate to achieve filter banks with less reconstruction error than the existing methods do.(2) Design of oversampled linear phase DFT modulated filter banks. Linear phase property is very important to some practical application case such as audio coding and image processing. The design of linear phase filter banks is an important and active issue. This paper proposes bi-iterative linear programming algorithm to design linear phase DFT modulated filter banks. The bi-iterative algorithm iteratively optimize the analysis and synthesis prototype filer to minize the maximal magnitudes of the frequency response of the prototype filter in the stopband region and the transfer function distortion and the aliasing transfer functions in frequency domain. Each step of the iteration is to solve a linear semi-infinite programming that is transformed into a linear programming by frequency discretization. The merit of the proposed algorithm is that the simple and efficient Chebyshev constraints assure the transfer function distortion and aliasing transfer functions are comparable in magnitude. As a result, the filter bank achieves less reconstruction error. Simulation show that the proposed algorithm can obtain better DFT modulated filter banks in overall performance than the existing algorithms.(3) Design of 2M channel critically sampled biorthogonal interleaved DFT modulated filter banks. This paper proposes a new structure of complex modulated filter banks, called interleaved DFT modulated filter banks. In the filter banks, the analysis filters are generated by interlaced exponential modulating two different analysis prototype filters, and the synthesis filters are generated by two different synthesis prototype filters via the same manner. Similar to DFT modulated filter banks, the filter banks have a simple polyphase structure. The proposed interleaved DFT modulated filter bank can achieve critically sampled perfect reconstruction complex filter bank with FIR analysis and synthesis filters, which is impossible for DFT modulated filter banks. By the polyphase structure of analysis and synthesis filters, perfect reconstruction condition of 2M channel critically sampled interleaved DFT modulated filter banks with FIR analysis and synthesis filters is derived. Subsequently, a design method of the prototype filters of critically sampled biorthogonal interleaved DFT modulated filter banks is given. In additional, a design technique of the prototype filters is presented for critically sampled biorthogonal interleaved DFT modulated filter banks with FIR analysis filters and IIR synthesis filters.(4) Design of oversampled near perfect reconstruction interleaved DFT modulated filter banks. Three design algorithms are porposed to design oversampled near perfect reconstruction interleaved DFT modulated filter bank, which is called nonlinear optimization algorithm, bi-iterative quadratic programming algorithm and bi-iterative second order cone programming algorithm, respectively. In nonlinear optimization algorithm, design of interleaved DFT modulated filter banks is formulized as a non-convex optimization that minimizes the stopband energy of prototype filters subject to the maximal frequency spectrum component, the correlation and the passband flatness of the prototype filters. In bi-iterative quadratic programming, the design problem of interleaved DFT modulated filter bank is formulized to a noncovex optimization. The stopband energies of the prototype filters, the inband aliasing and the residual aliasing are optimized while constraining the level of the transfer function distortion for all of the frequencies within a fixed level. The bi-iterative algorithm is used to optimize iteratively over both the analysis and synthesis prototype filters. In each step of iteration, the objective of the optimization is a quadratic function on optimized variables and the constraints are linear constraints on optimized variables, which is correspond to solve a quadratic programming and the global optimal solution can be obtained. In bi-iterative second order cone programming algorithm, a non-convex optimization that minimizes the maximal amplitude of the transfer function distortion and the aliasing transfer functions subject to stopband and transition-band energy as well as the passband flatness of the prototype filters. A bi-iterative second-order cone programming is proposed to solve the optimization by iteratively optimize the synthesis and analysis prototype filter. Owing to more degrees of freedom for design, the three design algorithms can achieve oversampled near perfect reconstruction complex filter banks with satisfied performance indexes.The dissertation is organized as follows.Chapter 1 introduces the concepts of multirate filter bank, an overview of its development, and its classification and applications.Chapter 2 briefly overviews the elements of multirate filter bank, and the perfect reconstruction condition of M channel filter banks and the structures of M channel modulated filter banks.Chapter 3 provides a bi-iterative second order cone programming algorithm to design of analysis and synthesis prototype filter of oversampled DFT modulated filter bank with near perfect reconstruction..Chapter 4 provides a bi-iterative linear programming algorithm to design analysis and synthesis prototype filter of oversampled linear phase DFT modulated filter banks with near perfect reconstruction.Chapter 5 provides the structure and fast algorithm of 2M channel critically sampled interleaved DFT modulated filter banks, and proves the perfect reconstruction condition. Subsequently, design algorithms of FIR analysis prototype filters and FIR/IIR synthesis prototype filters are presented, respectively. Chapter 6 provides the structure oversampled near perfect reconstruction interleaved DFT modulated filter banks, and gives three design algorithms (the nonlinear optimization algorithm, the bi-iterative quadratic programming algorithm, and bi-iterative second order cone programming algorithm) to obtain analysis and synthesis prototype filters.Chapter 7 summaries the dissertation, and gives an outlook on the future development of multirate filter banks.