Design Algorithms Of DFT Modulated Filter Banks

Recently, theory and design of filter banks have been widely investigated. Filter banks have been applied in many fields, such as speech signal processing, subband coding, image processing, radar signal processing and multimedia signal processing. Among the general filter banks, modulated ones have gained much popularity, due to their ease design and realization. The modulated filter banks can be classified into cosine-modulated filter banks and Discrete Fourier Transform (DFT) modulated filter banks. In the one-dimensional (1D) case, different from the 1D cosine-modulated filter banks, the 1D DFT modulated filter banks can separate the positive and negative frequency components of the input signal into different subbands for separate processing, which is very useful for constructing directional filter banks and processing complex-valued signal. In the two-dimensional (2D) case, the 2D DFT modulated filter banks can separate frequency components of the input signal in the first and third quadrant (or in the second and forth quadrants) into different subbands for separate processing, which makes the 2D DFT modulated filter banks more suitable to process 2D complex-valued signals than the 2D cosine-modulated filter banks. Therefore, it is very meaningful to study the theory and design of the DFT modulated filter banks.There are lots of algorithms proposed for designing 1D DFT modulated filter banks. However, most of these algorithms are based on the semi-definite program (SDP) algorithm, which incurs heavy computational cost. Therefore, design algorithms with low computational complexity are desired intensively. On the other hand, there is lack of articles on the 2D nonseparable DFT modulated filter banks. In this dissertation, the theory, structure and design methods for the 1D and 2D DFT modulated filter banks are thoroughly studied.This dissertation studies the theory and design of the 1D and 2D modulated filter banks. The main work is listed as follows.(1) The structure and design of 1D low-delay DFT modulated filter banks with single-prototype. Firstly, the structure of the filter banks is analyzed. Since the traditional modulation structure for single-prototype does not allow low-delay, a new modulation structure is utilized. Secondly, the design method of the filter banks is presented. The design problem is formulated into an unconstrained optimization problem that minimizes the weighted sum of the transfer function distortion and stopband energy, the objective function is a forth-degree function with respect to the prototype filter (PF). Based on the analytic expression of the gradient vector and Hessian matrix of the objective function, the modified Newton's method with exact line search is proposed to solve the design problem. Since the PF is updated by solving linear systems in each iteration, the design method has very low computational cost. Thus, the method is suitable for designing 1D high-complexity DFT modulated filter banks, i.e., filter banks with a very large number of channels and PFs with very large length. The fast convergence of the algorithm is proved. Numerical examples show that the proposed algorithm can lead to filter banks with satisfactory overall performance.(2) The structure and design of 2D double-prototype oversampled DFT modulated filter banks. Firstly, the basic structure of the filter banks is reviewed. Secondly, we have investigated the permissibility of the filter banks, i.e., the necessary condition for the filter banks to have fine overall performance. The concept fully oversampled is primarily introduced. Thirdly, the design problem of the filter banks is formulized into a constrained optimization problem, where the objective function is the transfer function distortion and aliasing distortion of the filter banks and the constraints are the frequency selectivity of the PFs, such as the passband flatness and stopband attenuation, etc. Finally, the bi-iterative second-order cone program (BI-SOCP) algorithm is presented to solve the optimization problem. In the algorithm, when one of the PF is fixed, the design problem is reduced to be a SOCP with respect to the other PF. The analysis and synthesis PFs are alternately optimized by the bi-iterative scheme. Although the subproblem in each iteration is convex, the final solution of the algorithm is dependent on the initial point. Therefore, we propose a method for designing the initial PF. Numerical results demonstrate that the BI-SOCP algorithm can generate filter banks with fine overall performance.(3) The structure and design of 2D single-prototype oversampled DFT modulated filter banks. First, the essential structure of the filter banks is analyzed and a spatial condition for the filter bank to be without transfer function distortion is derived. Second, we propose a design algorithm based on modified Newton's method. In the algorithm, the design problem of the 2D single-prototype oversampled DFT modulated filter banks is formulated into an unconstrained optimization problem whose objective function is the weighted sum of the transfer function distortion and the stopband energy of the prototype filter (PF). Based on the gradient vector and Hessian matrix of the objective function, the modified Newton's method with exact line search is proposed to solve the design problem. At each iteration, the PF is calculated with closed formula. Also, it is proved that the algorithm is of fast convergence. Finally, numerical examples suggest that the proposed algorithm is of high computational efficiency and it can lead to filter banks with good overall performance.(4) Structure and design of two-dimensional 2×oversampled DFT modulated and critically sampled modified DFT (MDFT) modulated filter banks. Some applications require 2D critically sampled filter banks, such as image compression. However, the general 2D critically sampled DFT modulated filter banks can not simultaneously possess small reconstruction error and high stopband attenuation. Here, 2D critically sampled MDFT modulated filter banks are constructed, which can behave small reconstruction error and high stopband attenuation at the same time. The 2D MDFT modulated filter banks are rebuilt from the 2D 2×oversampled DFT modulated filter banks. First, the structure of the 2D 2×oversampled DFT modulated filter banks is analyzed. Based on the polyphase decomposition, the perfect-reconstruction (PR) condition is derived and the efficient implementation of the filter banks is given. Second, the critically sampled MDFT modulated filter banks are rebuilt from the oversampled ones. Meanwhile, the PR condition of the MDFT modulated filter banks is deduced. It is proved that, without regard to a scaling factor, the 2D MDFT modulated filter banks and the 2D 2×oversampled DFT modulated filter banks have the same input-output relationship in the case where the synthesis PF is identical to the analysis PF. Third, the design algorithms for the two classes of filter banks are proposed. On one hand, an algorithm based on lifting structure is proposed to design the 2D 2×oversampled DFT modulated filter banks with double-prototype. By parameterizing the PR condition of the filter banks, the design of the PFs is reduced to the design of the lifting filters. The PR property of the filter banks is assured by the parametric structure, which is verified by the numerical results. On the other hand, a method based on unconstrained optimization is presented for the design of 2D critically sampled MDFT modulated filter banks with single-prototype, where the nearly-perfect-reconstruction filter banks are designed by minimizing the reconstruction error of the filter banks. Numerical examples show that the obtained filter banks have enough small reconstruction error.