Jun-Zheng Jiang

Design of oversampled double-prototype DFT modulated filter banks via bi-iterative second-order cone program

Oversampled double-prototype DFT modulated filter banks can achieve better overall performance than their single-prototype counterparts, owing to more degrees of freedom available in design. In this paper, we reformulate their design problem into a non-convex optimization, which minimizes the maximal amplitude of the transfer function distortion and aliasing transfer functions of the filter bank subject to fixed bounds on stopband and transition-band energy and the passband flatness of the prototype filters (PFs), and the inband aliasing of the analysis filters. A bi-iterative second-order cone program (BI-SOCP) algorithm is proposed to solve the non-convex optimization. The BI-SOCP can always generate filter banks with satisfactory overall performance starting from appropriate initial PFs, though the iteration process cannot assure to converge to the optimal solution of the problem. The new formulization allows the component cancellation of each aliasing transfer function. The BI-SOCP can still design filter banks of nearly perfect reconstruction (NPR) in the case of less redundancy ratio and short PFs. Finally, several numerical examples are demonstrated to verify the effectiveness of the algorithm.

Fast design of 2D fully oversampled DFT modulated filter bank using Toeplitz-block Toeplitz matrix inversion

This letter presents a low-complexity algorithm to design two-dimensional (2D) DFT modulated filter bank (DMFB) with nearly perfect reconstruction (NPR). The design problem is formulated into an unconstrained optimization problem whose cost function consists of the overall distortion and stopband energy. By exploiting the gradient information, the prototype filter (PF) coefficients are iteratively optimized with closed-form formula. At each iteration, the computational complexity is dramatically reduced by evoking Toeplitz-block Toeplitz matrix inversion and matrix inversion lemma, which makes the algorithm suitable for design 2D DMFB with a large number of channels and PF coefficients. Numerical example and comparison are included to show the effectiveness of the proposed design algorithm. The stopband energy matrix is a Toeplitz block Toeplitz matrix.The algorithm complexity of is largely reduced via matrix inverse lemma..The proposed algorithm is suitable for designing 2D high-complexity DMFB.

Design of 2D oversampled linear phase DFT modulated filter banks via modified Newton's method

In this paper, the structure of the 2D oversampled DFT modulated filter banks is analyzed and a spatial-domain condition of a filter bank without transfer function distortion is derived. Based upon the spatial-domain condition, a modified Newtons method is presented for fast design of 2D oversampled linear phase (LP) DFT modulated filter banks with nearly perfect reconstruction (NPR). We formulate the design problem into an unconstrained optimization with a fourth-order objective function, which is the weighted sum of the transfer function distortion of the filter bank and the stopband energy of the prototype filter (PF). The optimization is solved by the modified Newtons method, where each of iterations updates the PF by a set of linear equations. It is proved that the iteration process fast converges to a stationary point of the objective function. Compared with the existing methods, the new method is fast in computation and can design 2D filter banks with a large number of subbands.

Design of 2D linear phase DFT modulated filter banks using bi-iterative second-order cone program

In this paper, the recent bi-iterative second-order cone program (BI-SOCP) to design one-dimensional (1D) DFT modulated filter banks is extended to the 2D case to design two-dimensional (2D) linear phase nonseparable DFT modulated filter banks. Some essential properties of 2D DFT modulated filter banks are presented and a permissibility of the configuration of filter banks is given. The design problem is formulated into a non-convex optimization, which minimizes the maximal amplitude of the transfer function distortion and aliasing transfer functions of the filter bank subject to predefined bounds on the stopband energy, the transition-band energy, and a passband flatness measure of the prototype filters (PFs). The optimization problem is solved by the BI-SOCP algorithm, where 2D linear phase analysis and synthesis PFs are alternately optimized via second-order cone programs (SOCPs). Numerical examples are included to show that the BI-SOCP can yield 2D linear phase nonseparable DFT modulated filter banks with good overall performance.

Optimization Design of Two-Channel Biorthogonal Graph Filter Banks

Narang and Ortega have constructed a two-channel biorthogonal graph filter bank with compact support. The design method does not consider the spectral response of the kernels. In this letter, we employ optimization approach to design the spectral kernels. The analysis and synthesis kernels are, respectively, optimized with constrained optimization problems, in which the reconstruction error and spectral selectivity are controlled simultaneously. The optimization problems are semidefinite programming (SDP), which can be solved effectively. Numerical examples and comparison are included to show that the proposed approach is more flexible in making trade-off between the spectral selectivity and reconstruction error over the existing method.

Iterative design of two-dimensional critically sampled MDFT modulated filter banks

In this paper, an efficient design algorithm is proposed to design the recently introduced two-dimensional (2D) critically sampled modified DFT (MDFT) modulated filter bank. First of all, the original perfect-reconstruction (PR) condition in frequency-domain of the filter bank is transformed into a spatial-domain condition, which is a set of quadratic equations with respect to the prototype filter (PF). Then, with the derived PR equations, the design problem is formulated into an unconstrained optimization problem that involves PR condition and stopband energy of the PF. An iterative algorithm is proposed to solve the optimization problem. The convergence of the algorithm is proved. Numerical results and comparison with existing method are included to show the effectiveness of the proposed design algorithm.

Efficient design of high-complexity interleaved DFT modulated filter bank

This paper presents several novel properties of the interleaved DFT modulated filter bank and efficient algorithm for designing the filter bank with a very large number of channels and very long filters (also known as high-complexity). For interleaved DFT modulated filter bank, we find a new property of the transfer function and aliasing transfer functions, by which the complexity of the calculation of these functions can be dramatically reduced over that by the traditional way. Also, a time-domain condition is derived for the filter bank to have no transfer function distortion. Based on new properties, a bi-iterative unconstrained quadratic program (BI-UQP) algorithm is proposed to design the filter bank. Numerical examples and comparisons with other method are included to demonstrate the effectiveness of the proposed method.

Design of two-dimensional large-scale DFT-modulated filter bank

This study presents a novel property of the two-dimensional DFT-modulated filter bank and an efficient algorithm for designing the filter bank with large scale (with a large number of subbands and filters of large spatial support). It is firstly shown that the overall transfer function and aliasing transfer functions can be simply represented with the multiplication of the prototype filters and their modulated filters; a new property used to remarkably reduce the calculation of these functions. On the other hand, the design problem of the filter bank is formulated into an unconstrained optimisation problem. Based on the gradient vector, the conjugate gradient method is utilised to solve the design problem. The convergence of the proposed algorithm is analysed. Numerical examples and comparisons with other methods are included to show the performance of the algorithm.

Design of NPR DFT-Modulated Filter Banks via Iterative Updating Algorithm

In this paper, an efficient algorithm is proposed to design nearly-perfect-reconstruction (NPR) DFT-modulated filter banks. First, the perfect-reconstruction (PR) condition of the oversampled DFT-modulated filter banks in the frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF) in the time domain. Second, the design problem is formulated as an unconstrained optimization problem that involves PR condition and stopband energy of the PF. With the gradient vector of the objective function, an efficient iterative algorithm is presented to design the PF, which is updated with linear matrix equations at each iteration. The algorithm is identified as a modified Newton’s method, and its convergence is proved. Numerical examples and comparison with many other existing methods are included to demonstrate the effectiveness of the proposed method.

Efficient design of prototype filter for large scale filter bank-based multicarrier systems

This study presents a new property of the filter bank-based multicarrier (FBMC) system. Also, an efficient iterative algorithm for designing the system with a large number of subcarriers and a prototype filter with a very long length are proposed. For the system, the compact from conditions are derived for both the intersymbol interference free and the interchannel interference (ICI) free. Based on these new conditions, the design of the prototype filter is formulated as an unconstrained optimisation problem where the objective function is the weighted sum of the total distortion of the system and the stopband energy. By deriving the gradient vector of the objective function, an efficient iterative algorithm is proposed for finding the solution of the optimisation problem. In addition, an efficient matrix inversion approach is presented to greatly reduce the computational complexity of the iterative algorithm. As a result, it is feasible to design the FBMC system with thousands of subcarriers. The convergence of the iterative algorithm is proved. Computer numerical simulation results with the comparisons to the existing methods are presented. It is shown that the proposed design algorithm is more effective and efficient than the existing methods.