In this paper, an optimal method is proposed to design sparse-coefficient notch filters with principal basic vectors in the column space of a matrix constituted with frequency samples. The proposed scheme can perform in two stages. At the first stage, the principal vectors can be determined in the least-squares sense. At the second stage, with some components of the principal vectors, the notch filter design is formulated as a linear optimization problem according to the desired specifications. Optimal results can form sparse coefficients of the notch filter by solving the linear optimization problem. The simulation results show that the proposed scheme can achieve better performance in designing a sparse-coefficient notch filter of small order compared with other methods such as the equiripple method, the orthogonal matching pursuit based scheme and the L1-norm based method.

The existing literature focuses on the applications of fast filter bank due to its excellent frequency responses with low complexity. However, the topic is not addressed related to the general transfer function expressions of the corresponding subfilters for a specific channel. To do this, in this paper, general closed-form transfer function expressions for fast filter bank are derived. Firstly, the cascaded structure of fast filter bank is modelled by a binary tree, with which the index of the subfilter at each stage within the channel can be determined. Then the transfer functions for the two outputs of a subfilter are expressed in a unified form. Finally, the general closed-form transfer functions for the channel and its corresponding subfilters are obtained by variables replacement if the prototype lowpass filters for the stages are given. Analytical results and simulations verify the general expressions. With such closed-form expressions lend themselves easily to analysis and direct computation of the transfer functions and the frequency responses without the structure graph.

In software defined radio systems, a channelizer plays an important role in extracting the desired signals from a wideband signal. Compared to the conventional methods, the proposed scheme provides a solution to design a digital channelizer extracting the multiple subband signals at different center frequencies with low complexity. To do this, this paper formulates the problem as an optimization problem, which minimizes the required multiplications number subject to the constraints of the ripple in the passbands and the stopbands for single channel and combined multiple channels. In addition, a solution to solve the optimization problem is also presented and the corresponding structure is demonstrated. Simulation results show that the proposed scheme requires smaller number of the multiplications than other conventional methods. Moreover, unlike other methods, this structure can process signals with different bandwidths at different center frequencies simultaneously only by changing the status of the corresponding multiplexers without hardware reimplementation.