Design of semi-sparse multi-band digital filters using branch and bound method

Recently, design methods for digital filters with sparse coefficients (sparse filters) are well studied. The sparse filters mean the filter contain some zero coefficients. Thanks to zero coefficients, the number of multipliers of the filter can be reduced. However, the performance (or accuracy) of the filter is degraded at the cost of less multipliers. In this work, we propose a design method for multi-band digital filters with "semi-sparse coefficients" which take not only 0 but also -1 and 1. It is expected that the performance of the filter with semi-sparse coefficients can be better than that with sparse coefficients. The design problem is to optimize the combination of the semi-sparse coefficients and compute the non-sparse (real value) coefficients which is not semi-sparse coefficients. Hence, the design problem is a mixed integer programming problem (MIP), and the design procedure of semi-sparse filter is more difficult than that of sparse filter. In order to optimize the combination of the semi-sparse coefficients, we use an algorithm which is based on the branch and bound method. Also, the non-sparse coefficients can be computed with the Lagrange multiplier method. Finally, we present the design example in order to demonstrate the effectiveness of our method.