Efficient Design of High-Complexity Cosine Modulated Filter Banks Using $2 M$th Band Conditions

This paper presents several new properties of biorthogonal cosine modulated filter banks (CMFBs) and efficient algorithms for designing CMFBs with a very large number of subbands and very long filters. For a biorthogonal CMFB, we find the periodicity and symmetry of its overall transfer function and aliasing transfer functions which can be efficiently computed based on a decimated uniform discrete Fourier transform (DFT) analysis filter bank. By exploiting gradient information and $2 M$th band conditions, efficient algorithms are proposed for designing both orthogonal and biorthogonal CMFBs. In addition, an efficient matrix inversion algorithm with $O(N^{2} )$ complexity is also presented. Several numerical examples and comparisons with many other existing methods are included to demonstrate the design performance and efficiency of the algorithms.