New methods for two-dimensional filter and filter bank design

Recently, a new algorithm for weighted least squares linear-phase FIR filter design has been presented. In this dissertation, that algorithm is extended in several directions. First, it is shown how to combine this new algorithm with a well-known algorithm used to solve quadratic programming problems. This combination results in a faster method for constrained least squares FIR filter design. The proposed algorithm is numerically stable and suitable for high order filter design. Second, a new method for complex linear-phase FIR filter design is presented. The problem of least squares filter design is solved by projecting the desired frequency response onto the subspace spanned by an appropriate basis that is computed in an efficient way. Third, a new method for weighted least squares 2D linear-phase FIR filter design is developed. It is shown how to compute the orthonormal basis efficiently in the cases of quadrantally-symmetric filter design and centro-symmetric filter design. The usefulness of the proposed method is demonstrated through examples. These examples show that the proposed method is faster than a conventional weighted least squares filter design method. Also, the amount of storage required to compute the filter co-efficients is greatly reduced. This method is also extended to the two-dimensional complex FIR linear-phase case. Next, the problem of designing two-dimensional filter banks with linear-phase analysis filters and rectangular sampling grids is studied. The perfect reconstruction constraints are written in terms of the analysis filter coefficients and these filter coefficients are directly optimized in order to approximate a desired frequency response. Sufficient conditions for the existence of a solution are derived. The nonlinear optimization problem that must be solved in order to optimize the filter coefficients is studied in detail. This optimization problem is more difficult than its one-dimensional counterpart and some suggestions to overcome the difficulties that arise in practice are presented. Finally, the previous results are generalized to the design of two-dimensional FIR linear-phase filter banks with arbitrary sampling grids. The filter bank design examples demonstrate the applicability and usefulness of the proposed method.