Design of two-dimensional digital filters using singular-value decomposition




Wang, Hui Ping

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This thesis presents a study on the design of two-dimensional (2-D) digital filters by using the singular-value decomposition (SVD). A new method for the design of 2-D quadrantally symmetric FIR filters with linear phase response is proposed. It is shown that three realizations are possible, namely, a direct realization, a modified version of the direct realization, and a realization based on the combined application of the SV and LU decompositions. Each of the three realizations consists of a parallel arrangement of cascaded pairs of 1-D filters; hence extensive parallel processing and pipelining can be applied. The three realizations are compared and it is shown that the realization based on the SV and LU decomposition leads to the lowest approximation error and involves the smallest number of multiplications. It is shown that the SVD of the sampled amplitude response of a 2-D digital filter with real coefficients possesses a special structure: every singular vector is either mirror-image symmetric or antisymmetric with respect to its midpoint. Consequently, the SVD method can be applied along with 1-D FIR techniques for the design of linear-phase 2-D filters with arbitrary prescribed amplitude responses which are symmetrical with respect to the origin of the (ω1, ω2) plane. A method for the design of 2-D IIR digital filters based on the combined application of the SVD and the balanced approximation (BA) is proposed. It is shown that the approximation error in the phase angle is bounded by the sum of the neglected Hankel singular values of the filter. Consequently, the phase response of the resulting filter is approximately linear over the passband region provided that only small Hankel singular values are neglected. It is also shown that the resulting 2-D filter is nearly balanced, which implies that the filter has low roundoff noise as well as low parameter sensitivity. Furthermore, the 2-D filter obtained is more economical and computationally more efficient than the original 2-D FIR filter, and in the case where an IIR filter is obtained the stability of the filter is guaranteed. Efficient general algorithms for the evaluation of the 1-D and 2-D gramians for 1-D and 2-D, causal, stable, recursive digital filters are proposed, which facilitate the application of the BA method in the design of digital filters. The algorithms obtained are based on a two-stage extension of the Astrom-Jury-Agniel (AJA) algorithm. It is shown that the AJA algorithm can be modified to solve a 1-D Lyapunov equation in a recursive manner. The recursive algorithm is then extended to the case where the rational function vector involved depends on two complex variables. It is shown that the two algorithms obtained can be combined to evaluate the 2-D gramians. The proposed algorithms are also useful in obtaining optimal digital filter structures that minimize the output-noise power due to the roundoff of products.



Electric filters, Digital, Digital filters (Mathematics), Signal processing, Digital techniques