A Systematic Algorithm for the Synthesis of Multiplierless Lattice Wave Digital Filters

Among the best structures for implementing recursive digital filters are lattice wave digital (LWD) filters (parallel connections of all-pass filters). They are characterized by many attractive properties, such as a reasonably low coefficient sensitivity, a low roundoff noise level, and the absence of parasitic oscillations. This book chapter describes an efficient algorithm for the design of multiplierless LWD filters in the following three cases. In the first case, the overall filter is constructed as a cascade of low-order LWD filters. As a consequence, the number of bits required for both the data and coefficient representations are significantly reduced compared with the conventional direct-form LWD filter. In the second case, approximately linear-phase LWD filters are constructed as a single block because it has been observed that in this case the use of a cascade of several filter blocks does not provide any benefits over the direct-form LWD filter design. The third case concentrates on the design of special recursive single-stage and multistage Nth-band decimators and interpolators providing the sampling rate conversion by the factor of N. For this filter class, the decimation and interpolation filter in the single-stage design (the kth decimation and interpolation filter in the multistage design, where N is factorizable as a product of K integers as N = N1N2 · · · NK) is characterized by the fact that it can be decomposed into parallel connection of N (Nk) polyphase components that are obtainable from cascades of first-order all-pass filters by substituting for each unit delay N (Nk) unit delays. The coefficient optimization is performed using the following three steps. First, an initial infinite-precision filter is designed such that it exceeds the given criteria in order to provide some tolerance for coefficient quantization. Second, a nonlinear optimization algorithm is used for determining a parameter space of the infinite-precision coefficients including the feasible space where the filter meets the given criteria. The third step involves finding the filter parameters in this space so that the resulting filter meets the given criteria with the simplest coefficient representation forms. The proposed algorithm guarantees that the optimum finiteprecision solution can be found for the multiplierless coefficient representation forms. Filters of this kind are very attractive in very large-scale integration implementations because the realization of these filters does not require the use of very costly general multiplier elements. Several examples are included to illustrate the benefits of the proposed synthesis scheme as well as the resulting filters. 11