FRM-Based FIR Filters With Optimum Finite Word-Length Performance

Abstract

It is well known that filters designed using the frequency response masking (FRM) technique have very sparse coefficients. The number of nontrivial coefficients of a digital filter designed using the FRM technique is only a very small fraction of that of a minimax optimum design meeting the same set of specifications. A digital filter designed using FRM technique is a network of several subfilters. Several methods have been developed for optimizing the subfilters. The earliest method optimizes the subfilters separately and produces a network of subfilters with excellent finite word-length performance. Subsequent techniques optimize the subfilters jointly and produce filters with significantly smaller numbers of nontrivial coefficients. Unfortunately, these joint optimization techniques, that optimize only the overall frequency response characteristics, may produce filters with undesirable finite word-length properties. The design of FRM-based filters that simultaneously optimizes the frequency response and finite wordlength properties had not been reported in the literatures. In this paper, we develop several new optimization approaches that include the finite word-length properties of the overall filter into the optimization process. These new approaches produce filters with excellent finite word-length performance with almost no degradation in frequency response performance.

Design of frequency-response-masking FIR filters using SOCP with coefficient sensitivity constraint

“…Since x * is directly related to the impulse responses of the optimized subfilters (see (2b)-(2d) and (3)), by (7) this implies that the CS of the FRM filter obtained is always low as long as the initial FRM filter has a low CS. We demonstrate the validity of the analysis by applying the SOCP-based algorithm in [12] to design an FRM filter with N = 45, N a = 27, N c = 19, M = 9, ω p = 0.3, and ω a = 0.305 (here the design specifications are identical to those in the example described in [22], [23]). The initial x 0 was generated using the method in [1], the value of β in (6c) was set to 0.1533, and no constraints on CS were imposed.…”

“…We begin our examination of the CS of the above SOCP algorithm by recalling the definition of a sensitivity measure S 2 1 introduced in [22], [23], which is given in our notation as…”

“…2, the CS of the FRM filters designed using the method in [12] is inherently low. However, the design example presented in [22], [23] shows that with the same design specifications, the sensitivity measure S 2 1 can be made as low as 26.34. In this section, the SOCP-based algorithm in [12] is enhanced by imposing a constraint on S 2 1 .…”

“…references [2]- [21] for the subsequent development in analysis and design of various types of FRM filters. The finite word-length (FWL) performance of FRM filters, which seems to have been overlooked in the past, has been examined recently in [22], [23]. In those papers, it is demonstrated that unless the coefficient sensitivity issue is taken into consideration by the design algorithm, FRM filters with excellent performance in terms of approximation accuracy but also with exceedingly high coefficient sensitivity (CS) may be produced.…”

“…Our examination shows that the SOCP-based design method is guaranteed to produce FRM filters with low CS as long as the CS of the initial FRM filter is low. Second, we present an enhanced SOCP-based design method that incorporates a constraint on the CS measure S 2 1 introduced in [22], [23]. Our formulation shows that SOCP provides a suitable design setting for FRM filters, where the CS of the FRM filter must be taken into account.…”