New initialization methods for discrete coefficient FIR filter design with coefficient scaling and the use of scale factor in the design process

The initialization of filter coefficients in discrete-coefficient finite-impulse-response (FIR) filter design (with coefficient scaling) using coefficient-value-assignment-based optimization techniques is considered. A common weakness of existing initialization measures, a total-square-error (TSE) measure and a maximum-error (ME) measure, is described. New TSE and ME measures that overcome the weakness are introduced. As opposed to the current knowledge, it is revealed that TSE and ME measures do not necessarily provide different initial coefficient set information and that the initial coefficient set information by the ME measure is not unique in general. On a parallel track, the treatment of coefficient scaling in the optimization process is considered. The closed-form expression of the objective function normalized peak ripple magnitude (NPRM), which was defined implicitly in the literature, is given. In contrast to existing methods, the computation of NPRM accordingly does not require a scale-factor parameter independent of filter coefficients although coefficient scaling persists. Then, TSE and ME measures become practically equivalent (although, in general, they indicate different initial scale factors) since an explicit scale factor parameter is no longer needed.