Lagrange multiplier approach to the design of FIR filters for multirate applications

The method of Lagrange multipliers is presented for the design of optimal finite-duration impulse-response (FIR) digital filters for multirate signal processing applications. The filter passbands are constrained to be maximally flat, and the integral of the aliasing error is minimized. The use of these optimality criteria for practical applications is justified. It is shown that the overall frequency response can be represented as a linear combination of individual frequency responses corresponding to the separate design constraints, with the Lagrange multipliers indicating the sensitivity of the integrated aliasing error to changes in the constrained values. The constraints can be modified and the filter can be updated with only a small number of computations. In the case of maximally flat filters it is established that the constraint matrix is of the Vandermonde type, and, using this result, it is shown that a unique, optimal filter always exists. Its coefficients can be determined by solving a linear system of equations