An L2-based method for the design of 1-D zero phase FIR digital filters

Finite impulse response (FIR) filters obtained with the classical L₂ method have performance that is very sensitive to the form of the ideal response selected for the transition region. It is known that design requirements do not constrain in any way the ideal response inside this region. Most existing techniques utilize this flexibility. By selecting various classes of functions to describe the undefined part of the ideal response they develop methods that improve the performance of the L₂ based filters. In this paper we propose a means for selecting the unknown part of the ideal response optimally. Specifically by using a well-known property of the Fourier approximation theory we introduce a suitable quality measure. The proposed measure is a functional of the ideal response and depends on its actual form inside the transition region. Using variational techniques we succeed in minimizing the introduced criterion with respect to the ideal response and thus obtain its corresponding optimum form. The complete solution to the problem can be obtained by solving a simple system of linear equations suggesting a reduced complexity for the proposed method. An extensive number of design examples show the definite superiority of our method over most existing non min-max design techniques, while the method compares very favorably with min-max optimum methods. Finally we prove that the approximation error function of our filter has the right number of alternating extrema, required by the L_∞ criterion, in the passband and stopband. This results in a significant convergence speed up, if our optimum solution is used as an initialization scheme, of the Remez exchange algorithm