An $L_1$-Method for the Design of Linear-Phase FIR Digital Filters

Abstract

Abstract-This paper considers the design of linear-phase finite impulse response digital filters using an 1 optimality criterion. The motivation for using such filters as well as a mathematical framework for their design is introduced. It is shown that 1 filters possess flat passbands and stopbands while keeping the transition band comparable to that of least-squares filters. The uniqueness of 1 -based filters is explored, and an alternation type theorem for the optimal frequency response is derived. An efficient algorithm for calculating the optimal filter coefficients is proposed, which may be viewed as the analogue of the celebrated Remez exchange method. A comparison with other design techniques is made, demonstrating that the 1 approach may be a good alternative in several applications.

Efficient Design of Digital FIR Differentiator using $L_1$-Method

“…The L 1 algorithm developed for the design of FIR linear phase filters [22][23][24][25], is successful in optimizing the type-1 FIR filter coefficients and the designed filter yields a smallest overshoot around the discontinuity along with flattest response in the passband as compared to least-square, minimax and window technique. The L 1 algorithm for the design of (N − 1)th order type-4 FIR differentiator with antisymmetric coefficients is described in this section.…”

“…Step 3: Determine the Hessian matrix H k (second order derivative of error function) of size M × M, based on the value of t, where t represents the number of zeros of ε and the zeros are considered to be simple [22].…”

“…Step 6: Calculate step-size, α k according to the Armijo rule [22], to guarantee sufficient decrease of ε at the kth iteration.…”

“…The L 1 -method is formulated using the fact that the L 1 -error function is certainly differentiable and this modified Newton method is developed for computing the differentiator coefficients [22]. The zeros of the differentiable L 1 -error function are replaced with a different set of zeros at each iteration in order to decrease the error.…”

The Design of Optimal L>inf<1>/inf<Linear Phase FIR Digital Filters

“…(11) As a consequence of this theorem, it can be shown that the L1 norm in (8) is differentiable for all a ∈ R M +1 , except for two particular points, a = (1, 0, ..., 0), and a = (0, ..., 0); see [11]. However, these points refer to the degenerate case of a constant filter, and therefore can be ignored.…”