Exact synthesis of LDI and LDD ladder filters

Abstract

The low sensitivity property of lossless discrete integrator (LDI) low-pass ladder filters is shown to be preserved in lossless discrete differentiator (LDD) high-pass ladder filters 111. The exact design method for LDI ladder filters given in [Z] is further developed by introducing a set of closed-form design formulas for digital ah-pole Chebyshev transfer functions. A new technique for improving the numerical accuracy in the synthesis procedure is introduced. Finally, a comprehensive LDI ladder filter design program is presented.

Suppression of overflow limit cycles in LDI all-pass/lattice filters

“…[18], [20], and [24]). For a linear system, the quadratic form V (x(n)) = x T (n)Px(n) (7) is a Lyapunov function, which implies that the system is stable, if and only if V (x(n)) > 0, x(n) 6 = 0, V (0) = 0, and 1V (x(n)) = V (x(n+1))0V (x(n)) = x T (n)[A T PA0P ]x(n) = 0x T (n)Qx(n) 0, i.e., if the matrices P and Q = P 0 A T PA are positive definite and positive semidefinite, respectively. While Lyapunov functions for stable linear systems always exist, finding an analytically parametrized matrix P is, in general, a difficult task.…”

“…Hence, if H(z) is low pass, H(0z) is high pass. Making the above substitution in an LDI filter, a lossless digital differentiator (LDD) filter is obtained [7] in which delayed and nondelayed integrators become delayed and nondelayed differentiators, 01=(z + 1) and z=(z + 1), respectively.…”

“…Condition (2) is based on the sector information of the overflow nonlinearities, namely, the sector [01; 1]. According to yet another approach [2] (taking into account a recent observation [3], [4], namely, only with P in block diagonal form, one can prove that the pseudo-energy along the consecutive diagonals is decreasing [1], [5]- [7]) (1) using saturation nonlinearities is free of overflow oscillations if there exists a positive definite block diagonal matrix…”

“…To derive (3) it is assumed that the nonlinearities satisfy f T (y(k; l))D[y(k; l) 0 f (y(k; l))] 0. Thus, condition (3) Recently, Liu and Michel [7] presented a criterion for the global asymptotic stability of (1) using the generalized overflow nonlinearities …”

Delta operator based design of 1-D and 2-D filters: An overview

“…The traditional ladder synthesis procedure in zdomain using the lossless discrete-time integrator (LDI) suffers from numerical problems if the poles of the filter are clustered around 1 z = as in the case of narrowband filters or when high sampling rates are involved [2,3]. The delta operator based ladder realization procedure helps to overcome such numerical problems [4].…”