This letter theoretically analyzes and minimizes the L2-sensitivity for all-pass fractional delay digital filters of which structure is given by the normalized lattice structure. The L2-sensitivity is well known as one of the useful evaluation functions for measuring the performance degradation caused by quantizing filter coefficients into finite number of bits. This letter deals with two cases: L2-sensitivity minimization problem with scaling constraint, and the one without scaling constraint. It is proved that, in both of these two cases, any all-pass fractional delay digital filter with the normalized lattice structure becomes an optimal structure that analytically minimizes the L2-sensitivity.

In this paper, we propose Affine Combination Lattice Algorithm (ACLA) as a new lattice-based adaptive notch filtering algorithm. The ACLA makes use of the affine combination of Regalia's Simplified Lattice Algorithm (SLA) and Lattice Gradient Algorithm (LGA). It is proved that the ACLA has faster convergence speed than the conventional lattice-based algorithms. We conduct this proof by means of theoretical analysis of the mean update term. Specifically, we show that the mean update term of the ACLA is always larger than that of the conventional algorithms. Simulation examples demonstrate the validity of this analytical result and the utility of the ACLA. In addition, we also derive the step-size bound for the ACLA. Furthermore, we show that this step-size bound is characterized by the gradient of the mean update term.

This paper proposes the Gramian-preserving frequency transformation for linear discrete-time state-space systems. In this frequency transformation, we replace each delay element of a discrete-time system with an allpass system that has a balanced realization. This approach can generate transformed systems that have the same controllability/observability Gramians as those of the original system. From this result, we show that the Gramian-preserving frequency transformation gives us transformed systems with different magnitude characteristics, but with the same structural property with respect to the Gramians as that of the original system. This paper also presents a simple method for realization of the Gramian-preserving frequency transformation. This method makes use of the cascaded normalized lattice structure of allpass systems.