This paper discusses the behavior of the second-order modes (Hankel singular values) of linear continuous-time systems under typical frequency transformations, such as lowpass-lowpass, lowpass-highpass, lowpass-bandpass, and lowpass-bandstop transformations. Our main result establishes the fact that the second-order modes are invariant under any of these typical frequency transformations. This means that any transformed system that is generated from a prototype system has the same second-order modes as those of the prototype system. We achieve the derivation of this result by describing the state-space equations and the controllability/observability Gramians of transformed systems.

This paper discusses the behavior of the second-order modes (Hankel singular values) of linear discrete-time systems under bounded-real transformations, where the transformations are given by arbitrary transfer functions with magnitude bounded by unity. Our main result reveals that the values of the second-order modes are decreased under any of the above-mentioned transformations. This result is the generalization of the theory of Mullis and Roberts, who proved that the second-order modes are invariant under any allpass transformation, i.e. any lossless bounded-real transformation. We derive our main result by describing the controllability/observability Gramians of transformed systems with the help of the discrete-time bounded-real lemma.

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.

This letter presents a simple and explicit formulation of non-unique Wiener filters associated with the linear predictor for processing of sinusoids. It was shown in the literature that, if the input signal consists of only sinusoids and does not include a white noise, the input autocorrelation matrix in the Wiener-Hopf equation becomes rank-deficient and thus the Wiener filter is not uniquely determined. In this letter we deal with this rank-deficient problem and present a mathematical description of non-unique Wiener filters in a simple and explicit form. This description is directly obtained from the tap number, the frequency of sinusoid, and the delay parameter. We derive this result by means of the elementary row operations on the augmented matrix given by the Wiener-Hopf equation. We also show that the conventional Wiener filter for noisy input signal is included as a special case of our description.

In the field of adaptive notch filtering, Monotonically Increasing Gradient (MIG) algorithm has recently been proposed by Sugiura and Shimamura [1], where it is claimed that the MIG algorithm shows monotonically increasing gradient characteristics. However, our analysis has found that the underlying theory in [1] includes crucial errors. This letter shows that the formulation of the gradient characteristics in [1] is incorrect, and reveals that the MIG algorithm fails to realize monotonically increasing gradient characteristics when the input signal includes white noise.

This paper presents FIR digital filters based on stochastic/binary hybrid computation with reduced hardware complexity and high computational accuracy. Recently, some attempts have been made to apply stochastic computation to realization of digital filters. Such realization methods lead to significant reduction of hardware complexity over the conventional filter realizations based on binary computation. However, the stochastic digital filters suffer from lower computational accuracy than the digital filters based on binary computation because of the random error fluctuations that are generated in stochastic bit streams, stochastic multipliers, and stochastic adders. This becomes a serious problem in the case of FIR filter realizations compared with the IIR counterparts because FIR filters usually require larger number of multiplications and additions than IIR filters. To improve the computational accuracy, this paper presents a stochastic/binary hybrid realization, where multipliers are realized using stochastic computation but adders are realized using binary computation. In addition, a coefficient-scaling technique is proposed to further improve the computational accuracy of stochastic FIR filters. Furthermore, the transposed structure is applied to the FIR filter realization, leading to reduction of hardware complexity. Evaluation results demonstrate that our method achieves at most 40dB improvement in minimum stopband attenuation compared with the conventional pure stochastic design.

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.

This paper discusses the behavior of the second-order modes (Hankel singular values) of linear continuous-time systems under variable transformations with positive-real functions. That is, given a transfer function H(s) and its second-order modes, we analyze the second-order modes of transformed systems H(F(s)), where 1/F(s) is an arbitrary positive-real function. We first discuss the case of lossless positive-real transformations, and show that the second-order modes are invariant under any lossless positive-real transformation. We next consider the case of general positive-real transformations, and reveal that the values of the second-order modes are decreased under any general positive-real transformation. We achieve the derivation of these results by describing the controllability/observability Gramians of transformed systems, with the help of the lossless positive-real lemma, the positive-real lemma, and state-space formulation of transformed systems.

This paper presents a new analysis of power complementary filters using the state-space representation. Our analysis is based on the bounded-real Riccati equations that were developed in the field of control theory. Through this new state-space analysis of power complementary filters, we prove that the sum of the controllability/observability Gramians of a pair of power complementary filters is represented by a constant matrix, which is given as a solution to the bounded-real Riccati equations. This result shows that power complementary filters possess complementary properties with respect to the Gramians, as well as the magnitude responses of systems. Furthermore, we derive new theorems on a specific family of power complementary filters that are generated by a pair of invertible solutions to the bounded-real Riccati equations. These theorems show some interesting relationships of this family with respect to the Gramians, zeros, and coefficients of systems. Finally, we give a numerical example to demonstrate our results.

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.