Shunsuke Yamaki

A Closed Form Solution to L2-Sensitivity Minimization of Second-Order State-Space Digital Filters

This paper proposes a closed form solution to L2-sensitivity minimization of second-order state-space digital filters. Restricting ourselves to the 2nd order case of state-space digital filters, we can formulate L2-sensitivity minimization problem by hyperbolic functions. As a result, L2-sensitivity minimization problem can be converted into a problem to find the solution to a 4th degree polynomial equation of constant coefficients, which can be algebraically solved in closed form without iterative calculations

On the Absence of Limit Cycles in State-Space Digital Filters With Minimum

This brief proposes a systematic approach to synthesis of limit cycle free state-space digital filters with minimum L2-sensitivity. We synthesize the minimum L2-sensitivity realization adopting the balanced realization as an initial realization. The coordinate transformation matrix which transforms the balanced realization into the minimum L2-sensitivity realization is expressed as the product of a positive definite symmetric matrix and arbitrary orthogonal matrix. We show that the controllability and observability Gramians of the minimum L2-sensitivity realization satisfy a sufficient condition for the absence of limit cycles when we select an appropriate orthogonal matrix. As a result, the minimum L2-sensitivity realization without limit cycles can be synthesized by selecting an appropriate orthogonal matrix.

L_{2}

This paper proposes a closed form solution to L2-sensitivity minimization of second-order state-space digital filters subject to L2-scaling constraints. The proposed approach reduces the constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. Furthermore, restricting ourselves to the case of second-order state-space digital filters, we can express the L2-sensitivity by a simple linear combination of exponential functions and formulate the L2-sensitivity minimization problem by a simple polynomial equation. As a result, L2-sensitivity is expressed in closed form, and its minimization subject to L2-scaling constraints is achieved without iterative calculations.

-Sensitivity

This paper proposes a closed form solution to L/sub 2/-sensitivity minimization of second-order state-space digital filters. Restricting ourselves to the 2nd order case of state-space digital filters, we can formulate L/sub 2/-sensitivity minimization problem by hyperbolic functions. As a result, L/sub 2/-sensitivity minimization problem can be converted into a problem to find the solution to a 4th degree polynomial equation of constant coefficients, which can be algebraically solved in closed form without iterative calculations.

A Closed Form Solution to L2-Sensitivity Minimization of Second-Order State-Space Digital Filters Subject to L2-Scaling Constraints

This paper analyzes effects of stochastic phase spectrum differences on phase-only correlation (POC) functions. We assume phase spectrum differences between two signals are statistically constant for frequency indices. That is, they have identical probability density function for all frequency indices. We derive the general expressions of the expectation and variance of the POC functions. Relationships between the POC functions and the phase spectrum differences are formulated. This result mathematically guarantees the validity of the POC functions used for similarity measure in matching techniques.

A closed form solution to L/sub 2/-sensitivity minimization of second-order state-space digital filters

This paper reveals the class of digital filters with all second-order modes equal. We first prove that if the second-order modes of a digital filter are all equal, the L2-sensitivity minimization problem of the digital filter can be solved analytically. We derive a general expression of the transfer function of digital filters with all second-order modes equal. Furthermore, we show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter.

Effects of stochastic phase-spectrum differences on phase-only correlation functions: Part II: Statistically proportional phase spectrum differences to frequency indices

This letter presents explicit expressions of the balanced realization of second-order digital filters with real poles. We consider two cases of second-order digital filters: that of real and distinct poles and that of real and multiple poles. Simple formulas are derived for the synthesis of the balanced realizations of these second-order digital filters.

Derivation of the Class of Digital Filters With All Second-Order Modes Equal

This paper proposes a fast convergence algorithm for L2-sensitivity minimization problem of two-dimensional (2-D) separable-denominator state-space digital filters subject to L2-scaling constraints. The proposed algorithm reduces a constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation, and minimizes the L2-sensitivity by iterative calculation called successive substitution method. Our novel algorithm can achieve the L2-sensitivity minimization with quite fast convergence behavior.

Explicit Expressions of Balanced Realizations of Second-Order Digital Filters With Real Poles

L2-sensitivity is one of the functions that evaluates the coefficient quantization effects of state-space digital filters. This paper proposes a closed form solution to L2-sensitivity minimization of 2nd-order state-space digital filters subject to L2 -scaling constraints. The proposed solution reduces a constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. Futhermore, by restricting ourselves to the case of 2nd-order state-space digital filters, we can formulate the L2-sensitivity minimization problem via hyperbolic functions. As a result, we can express the L2-sensitivity in closed form, whose minimization subject to L2-scaling constraints is achieved without iterative calculations

A Fast Convergence Algorithm for L2-Sensitivity Minimization of 2-D Separable-Denominator State-Space Digital Filters

This paper gives conjecture on the absence of limit cycles of the minimum L2-sensitivity realizations subject to L2-scaling constraints for second-order digital filters. We design second-order digital filters with various pole-zero configurations, synthesize the minimum L2-sensitivity realizations subject to L2-scaling constraints, and examine if their coefficient matrices satisfy a sufficient condition for the absence of limit cycles. As a result, in the range of practical pole radii, it is shown that the minimum L2-sensitivity realizations subject to L2-scaling constraints of second-order digital filters satisfy a sufficient condition for the absence of limit cycles. Furthermore, we demonstrate the absence of limit cycles of the minimum L2-sensitivity realizations of a second-order digital filter by observing its zero-input response.