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.

In the design of two-dimensional digital filters (2 DDF's), if the given 2 DDF design specifications can be decomposed into one-dimensional digital filter (1 DDF) specifications, the 2 DDF design problems can be reduced to 1 DDF ones. Thus the 2 DDF design problems can be made simpler. However, in the frequency domain design, the conventional decomposition methods can not avoid the problem that the 1 DDF magnitude specifications obtained from 2 DDF magnitude specification decomposition are not always nonnegative. Since negative values can not be regarded as magnitude specifications, design problem become intricate. Therefore, it is desirable in practice to develop a 2 DDF magnitude decomposition method which can guarantee the resulting 1 DDF magnitude specifications to be always nonnegative. Unfortunately, up to now, no such a method has ever been proposed. In this paper, we propose a new decomposition method called the Iterative Singular Value Decomposition (ISVD) for the decomposition of the given 2 DDF magnitude specification matrix. By the ISVD, the prescribed 2 DDF magnitude specification is decomposed into a pair of 1DDF ones, one of which is the magnitude response of a one-input/multi-output 1 DDF and the other is that of a multi-input/one-output 1 DDF. The ISVD guarantees that the resultant 1 DDF magnitude specifications are always nonnegative. The problem of designing a 2 DDF is then simplified through designing a pair of 1 DDF's with different delay elements. In our design method, 1 DDF's are designed by utilizing nonlinear optimization method to minimize the weighted magnitude square error functions. In the optimization process, a variable substitution method is proposed for transforming constrained optimization problems to unconstrained ones. As a result, no attentions should be paid to 1 DDF stability during the optimization process, and the stability of the resulting 1 DDF's is ensured. Three design examples are given to illustrate the effectiveness of the proposed method.

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.

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.

In this paper a new general method for approximation of arbitrary multiband filter loss specifications, including all classical, maximally flat and equiripple approximations as special cases, is proposed. It is possible to specify different magnitude behavior (flat or equiripple of given degree) and different maximal losses in the different passbands and to optimize all transmission and attenuation zeroes positions or to have some of them fixed. The optimization procedures for adjustment of the filter response are based on modified Remez algorithm and are performed in s-domain what is regarded since recently as an advantage in the case of design of parallel allpass structures based IIR digital filters. A powerful algorithm and appropriate software are developed following the method and their efficiency is verified through design examples.

This paper proposes a fast encoding algorithm for iterated function system (IFS) coding of gray-level homogeneous fractal images. In order to realize IFS coding of high order fractal images, it is necessary to solve a set of simultaneous equations with many unknowns. Solving the simultaneous equations using a multi-dimensional, numerical root-finding method is however very time consuming. As preprocessing of numerical computation, the proposed algorithm employs univariate polynomial manipulation, which requires less computation time than multivariate polynomial manipulation. Moreover, the symmetry of the simultaneous equations with respect to the displacement coefficients enables us to derive an equation with a single unknown from the simultaneous equations using univariate polynomial manipulation. An experimental result is presented to illustrate that the encoding time of the proposed algorithm is about 5 seconds on a personal computer with a 400 MHz Pentium II processor.

This paper proposes a measure of coefficient quantization errors for linear discrete-time state-space systems. The proposed measure of state-space systems agrees with the actual output error variance since it is derived from the exact evaluation of the output error variance due to coefficient deviation. The measure in this paper is represented by the controllability and the observability gramians and the state covariance matrix of the system. When the variance of coefficient variations is very small, the proposed measure is identical to the conventional statistical sensitivity of state-space systems. This paper also proposes a method of synthesizing minimum measure structures. Numerical examples show that the proposed measure is in very good agreement with the actual output error variance, and that minimum measure structures have a very small degradation of the frequency characteristic due to coefficient quantization.

This paper proposes a design method of 2-D variable IIR digital filters with high frequency tuning accuracy. In the proposed method, a parallel complex allpass structure is used as the prototype structure of the 2-D variable digital filters in order to obtain low sensitivity characteristic. Because the proposed 2-D variable digital filter is composed of first-order complex allpass sections connected in parallel, the proposed variable digital filter possesses several advantages such as low sensitivity characteristic in the passband, simple stability monitoring and high parallelism. In order to improve the frequency tuning accuracy of the proposed variable digital filter, each first-order complex allpass section is substituted by a new first-order complex allpass section with low sensitivity characteristic. Moreover, the coefficient sensitivity analysis of a 2-D parallel complex allpass structure is presented. Numerical examples show that the proposed 2-D variable IIR digital filter has high tuning accuracy under the finite coefficient wordlength.

In this paper, we consider the steady state mean square error (MSE) analysis for 2-D LMS adaptive filtering algorithm in which the filter's weights are updated along both vertical and horizontal directions as a doubly-indexed dynamical system. The MSE analysis is conducted using the well-known independence assumption. First we show that computation of the weight-error covariance matrix for doubly-indexed 2-D LMS algorithm requires an approximation for the weight-error correlation coefficients at large spatial lags. Then we propose a method to solve this problem. Further discussion is carried out for the special case when the input signal is white Gaussian. It is shown that the convergence in the MSE sense occurs for step size range that is significantly smaller than the one necessary for the convergence of the mean. Simulation experiments are presented to support the obtained analytical results.

This paper studies multidimensional linear periodically shift-variant digital filters (LPSV filters). The notion of a generalized multidimensional transfer function is presented for LPSV filters. The frequency characteristic of the filters is discussed in terms of this transfer function. Since LPSV filters can decompose the spectrum of an input signal into some spectral partitions and rearrange the spectrum, LPSV filters can serve as a frequency scrambler. To show the effect of multidimensional frequency scramble, 2-D LPSV filters are designed based on the 1-D Parks-McClellan algorithm. The resultant LPSV filters divide the input spectrum into some components that are permuted and possibly inverted with keeping the symmetric of the spectrum. Experimental results are presented to illustrate the effectiveness of frequency scramble for real images.

This letter derives frequency dependent Lyapunov equations for the 2-D observability and controllability Gramians which are the most important matrices in the study on the structural property of 2-D systems. These equations provide a new and efficient method to compute the 2-D Gramians.

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.

This paper proposes statistical analysis of phase-only correlation functions based on linear statistics and directional statistics. We derive the expectation and variance of the phase-only correlation functions assuming phase-spectrum differences of two input signals to be probability variables. We first assume linear probability distributions for the phase-spectrum differences. We next assume circular probability distributions for the phase-spectrum differences, considering phase-spectrum differences to be circular data. As a result, we can simply express the expectation and variance of phase-only correlation functions as linear and quadratic functions of circular variance of phase-spectrum differences, respectively.

This paper proposes a moment based encoding algorithm for iterated function system (IFS) coding of non-homogeneous fractal images with unequal probabilities. Moment based encoding algorithms for IFS coding of non-homogeneous fractal images require a solution of simultaneous algebraic equations that are difficult to handle with numerical root-finding methods. The proposed algorithm employs a variable elimination method using Grobner bases with floating-point coefficients in order to derive a numerically solvable equation with a single unknown. The algorithm also employs a varying associated-probabilities method for the purpose of decreasing the computational complexity of calculating Grobner bases. Experimental results show that the average computation time for encoding a non-homogeneous fractal image of 256256 pixels and 256 gray levels is about 200 seconds on a PC with a 400 MHz AMD K6-III processor.

In this paper, we propose to employ an extension to the natural gradient algorithm for robust Independent Component Analysis against outliers. The standard natural gradient algorithm does not exhibit this property since it employs nonrobust sample estimates for computing higher order moments. In order to overcome this drawback, we propose to use robust alternatives to higher order moments, which are comparatively less sensitive to outliers in the observed data. Some computer simulations are presented to show that the proposed method, as compared to the standard natural gradient algorithm, gives better performance in the presence of outlying data.

This letter proposes performance evaluation of phase-only correlation (POC) functions using signal-to-noise ratio (SNR) and peak-to-correlation energy (PCE). We derive the general expressions of SNR and PCE of the POC functions as correlation performance measures. SNR is expressed by simple fractional function of circular variance. PCE is simply given by squared peak value of the POC functions, and its expectation can be expressed in terms of circular variance.

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.

In this paper, we use a modified Gaussian filter to improve enlargement accuracy of the arbitrary scale LP enlargement method, which is based on the Laplacian pyramid representation (so called "LP method"). The parameters of the proposed algorithm are extracted through a theoretical analysis and an experimental estimation. Experimental results show that the proposed modified Gaussian filter is effective for the arbitrary scale LP enlargement method.