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In this paper, a new class of reciprocal-orthogonal parametric (ROP) transforms having 3N/2 independent parameters for a sequence length N that is a power of two is proposed. The basic idea behind the proposed transforms is to appropriately combine a new parametric kernel with that of the well-known Walsh-Hadamard transform that results in a square parametric matrix operator of order N with some very interesting properties. It is shown that the inverse matrix operator of the proposed class of transforms can be easily obtained by taking the reciprocal of each of the entries of the forward matrix and then transposing the resulting matrix. In addition, a simple method is introduced in order to facilitate the generation of the matrix operator of the proposed ROP transforms. This method is then used specifically to construct new classes of unitary and multiplication-free transforms. Many other new transforms, as well some of the existing ones, can be derived from these proposed ROP transforms. An efficient algorithm is developed for a fast computation of the proposed transforms. In view of the availability of this fast algorithm and the property of easily computable inverse transform, the proposed ROP transforms can be used in many transform-based applications, with their independent parameters providing more degrees of freedom such as affording an additional secret key in watermarking and encryption applications. Index Terms--Complex Hadamard matrices, reciprocal-orthogonal parametric (ROP) transforms and matrices, unitary transforms, Waish-Hadamard transform (WHT).

In this paper, we focus on the problem of driving a general network to a selected cluster synchronization pattern by means of a pinning control strategy. Sufficient conditions are presented to guarantee the realization of the cluster synchronization pattern for all initial values. We also show the detailed steps on how to construct the coupling matrix and to modify the control strengths. Moreover, the method of adapting the coupling strength is provided to refine the result. Index Terms--Adaptive adjustment, cluster synchronization, globally asymptotically stable, linearly coupled networks, pinning control.

This paper is a tutorial on Grobner bases and a survey on the applications of Grobner bases in the broad field of signals and systems. A reasonably detailed review is given of several fundamental theoretical issues that occur in the use of Grobner bases in multidimensional signals and systems applications. These topics include the primeness of multivariate polynomial matrices, multivariate unimodular polynomial matrix completion, and prime factorization of multivariate polynomial matrices. A brief review is also presented on the wide-ranging applications of Griibner bases in multidimensional as well as one-dimensional circuits, networks, control, coding, signals, and systems and other related areas like robotics and applied mechanics. The impact and scope of Grobner bases in signals and systems are highlighted with respect to what has already been accomplished as a stepping stone to expanding future research. Index Terms--Circuits and systems, Grobner bases, matrix factorization, multidimensional systems, multivariate polynomial, signal processing.

In the simplest case, the matrix inversion lemma gives an explicit formula of the inverse of a positive-definite matrix A added to a rank-one matrix [bb.sup.H] as follows:[(A + [bb.sup.H]).sup.-1] = [A.sup.-1] - [A.sup.-1]b[(1 + [b.sup.H][A.sup.-1]b).sup.-1] [b.sup.H[ [A.sup.-1]. It is well known in the literature that this formula is very useful to develop a recursive least-squares algorithm for the recursive identification of linear systems or the design of adaptive filters. We extend this result to the case when the matrix A is singular and present a matrix pseudo-inversion lemma along with some illustrative examples. Such a singular case may occur in a situation where a given problem is overdetermined in the sense that it has more equations than unknowns. This iemma is important in its own right, but in order to show the usefulness of the lemma, we apply it to develop an adaptive super-exponential algorithm for the blind deconvolution of multi-input multi-output systems. Index Terms--Adaptive super-exponential algorithm (SEA), matrix pseudo-inversion lemma, recursive algorithms.

A constructive procedure for factoring a polynomial paraunitary matrix as a product of atomic paraunitary matrices is considered. The generic form of the paraunitary factors results from the use of projection matrices. In the univariate case, the factorization carried out in the z-transform domain collapses to other univariate paraunitary matrix factorization methods. In the multivariate case, the generic factor is constructed based on the assumption of its existence. Nontrivial examples for multiband univariate, bivariate, and trivariate cases are given to illustrate the simple and straightforward algorithmic implementation. Index Terms--Multivariate polynomial paraunitary matrix factorization, projection operators.

A criterion for the elimination of limit cycles in direct form digital filters utilizing single saturation overflow nonlinearity is presented. The criterion takes the form of linear matrix inequality and, hence, is computationally tractable. An example showing the effectiveness of the present criterion is given. Index Terms--Asymptotic stability, digital filters, finite wordlength effects, limit cycles, nonlinear systems.

In this note, we point out that the results given in the paper by Wu and Cui (2007) can be obtained by using the results given in Lu et al. (2003). Moreover, the results given in the second paper are much stronger under less restrictive conditions. Index Terms--Delayed neural networks, equilibrium point, global exponential stability, linear matrix inequality (LMI).