Multidimensional Systems and Signal Processing

In this paper, a time-recursive algorithm for the computation of discrete-time, discrete-frequency, auto-ambiguity function is given, which in turn is used to compute the Wigner time-frequency distribution. We use the Richman-Parks-Shenoy's definition of the discrete-time, discrete-frequency, auto-ambiguity function for finite data sequences. Block diagram implementation of the time-recursive algorithm is described, computational complexity with respect to increasing block size is compared with straight forward computation, and the proper time-recursive block size is suggested.

This paper develops a novel ellipse fitting algorithm by recovering a low-rank generalized multidimensional scaling (GMDS) matrix. The main contributions of this paper are: i) Based on the derived Givens transform-like ellipse equation, we construct a GMDS matrix characterized by three unknown auxiliary parameters (UAPs), which are functions of several ellipse parameters; ii) Since the GMDS matrix will have low rank when the UAPs are correctly determined, its recovery and the estimation of UAPs are formulated as a rank minimization problem. We then apply the alternating direction method of multipliers as the solver; iii) By utilizing the fact that the noise subspace of the GMDS matrix is orthogonal to the corresponding manifold, we determine the remaining ellipse parameters by solving a specially designed least squares problem. Simulation and experimental results are presented to demonstrate the effectiveness of the proposed algorithm.

Thephase functions of N-dimensional (N-D) digital all-pass filtersare investigated to approximate a prescribed phase response ina frequency region. The set of phase functions of the all-passfilters have common properties with some nonlinear approximatingfunctions. This similarity answers the question of characterizationof minimal approximation in the set of phase functions. The optimalapproximation is characterized by known theorems of TschebycheffApproximation Theory. Among the main tools of the theory, theGlobal and Local Kolmogoroff Criteria, are shown to give necessaryand sufficient conditions for best approximations in the phasefunctions of N-D all-pass filters. Moreover, this best approximationin the phase functions is shown to be a global minimum. The approximationon discrete point sets (H-sets) in a compact multidimensionaldomain is studied. Optimal N-dimensional approximation is notunique, an inherent property of functions of several variables.

Repetitive, or multipass, processes are a class of 2D systems characterized by a recursive action with interaction between successive outputs or pass profiles. This interpass interaction is the source of the unique control problem for these processes in that it can cause the output sequence to exhibit oscillations which increase in amplitude from pass to pass. Previous work has developed an abstract stability theory and applied it to subclasses, such as discrete nonunit memory linear processes which are considered here, to produce basic stability tests. This article begins by reviewing the known stability tests and concludes that, at best, they only produce highly qualitative indicators of relative stability and performance. Hence, unlike (say) Bode and Nyquist tests for standard linear systems, they are of limited appeal as a basis for computer-aided control systems design. To remove this difficulty, step response data is used to develop new simulation-based tests which yield, at no extra cost, unique computable performance measures. Further, the undoubted advantages of having such measures available is clearly shown by developing a (virtually) complete solution to controller design for one subclass.

We obtain a standard Bounded Real Lemma and a strict Bounded Real Lemma for the class of input/state/output linear systems with evolution along a free semigroup recently introduced by the authors. As an application we show the equivalence of this result with a recent result from robust control theory: the structured μ-singular value of an operator (taken in a certain uniform sense) is equal to its upper bound computed via diagonal scalings in case the structured singular value is taken with respect to the collection of time-varying perturbations given by a linear-fractional model.

In this paper, a novel 2-D Schur algorithm is developed as a natural extension of the 1-D Schur recursion. This lattice structure is based on Parker and Kayran's four-field lattice approach. Starting with given 2-D autocorrelation samples, four quarter-plane gapped functions are generated. Their linear combination is used to satisfy gap conditions and calculate 2-D lattice parameter reflection factors for the first stage. In order to determine the growing number of 2-D reflection coefficients at succesive stages, appropriately defined auxiliary gapped functions are introduced after the first order. The theory has been confirmed by computer simulations. In addition to developing the basic theory, the presentation includes a comparison between the proposed 2-D lattice structure and other existing four-field lattice structures.

As has been shown, attractive methods for numerically integrating partial differential equations (PDEs) resulting from physical problems can be obtained by simulating the actual physical passive (conservation of energy) dynamical system by means of a discrete passive dynamical system, and this in such a way that the full parallelism and the exclusively local nature of the interconnections (principle of action at proximity) are preserved. An alternative approach for developing such methods is presented which, while still using principles of the same type as those on which multidimensional wave digital filters (WDFs) are based, involves appropriate transformations of the original coordinates of the physical problem at hand. This alternative approach is not only easier to apply than the one referred to above but also more general; it is illustrated on the one hand by the same examples as those that have been used for the other approach, and on the other by showing the applicability to Maxwell's equations.

We extend a result of Napp Avelli, van der Put, and Rocha with a system-theoretic interpretation to the noncommutative case: Let $$P$$ be a f.g. projective module over a two-sided Noetherian domain. If $$P$$ admits a subdirect product structure of the form $$P \cong M \times _T N$$ over a factor module $$T$$ of grade at least $$2$$ then the torsion-free factor of $$M$$ (resp. $$N$$ ) is projective.