Kenneth Steiglitz

A technique for the identification of linear systems

An iterative technique is proposed to identify a linear system from samples of its input and output in the presence of noise by minimizing the mean-square error between system and model outputs. The model chosen has a transfer function which is a ratio of polynomials in z-1. Although the regression equations for the optimal set of coefficients are highly nonlinear and intractable, it is shown that the problem can be reduced to the repeated solution of a related linear problem. Computer simulation of a number of typical discrete systems is used to demonstrate the considerable improvement over the Kalman estimate which can be obtained in a few iterations. The procedure is found to be effective at signal-to-noise ratios less than unity, and with as few as 200 samples of the input and output records.

Operations on Images Using Quad Trees

A quad tree for representing a picture is a tree in which successively deeper levels represent successively finer subdivisions of picture area. An algorithm is given for superposing N quad trees in time proportional to the total number of nodes in the trees. Warnock-type algorithms are then presented for building the quad tree for the picture of the boundary of a polygon, and for coloring the interior of such a polygon. These algorithms take O(v + p + q) time, where v is the number of polygon vertices, p is the polygon perimeter, and q is a resolution parameter. When the resolution q is fixed, these algorithms are asymptotically optimal.

Eigenvectors and functions of the discrete Fourier transform

A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.

The Design of Minimum-Cost Survivable Networks

We consider the problem of designing a network which satisfies a prespecified survivability criterion with minimum cost. The survivability criterion demands that there be at least r_{ij} node disjoint paths between nodes i and j , where (r_{ij}) is a given redundancy requirement matrix. This design problem appears to be at least as difficult as the traveling salesman problem, and present techniques cannot provide a computationally feasible exact solution. A heuristic approach is described, based on recent work on the traveling salesman problem, which leads to a practical design method. Algorithms are described for generating starting networks, for producing local improvements in given networks, and for testing the redundancy of networks at each stage. This leads to networks which are locally optimum with respect to the given transformation. Randomizing the starting solution ensures that the solution space is widely sampled. Two theorems are given which allow great reduction in the amount of computation required to test the redundancy of a network. Finally, some design examples are given.

Computation of spectra with unequal resolution using the fast Fourier transform

The discrete Fourier transform of a sequence, which can be computed using the fast Fourier transform algorithm, represents samples of the z transform equally spaced around the unit circle. In this letter, a technique is discussed and illustrated for transforming a sequence to a new sequence whose discrete Fourier transform is equal to samples of the z transform of the original sequence at unequally spaced angles around the unit circle.

Computer-aided design of recursive digital filters

A practical method is described for designing recursive digital filters with arbitrary, prescribed magnitude characteristics. The method uses the Fletcher-Powell optimization algorithm to minimize a square-error criterion in the frequency domain. A strategy is described whereby stability and minimum-phase constraints are observed, while still using the unconstrained optimization algorithm. The cascade canonic form is used, so that the resultant filters can be realized accurately and simply. Design examples are given of low-pass, wide-band differentiator, linear discriminator, and vowel formant filters.

Adaptive step size random search

Fixed step size random search for minimization of functions of several parameters is described and compared with the fixed step size gradient method for a particular surface. A theoretical technique, using the optimum step size at each step, is analyzed. A practical adaptive step size random search algorithm is then proposed, and experimental experience is reported that shows the superiority of random search over other methods for sufficiently high dimension.

The complexity of analog computation

We ask if analog computers can solve NP-complete problems efficiently. Regarding this as unlikely, we formulate a strong version of Churchs Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation.

Suppression of near- and far-end crosstalk by linear pre- and post-filtering

Full-duplex data communication over a multi-input/multi-output linear time-invariant channel is considered. The minimum mean square error (MMSE) linear equalizer is derived in the presence of both near- and far-end crosstalk and independent additive noise. The MMSE equalizer is completely specified in terms of the channel and crosstalk transfer functions by using a generalization of previous work due to Salz (1985). Conditions are given under which the equalizer can completely eliminate both near- and far-end crosstalk and intersymbol interference. The MMSE transmitter filter, subject to a transmitted power constraint, is specified when the channel and crosstalk transfer functions are bandlimited to the Nyquist frequency. Also considered is the design of MMSE transmitter and receiver filters when the data signals are arbitrary wide-sense stationary continuous or discrete-time signals, corresponding to the situation where the crosstalk is not phase-synchronous with the desired signal. >

Soliton-like behavior in automata

Abstract We propose a new kind of automaton that uses newly computed site values as soon as they are available. We call them Filter Automata (FA); they are analogous to Infinite Impulse Response (IIR) digital filters, whereas the usual Cellular Automata (CA) correspond to Finite Impulse Response (FIR) digital filters. It is shown that as a class the FAs are equivalent to CAs, in the sense that the same array of space-generation values can be produced; they must be generated in a different order, however. A particular class of irreversible, totalistic FAs are described that support a profusion of persistent structures that move at different speeds, and these particle-like patterns collide in nondestructive ways. They often pass through one another with nothing more than a phase jump, much like the solitons that arise in the solution of certain nonlinear differential equations. Histograms of speed, displacement, and period are given for neighborhood radii from 2 to 6 and particles with generators up to 16 bits wide. We then present statistics, for neighborhood radii 2 to 9, which show that collisions which preserve the identity of particles are very common.