The power spectral measure, an informative feature of a stationary time-discrete stochastic process, describes the relative strength of uncorrelated frequency components that compose the process. In spectral estimation one wants to describe the spectral measures of processes having a prescribed initial block of autocorrelation coefficients. In the continuum of possibilities, two types are frequently distinguished: the one-parameter family of maximum-likelihood spectral measures, so named because they carry the largest possible weight at one specified frequency, and the maximum-entropy spectral measure which is often considered to be the most representative of the entire set of possible solutions. Although these choices are very different, representing, respectively, the most and least predictable of the eligible processes, we show that the maximum-entropy measure is exactly the uniform average of the family of maximum-likelihood measures over its parameter.

Phase-space decompositions for signals of finite energy have been used to formulate the intuitive but elusive idea that frequency content of a signal can vary with time as the signal evolves. The decompositions consist of subdividing the time and frequency axes into certain ranges and expanding signals stably in a two-parameter basis of fixed functions of time, in which the (k,m)th coefficient is viewed as describing the part of the signal that is concentrated in the kth time and mth frequency range. It is shown that in any such expansion, the time-frequency space must be subdivided, or, equivalently, the coefficients computed, at least at the Nyquist rate of two per unit of time and cycle of bandwidth.>

The authors reconsider the problem of determining the minimum distance between output sequences of an ideal band-limiting channel that are generated by uncoded binary sequences transmitted at a rate exceeding the Nyquist rate. For signaling rates up to about 25% faster than the Nyquist rate, it is shown that the minimum distance does not drop below the value which it would have in the ideal case wherein there is not intersymbol interference. Mathematically, the problem is to decide if the best L/sup 2/ Fourier approximation to the constant 1 on the interval (- sigma pi , sigma pi ), 00, with coefficients restricted to be =1 or =0, occurs when all coefficients are zero. This is shown to be optimal for 0.802...>

A band-limited signal of finite energy can be reconstructed from its samples taken at the Nyquist rate. Moreover, the reconstruction is stable, a feature crucial for implementation: a small error in the sample values generates only a correspondingly small error in the resulting signal. The Nyquist sample values are mutually independent, so that knowledge of them in a given interval- T \leq t \leq 0provides hardly any information about the behavior of the signal outside the interval. However, when the samples are taken at a greater rate--a case referred to as "over-sampling"--they are interrelated, and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure. A natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation; this problem is relevant as well to prediction theory. With oversampling, whenT = \infty, so that the samples are known on the entire half-linet < 0, they determine the signal everywhere, although the reconstruction is now no longer stable. Here we examine the case of finiteT;of course, a finite amount of data can yield only limited accuracy. We prove that the samples can be used to form an approximation to the signal outside the sampling interval, with an error which, asymptotically asT \rightarrow \infty, decreases exponentially inT, over a range which grows linearly withT. However, as in the limiting case, these approximations are not useful in practice, since they require the sample values to be known exactly. In the presence of measurement error, the nature of the results changes: good approximations are now available for only a bounded distance outside the interval of observation, regardless of its length, and their accuracy and range of validity can be increased only by improving the precision of sample reading. Since physical measurements are never perfect, it is this conclusion which counts for applications. The same results hold for the extrapolation of bounded band-limited signals.

It is desirable to choose the waveforms making up a signaling alphabet so that they are maximally separated one from another. This problem is considered, in the space of square-integrable functions, for signals which have finite duration, and are constrained in the ranges of their values as well as in energy. Corresponding to each of the following cases, we establish sharp bounds for the minimum distance and for the average distance between elements of a fixed size signal set, and construct sets of signals that attain both bounds simultaneously. \begin{list} \item {\em Case A (Energy Constraint Only):} The average energy of the waveforms in the signal set is at most\sigma, where0 \leq \sigma < \infty. \item {\em Case B (Energy and Peak Amplitude Constraints):} The average energy of the waveforms in the signal set is\leq \sigma (0 \leq \sigma < 1), and the absolute value of each waveform is at most1. \item {\em Case C (Energy and Value Constraints):} The average energy of the waveforms in the signal set is at mostb^{2}\sigma + a^{2}(1 - \sigma), and each waveform takes values in the set[a, b], where0 \leq a < b < \infty, and0 \leq \sigma \leq 1. \end{list} Cases A and B are applicable to signal design for communication in channels with additive noise (say Gaussian), and Case C is applicable to signal design for optical channels, where the signal represents the intensity of a photon stream. The general character of the results is that the minimum distance behaves like\gamma \sigmain Cases A and B, and like\gamma \sigma (1 - \sigma)in Case C, with\gammaa suitable constant.

Sets of unit vectors inN-dimensional Euclidean vector space whose constituent vectors are separated one from another by at least a fixed distanced, prescribed once for all and independent ofN, are of interest in theory and practice; they have fondly been called "porcupine codes." Although an elegant constructive proof of Gilbert shows that the number of vectors in a porcupine code (of givend) can increase exponentially withN, no systematic method is yet known for generating porcupine codes of this cardinality. Corresponding to a collection ofMvectors, we can partition the space into maximum-likelihood regions, thejth of which consists of those vectors that lie closer to thejth than to any other element of the collection. Each maximum-likelihood region is bounded by at most(M - 1)hyperplanes, and we denote byKthe total number of these bounding hyperplanes. Collections for whichKis small may be expected to have greater symmetry than those for whichKis large. In this paper we show that, for porcupine codes,K \geq (M/2)^{1/s}, withsdepending only ond, the minimum separation of the code vectors. Hence, for the number of vectors of a porcupine code to increase exponentially with dimension, the number of separating hyperplanes must do so as well. We conclude with, an application to the permutation codes introduced by Slepian, showing that the number of vectors of a porcupine code which is of permutation-modulation type can not increase exponentially withN.