The most common form of quantization is rounding-off, which occurs in all digital systems. A general quantizer approximates an observed value by the nearest among a finite number of representative values. In estimating weighted integrals of a time series with no quadratic mean derivatives, by means of samples at discrete times, it is known that the rate of convergence of the mean-square error is reduced from n/sup -2/ to n/sup -1.5/ when the samples are quantized. For smoother time series, with k=1, 2, ... quadratic mean derivatives, it is now shown that the rate of convergence is reduced from n/sup -2k-2/ to n/sup -2/ when the samples are quantized, which is a very significant reduction. The interplay between sampling and quantization is also studied, leading to (asymptotically) optimal allocation between the number of samples and the number of levels of quantization.

Some second-order properties of random processes such as periodic correlation, stationarity, harmonizability, self-similarity, are characterized via corresponding properties of their wavelet transform: any one of these properties of the wavelet transform characterizes the corresponding property of the increments of the random process, of order equal to the order of regularity of the analyzing wavelet. These results are then specialized to fractional Brownian motion and other self-similar processes.>

The multiresolution decomposition of deterministic and random signals and the resulting approximation at increasingly finer resolution is examined. Specifically, an nth-order expansion is developed for the error in the wavelet approximation at resolution 2/sup -l/ of deterministic and random signals. The deterministic signals are assumed to have n continuous derivatives, while the random signals are only assumed to have a correlation function with continuous nth-order derivatives off the diagonal-a very mild assumption. For deterministic signals square integrable over the entire real line, for stationary random signals over finite intervals, and for nonstationary random signals with finite mean energy over the entire real line, the smoothness of the scale function can be matched with the signal smoothness to substantially improve the quality of the approximation. In sharp contrast, this is feasible only in special cases for nonstationary random signals over finite intervals and for deterministic signals which are only locally square integrable.>

Explicit expression are derived for the conditional expectation and variance of the encoder in a predictive DPCM coder with an N-level quantizer, when a stationary Ornstein-Uhlenbeck process is a source. A representation of the encoder in terms of a stochastic integral is presented. These expressions yield a nonlinear stochastic difference equation for the decoding error process and a stochastic differential equation (SDE) as a weak limit for the error process. The statistical properties of the error obtained as a solution of the limiting SDE are interpreted in terms of the slope overload error.>

The problem of estimating a weighted average of a random process from noisy observations at a finite number of sampling points is considered. The performance of sampling designs with optimal or suboptimal, but easily computable, estimator coefficients is studied. Several examples and special cases are studied, including additive independent noise, nonlinear distortion with noise, and quantization noise.>

The asymptotic performance of linear predictors of continuous-time stationary processes is studied from observations at n sampling instants on a fixed observation interval. Both optimal and simpler choices of predictor coefficients are considered, using uniform sampling as well as nonuniform sampling tailored to the statistics of the process under prediction. The focus is on stationary processes with rational spectral densities and the asymptotic performance for cases with and without a quadratic-mean derivatives is obtained. The analytical results are supplemented by numerical examples depicting small- and large-sample-size performance.>

An adaptive matched differential pulse-code modulator (AMDPCM) is analyzed. The adaptation of the symmetric uniform quantizer parameter\Delta_{n}is performed by fixed multipliers assigned to the quantizer output levels. The input is stationary first-order Gauss-Markov. The correlation of the samples is used as the leakage parameter in the matched integrator, with the predictive reconstruction similarly matched. For a4-level quantizer and multipliers(\gamma^{-1}, \gamma)the limiting joint distribution of the prediction error and\Delta_{n}is derived and the asymptotic sample-point and time-averaged mean-square error (rose) and mean and variance of\Delta_{n}as functions of\gamma \in (1,2]are computed and plotted. It is found that the asymptotic performance of AMDPCM does not depend on the choice of\Delta_{0}, that the increase in mse incurred by using A(M)DPCM instead of (M)DPCM with\Delta_{opt}is small, with mse(A(M)DPCM)\downarrow \min_{\Delta}mse ((M)DPCM) as\gamma \downarrow 1, and that the signal-to-noise ratio of AMDPCM does not depend on the input power.

While delta modulation (DM) simply compares the current predictive estimate of the input with the current sample, delayed delta modulation (DDM) also compares with the upcoming sample so as to detect and anticipate slope overloading. Since this future sample must be available before the present output is determined and the estimate updated, delay is introduced at the encoding. The performance of DDM with perfect integration and step-function reconstruction is analyzed for each of three random input signals. In every case, the stochastic stability of the system is established. For a discrete time, independent and identically distributed input, the (limiting) joint distribution of input and output is derived, and the (asymptotic) mean-square sample point error mse(SP) is computed when the input is Gaussian. For a Wiener input, the joint distribution of the sample point and prediction errors is derived, and mse(SP) and the time-averaged mse (mse(TA)) are computed. For a stationary first-order Gauss-Markov input, the joint distribution of input and output is derived and mse(SP) and mse(TA) computed. Graphs of the mse's illustrate the improvement attainable by using DDM instead of DM. With optimal setting of parameters, mse(SP) (mse(TA)) is reduced about15percent (35percent).

A simple class of quantizers is introduced which are asymptotically optimal, as the number of quantization levels increases to infinity, with respect to a meanrth power absolute error distortion measure. These asymptotically optimal quantizers are very easy to compute. Their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed. In addition their asymptotic robustness is studied under location, scale, and shape mismatch for several families of distributions.

Sampling designs for the detection of sure signals in Gaussian noise are considered. Both deterministic and random sampling schemes, using optimal and nonoptimal detectors, are presented and their performance is studied. The analytical results are supplemented by comparison of performance for small and large Sample size for some representative processes including the Gauss-Markov and Wiener processes.

Finite sampling approximations, along with bounds on the approximation error, are derived for certain deterministic and random signals which are not band-limited.

Bounds are derived for the truncation error of the cardinal sampling expansion for a large class of band-limited deterministic and random signals. These bounds extend and improve upon the bounds available in the literature.

A signal cannot in general be reconstructed from its sign, i.e., from its hard-limited version. However, by the deliberate addition of noise to samples of the signal prior to hard limiting, it is shown that the signal can be estimated consistently from its hard-limited noisy samples as the sampling rate tends to infinity. In fact, such estimates are shown to converge with probability one to the signal and to be asymptotically normal. Although the estimates are in general nonlinear, they can be made linear by a proper choice of the noise distribution. These rather unexpected results hold for all bounded and uniformly continuous signals. In addition to the hard-limiter, such results are also established for certain monotonic and nonmonotonic nonlinearities.

A nonrandom and unknown signals(t), in additive Gaussian noise with mean zero and known covariance function, is passed through a known memoryless nonlinearityf(x). The identification of the signal from the distributions or moments of the output process is considered. Arbitrary signals can be identified for monotonic and for certain odd, not necessarily monotonic, nonlinearities; these include hard limiters, quantizers, and infinite-interval windows. Arbitrary signals can be identified up to a global sign for two distinct classes of even nonlinearities; these include full-wave even\nuth-law devices and symmetric-interval windows.

The problem of reconstructing the varianceR(0)of a zero-mean stationary Gaussian process passed through a memoryless nonlinearityf(x)is considered whenfand the first two moments of the output

The problem of reconstructing the normalized covariance functionR(t)of a zero-mean stationary Gaussian process observed through a zero-memory nonlinearityf(x)is considered, when the nonlinearity and the correlation function or the second-order distribution of the output process are known. Three kinds of results are established. (i) Arbitrary covariances can be reconstructed for certain nonlinearities, including monotonicf, appropriate interval windows, and certain quite generalf. (ii) Certain covariances can be reconstructed for arbitrary nonlinearities: included here are positive covariances(\geq 0), covariances with rational spectral densities, and bandlimited covariances. (iii) Certain covariances, satisfying rather weak conditions, that can easily be checked in terms of the output correlation function, can be reconstructed for certain nonlinearities that include symmetric as well as nonsymmetricf.

The Shannon lower bound on the rate distortion function of spherically invariant random vectors and sequences is found, and a condition for its tightness is given. Under this condition, simple upper bounds which are valid over a certain range of distortions are given. Also included is a coding theorem for certain stationary discrete-time spherically invariant sources.

The analysis of the performance of a delta modulator with a Wiener process input is considered. The mean and the variance of the steady-state squared error are derived. In addition, the mean of the steady-state error of order four is calculated. It is shown that the behavior of all these moments as functions of the step size Δ and the sampling intervalTis similar: for each fixed Δ, the moments are monotonically increasing inT. More important, for each fixedT, there exists an optimal step size Δoptwhich minimizes each of these moments. It is shown that Δoptis a multiple ofT^{1/2}and that the corresponding minimum value of the moments is a multiple of eitherTor T2, depending on the order of the moment.

An explicit and easily implemented solution is given to the general problem of linear mean-square estimation of a signal or system process based upon noisy observations, under the assumption that the auto- and cross-correlation functions of the signal and the observation processes are known. Also a number of specific estimation problems are briefly discussed.

The series expansion of a bivariate distribution and the series expansion of the output of a nonlinearity are considered, as well as the relationship between these two problems. Three distinct expansions of bivariate distributions are presented along with a constructive procedure to obtain them explicitly. The cross-covarianee property and certain results on the expansion of the output of a nonlinearity are extended to a larger class of random processes.