Formulas for the expected zero-crossing rate of non-Gaussian mixtures and products of Gaussian processes are obtained. The approach we take is to first derive the expected zero-crossing rate in discrete time and then obtain the rate in continuous time by an appropriate limiting argument. The processes considered, which are non-Gaussian but derived from Gaussian processes, serve to illustrate the variability of the zero-crossing rate in terms of the normalized autocorrelation function p(t) of the process. For Gaussian processes, Rice's formula gives the expected zero-crossing rate in continuous time as 1//spl pi//spl radic/(-/spl rho/"(0)). We show processes exist with expected zero-crossing rates given by /spl kappa///spl pi//spl radic/(-/spl rho/"(0)) with either /spl kappa//spl Gt/1 or /spl kappa//spl Lt/1. Consequently, such processes can have an arbitrarily large or small zero-crossing rate as compared to a Gaussian process with the same autocorrelation function.

The estimation of power is problematic when the amplitudes in the discrete spectral component are random. An example of this is a mixed spectrum stationary Gaussian process with mean zero. In this regard, it is shown that the maximum likelihood estimator of the power from a random sample of peaks exceeding a relatively large threshold, the peak estimator is more efficient than the sample variance from a random sample of the same size of signal values. When evaluated from nonrandom samples, indications are that the peak estimator may still have a relatively small mean square error.

Some statistical properties in regard to the contraction mapping (CM) method are discussed. One of the requirements in this method is that the filter be parameterized to satisfy a certain fundamental property. The parameterization clearly depends on the normalized noise spectrum which theoretically has to be known or estimated a priori. If this information is available, one can first whiten the noise with a linear filter and then apply the CM method to the filtered data. In this way, the parameterization only needs to be done under the white noise assumption and filters like the AR(2) can be used by the CM method. In applications, however, prewhitening may not always be necessary.>

Formulas for the expected zero-crossing rates of random processes that are monotone transformations of Gaussian processes can be obtained by using two different techniques. The first technique involves derivation of the expected zero-crossing rate for discrete-time processes and extends the result of the continuous-time case by using an appropriate limiting argument. The second is a direct method that makes successive use of R. Price's (1958) theorem, the chain rule for derivatives, and S.O. Rice's (1954) formula for the expected zero-crossing rate of a Gaussian process. A constant, which depends on the variance of the transformed process and a second-moment of its derivative, is derived. Multiplying Rice's original expression by this constant yields the zero-crossing formula for the transformed process. The two methods can be used for the general level-crossing problem of random processes that are monotone functions of a Gaussian process.>

The precise effect of noise on HOC (higher-order crossing) sequences is examined, settling some issues not dealt with previously. It is shown that it is possible to construct HOC sequences that converge to true discrete frequencies in a mixed spectrum even for very low signal-to-noise ratios. The availability of the correlation function is then assumed. When the expected HOCs (from repeated differences) are known, the correlation function is automatically known as well. However, other quantities, not just the expected HOC, can serve the same purpose. For example, under the Gaussian assumption, the correlation function can be obtained from that of the binary process derived by clipping the original process at any level.>

We advance a coherent development of zero-crossing-based methods and theory appropriate for fast signal analysis. Quite a few ideas pertaining to zero-crossing counts found in the literature can be expressed and interpreted with the help of this more general setup. A central issue addressed in some detail is the fruitful connection which exists between zero-crossing counts and linear filtering. This connection is explored and interpreted with the help of a certain zero-crossing spectral representation, and is then applied in spectral analysis, detection, and discrimination. Zero-crossing counts in filtered time series are called higher order crossings. The theme of this work is that higher order crossings analysis provides a useful descriptive as well as an analytical tool that can in many respects match spectral analysis. To a great extent these two types of analysis are, in fact, equivalent, but each emphasizes a different point of view. Advantages offered by higher order crossings are great simplicity and a drastic data reduction.

The limiting forms of binary processes obtained via sequential filtering followed by clipping are discussed in the cases of one- and two-dimensional processes. In particular, repeated differencing followed by clipping displays a convergence phenomenon shared by stationary stochastic processes and fields. For this case, the higher order crossings are monotone under some conditions.

The pictorial information in a stationary time series as depicted by crossings of levels and crossings of random levels and related quantities is studied. It is shown that such graphical features are directly connected with the covariance function and hence with the spectral density. Many of these features can be actually applied in estimation and in the study of extremes. In the Gaussian case, the finite dimensional distributions are completely determined by the axis crossings and by the crossings of a random curve (to be defined) if the process is essentially bounded. Certain graphical patterns are suggested for a fast recognition of low-order ARMA models.

We consider the problem of maximum likelihood estimation of the parameter in the first-order autoregressive stationary process after loss of information due to hard limiting. For this particular transformation, the exact maximum likelihood estimator is found, and its distribution function is approximated. A numerical comparison with the common estimate obtained from the original data shows that, for moderate sample sizes and small variance of the error term, very little precision is lost as a result of the binary transformation. On the other hand, the suggested estimator is simple and easy to compute.