Steven Kay

Fundamentals of statistical signal processing: estimation theory

Minimum variance unbiased estimation Cramer-Rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the Bayesian philosophy general Bayesian estimators linear Bayesian estimators Kalman filters summary of estimators extension for complex data and parameters.

Spectrum analysis—A modern perspective

A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.

A fast and accurate single frequency estimator

A novel frequency estimator for a single complex sinusoid in complex white Gaussian noise is proposed. The estimator is more computationally efficient that the optimal maximum-likelihood estimator yet attains equally good performance at moderately high signal-to-noise ratios. The estimator is shown to be related to the linear prediction estimator. This relationship is used to reveal why the linear prediction estimator does not attain the Cramer-Rao bound even at high signal-to-noise ratios. >

Gaussian Random Processes

There are several types of random processes that have found wide application because of their realistic physical modeling yet relative mathematical simplicity. In this and the next two chapters we describe these important random processes. They are the Gaussian random process, the subject of this chapter; the Poisson random process, described in Chapter 21; and the Markov chain, described in Chapter 22. Concentrating now on the Gaussian random process, we will see that it has many important properties. These properties have been inherited from those of the N-dimensional Gaussian PDF, which was discussed in Section 14.3.

Theory of the Stochastic Resonance Effect in Signal Detection—Part II: Variable Detectors

In Part I of this paper [ldquoTheory of the Stochastic Resonance Effect in Signal Detection: Part I-Fixed Detectors,rdquo IEEE Transactions on Signal Processing, vol. 55, no. 7, pt. 1, pp. 3172-3184], the mechanism of the stochastic resonance (SR) effect for a fixed detector has been examined. This paper analyzes the stochastic resonance (SR) effect under the condition that the detector structure or its parameters can also be changed. The detector optimization problem with SR noise under both Neyman-Pearson and Bayesian criteria is examined. In the Bayesian approach when the prior probabilities are unknown, the minimax approach is adopted. The form of the optimal noise pdf along with the corresponding detector as well as the maximum achievable performance are determined. The developed theory is then applied to a general class of weak signal detection problems. Under the assumptions that the sample size N is large enough and the test statistics satisfies the conditions of central limit theorem, the optimal SR noise is shown to be a constant vector and independent of the signal strength for both Neyman-Pearson and Bayesian criteria. Illustrative examples are presented where performance comparisons are made between the original detector and the optimal SR noise modified detector for different types of SR noise.

Can detectability be improved by adding noise

It is shown that under certain conditions the performance of a suboptimal detector may be improved by adding noise to the received data. The reasons for this counterintuitive result are explained and a computer simulation example given.

The effects of noise on the autoregressive spectral estimator

The autoregressive power spectral density estimator possesses excellent resolution properties. However, it has been shown that for the case of a sinusoidal autoregressive process the addition of noise to the time series results in a decrease in spectral resolution. It is proven that, in general, the effect of white noise on the autoregressive spectral estimate is to produce a smoothed spectrum. This smoothing is a result of the introduction of spectral zeros due to the noise. Finally, the use of a large-order autoregressive model to combat the effects of noise is discussed.

Waveform Design for Multistatic Radar Detection

We derive the optimal Neyman-Pearson (NP) detector and its performance, and then present a methodology for the design of the transmit signal for a multistatic radar receiver. The detector assumes a Swerling I extended target model as well as signal-dependent noise, i.e., clutter. It is shown that the NP detection performance does not immediately lead to an obvious signal design criterion so that as an alternative, a divergence criterion is proposed for signal design. A simple method for maximizing the divergence, termed the maximum marginal allocation algorithm, is presented and is guaranteed to find the global maximum. The overall approach is a generalization of previous work that determined the optimal detector and transmit signal for a monostatic radar.

Accurate frequency estimation at low signal-to-noise ratio

A frequency estimator for sinusoids in white noise is described. Convergence results are obtained for the single sinusoid case and a simulation described for the multiple sinusoid case. The estimator is shown to be capable of providing accurate frequency estimates at lower SNRs than currently existing techniques. Furthermore, the simplicity of the algorithm lends itself to a simple and efficient implementation.

Exponentially embedded families - new approaches to model order estimation

The use of exponential embedding of two or more probability density functions (pdfs) is introduced. Termed the exponentially embedded family (EEF) of pdfs, its properties are first examined and then it is applied to the problem of model order estimation. The proposed estimator is compared with the minimum description length (MDL) and is found to be superior for cases of practical interest. Also, we point out there is a relationship between the embedded family model order estimator and the generalized likelihood ratio test (GLRT). The embedded family estimator appears to extend the GLRT to the case of multiple alternative hypotheses that have differing numbers of unknown parameters.