On Detecting a Signal in N Stationarily Correlated Noise Series

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Abstract
A useful model in the analysis of multiple time series is to assume that each series is composed of a fixed signal common to all series and a wide sense stationary normal noise process. If the noise series are uncorrelated with identical autocorrelation functions, we have in the time domain a collection of independent normal vectors with a common covariance matrix. The likelihood ratio statistic for detecting the signal is then a singular version of Hotelling's T 2. In this paper, a likelihood ratio test utilizing the asymptotic properties of the finite Fourier transform leads to an asymptotically non-singular frequency domain version of Hotelling's T 2. It is shown that the F statistic for detecting the signal depends only upon certain spectral power measurements with the Type I error and signal detection probabilities determined from existing tables for the central and non-central F distributions. Several examples are presented illustrating the detection procedure.

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