Time series: data analysis and theory
Time series: data analysis and theory
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All signals that are normally called "periodic" have some amplitude and phase variation from period to period. For example an active sonar system transmits a periodic pulse train to detect targets. The received pulses are not perfectly periodic due to random modulation of the pulses from scattering and attenuation. Target scattering and propagation distortion of the transmitted and received signals produce seemingly random variation from pulse to pulse. A stochastic nonparametric model of period-to-period variation is presented in this paper. This model is used to derive an asymptotic likelihood-ratio test for wide bandwidth pulses that undergo a random modulation from pulse to pulse. It is shown that the test statistic is a linear combination of the matched filter using the expected value of the received signal as the matching vector and the signal's periodogram.