On the Description of Spectrogram Probabilities with a Chi-Squared Law
Résumé
Given a correlated Gaussian signal, may a chi-squared law of probability always be used to describe a spectrogram coefficient distribution? If not, would a "chi-squared description" lead to an acceptable amount of error when detection problems are to be faced in the time-frequency domain? The two questions prompted the study reported in this paper. After deriving the probability distribution of spectrogram coefficients in the context of a non-centred Gaussian correlated signal, the Kullback-Leibler divergence is first used to evaluate to what extent the non-whiteness of the signal and the Fourier analysis window impact the probability distribution of the spectrogram. To complete the analysis, a detection task formulated as a binary hypothesis test is considered. We evaluate the error committed on the probability of false alarm when the likelihood ratio test is expressed with chi-squared laws. From these results, a chi-squared description of the spectrogram distribution appears accurate when the analysis window used to construct the spectrogram decreases to zero at its boundaries, regardless of the level of correlation contained in the signal. When other analysis windows are used, the length of the window and the correlation contained in the analysed signal impact the validity of the chi-squared description.
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