Circularity of the STFT and Spectral Kurtosis for Time-Frequency Segmentation in Gaussian Environment

This paper investigates the circularity of short time Fourier transform (STFT) coefficients noise only, and proposes a modified STFT such that all coefficients coming from white Gaussian noise are circular. In order to use the spectral kurtosis (SK) as a Gaussianity test to check if signal points are present in a set of STFT points, we consider the SK of complex circular random variables, and its link with the kurtosis of the real and imaginary parts. We show that the variance of the SK is smaller than the variance of the kurtosis estimated from both real and imaginary parts. The effect of the noncircularity of Gaussian variables upon the spectral kurtosis of STFT coefficients is studied, as well as the effect of signal presence. Finally, a time-frequency segmentation algorithm based on successive iterations of noise variance estimation and time-frequency coefficients detection is proposed. The iterations are stopped when the spectral kurtosis on nondetected points reaches zero. Examples of segmented time-frequency space are presented on a dolphin whistle and on a simulated signal in nonwhite and nonstationary Gaussian noise.