Development of a linear time-scale coherence spectrum

In spectral analysis of two or more signals, it is desirable to quantify the statistical relationships among them. The linear coherence spectra based on Fourier analysis have been widely used to quantify the linear relationships between two stationary fluctuating signals. However, in the case of nonstationary signals, the Fourier-based linear coherence spectrum has a limited utility, since temporal information is now required. It is the purpose of this paper to extend the notion of the classical linear coherence spectra to the time-scale coherence spectrum. it will be shown that not only is the new time-scale coherence spectrum capable of measuring linear relationships between two nonstationary signals in the time-scale domain but also of detecting their phase differences as well. Using the Cauchy-Schwarz inequality, it can be shown that the time-scale coherence spectrum is bounded by zero and one. The latter number indicates a perfect linear relation between the two nonstationary signals at a particular time and scale. The efficacy of the new method is demonstrated and compared with that of the classical coherence spectrum through numerical examples.