On Selected Properties of Time- frequency Analysis

Time-frequency signal representation is very useful form for various methods for signal analysis and processing. This form is especially suitable as input for convolution neural networks. We show this on electroencephalography data and Brain Computer Interface. However, the contribution focuses on more fundamental properties of the time-frequency representation and accompanying properties. We start with time-frequency properties of signals such an instant frequency in time. We show how the Fourier transform can be used to perform the opposite analysis, the signal instant time in frequency. The connection with axis swapping of time-frequency plane is shown. The overview of different centers of signal in time domain is provided and extended into time-frequency domain. The analysis is extended to time-frequency relations in the structure of wavelet orthogonal filter banks. We focus also on skewness and analyze in more detail the asymmetric wavelets and focus on their centers and skewness and relation to their scaling and wavelet coefficients. We also concentrate on time-frequency properties of the wavelet function themselves. We compare various methods for instant frequency and envelope estimation, including Hilbert transform, Fourier transform, Wigner-Ville distribution and analytic complex wavelet transform based approaches.