Generalized Singular Spectrum Analysis Associated With Fractional Fourier Transform

In this study, an adaptive nonstationary signal decomposition technique is developed, which is associated with the singular spectrum analysis (SSA) and the discrete fractional Fourier transform (DFrFT). The SSA is a data-adaptive discrete signal analysis method that does not require a parameter model in advance, of which the core operation is singular value decomposition (SVD). It has been proven that the singular values of the constructed Hankel matrix are equally distributed to the power spectrum of the discrete signal, which bridges the discrete signal and time-frequency analysis. However, the power spectrum of a nonstationary signal is often wide-band or even non-bandlimited, which degrades the performance of the SSA. The DFrFT contains the Fourier transform as a special case and can be regarded as a linear time-frequency representation. It has achieved significant success in non-stationary signal processing, especially in radar and communications signal processing. A wide-band (or non-bandlimited) signal may become narrow-band (or bandlimited) in the DFrFT domain. To enhance the SSA with the help of this property, a new Hankel matrix is constructed using the fractional time-shift operation, and its singular values are proven to be related to the fractional power spectrum of the sequence. To handle nonstationary signals, the generalized SSA is designed associated with DFrFT. It is more flexible and suitable in nonstationary signal processing than the SSA due to its free parameter. Its superior performances over other popular data-driven methods are verified and displayed using simulations.