Multiple WindowTime - Varying Spectrum

We overview a new non-parametric method for estimating the time-varying spectrum of a non-stationary random process. Our method extends Thomson's powerful multiple window spectrum estimation scheme to the time-frequency and timescale planes. Unlike previous extensions of Thomson's method, we identify and utilize optimally concentrated Hermite window and Morse wavelet functions and develop a statistical test for extracting chirping line components. Examples on synthetic and real-world data illustrate the superior performance of the technique. 2 Introduction Many methods exist for estimating the power spectra of stationary signals 1]. However, these methods are insuucient for the non-stationary signals that occur in important applications such as radar, sonar, acoustics, biology, and geophysics. These applications demand time-frequency representations that indicate how the power spectrum changes over time. To date, research in time-frequency analysis has mainly focused on deterministic signals. Only more recently has attention turned to non-stationary random processes 2{13]. Unlike the power spectrum for stationary random processes, there is no unique deenition for the time-varying spectrum of a nonstationary random process x. Perhaps the best compromise is the Wigner-Ville spectrum (WVS) W x 9]. Given the instantaneous auto-correlation function r x (t;) := Ex (t ? =2) x(t + =2)]; (2.1) the WVS is deened as its Fourier transform W x (t; f) := Z r x (t;) e ?j2f dd: (2.2) Alternatively, the WVS can be deened as the expected value of the empirical Wigner distributions (WDs) W x 14,15] of the realizations of the process W x (t; f) = EW x (t; f)] = E Z x (t ? =2) x(t + =2)e ?j2f dd : (2.3)