MORE ON TIME-FREQUENCY ANALYSIS AND RANDOM PROCESS R 11/26 Basics of random process Definition : random variable is a mapping from probability space to a number Definition : random process is a mapping from probability space to set of function indexed by t Basics of random process Auto-correlation Function If we set We get Basics of random process Power spectral density Relation with auto-correlation function Basics of random process Stationary Process Strict-Sense Stationary (SSS) first order n-th order Wide-Sense Stationary (WSS) TF Analysis Fractional Fourier Transform (FRFT) Operator form is denoted as Linear Canonical Transform (LCT) Operator form is denoted as Property: If LCT becomes FRFT TF Analysis and random process Define g(t) is a stationary random process, is the FRFT of g(t), auto-correlation function of is Its no longer stationary TF Analysis and random process For LCT Generally speaking, signal after LCT usually not stationary, but with (Fresnel transform) TF Analysis and random process PSD of FRFT and LCT of a signal FRFT LCT TF Analysis and random process For white noise using equation and by sifting property TF Analysis of nonstationary random process Generally speaking, nonstationary r.p. analysis is far more complicated TF Analysis of nonstationary random process Definition: If g(t) is a nonstationary random process and is stationary and autocorrelation function of it is independent of u. We can call it - order fractional stationary random process TF Analysis of nonstationary random process WDF and AF of r.p. and FRFT of r.p. For a nonstationary random process mean of its WDF is invariant along (cos(a),sin(a)), AF is not zero when TF Analysis of nonstationary random process It can be shown that we can decompose a nonstationary random process h(t) into - order fractional stationary random process So TF Analysis of nonstationary random process Fractional filter design Consider (i.e. signal and noise) H(u) is a bandpass filter Consider white noise Fractional filter design For white noise Fractional filter design To minimize the energy noise, we can select the cutoff-lines that make area as small as possible Fractional filter design For white noise Reference [1]Lecture notes of Random Process And Its Application, Char Dir Chung [2] S. -C. Pei and J. -J. Ding, Fractional Fourier transform, wigner distribution,and filter design for stationary and nonstationary random processes, IEEE Trans. Signal Process., vol. 58, no. 8, pp. 4079 4092, Aug
