MORE ON TIME-FREQUENCY ANALYSIS AND RANDOM PROCESS R 電信一 吳卓穎 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 It’s 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