Fractional Fourier Transform, Wigner Distribution, and Filter Design for Stationary and Nonstationary Random Processes

• Published in: IEEE transactions on signal processing Vol. 58; no. 8; pp. 4079 - 4092
• Main Authors: Pei, Soo-Chang, Ding, Jian-Jiun
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2010; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 1f noise; Ambiguity function (AF); Autocorrelation; Autocorrelation functions; Distribution functions; Estimates; filter design; Fourier transforms; fractional Fourier transform (FRFT); Invariants; linear canonical transform (LCT); Noise; Nonlinear filters; Optical filters; Pattern analysis; Process design; Random processes; Segments; stationary and nonstationary random processes; Studies; White noise; Wigner distribution; Wigner distribution function (WDF)
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2010.2048206

In this paper, we derive the relationship among the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the stationary and nonstationary random processes. We find many interesting properties. For example, if we perform the FRFT for a stationary process, although the result is no longer stationary, the amplitude of the autocorrelation function is still independent of time. We also find that the LCT of a white noise is still a white one. For the FRFT of a stationary process, the ambiguity function (AF) is a tilted line and the Wigner distribution function (WDF) is invariant along a certain direction. We also define the "fractional stationary random process" and find that a nonstationary random process can be expressed by a summation of fractional stationary random processes. In addition, after performing the filter designed in the FRFT domain for a white noise, we can use the segment length of the ω -axis on the WDF plane to estimate the power of the noise and use the area circled by cutoff lines to estimate its energy. Thus, in communication, to reduce the effect of the white noise, the "area" of the WDF of the transmitted signal should be as small as possible.