On discrete Fourier analysis of amplitude and phase modulated signals

Waldemar Popiński¹

¹ Space Research Centre Polish Academy of Sciences Bartycka 18a 00-716 Warszawa, Poland

Applicationes Mathematicae, Tome 39 (2012) no. 1, pp. 57-69.

Résumé

In this work the problem of characterization of the Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal $o_t$, $t=0,1,\dots,n-1$, modulated by random fluctuations of its amplitude and/or phase is investigated. It is assumed that the amplitude and/or phase of the signal at discrete times of observation are distorted by realizations of uncorrelated random variables or randomly permuted sequences of complex numbers. We derive the expected values and bounds on the variances of such distorted signal DFT spectra. It is shown that the modulation considered in general entails changes in the amplitude and/or phase of the DFT spectra expected values, which together with imposed random deviations with finite variances can vary the amplitudes of peaks existing in the original signal spectrum, and consequently similarity to the original signal spectrum can be significantly blurred.

Détail

DOI : 10.4064/am39-1-4

Keywords: work problem characterization discrete fourier transform dft spectrum original complex valued signal dots n modulated random fluctuations its amplitude phase investigated assumed amplitude phase signal discrete times observation distorted realizations uncorrelated random variables randomly permuted sequences complex numbers derive expected values bounds variances distorted signal dft spectra shown modulation considered general entails changes amplitude phase dft spectra expected values which together imposed random deviations finite variances vary amplitudes peaks existing original signal spectrum consequently similarity original signal spectrum significantly blurred