Fourier transform of a random process
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From: fisico32 on 5 May 2010 09:18 Hello Forum, here I am again with my questions... It is said that a Fourier transform is not defined for a random process.... A random process is defined as a infinite collection or realizations, signals y_n(t) which are different from each other. We could perform the FT of each realization and obtain many different power spectra and phase spectra, one for each realization. We could then take an average of them and get the FT of the process.... What is wrong with that? In real life we have a finite amount of realizations which last a finite amount of time. Instead, we take the FT of the autocorrelation function and get the PSD, squared modulus of the Fourier transform.... thanks fisico32
From: robert bristow-johnson on 5 May 2010 12:06 On May 5, 9:18 am, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > Hello Forum, > > here I am again with my questions... > > It is said that a Fourier transform is not defined for a random > process.... > A random process is defined as a infinite collection or realizations, > signals y_n(t) which are different from each other. and each would have their own particular Y_n(f) > We could perform the FT of each realization and obtain many different power > spectra and phase spectra, one for each realization. > We could then take an average of them and get the FT of the process.... > What is wrong with that? if you include phase in that average, that is you are averaging Y_n(f) rather than |Y_n(f)|^2, then you might average things to zero, just as averaging an infinite number (or very large number) of y_n(t) will get you to zero (assuming zero mean and finite variance). but if you toss phase and average the magnitude-squared, you should get in the limit, the same power spectrum that you would get from the FT of the autocorrelation. > In real life we have a finite amount of realizations which last a finite > amount of time. > > Instead, we take the FT of the autocorrelation function and get the PSD, > squared modulus of the Fourier transform.... which is the power spectrum. r b-j
From: Tim Wescott on 5 May 2010 13:23
fisico32 wrote: > Hello Forum, > > here I am again with my questions... > > It is said that a Fourier transform is not defined for a random > process.... > A random process is defined as a infinite collection or realizations, > signals y_n(t) which are different from each other. > > We could perform the FT of each realization and obtain many different power > spectra and phase spectra, one for each realization. > We could then take an average of them and get the FT of the process.... Better, we could start with the known properties of the process and we could find the expected value of the Fourier transform of it. The Fourier transform of a random process is itself a random process, so we can find its statistics. > What is wrong with that? Mathematically, nothing. The real problem is that the expected value of the Fourier transform of a random, stationary, zero-mean process is zero. This isn't very informative. > In real life we have a finite amount of realizations which last a finite > amount of time. Correct, but we are still working with a random process. > Instead, we take the FT of the autocorrelation function and get the PSD, > squared modulus of the Fourier transform.... --- because this _is_ useful information, easy to get, and easy to determine from measured data. Deciding just what the measurements _mean_ can be pretty problematical -- it seems that every one has a different method of estimating the PSD of a real signal -- but at least you're starting from a firm mathematical footing. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com |