Re: What can we expect by taking Fourier Transform of noise or random samples?

From: Isaac Gerg (isaac.gergNOSPAM_at_adelphia.net)
Date: 12/06/04 

Date: Mon, 06 Dec 2004 08:36:38 -0500

http://www.imageprocessingbook.com/index_dip2e.htm

has a section in about Weiner filtering and how straight inverserve
filter breaks down. For some real life examples, see this paper:

http://www.gergltd.com/users/isaac.gerg/cse585/finalpj/FinalReport.doc

Paul Sanchez wrote:
> In article <co6nq8$oaj$1@news.Stanford.EDU>,
> "lucy" <losemind@yahoo.com> wrote:
>
>
>>Hi all,
>>
>>I am trying to understand using Fourier Transform to do denoising.
>>
>>I heard many times that people say suppose I have z(t)=y(t)+n(t), where z is
>>the observations of the useful signal y and n is the noise... then we can
>>look at the spectrum of z(t) and then if we know our useful signal is in
>>some certain frequency range then we can do some filtering to recover the
>>useful signal and get rid of the noise.
>>
>>I want to understand this approach.
>>
>>I feel n(t) is a random process, y(t) is a deterministic signal, then is
>>z(t) wide sense stationary? My guess is that z(t) is a non-WSS random
>>process... what is the meaning of taking Fourier Transform of a random
>>process?
>>
>>If we take FT of z(t)=y(t)+n(t), what is the meaning of taking FT of a
>>random process n(t)? Commonly people say noise n(t) is white... but this
>>"white" is talking about the power spectral density of the random process,
>>it has nothing to do with taking the FT of n(t) and consequently z(t)... Am
>>I right?
>>
>>I did some experiments in Matlab:
>>
>>
>>>>plot(abs(fftshift(fft(rand(1, 20000)))))
>>
>>gives a huge peak/impulse at DC frequency f=0... what does this mean?
>
>
> It means that the sample mean of the data is non-zero. That's what the
> FFT is estimating at f=0. Many people preprocess the data by
> subtracting the average from each observation to center at zero.
>
> By the way, to have a true spectrum you should be squaring the transform
> results, not abs'ing.
>
>
>>>>plot(abs(fftshift(fft(rand(1, 20000)))))
>>
>>gives a uniformly chaotic noisy spectrum, looks like white...
>
>
> Uniform spectrum indicates no correlation. This follows directly from
> the Weiner-Khintchine theorem, which says that the spectrum can be found
> as the Fourier transform of the autocovariance function. If your noise
> terms are independent, then the autocovariance is zero for all lags
> other than zero, yielding a flat spectrum. (A common misconception is
> that the noise needs to be Gaussian for white noise, but uncorrelated is
> sufficient.)
>
>
>>>>plot(abs(fftshift(fft(random('rayleigh', 10, 1, 20000)))))
>>
>>also gives a huge peak/impulse at DC frequency f=0... what does this mean?
>>
>>--------------------
>>
>>Maybe I should ask what shall we expect to see if we take FT of random
>>samples?
>>
>>Any thoughts?
>>
>>Thanks a lot!
>
>
> Chatfield has written a very nice text called "Time Series Analysis".
> It's not as comprehensive or theoretical as many others, but it's quite
> accessible.

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