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

From: John O'Flaherty (quiasmox_at_yahoo.com)
Date: 11/26/04 

Date: Fri, 26 Nov 2004 10:13:49 -0600

John O'Flaherty wrote:
> lucy 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?
>>
>>
>>>> plot(abs(fftshift(fft(rand(1, 20000)))))
>>
>>
>>
>> gives a uniformly chaotic noisy spectrum, looks like white...
>>
>>
>>>> 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?
>
>
> Is the random function balanced around 0? If not, the transform will
> have a dc peak because it's all positive. Maybe you could do
> (rand(1,20000)-0.5)?

Or use randn(1,20000), which give a set of normally distributed numbers
with mean zero. 

--
john

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