[R] noise poser spectral density

[Bill Simpson]

I know there are some time series experts on this list, and I am hoping
one of you might help me with this problem. I am trying to understand the
relationship between No (noise power spectral density) and Gaussian white
noise sigma^2.

I want to check numerically the theoretical relationship (from Selin,1965)
No = sigma^2/(2*W) = sigma^2 * delta t, where
No is power spectral density
delta t is sampling interval in sec = 1/(2*W)
W is the bandwidth: the power spectral density (psd) is flat (equal to No)
for freqs between -W and W Hz

I generate a vector x of n samples from N(0, sigma^2).
n<-2048
sigma<-10
x<- rnorm(n=n,sd=sigma)

I checked R's fft against the examples in Bracewell 1978 p. 364, e.g.
{1,0,0,0} gives fft 1/4*{1,1,1,1}
To make R's fft give this result I do fft(x)/n. Then

psd<-Mod(fft(x)/n)^2

The psd varies wildly across freqs, as expected. I do an average over
freqs
mean(psd)
It turns out that this is roughly equal to sigma^2/n. But
n = duration/(delta t), so I find:
			mean(psd) = [sigma^2 * (delta t)]/duration
However theory says:	mean(psd)=sigma^2*(delta t)
The numerical result is too small by a factor of duration.

Is the theory wrong?
Should I just multiply mean(psd) by duration?

I am confused. Thanks for any help.

Bill Simpson

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