[R] fft: characteristic function to distribution

[Thomas Steiner]

Thank you Prof Ripley for your answer.

> > The characteristic function is the inverse Fourier transform of the
> > distribution function. The characteristic function of a normaly
> > distributed random variable is exp(-t^2/2).
> >
>
> The fft is a discrete Fourier transforn, not a continuous one.

This is correct. I try to approximate the continous normal
distribution with infinite support by a set of discrete and bounded
points. A real discrete baby example would be the bernoulli
distribution:

p=0.4
t=seq(-0.01,1.001,length=100)
char=1-p+p*exp(1i*t)
cdf=stepfun(c(0,1),c(0,1-p,1))
plot(t,cdf(t),type="l",col="red",ylim=range(cdf(t),Re(fft(char)[2:99])))
lines(t,fft(char),col="blue")

This is more or less like the normal example.

> Further in each case where the normalizing constants are placed and the
> units of frequecy differ from source to source.
>
> ?fft has references to exactly what it computes: please consult them.

I read the documentation/help page. More details there would be
helpful. For example an example (it says "example*s*") something where
explicit expressions are known (as I tried it here).
Another possible improvement could be to make for example the
following sentence nicer/clearer: "(the inverse has a + in the
exponent of e, but here, we do not divide by 1/length(x))."
I did not consult the two given references (two old but surely valuable books).

Enough prattled. Can you give a working example where the cummulative
distribution function and the fourier transform are explicitly known?

I cannot add any value neither to wonderful R nor to this helpful
function. But perhaps my question isn't that stupid and you can give a
hint to proceed. Thank you very much in advance,
Thomas