The Fourier transform is used to map functions to and from time/frequency domains. I can make sense of what it means to calculate the Fourier transform of, say:
$$y(t) = e^{j\omega_0t}$$
which is
$$Y(\omega) = 2\pi\delta(\omega - \omega_0)$$
But, what does it mean when I use, for example, R language's FFT function stats::fft(x)
for an arbitrary x
array of real values, say, x=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
?
> x <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
> stats::fft(x)
[1] 55+ 0.000000i -5+15.388418i -5+ 6.881910i -5+ 3.632713i -5+ 1.624598i
[6] -5+ 0.000000i -5- 1.624598i -5- 3.632713i -5- 6.881910i -5-15.388418i
Note: I used R, but it was just an example. Could be Python's NumPy/SciPy, MATLAB's, Julia, etc.
Now, I know that the FFT is basically a fast implementation of the discrete Fourier transform (DFT). Assuming that, in this example, x
is a discrete-valued function in the time domain, I don't have any information on its frequency (or sample frequency, for that matter).
So what does that output means, in the more "essential/core" sense of the Fourier transform? If I interpret those results as the spectrum of x
, how can that be? Considering that I only passed amplitudes and not much else.
x
sequence with? $\endgroup$