Convert time series to frequency domain

Question

I have a fundamental question:

I would like to know, why this time series:

k<-c(4,5,6,2,3,1)

is equal to:

21.0+0.000000i  0.5-6.062178i -1.5-0.866025i  5.0-0.000000i -1.5+0.866025i  0.5+6.062178i

In time series I have a set of points, but what is the resault of fft , are there points?

Answer 1

Fourier says that any (non-pathological) waveform can be decomposed into a bunch of sinewaves. The FFT does that for reasonable samples of a given waveform.

So your FFT results are the coefficients of each sinewave sub-component: the first for 0 Hz (or DC, or sum), the 2nd for a sinewave of 1 period per aperture, the next: 2 cycles per aperture, and etc. You can consider each coefficient pair x+iy, as either a vector in the complex plane for a sinewave's magnitude and phase, or as multipliers for a cosine and a sine that sum up to another sinewave of a specified phase.