What does it mean to calculate the FFT of an integer sequence?

Question

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.

Answer 1

(edited): Considering that I only passed amplitudes and not much else, if I interpret those results as the spectrum of $x$, how can that be?

You passed it amplitudes in a specific order. That means that you can then do any transform on it that depends on the ordering -- which the DFT does.

Without making further assumptions about how the data set was arrived at, the spectrum of $x$ is pretty much a stand-alone thing, meaning whatever it means to your problem at hand to have that spectrum.

Where Fourier analysis becomes useful is if you can make assumptions about the systems that are generating the data, or that are operating on that signal. If you assume that those systems are linear, and even more so if you assume that those systems are time-invariant*, then it gets really handy, because linear systems behave in predictable ways to sinusoidal inputs, and because linear systems have the property of superposition.

Even where the problem at hand isn't explicitly citing linear systems, it's an implicit assumption to Fourier analysis that you're dealing with systems that are linear, nearly linear, or that at least have nearly linear components.

* Although that's not necessary for Fourier analysis.

Answer 2

Hi, eduardokapp. I think you can understand Fourier transform,dft and fft. so, i am going to explain about the meaning of your time domain integer sequence and its fft result by using following simple image from matlab.

here, red line is the spectrum and time-graph when Fs is 500 and blue line is when Fs is 1000. so, your integer sequence is the sample values that have any sample-rate you can select. therefore, fft result is specified by the sample-rate. so, one value of fft result sequence is the ft result for the frequency operated from the index of the value.

here, my matlab code :

Fs1 = 1000;            % Sampling frequency  
Fs2 = 500;
T1 = 1/Fs1;             % Sampling period 
T2 = 1/Fs2;
L = 10;             % Length of signal
t1 = (0:L-1)*T1;        % Time vector
t2 = (0:L-1)*T2;

%S = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);
%X = S + 2*randn(size(t));

X = (0:L-1);     % your integer sequence
subplot(2,1,1)
plot(1000*t1(1:10),X(1:10),1000*t2(1:10),X(1:10))
title('Signal Corrupted with Zero-Mean Random Noise')
xlabel('t (milliseconds)')
ylabel('X(t)')

Y = fft(X); % fft result
P2 = abs(Y/L);
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);
f1 = Fs1*(0:(L/2))/L;
f2 = Fs2*(0:(L/2))/L;
subplot(2,1,2)
plot(f1,P1,f2,P1) 
title('Single-Sided Amplitude Spectrum of X(t)')
xlabel('f (Hz)')
ylabel('|P1(f)|')

thanks.