Fourier playground

Tool

Presets

Keep signal real

Window length T s start t₀ s log₂ N = 256 samples

Show continuous curves Points only (no line)

Discrete vs continuous: what changes?

Dots and stems are the N values you actually have. The faint curves are what lies between them: the band-limited signal through the samples (left) and the DTFT, the smooth spectrum of the windowed samples (right).
Sampling in time makes the spectrum repeat every fs = N/T. Anything above the Nyquist frequency fs/2 folds back. Try “Sine 50 Hz” and lower N to 64.
A finite window means the DFT bins, spaced 1/T apart, only sample the smooth DTFT. A tone that does not fit a whole number of cycles into T leaks into neighbouring bins. Try T = 1.1 with “Sine 5 Hz” and switch the curves on.
Sampling in frequency treats the window as one period of an endlessly repeating signal, so the left and right edges are neighbours.
Changing t₀ only shifts the time labels and adds a linear phase to the spectrum; the magnitude stays put.
The DFT only samples the spectrum at whole multiples of 1/T. A Gaussian centred at t = T/2 picks up the phase e^−iπk = ±1 at every bin, so its imaginary part is exactly 0 there even though the continuous spectrum is not. Turn on the continuous curves, or try “Shifted Gaussian”.
Presets are re-evaluated when you change the window, so a 5 Hz tone stays 5 Hz. A hand-drawn signal keeps its sample values, so it stretches with T and is interpolated when N changes.

Signalx(t)

Show ± bins x scale