Explore how time-domain sampling mathematically replicates spectra, and how low-pass filtering reconstructs the signal.
Energy vs. Power Signals: In the presets, the Sinc Pulse is an Energy Signal (finite energy, decays to 0). Its Fourier Transform is a continuous rectangular function. The Cosines are Power Signals (infinite energy, finite average power, never decay). Because they do not decay, the standard Fourier integral diverges. We mathematically represent them in the frequency domain using Dirac delta functions (impulses), shown here as arrows.
The Mechanism of Sampling: Sampling in the time domain is mathematically modeled as multiplying $x(t)$ by an impulse train. Multiplication in time corresponds to convolution in frequency. Convolving a spectrum with an impulse train creates infinitely many shifted copies (replicas) of the original spectrum spaced by $f_s$.
Nyquist Criterion & Aliasing: To recover the signal, we apply an ideal Low-Pass Filter (LPF) from $-f_s/2$ to $f_s/2$. If $f_s > 2B$, the central replica is isolated, and the LPF perfectly extracts $X(f)$. If $f_s \le 2B$, the replicas overlap (Aliasing). Try the 6 Hz Cosine preset with $f_s = 5$ Hz. Turn on the "Show Impersonator Signal" toggle. You will see that the samples of a 6 Hz cosine perfectly match a 1 Hz cosine. Because the LPF only extracts frequencies up to $2.5$ Hz, it outputs the 1 Hz "impersonator" instead of the 6 Hz original.
Time-Domain Reconstruction: Passing impulses through a brick-wall LPF in frequency corresponds to convolving the samples with a $\text{sinc}$ function in time. Turn on "Show Time-Domain Sincs" to see how the mathematical formula $\hat{x}(t) = \sum x(nT_s)\text{sinc}(f_s(t - nT_s))$ physically rebuilds the smooth analog waveform.