Fourier transform of Unit Pulse

Reconstructed a unit pulse using weighted cosine components of sinc(πf); higher frequency resolution yields better accuracy.

In this exploration, we reconstructed a unit pulse by synthesizing it from its Fourier Transform, which is the sinc(πf) function. The sinc function represents the frequency-domain spectrum of an ideal rectangular pulse in the time domain.

By weighting cosine components with the values of sinc(πf) and summing them over a range of frequencies, we approximated the original pulse. The process demonstrates how finite frequency components with limited resolution can approximate a pulse, though not perfectly.

If the frequency range or resolution is too limited, the result may resemble a pulse train rather than a single pulse, due to spectral truncation and aliasing effects. This is because an ideal pulse theoretically contains infinite frequencies with infinitesimally small spacing.

Increasing the number of frequency points and the frequency span improves the reconstruction by approximating this ideal condition. The use of cosine terms is due to the real and even nature of the spectrum for a symmetric real-valued pulse. Graphs showing the individual frequency components and their weighted sum illustrate how the time-domain pulse emerges from the frequency domain.