Visualize the Fourier Transform of Common Energy Signals

Interactive Fourier transform visualization: pick a rectangular, triangular, exponential, Gaussian, or gated-cosine signal, tweak its amplitude, width, and time shift, then sweep frequency and watch the magnitude spectrum and phase spectrum build up point by point, in either hertz or radians per second.

This interactive Fourier transform visualization shows the transform as something you watch being computed rather than a formula you look up. Pick one of six energy signals, a rectangular pulse, a triangular pulse, a causal exponential, a two-sided exponential, a Gaussian, or a gated cosine burst, then set its amplitude, its width or decay rate, and how far it sits from the origin. Three plots share the stage: the signal itself on top, the magnitude spectrum in the middle, and the phase spectrum below it, with the two spectrum plots aligned on one frequency axis so you can read both off at the same frequency. Checkboxes let you plot the magnitude alone, the phase alone, or both, and dropping one gives the other the full height of the stage. Every parameter applies live, so you can drag a width or an amplitude while the sweep is running and watch the spectrum deform under it rather than starting over.

The driver is the frequency itself. At every value the transform integral returns one complex number, whose length is plotted on the magnitude curve and whose angle is plotted on the phase curve, so both fill in from the centre outward as the sweep advances. Because every signal here is real, the negative and positive halves fill in together, which is conjugate symmetry made visible. Playback pauses briefly on each landmark frequency, and stepping moves between those landmarks rather than in fixed increments, so every null and every peak gets its own press.

Six presets isolate one property each. Wide pulse and narrow pulse use the same frequency axis on purpose, so widening the pulse visibly pulls the nulls inward and the time-bandwidth tradeoff is something you see rather than something you are told. Shift only changes nothing but the time shift, and the magnitude curve stays pixel for pixel identical while the phase turns into a ramp, which is the shift theorem in one control. The causal and even exponentials share a decay envelope and differ only in symmetry, and that difference alone is what gives one of them a phase and leaves the other flat. The cosine burst splits the single sinc in two and moves a half-height copy to each carrier frequency.

You can read the whole thing in hertz or in radians per second. Switching rewrites the transform pair shown below the plots, which is the fastest way to see why one variable gives tidier formulas than the other and where the factor of two pi goes. Two numeric checks keep the page honest: the closed form driving the curves is compared against an independent numerical integration of the signal at the current frequency, and Parseval's relation is checked by measuring the energy in the time domain and in the frequency domain separately and showing that the two agree.