Fourier operator

The Fourier operator defines a continuous two-dimensional function that extends along time and frequency axes, outwards to infinity in all four directions.

This is analogous to the DFT matrix but, in this case, is continuous and infinite in extent.

Along any fixed value of time, the value of the function varies as a complex exponential in frequency.

Likewise along any fixed value of frequency the value of the function varies as a complex exponential in time.

Diagonal slices through the Fourier operator give rise to chirps.

Depiction of how the Fourier operator acts on an input rectangular pulse (on the far right) to generate its Fourier transform (on the left-hand side), a sinc function.