Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 4 / b / Solution

A direct way to avoid the singular transform of $1/t^2$ is to invert a triangular function. Evenness givesAt $t=0$ the limit is $1/2$, so there is no singularity in the original signal. ThusAlternatively, the hint uses the Hadamard finite-part reciprocal-power distribution, whose distributional Fourier transform is $\mathcal{F}[\mathrm{fp}(1/t^2)]=-\pi |\omega |$; combining its three translates gives $\pi (|\omega -1|+|\omega +1|-2|\omega |)/2$, the same triangle. Treating $1/t^2$ as an ordinary integrable function would be invalid.

The message spectrum has height $\pi$ at zero, vanishing at $\omega =\pm 1$. For $\Omega >1$, the modulated continuous spectrum consists of disjoint triangles of height $\pi /2$, centred at $\pm \Omega$, each with half-width one. In addition there is a Dirac delta carrier line of weight $\pi$ at each centre. These lines have specified area, not a finite height; arrows mark them schematically.