I will describe a special class of complex valued functions (digital signals) on the finite line $F_p$, that we call the oscillator system. These functions satisfy many interesting properties which seem to be ideal for applications in various fields of digital signal processing, including radar and communication.

I will begin my lecture with a brief exposition of the theory of continuous and discrete radar. I will explain why eigenfunctions of the harmonic oscillator $D=\partial_t^2-t^2$ are ideal signals for continuous radar. I will then sketch the theory of CDMA (Code Division Multiple Access) and show that the oscillator system suggest a solution to the discrete radar and CDMA problems. This is a new approach to these problems which usually require random vectors for their solutions.

My main goal, is to describe the oscillator functions as a discrete analog for the eigenfunctions of $D$. In the course, I will introduce the Weil representation of the group $SL_2(F_p)$ and hint towards its fundamental role in harmonic analysis, both in the continuous and discrete settings.

Joint work with Ronny Hadini (U of Chicago) and Nir Sochen (Tel Aviv).