We will explore basic group representation symmetries of the space $C^N$ of sequences, of N complex numbers, that are used in digital signal processing (DSP). There are two types of symmetries: (1) The Heisenberg representation, which generalize the time-shift and frequency-shift operators. (2) The Weil
representation which generalize the discrete Fourier transform (DFT). The Heisenberg—Weil symmetries can be used to obtain efficient radar detection algorithms. Here, we would like to know the distances to moving objects and their velocities. The radar system is built to fulfill this task. The radar transmits a waveform S which bounds back from the objects and the echo R is received. In practice, we can work in the digital model, namely $S$ and $R$ are sequences of $N$ complex numbers.
THE RADAR PROBLEM IS: Design $S$, and an effective method of extracting, using $S$ and $R$, the distances and velocities of all targets.
In many applications the current sequences S which are used are pseudo-random and the algorithm they support takes $O(N2\log N)$ arithmetic operations. In the lecture we will use the group representation symmetries to introduce the Heisenberg sequences, and a much faster detection algorithm called the Cross Method. It solves the Radar Problem in $O(N\log N+m2)$ operations for $m$ objects.
The lecture is based on parts from a joint project with Alexander Fish (Math, Sydney), A. Sayeed (EE, Madison), K. Scheim (General Motors, Herzeliya), O. Schwartz (EECS, Berkeley).