IDA: Interactive Document on Algebra

Section 2.5
Radar detection

Radar is used to determine the position and velocity of a given object. Greater precision can be achieved if more radio pulses are sent at different times over a very short time interval. Each of these pulses within such a package may have a different frequency.

Let us assume that the pulses within one package are sent out at constant time intervals. A detection pattern records which frequency is being transmitted at which time.

If the object to be detected travels at constant speed, each of the pulses will come back at the same shift in time and frequency. Therefore, the position, velocity information from the radar can be read off as the vector in the 2-dimensional grid representing time, frequency coordinates from the pulse sent out to the corresponding pulse coming back.

However, due to physical circumstances, the pattern is likely to be blurred. Our goals are to design a detection pattern that optimizes the chance of recognizing the time, frequency shift (also called translation).

We formalize the discrete mathematical model for this pattern as follows, thereby assuming that no frequency occurs twice.

Definition

A detection pattern of size n is a square n×n grid of integer points in the plane such that in each row and in each column one point is black and all others are white.
The overlap of a detection pattern D is the maximal number of black points that D has in common with a proper translation over an integral vector.
A detection pattern is called optimal if its overlap is 1.

The question now is for which sizes can we find an optimal detection pattern. We claim that such patterns are optimal for detection.