# The Football Positioning System

It’s NFL playoff time, which means that sports fans will be treated to the sight of the most high-stakes farce in sports, namely the ritual of “bringing out the chains” to determine whether a team has gained enough yards for a first down. We’ve all seen this: the play is whistled dead, a referee un-stacks the pile of players, picks up the ball, and puts it down more or less where the player was stopped. Then he tosses the ball into the middle of the field, to a second referee, who tries to replicate the spot closer to the center of the field. Then a guy on the sideline carrying a big stick (connected by a ten-yard chain to another stick held by another guy) tries to put the end of the stick at the same position as the ball.

Three plays later, the spotting procedure is repeated, and then the sticks are bought out to the center of the field, the chain is stretched taut, and they measure the position of the ball to the nearest millimeter. Because, of course, there’s absolutely no error in placing the sticks.

The whole ritual is preposterous, and anybody with the slightest scientific inclination has to wonder: “Isn’t there a better way of doing this?” So, what would be required to do a better job of this?

There are a bunch of ways you might choose to approach this problem, but let’s take the Global Positioning System as a model. GPS works by using a network of satellites orbiting the earth in well-defined orbits. Each satellite contains an atomic clock, and it constantly broadcasts the time according to that clock. Your GPS receiver picks up the time signals from at least three satellites, and compares the times. The signals from satellites that are farther away take longer to arrive, and thus the difference between times gives you the distance from any given satellite to your position on Earth. Calculating that distance for three satellites specifies your position on the face of the earth, assuming that you know the orbits of the satellites very well.

GPS works well enough to prevent even the most directionally challenged person from getting lost, so it ought to be good enough for football. We could imagine setting up receivers at the four corners of the field, putting a small chip in the ball that sends out a signal, and using the time delay between the signals reaching each of the four receivers to determine the exact position of the ball on the field.

(Actually, it might be easier to put both transmitters and receivers on the four corners, and some passive resonant circuit in the ball, then look for the echo at the receivers. The math works out the same either way, so it doesn’t make much difference.)

So, what would such a football positioning system require in terms of timing accuracy? Well, an American football field is 300 feet long and 160 feet wide, and radio waves travel roughly one foot per nanosecond. If we imagine putting the ball exactly in the center of the field, we can use the Pythagorean Theorem to find the distance to each corner, which is 170 feet. If we placed the ball at the precise center of the field, a signal from the ball would arrive at each detector simultaneously, 170 ns after it was emitted.

Now, let’s imagine moving the ball one foot (about the length of the ball) closer to one end zone or the other. The distance from the closer receivers to the ball is now 169.12 feet, and from the more distant receivers, it’s 170.88 feet. The signal from the ball to the corners of the field would arrive at one end zone about 1.8 nanoseconds later after the other.

So, getting the position of the ball on the field to within one length of the ball would require a system that could detect differences of about one nanosecond in the arrival times of pulses from the ball. Other positions on the field are pretty comparable. I calculated positions for a few different cases, and the smallest difference I got was for a ball exactly on the goal line, at the center of the field. Moving that ball back one foot changes the difference between arrival time by a hair under a nanosecond (unless I made a horrible math error, which is possible).

That’s challenging, but might be doable. Of course, the length of the ball is pretty coarse by the standards of farcical distance measurements in football, so you’d like a system that could do much better. And that starts to get tough– getting the position to a centimeter would require time resolution of about three one-hundredths of a nanosecond. Getting transmitters and receivers that are stable to that precision is difficult proposition.

This is, of course, an incredibly naive way of approaching the problem, and I’m sure somebody with more knowledge of signal processing and the workings of GPS than I have could do better. But it should serve to give you some idea of what would be required to eliminate the ritual bringing out of the chains in football games.