Measuring distance from a satellite
We saw in the last section that a position is calculated from distance measurements to at least three satellites.
The Big Idea Mathematically:
In a sense, the whole thing boils down to those “velocity times travel time” math problems we did in high school. Remember the old: “If a car goes 60 miles per hour for two hours, how far does it travel?”
Velocity (60 mph) x Time (2 hours) = Distance (120 miles)
In the case of GPS we’re measuring a radio signal so the velocity is going to be the speed of light or roughly 186,000 miles per second.
The problem is measuring the travel time.
- Timing is tricky
- We need precise clocks to measure travel time
- The travel time for a satellite right overhead is about 0.06 seconds
- The difference in sync of the receiver time minus the satellite time is equal to the travel time
The timing problem is tricky. First, the times are going to be awfully short. If a satellite were right overhead the travel time would be something like 0.06 seconds. So we’re going to need some really precise clocks. We’ll talk about those soon.
But assuming we have precise clocks, how do we measure travel time? To explain it let’s use a goofy analogy:
Suppose there was a way to get both the satellite and the receiver to start playing “The Star Spangled Banner” at precisely 12 noon. If sound could reach us from space (which, of course, is ridiculous) then standing at the receiver we’d hear two versions of the Star Spangled Banner, one from our receiver and one from the satellite.
These two versions would be out of sync. The version coming from the satellite would be a little delayed because it had to travel more than 11,000 miles.
If we wanted to see just how delayed the satellite’s version was, we could start delaying the receiver’s version until they fell into perfect sync.
The amount we have to shift back the receiver’s version is equal to the travel time of the satellite’s version. So we just multiply that time times the speed of light and BINGO! we’ve got our distance to the satellite.
That’s basically how GPS works.
Only instead of the Star Spangled Banner the satellites and receivers use something called a “Pseudo Random Code” – which is probably easier to sing than the Star Spangled Banner.
Distance to a satellite is determined by measuring how long a radio signal takes to reach us from that satellite.
- To make the measurement we assume that both the satellite and our receiver are generating the same pseudo-random codes at exactly the same time.
- By comparing how late the satellite’s pseudo-random code appears compared to our receiver’s code, we determine how long it took to reach us.
- Multiply that travel time by the speed of light and you’ve got distance.