# GPS timing and computational complexity

I have been using Garmin GPS device for years. Just curious which algorithm is used in the device:

1. Navigation equations that can calculate both GPS device's position and its clock bias. In this way, the time display should be accurate with the clock in the satellite. However, the computational complexity could be very high. See https://en.wikipedia.org/wiki/Global_Positioning_System.

2. Use TDOA method similar to what is explained at https://en.wikipedia.org/wiki/Multilateration. The computational complexity could be much lower, but you will not be able to obtain the receiver's time bias.

Note: This question is based on the assumption that the satellites' positions and TOFs have already been extracted from the received satellite signals.

• Computational complexity may be high for 1980, but such GPS solutions are available in some pretty inexpensive devices here and now in 2020. Commented May 30, 2020 at 23:48

Computational complexity lies in the correlator banks you need to despread the GPS signal, not in the position and time calculation (these are not separate things. You can't have one without the other).
In fact, the TDOA system is very likely far more computationally intense, since it needs to do more cross-correlations.

You need to use the first method; 2. can't work, you'd need a clock that is about as stable and thus as large, power-hungry and expensive as that on the GPS satellites. Even with that clock, it wouldn't work well, because you still need to account for a lot of effects, including special relativity, that you can't solve via simple TDOA.

Computational complexity is still rather high, indeed, but really, not a problem since the late 1990s, where highly integrated chips for GPS became the norm. What it does do is drain the battery of your smartphone!

• Comments are not for extended discussion; this conversation has been moved to chat.
– Peter K.
Commented Jun 7, 2020 at 21:15
• @PeterK. I know it's the rules, but be aware that I'm not using the chat (I've got enough chats that I have to watch for job, business, project leadership and private reasons already) Commented Jun 7, 2020 at 21:17
• Understood. Then perhaps they OP can update their question rather than burying updates in the comments.
– Peter K.
Commented Jun 7, 2020 at 23:46
• I think I have finalized my opinion on the answer to the original question. Again, receiver clock bias is the same for all the satellite signals. Method one is used with much less complexity due to the Linearization of the pseudorange equations. Besides, correlation operations used to measure TOAs shouldn't make any difference here. Commented Jun 8, 2020 at 1:54