Turn signal time into a position

Navigation broadcasts provide timing and orbital information. A receiver compares the signal’s timing with its own clock. Multiplying a travel-time estimate by the speed of light produces a range estimate. Clock offsets and propagation effects make this a pseudorange rather than a perfectly measured geometric distance.

Try an original scale calculation: approximate light speed as 300,000,000 meters per second. An uncompensated timing error of one microsecond is 0.000001 seconds, so it corresponds to about 300 meters of range error. This is not a claimed BeiDou user error. It shows why a small time discrepancy matters when a signal travels so quickly.

In a basic three-dimensional, single-system solution, the receiver has four unknowns: three position coordinates and its clock offset. Measurements from at least four suitably arranged satellites allow those quantities to be estimated together. A particular fourth satellite does not independently hand the receiver the missing time.

Sketch a receiver with four incoming measurements, then label the four unknowns beside one shared calculation. Do not assign one satellite to longitude and another to latitude. Each measurement constrains the combined solution. Extra usable observations can provide redundancy, but counting signals does not replace checking geometry and measurement quality.

Estimate position and clock offset together

For this teaching model, write measured range as geometric range plus a receiver-clock term plus remaining errors. A common receiver-clock error contributes to all measurements, while reflection from a nearby wall may affect only certain signal paths. Real processing must also account for satellite clocks and other effects omitted from the simple exercise.

Suppose all four range readings acquire an extra 300 meters because of the same receiver-clock shift. The model should estimate a clock change rather than forcing every range discrepancy into a position change. Now suppose only one reading acquires an extra path delay. Compare the residuals—the differences between observed and predicted values—to investigate that inconsistent measurement.