Pseudorange Measurement

The pseudorange \(p_{r,s}\) is calculated by multiplying the travel time \(\tau_{r,s}\) by the speed of light \(c\):

\[ p_{r,s} = c \cdot \tau_{r,s} \; \text{where} \; \tau_{r,s} = t_{r} - t_{s} \]

with:

\(t_{s}\): signal transmission time (from satellite s)
\(t_{r}\): time of signal arrival (determined by receiver clock)

There are then two situations:

Signal acquisition: pseudo-range prediction unknown, receiver-generated spreading code searched until correlation peak is found
Signal tracking: pseudo-range prediction known, only vary the receiver-generated code phase slightly

Perceived carrier frequency varies due to: Doppler effect and receiver clock drift.

A GNSS navigation solution is 4D with three position dimensions and one time dimension. For any satellite, the pseudo-range measurement, corrected for satellite clock error (and other known errors):

\[ \rho(t_{s,a}) = \sqrt{(r_s(t_{s,t}) - r_a(t_{s,a}))^T \cdot (r_s(t_{s,t}) - r_a(t_{s,a}))} + \delta \rho(t_{s,a}) \]

with:

\(r_s(t_{s,t})\): satellite position at time of signal transmission
\(r_a(t_{s,a})\): user antenna position at time of signal arrival
\(\delta \rho(t_{s,a})\): receiver clock offset