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The definition for the time of arrival is: $TOA=\frac{d}{c}+t_e$ where $c$ the wave speed, $d$ the range between the transmitter and the receiver and $t_e$ the transmission time of the signal to the transmitter.

My question: I don't understand why there is the term $t_e$ for me the TOA is just $\frac{d}{c}$

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  • $\begingroup$ As Dan Boschen very well answered, you can think of $t_{e}$ as the time elapsed after the "initialisation of the system". For example, if you started the system at midnight, then $t_{e}$ would be the time your watch shows when the signal left the transmitter. While the Time of Flight (ToF) is what is given by the "Newtonian" formulation of speed, when solved for time, $\frac{d}{c}$ as you state. $\endgroup$
    – ZaellixA
    Commented May 27, 2023 at 18:05

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$t_e$ is the absolute time when the signal was transmitted from the transmitter. $\frac{d}{c}$ is the Time of Flight (TOF). The Time of Arrival (TOA) is the absolute time when the signal reaches the receiver and therefore $TOA = TOF + t_e$.

Thus $t_e$ and $TOA$ represent absolute time such as May 21, 1997, 6:00:00 AM (or simply a number of time units from any agreed absolute reference), while $\frac{d}{c}$ represents a time difference as the TOF, in the same units as $t_e$ (for example, seconds).

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