# OFDM Times of Arrival of multipath

Assume i have received a known analog OFDM symbol (such the Short-Training-Field (STF) in WiFi systems), call it $s(t)$ with a known DFT per subcarrier, in the presence of multipath and absence of noise.

$$r(t) = \sum\limits_{i=1}^p \gamma_i s(t-\tau_i)$$ where $\gamma_i$ and $\tau_i$ are complex gains and the delays of the multipath, respectively. Without loss of generality, let $\tau_1 > \ldots > \tau_p$.

My question is simply, what is $\tau_1$? I could imagine what $\tau_2 \ldots \tau_p$ are. They are just relative delays with respect to $\tau_1$. But i don't seem to quite grasp how to quantify or understand what $\tau_1$ means. Is it absolute ? Is it with respect to something ?

Any suggestion is much appreciated.

Thanks

• In addition to the given answer, $\tau_1$ represents the smallest delay. It can be associated with the shortest total distance that a component travels. Can be LOS, for instance. – msm Oct 26 '16 at 0:00

$\tau_i$ are delays of the multipath
therefore, $\tau_1$ is simply the delay incurred by the shortest path.
And no, $\tau_{i\ge2}$ are not relative delays; look at the formula: you just add up different independent paths with different delays; every path has its own delay, which is totally independent from $\tau_1$.