= Multipath propagation
{wiki}
<Multipath propagation> occurs when a signal reaches a receiver along several routes, for example a direct <acoustic wave> and a <wave reflection> from a wall. In a linear, stationary model the received field can be represented as
$$
s_r(t)=\sum_j a_j s(t-\tau_j),
$$
where each path has delay $\tau_j$ and an appropriate amplitude factor $a_j$. More general paths can filter the signal as well as delay it.
A short <pulse> can produce distinct echoes when the delays are separated enough to resolve. For a narrowband tone, the copies instead overlap with different <wave phases> and can reinforce or cancel. Moving the receiver changes the path differences and can cause large changes in measured <wave intensity> even if source <power> stays constant. Comparing an early direct arrival with later echoes separates propagation time from reflection geometry. This is useful in <sonar>, room acoustics and <radio communication>, where delayed copies can obscure the intended signal.
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