The Uncertainty Principle

Abstract

It has thus far been assumed that the received signal is reflected from a stationary target so that the carrier frequency of the transmitted and reflected signals coincide. The general case will now be considered. Assume that the signal is reflected from a target whose range is r₀, and whose range rate is v₀. Thus the reflected signal is delayed by the time t₀ = 2r₀/c and acquires a Doppler shift of Q = (2v₀/c) ω₀:

$$ {s_{1}}\left( t \right) = s\left( {t - {t_{0}}} \right){e^{{j\Omega \left( {t - {t_{0}}} \right)}}} $$

.