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I am a bit confused about Doppler spread and Doppler Spectrum. I believe the Doppler spread is related to channel coherence time T_c, so that . Due to inconsistencies and incoherency of the channel, its behavior may change causing a change in the frequency and phase of the signal propagating through it.

If f_c is the frequency of the signal, and f_d is Doppler shift due to the movement of either source or the target or both, then Doppler Spectrum would be the range of frequencies over the incoming reflection from a moving target (or source), i.e. \ between f_c - f_d and f_c + f_d. Similarly, Doppler Bandwidth would be (2 * f_)m

How correct am I in understanding the two terms?

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