Short note: Doppler spread and coherence time

Doppler spread (B_D) is the width of the received spectrum when a single pure tone of frequency f_c is transmitted over a mobile channel. Because of relative motion, each multipath wave arrives with a different Doppler shift f_d = (v/λ) cos θ, so the received spectrum spreads over f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. B_D measures how fast the channel changes in time.

Coherence time (T_c) is the time-domain dual of Doppler spread. It is the time duration over which the channel impulse response is nearly constant, i.e. two received signals separated by less than T_c are strongly correlated.

T_c ≈ 1 / f_m
T_c ≈ 9 / (16π f_m)      (correlation > 0.5)
T_c ≈ 0.423 / f_m        (geometric mean, common rule)

Use: If symbol period T_s < T_c (B_s > B_D), the channel is slow fading; if T_s > T_c, it is fast fading.

Example: v = 60 km/h (16.67 m/s), f = 900 MHz (λ = 0.333 m) gives f_m = 50 Hz and T_c ≈ 0.423/50 = 8.46 ms.