Significance of coherence bandwidth and time

Coherence bandwidth (Bc)

Coherence bandwidth is the range of frequencies over which the channel can be considered "flat", i.e. it passes all spectral components with nearly equal gain and linear phase. Two sinusoids separated by less than Bc have strongly correlated amplitudes.

It is inversely proportional to the rms delay spread στ:

Bc ≈ 1/(50στ)   (correlation ≥ 0.9)
Bc ≈ 1/(5στ)    (correlation ≥ 0.5)

Significance:

• If signal bandwidth Bs < Bc: flat fading, no ISI, simple receiver.
• If Bs > Bc: frequency selective fading, causing ISI; equalizers, RAKE receivers or OFDM are needed.
• It sets the maximum symbol rate without equalization and the frequency spacing needed for frequency diversity.

Coherence time (Tc)

Coherence time is the time duration over which the channel impulse response is essentially unchanged. Two signals received within Tc have strongly correlated amplitudes. It is the time-domain dual of Doppler spread:

Tc ≈ 1/fm
Tc ≈ 9/(16π fm) ≈ 0.179/fm   (correlation 0.5)
Tc ≈ √(9/(16π)) / fm = 0.423/fm  (common rule)
fm = v/λ (maximum Doppler shift)

Significance:

• If symbol period Ts < Tc: slow fading, channel constant over a symbol.
• If Ts > Tc: fast fading, the channel changes within a symbol and the signal is distorted.
• It sets the time spacing for time diversity and interleaving depth, and how often channel estimates must be updated.

Example: v = 60 mph (26.82 m/s), fc = 1900 MHz → fm = 169.8 Hz, Tc = 0.423/169.8 ≈ 2.5 ms. A symbol rate above about 1/Tc = 400 symbols/s avoids distortion from motion.