Coherence bandwidth/time and types of fading

Coherence bandwidth

Coherence bandwidth (Bc) is the statistical range of frequencies over which the channel passes all spectral components with approximately equal gain and linear phase. It is inversely proportional to the rms delay spread:

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

Significance: if the signal bandwidth Bs < Bc, the signal sees flat fading with no ISI; if Bs > Bc, it sees frequency selective fading and ISI, so equalizers or OFDM are needed.

Coherence time

Coherence time (Tc) is the time over which the channel impulse response is essentially invariant. It is inversely proportional to maximum Doppler shift fm = v/λ:

Tc ≈ 9/(16π fm)  ≈ 0.179/fm
Tc ≈ 0.423/fm    (geometric mean rule, used in practice)

Significance: if the symbol period Ts < Tc, the channel is slow fading; if Ts > Tc, it is fast fading and the signal is distorted. It also sets the spacing needed for time diversity and interleaving.

Fading and its types

Fading is the rapid variation of the received signal amplitude and phase over short time or distance, caused by constructive and destructive addition of multipath waves and by motion.

Based on multipath time delay spread:

1. Flat fading: Bs ≪ Bc, Ts ≫ στ. All frequencies fade together, no ISI; spectral shape preserved.
2. Frequency selective fading: Bs > Bc, Ts < στ. Different frequencies fade differently; causes ISI.

Based on Doppler spread: 3. Fast fading: Ts > Tc, Bs < BD. Channel varies within a symbol; time-selective distortion. 4. Slow fading: Ts ≪ Tc, Bs ≫ BD. Channel constant over many symbols.

Large-scale fading (shadowing by hills and buildings) is the slower variation of the mean signal over hundreds of wavelengths; it is usually log-normally distributed.