Doppler spread, coherence bandwidth, fading classes

Doppler spread

Doppler spread (BD) is the measure of spectral broadening caused by the time rate of change of the mobile channel. When a pure tone fc is sent, the received spectrum (Doppler spectrum) extends from fc − fm to fc + fm, because multipath waves arrive from different angles θ and each gets a Doppler shift fd = (v/λ)cosθ. BD is taken as the maximum Doppler shift:

BD = fm = v/λ

If the signal bandwidth is much larger than BD, Doppler spread has negligible effect (slow fading). Its time-domain dual is coherence time Tc ≈ 0.423/fm.

Coherence bandwidth

Coherence bandwidth (Bc) is the range of frequencies over which the channel can be considered flat, i.e. two frequency components have strongly correlated amplitudes. It is derived from rms delay spread στ:

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

Classification of fading

Based on rms delay spread (multipath time dispersion):

1. Flat fading: Ts ≫ στ (Bs ≪ Bc). All frequency components fade equally; no ISI. Common rule: flat if Ts ≥ 10στ.
2. Frequency selective fading: Ts < στ (Bs > Bc). Received signal has multiple delayed copies, causing ISI; equalization needed.

Based on coherence time (Doppler spread): 3. Fast fading: Ts > Tc (Bs < BD). Channel changes within a symbol; causes signal distortion. 4. Slow fading: Ts ≪ Tc (Bs ≫ BD). Channel stays constant over many symbols.

 Ts (symbol period)
  ^
  | Flat-slow   | Flat-fast
10στ+-----------+-----------
  | Freq.sel.-  | Freq.sel.-
  |   slow      |   fast
  +-------------+----------> Ts
               Tc

The two classifications are independent: e.g. a channel can be flat and slow fading at the same time.