Parameters of mobile multipath channel

The parameters of a mobile multipath channel describe how the channel spreads the signal in time (because of multipath delays) and in frequency (because of motion). They are obtained from the power delay profile (PDP) P(τ) and the Doppler spectrum.

 P(τ) dB
  |  |
  |  |  |
  |  |  |     |
  |  |  |  |  |   |  <- threshold (e.g. −30 dB)
  +--+--+--+--+---+----> τ
     τ0 τ1 τ2 ...

1. Time dispersion parameters

• Mean excess delay (τ̄) – the first moment of the PDP: τ̄ = Σ a_k² τ_k / Σ a_k² = Σ P(τ_k) τ_k / Σ P(τ_k).
• RMS delay spread (σ_τ) – square root of the second central moment: σ_τ = √(τ̄² − (τ̄)²), where τ̄² = Σ P(τ_k) τ_k² / Σ P(τ_k). It is the most useful single measure of how much a symbol is spread. Typical values: ns indoors, µs outdoors.
• Maximum excess delay (X dB) – the time after the first arrival in which the multipath energy falls X dB below the maximum: τ_X − τ₀.

2. Coherence bandwidth (B_c)

The range of frequencies over which the channel is "flat" (passes all components with nearly equal gain and linear phase). It is inversely proportional to σ_τ:

• B_c ≈ 1/(50 σ_τ) for frequency correlation above 0.9.
• B_c ≈ 1/(5 σ_τ) for correlation above 0.5.

3. Doppler spread (B_D)

The width of the received spectrum when a pure tone f_c is sent. Due to motion, the spectrum spreads from f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. If the signal bandwidth is much larger than B_D, the Doppler effect is negligible.

4. Coherence time (T_c)

The time over which the channel impulse response stays nearly unchanged; it is the time-domain dual of Doppler spread.

• T_c ≈ 9/(16π f_m) (correlation 0.5)
• Common rule of thumb: T_c = √(9/(16π f_m²)) = 0.423 / f_m.

Summary

Example: σ_τ = 1 µs gives B_c(50%) ≈ 200 kHz; a 200 kHz GSM signal would then see some frequency selectivity.