Path loss and multipath channel parameters

Path loss

Path loss is the reduction in signal power as a wave travels from transmitter to receiver. It is the difference (in dB) between effective transmitted power and received power, and may include antenna gains:

PL(dB) = 10 log(Pt/Pr)
       = −10 log[Gt·Gr·λ² / ((4π)²d²)]  (free space)

In practice PL(d) ∝ dⁿ, where n is the path loss exponent (2 in free space, 2.7–5 in urban areas).

Parameters of mobile multipath channels

These parameters, obtained from the power delay profile and Doppler spectrum, are used to classify small-scale fading.

A. Time dispersion parameters (from the power delay profile P(τ))

1. Mean excess delay τ̄: first moment of the power delay profile.

τ̄ = Σ ak²τk / Σ ak² = Σ P(τk)τk / Σ P(τk)

2. RMS delay spread στ: square root of the second central moment; the main measure of time dispersion.

στ = √(τ²̄ − (τ̄)²),  τ²̄ = Σ P(τk)τk² / Σ P(τk)

3. Maximum excess delay (X dB): time during which multipath energy falls to X dB below the maximum, τX − τ0.

B. Coherence bandwidth (Bc) Range of frequencies over which the channel is flat. Bc ≈ 1/(50στ) (0.9 correlation) or 1/(5στ) (0.5 correlation). Compared with signal bandwidth, it decides flat vs frequency selective fading.

C. Doppler spread (BD) Width of the spectrum when a pure tone is sent; BD ≈ fm = v/λ. It measures how fast the channel changes.

D. Coherence time (Tc) Time over which the channel is constant; Tc ≈ 0.423/fm. Compared with symbol period, it decides fast vs slow fading.