Coherence bandwidth and delay spread relation

Frequency selective fading

Frequency selective fading occurs when the bandwidth of the transmitted signal is greater than the coherence bandwidth of the channel (Bs > Bc), or equivalently the symbol period is less than the rms delay spread (Ts < στ). Different frequency components fade differently; in time, the received signal contains multiple delayed, attenuated copies of each symbol, causing ISI and distortion.

Relationship between coherence bandwidth and delay spread

Delay spread describes the time dispersion of the channel; coherence bandwidth describes the same effect in the frequency domain. They are inversely related because the channel's frequency response is the Fourier transform of its impulse response.

Time domain parameters (from power delay profile P(τ)):

Mean excess delay  τ̄  = Σ P(τk)τk / Σ P(τk)
Second moment      τ²̄ = Σ P(τk)τk² / Σ P(τk)
RMS delay spread   στ = √(τ²̄ − τ̄²)

Coherence bandwidth: the range of frequencies over which two frequency components have strong amplitude correlation.

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

 Time domain               Frequency domain
 P(τ)                      |H(f)|
 |  |                      |~~~~~~~~~~~  small στ
 |  | |                    |   → wide Bc (flat)
 +--+-+-- τ (short)        +------------- f

 |  |   |    |             |~\_/~\_/~\_  large στ
 +--+---+----+-- τ (long)  +------------- f
                              → narrow Bc (selective)

Why: if the channel has paths with delay difference Δτ, their phases differ by 2πfΔτ. As frequency changes by about 1/Δτ, the relative phase changes by 2π, so the combined gain goes through peaks and nulls. Larger delay spread gives faster variation with frequency, i.e. smaller Bc.

Consequences:

• Bs ≪ Bc (Ts ≫ στ): flat fading.
• Bs > Bc (Ts < στ): frequency selective fading, ISI.
• Maximum symbol rate without equalizer is limited by στ (e.g. στ/Ts ≤ 0.1).

Example: στ = 1.37 µs → Bc(0.5) = 1/(5×1.37 µs) = 146 kHz, Bc(0.9) = 14.6 kHz. A 30 kHz AMPS channel would be flat for 0.5 correlation but a 200 kHz GSM signal would need equalization.

The relation is approximate; the exact coherence bandwidth depends on the actual impulse response, so Bc and στ are linked only statistically.