--- title: "Coherence bandwidth and delay spread relation" subject: "Wireless Communication" chapter: 3 key: "semester_7/wireless_communication:frequency-selective-coherence-delay" source: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-selective-coherence-delay site: BCT Engineering --- # Coherence bandwidth and delay spread relation Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks - Asked in: 2079 Chaitra, 2+7 marks - Video: [Coherence Bandwidth | Frequency Selective vs Frequency Flat Fading | Wireless Communication |Lec 21](https://www.youtube.com/watch?v=FPGCvdCZNYQ) — Quantum Study Circle - Clear your Concepts - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-selective-coherence-delay - Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3 ## Question What is frequency selective fading? Explain the relationship between coherence bandwidth and delay spread in time domain. ## Short answer Frequency selective fading occurs when signal bandwidth exceeds coherence bandwidth (Bs > Bc) or symbol period is smaller than delay spread (Ts < στ), causing ISI and frequency-dependent fading. Coherence bandwidth Bc is inversely related to rms delay spread στ; larger delay spread yields smaller Bc and faster frequency variation. ## Answer #### 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)): ```text 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. ```text Bc ≈ 1/(50στ) (correlation above 0.9) Bc ≈ 1/(5στ) (correlation above 0.5) ``` ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A channel is frequency selective when: A. Bs < Bc B. Bs > Bc, equivalently Ts < στ C. Ts is much greater than στ D. Tc is much greater than Ts Answer: B. The signal bandwidth exceeds the coherence bandwidth, so different frequency components fade differently. (Key point: Condition for frequency selective fading) 2. How are coherence bandwidth and rms delay spread related? A. They are directly proportional B. They are independent C. They are inversely related (Bc roughly 1/(5στ) to 1/(50στ)) D. Bc equals στ Answer: C. The frequency response is the Fourier transform of the impulse response, so larger delay spread gives smaller Bc. (Key point: Bc and delay spread relation) 3. Why does larger delay spread give a smaller coherence bandwidth? A. Path phases differ by 2πfΔτ, so the combined gain cycles faster with frequency B. Larger delay lowers the carrier frequency C. Delayed paths have no phase effect D. Larger delay spread removes the nulls Answer: A. Relative phase changes by 2π when frequency shifts by about 1/Δτ, giving peaks and nulls closely spaced in frequency. (Key point: Why Bc shrinks) 4. With στ = 1.37 µs, Bc for 0.5 correlation is about: A. 14.6 kHz B. 1.46 MHz C. 146 kHz D. 7.3 kHz Answer: C. Bc = 1/(5 x 1.37 µs) = 146 kHz; 0.9 correlation gives 14.6 kHz. 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