---
title: "Doppler spread, coherence bandwidth, fading classes"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:doppler-coherence-fading-classification"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/doppler-coherence-fading-classification
site: BCT Engineering
---

# Doppler spread, coherence bandwidth, fading classes

Wireless Communication · Chapter 3: Radio Wave Propagation in Mobile Networks

- Asked in: 2074 Magh, 4+4 marks
- Video: [What are Coherence Bandwidth and Coherence Time?](https://www.youtube.com/watch?v=bNMsizyWnK0) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/doppler-coherence-fading-classification
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Explain Doppler spread and coherence bandwidth. Classify fading on the basis of RMS delay spread and coherence time.

## Short answer

Doppler spread BD = v/λ; coherence bandwidth Bc is about 1/(5στ) or 1/(50στ). Fading is flat if Ts >> στ and frequency-selective if Ts < στ; it is fast if Ts > Tc and slow if Ts << Tc.

## Answer

#### 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:

```text
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:

```text
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.

| Basis | Type | Condition |
|---|---|---|
| στ | Flat | Ts ≫ στ, Bs ≪ Bc |
| στ | Frequency selective | Ts < στ, Bs > Bc |
| Tc | Fast | Ts > Tc, Bs < BD |
| Tc | Slow | Ts ≪ Tc, Bs ≫ BD |

```text
 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Doppler spread BD is:
   A. The spread of delays of multipath
   B. The spectral broadening, taken as maximum Doppler shift fm = v/λ
   C. The frequency range over which the channel is flat
   D. The duration of a symbol
   Answer: B. A tone fc spreads over fc ± fm; BD ≈ fm. (Key point: Doppler spread)

2. Coherence bandwidth with 0.5 correlation is approximately:
   A. 1/(50 στ)
   B. 50 στ
   C. 1/(5 στ)
   D. 0.423/fm
   Answer: C. Bc ≈ 1/(5στ) for 0.5 correlation and 1/(50στ) for 0.9. (Key point: Coherence bandwidth)

3. Which condition gives flat fading?
   A. Ts much greater than στ (Bs much less than Bc)
   B. Ts less than στ
   C. Bs greater than Bc
   D. Ts greater than Tc
   Answer: A. When Ts is much greater than στ, all components fade equally with no ISI. (Key point: Flat versus frequency selective)

4. Fast fading occurs when:
   A. Ts much less than Tc
   B. Ts greater than Tc (Bs less than BD)
   C. Bs much greater than BD
   D. Bs greater than Bc
   Answer: B. The channel changes within a symbol when Ts exceeds Tc. (Key point: Fast versus slow fading)

5. Are the delay-spread and Doppler classifications independent?
   A. No, flat always means fast
   B. No, selective always means slow
   C. Yes, e.g. a channel can be flat and slow at once
   D. Only for Rayleigh
   Answer: C. Delay spread and Doppler spread are separate effects. (Key point: Independence of classifications)

## Related in Wireless Communication

- [Doppler spread fading; delay spread numerical](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adoppler-fading-delay-spread-profile) — Wireless Communication, chapter 3
- [Coherence bandwidth and delay spread relation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afrequency-selective-coherence-delay) — Wireless Communication, chapter 3
- [Fading types by delay spread and Doppler spread](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afading-types-delay-doppler) — Wireless Communication, chapter 3, asked 2×
- [Short note: Doppler spread and coherence time](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adoppler-spread-coherence-time-note) — Wireless Communication, chapter 3
- [Coherence bandwidth/time and types of fading](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acoherence-significance-fading-types) — Wireless Communication, chapter 3, asked 3×
- [Numerical: rms delay spread; flat or selective for AMPS/GSM](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arms-delay-spread-amps-gsm) — Wireless Communication, chapter 3