---
title: "Short note: Doppler spread and coherence time"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:doppler-spread-coherence-time-note"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/doppler-spread-coherence-time-note
site: BCT Engineering
---

# Short note: Doppler spread and coherence time

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

- Asked in: 2070 Bhadra, 3 marks
- Video: [What are Doppler Shift, Doppler Spread, and Doppler Spectrum?](https://www.youtube.com/watch?v=LLr3-kotbz4) — 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-spread-coherence-time-note
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Write a short note on Doppler spread and coherence time.

## Short answer

Doppler spread is spectral broadening due to motion, spanning f_c ± f_m (f_m = v/λ). Coherence time T_c ≈ 0.423/f_m is the duration of near-constant channel response. Slow fading if symbol period << T_c; fast fading if symbol period > T_c.

## Answer

**Doppler spread (B_D)** is the width of the received spectrum when a single pure tone of frequency f_c is transmitted over a mobile channel. Because of relative motion, each multipath wave arrives with a different Doppler shift f_d = (v/λ) cos θ, so the received spectrum spreads over f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. B_D measures how fast the channel changes in time.

**Coherence time (T_c)** is the time-domain dual of Doppler spread. It is the time duration over which the channel impulse response is nearly constant, i.e. two received signals separated by less than T_c are strongly correlated.
```text
T_c ≈ 1 / f_m
T_c ≈ 9 / (16π f_m)      (correlation > 0.5)
T_c ≈ 0.423 / f_m        (geometric mean, common rule)
```

**Use:** If symbol period T_s < T_c (B_s > B_D), the channel is **slow fading**; if T_s > T_c, it is **fast fading**.

Example: v = 60 km/h (16.67 m/s), f = 900 MHz (λ = 0.333 m) gives f_m = 50 Hz and T_c ≈ 0.423/50 = 8.46 ms.

## Check yourself

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

1. The maximum Doppler shift f_m is:
   A. v lambda
   B. v/lambda
   C. lambda/v
   D. c/v
   Answer: B. f_m = v/lambda. (Key point: Maximum Doppler shift)

2. Coherence time is:
   A. Time over which the channel impulse response is nearly constant
   B. Time delay of the longest path
   C. Symbol duration
   D. Time between handoffs
   Answer: A. It is the time-domain dual of Doppler spread; signals closer than T_c are strongly correlated. (Key point: Coherence time meaning)

3. When is the channel slow fading?
   A. T_s > T_c
   B. T_s < T_c
   C. B_s < B_D
   D. Never at 900 MHz
   Answer: B. If the symbol period is less than coherence time (B_s > B_D), the channel is static over a symbol. (Key point: Slow vs fast fading)

4. For v = 16.67 m/s at 900 MHz, what is f_m?
   A. 5 Hz
   B. 500 Hz
   C. 50 Hz
   D. 5 kHz
   Answer: C. f_m = v/lambda = 16.67/0.333 = 50 Hz, so T_c about 0.423/50 = 8.46 ms. (Key point: Example calculation)

## Related in Wireless Communication

- [Doppler spread, coherence bandwidth, fading classes](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adoppler-coherence-fading-classification) — Wireless Communication, chapter 3
- [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×
- [Frequency dispersion and Doppler spread derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afrequency-dispersion-doppler-spread) — Wireless Communication, chapter 3
- [Doppler spread and Doppler shift derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adoppler-spread-shift-derivation) — Wireless Communication, chapter 3