---
title: "Doppler spread fading; delay spread numerical"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:doppler-fading-delay-spread-profile"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/doppler-fading-delay-spread-profile
site: BCT Engineering
---

# Doppler spread fading; delay spread numerical

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

- Asked in: 2070 Magh, 4+4 marks
- Video: [RMS Delay Spread | Average delay spread | Wireless communication | Lec 20](https://www.youtube.com/watch?v=7uwprEXAMSQ) — 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/doppler-fading-delay-spread-profile
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Define Doppler spread. Describe the types of small scale fading based on Doppler spread. Calculate the mean excess delay and rms delay spread for the multipath profile given below. Estimate the 90% and 50% coherence bandwidth of the channel.
[Figure: power delay profile Pr(τ) with impulses at τ = 0 µs (about 0 dB), τ = 1 µs (about −10 dB) and τ = 2 µs (about 0 dB); the vertical axis is marked 0 dB, −10 dB, −20 dB, −30 dB and the time axis is in µs.]

## Short answer

Doppler spread is spectral broadening due to motion: f_c ± f_m. Fast fading if symbol period > coherence time (B_s < B_D); slow fading if symbol period << coherence time. From PDP: τ̄ = 1 µs, σ_τ = 0.976 µs; B_c(90%) = 1/(50σ_τ) = 20.5 kHz, B_c(50%) = 1/(5σ_τ) = 205 kHz.

## Answer

#### Doppler spread
**Doppler spread (B_D)** is the range of frequencies over which the received Doppler spectrum is non-zero when a pure sinusoid is sent. Due to motion, the received spectrum occupies f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. Its time-domain dual is the coherence time T_c ≈ 0.423/f_m.

#### Types of small scale fading based on Doppler spread
| Point | Fast fading | Slow fading |
|---|---|---|
| Condition | T_s > T_c, B_s < B_D | T_s ≪ T_c, B_s ≫ B_D |
| Channel change | Within one symbol | Over many symbols |
| Effect | Time distortion of the symbol | Channel is static per symbol |
| Typical case | Very low data rate, high speed | Most digital systems |

#### Numerical (power delay profile)
From the figure: P(0) = 0 dB, P(1 µs) = −10 dB, P(2 µs) = 0 dB.

Linear powers: 1, 0.1, 1 → ΣP = 2.1

**Mean excess delay**
```text
τ̄ = (1×0 + 0.1×1 + 1×2) / 2.1 = 2.1 / 2.1 = 1 µs
```
**Second moment**
```text
τ̄² = (1×0² + 0.1×1² + 1×2²) / 2.1 = 4.1 / 2.1 = 1.952 µs²
```
**RMS delay spread**
```text
σ_τ = √(1.952 − 1²) = √0.952 = 0.976 µs
```
**Coherence bandwidth**
```text
B_c(90%) ≈ 1/(50 σ_τ) = 1/(50 × 0.976 µs) = 20.49 kHz
B_c(50%) ≈ 1/(5 σ_τ)  = 1/(5 × 0.976 µs)  = 204.9 kHz
```

**Answer:** τ̄ = 1 µs, σ_τ = 0.976 µs, B_c(90%) ≈ 20.5 kHz and B_c(50%) ≈ 205 kHz.

## Check yourself

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

1. Fast fading occurs when:
   A. T_s > T_c (B_s < B_D)
   B. T_s << T_c
   C. B_s >> B_D
   D. sigma_tau is small
   Answer: A. If the channel changes within a symbol (T_s > T_c), fading is fast. (Key point: Fast fading condition)

2. For powers 1, 0.1, 1 at 0, 1, 2 us, the mean excess delay is:
   A. 0.5 us
   B. 1 us
   C. 2 us
   D. 0.1 us
   Answer: B. (0 + 0.1 + 2)/2.1 = 1 us. (Key point: Mean excess delay)

3. The rms delay spread is:
   A. 0.976 us
   B. 1.952 us
   C. 0.952 us
   D. 1.4 us
   Answer: A. sqrt(1.952 - 1) = sqrt(0.952) = 0.976 us. (Key point: RMS delay spread)

4. The 90% and 50% coherence bandwidths are about:
   A. 205 kHz and 20.5 kHz
   B. 20.5 kHz and 205 kHz
   C. 2.05 MHz and 205 kHz
   D. 1 MHz and 100 kHz
   Answer: B. 90%: 1/(50 sigma_tau) = 20.5 kHz; 50%: 1/(5 sigma_tau) = 205 kHz. (Key point: Coherence bandwidths)

## 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
- [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
- [Okumura vs Hata; delay spread numerical](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-hata-delay-spread) — 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×
- [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
- [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