---
title: "Frequency dispersion and Doppler spread derivation"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:frequency-dispersion-doppler-spread"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-dispersion-doppler-spread
site: BCT Engineering
---

# Frequency dispersion and Doppler spread derivation

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

- Asked in: 2082 Bhadra, 2+4 marks
- Video: [Doppler Shift - Mobile Radio Propagation - Mobile Communication System](https://www.youtube.com/watch?v=nfk_9B3ZDFk) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/frequency-dispersion-doppler-spread
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Define the significance of term frequency dispersion in wireless communication. Derive the relation for doppler spread.

## Short answer

Frequency dispersion is spectral spreading caused by motion, quantified by Doppler spread BD. Path difference Delta l = v Delta t cos theta gives phase change 2πDelta l/λ, so Doppler shift fd = v/λ cosθ; maximum shift fm = v/λ, and Doppler spread BD = fm = v/λ.

## Answer

#### Significance of frequency dispersion
**Frequency dispersion** is the spreading of the signal spectrum in the frequency domain, caused by the relative motion between the mobile and the base station (or moving objects). Each multipath wave arrives from a different angle, so it gets a different Doppler shift, and a single tone is received as a band of frequencies of width up to 2fm.

- It shows that the channel is **time-variant**; it is measured by **Doppler spread** BD and its dual, **coherence time** Tc ≈ 1/BD.
- If BD is comparable to the signal bandwidth (Ts > Tc), the channel causes **fast fading** and distortion.
- It limits how slowly data can be sent and how often the receiver must track the channel; it decides interleaver depth and time diversity design.

#### Derivation of Doppler shift and Doppler spread
A mobile moves at speed *v* from point X to Y (distance *l*) in time Δt, while receiving from a distant source S. The wave arrives at angle θ to the direction of motion.

```text
                    S (far source)
                   /
                  /  waves nearly parallel
                 /
         X ----θ------------ Y    → v
           \___ Δl = l cosθ
           l = vΔt
```

Since the source is far, rays at X and Y are parallel. The difference in path length is:

```text
Δl = l cosθ = vΔt cosθ
```
The phase change of the received signal:

```text
Δφ = 2πΔl/λ = 2πvΔt cosθ / λ
```
Frequency change (Doppler shift) is the rate of change of phase:

```text
fd = (1/2π)·(Δφ/Δt) = (v/λ)·cosθ
```

- θ = 0° (moving towards source): fd = +v/λ (maximum, frequency rises)
- θ = 180° (moving away): fd = −v/λ
- θ = 90°: fd = 0

The maximum Doppler shift is fm = v/λ. Since the multipath waves arrive from all angles θ (0 to 2π), the received spectrum spreads over fc − fm to fc + fm. The **Doppler spread** BD is the measure of this width, taken as

```text
BD = fm = v/λ   (spectrum occupies 2fm in total)
```

**Example:** fc = 900 MHz, v = 72 km/h = 20 m/s → λ = 0.333 m, fm = 20/0.333 = 60 Hz.

## Check yourself

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

1. Frequency dispersion is caused by:
   A. Antenna mismatch
   B. Relative motion giving different Doppler shifts to multipath waves
   C. Delay spread
   D. Path loss
   Answer: B. A single tone is received as a band up to 2fm wide. (Key point: Frequency dispersion)

2. The Doppler shift formula derived is:
   A. fd = v lambda cos(theta)
   B. fd = (lambda/v) cos(theta)
   C. fd = (v/lambda) cos(theta)
   D. fd = v sin(theta)/lambda
   Answer: C. Phase change 2 pi v dt cos(theta)/lambda divided by 2 pi dt. (Key point: Doppler shift)

3. For what arrival angle is the Doppler shift zero?
   A. 0 degrees
   B. 45 degrees
   C. 180 degrees
   D. 90 degrees
   Answer: D. cos 90 = 0. (Key point: Zero shift)

4. Doppler spread BD is approximately:
   A. v/lambda
   B. lambda/v
   C. 1/Tc squared
   D. 2 v lambda
   Answer: A. BD ~ fm = v/lambda. (Key point: Doppler spread)

5. Frequency dispersion leads to fast fading when:
   A. BD is zero
   B. BD is comparable to signal bandwidth (Ts > Tc)
   C. Ts << Tc
   D. Bs < Bc
   Answer: B. The channel varies within a symbol. (Key point: Fast fading link)

## Related in Wireless Communication

- [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
- [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×
- [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
- [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
- [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
- [Fading factors and Doppler shift derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afading-factors-doppler-shift) — Wireless Communication, chapter 3

## Related in other subjects

- [Wired and wireless communication systems](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Awired-wireless-systems) — Basic Electronics Engineering, chapter 5, asked 2×
- [Spread spectrum frequency hopping in Bluetooth](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Afreq-hopping-bt) — Instrumentation II, chapter 5, asked 2×
- [MUF and critical frequency relation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Amuf-critical-frequency-relation) — Propagation and Antenna, chapter 4
- [Derive Mohr-Coulomb principal stress relation](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-coulomb-derive-principal-stresses) — Soil Mechanics, chapter 10, asked 4×