---
title: "Parameters of mobile multipath channel"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:mobile-multipath-channel-parameters"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/mobile-multipath-channel-parameters
site: BCT Engineering
---

# Parameters of mobile multipath channel

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

- Asked in: 2071 Magh, 7 marks
- Video: [Parameters of Mobile Multipath Channels | Wireless Communication](https://www.youtube.com/watch?v=dGbSagmarvk) — Er Sahil ka Gyan
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/mobile-multipath-channel-parameters
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

What are the parameters of mobile multipath channel? Explain.

## Short answer

Key parameters: mean excess delay, RMS delay spread, maximum excess delay (from power delay profile); coherence bandwidth (inversely proportional to RMS delay spread); Doppler spread and coherence time (from motion). These determine flat/frequency-selective and slow/fast fading.

## Answer

The parameters of a mobile multipath channel describe how the channel spreads the signal in **time** (because of multipath delays) and in **frequency** (because of motion). They are obtained from the **power delay profile (PDP)** P(τ) and the Doppler spectrum.

```text
 P(τ) dB
  |  |
  |  |  |
  |  |  |     |
  |  |  |  |  |   |  <- threshold (e.g. −30 dB)
  +--+--+--+--+---+----> τ
     τ0 τ1 τ2 ...
```

#### 1. Time dispersion parameters
- **Mean excess delay (τ̄)** – the first moment of the PDP:
  τ̄ = Σ a_k² τ_k / Σ a_k² = Σ P(τ_k) τ_k / Σ P(τ_k).
- **RMS delay spread (σ_τ)** – square root of the second central moment:
  σ_τ = √(τ̄² − (τ̄)²), where τ̄² = Σ P(τ_k) τ_k² / Σ P(τ_k).
  It is the most useful single measure of how much a symbol is spread. Typical values: ns indoors, µs outdoors.
- **Maximum excess delay (X dB)** – the time after the first arrival in which the multipath energy falls X dB below the maximum: τ_X − τ₀.

#### 2. Coherence bandwidth (B_c)
The range of frequencies over which the channel is "flat" (passes all components with nearly equal gain and linear phase). It is inversely proportional to σ_τ:
- B_c ≈ 1/(50 σ_τ) for frequency correlation above 0.9.
- B_c ≈ 1/(5 σ_τ) for correlation above 0.5.

#### 3. Doppler spread (B_D)
The width of the received spectrum when a pure tone f_c is sent. Due to motion, the spectrum spreads from f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. If the signal bandwidth is much larger than B_D, the Doppler effect is negligible.

#### 4. Coherence time (T_c)
The time over which the channel impulse response stays nearly unchanged; it is the time-domain dual of Doppler spread.
- T_c ≈ 9/(16π f_m) (correlation 0.5)
- Common rule of thumb: T_c = √(9/(16π f_m²)) = 0.423 / f_m.

#### Summary
| Parameter | Caused by | Decides |
|---|---|---|
| σ_τ, B_c | Multipath delay | Flat vs frequency selective |
| B_D, T_c | Motion (Doppler) | Slow vs fast fading |

Example: σ_τ = 1 µs gives B_c(50%) ≈ 200 kHz; a 200 kHz GSM signal would then see some frequency selectivity.

## Check yourself

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

1. Time dispersion parameters are derived from the:
   A. Power delay profile
   B. Antenna pattern
   C. Link budget
   D. Cell layout
   Answer: A. Mean excess delay, rms delay spread and maximum excess delay come from the power delay profile P(tau). (Key point: Source of parameters)

2. Coherence bandwidth is related to rms delay spread as:
   A. Directly proportional
   B. Inversely proportional
   C. Independent
   D. Proportional to the square
   Answer: B. B_c is about 1/(50 sigma_tau) or 1/(5 sigma_tau), inversely proportional. (Key point: Coherence bandwidth relation)

3. Doppler spread is the width of the spectrum:
   A. From f_c - f_m to f_c + f_m when a pure tone is sent
   B. Of the delay profile
   C. Of the antenna beam
   D. Of the noise
   Answer: A. Motion spreads a pure tone over f_c +/- f_m with f_m = v/lambda. (Key point: Doppler spread)

4. Which parameters decide slow versus fast fading?
   A. sigma_tau and B_c
   B. B_D and T_c
   C. Path loss exponent and sigma
   D. Maximum excess delay only
   Answer: B. Doppler spread and coherence time (motion) decide slow/fast; sigma_tau and B_c decide flat/frequency selective. (Key point: Which decides what)

## Related in Wireless Communication

- [Path loss and multipath channel parameters](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Apath-loss-multipath-parameters) — Wireless Communication, chapter 3

## Related in other subjects

- [Mobile boundary design: three approaches](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Adesign-three-methods-mobile-boundary) — Hydraulics, chapter 10