---
title: "Path loss and multipath channel parameters"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:path-loss-multipath-parameters"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/path-loss-multipath-parameters
site: BCT Engineering
---

# Path loss and multipath channel parameters

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

- Asked in: 2080 Bhadra, 2+4 marks
- Video: [3.1 Pathloss Shadowing and Multipath](https://www.youtube.com/watch?v=s1fy_n2wN0I) — Ali Muqaibel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/path-loss-multipath-parameters
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Define path loss. Explain the parameters of mobile multipath channels used to classify various types of fading.

## Short answer

Path loss is the dB reduction between transmitted and received power, often modeled as proportional to d^n. Multipath channel parameters are mean excess delay, RMS delay spread and coherence bandwidth, which classify flat versus frequency-selective fading, and Doppler spread and coherence time, which classify fast versus slow fading.

## Answer

#### Path loss
**Path loss** is the reduction in signal power as a wave travels from transmitter to receiver. It is the difference (in dB) between effective transmitted power and received power, and may include antenna gains:

```text
PL(dB) = 10 log(Pt/Pr)
       = −10 log[Gt·Gr·λ² / ((4π)²d²)]  (free space)
```
In practice PL(d) ∝ dⁿ, where n is the path loss exponent (2 in free space, 2.7–5 in urban areas).

#### Parameters of mobile multipath channels
These parameters, obtained from the power delay profile and Doppler spectrum, are used to classify small-scale fading.

**A. Time dispersion parameters** (from the power delay profile P(τ))

1. **Mean excess delay τ̄:** first moment of the power delay profile.
```text
τ̄ = Σ ak²τk / Σ ak² = Σ P(τk)τk / Σ P(τk)
```
2. **RMS delay spread στ:** square root of the second central moment; the main measure of time dispersion.
```text
στ = √(τ²̄ − (τ̄)²),  τ²̄ = Σ P(τk)τk² / Σ P(τk)
```
3. **Maximum excess delay (X dB):** time during which multipath energy falls to X dB below the maximum, τX − τ0.

**B. Coherence bandwidth (Bc)**
Range of frequencies over which the channel is flat. Bc ≈ 1/(50στ) (0.9 correlation) or 1/(5στ) (0.5 correlation). Compared with signal bandwidth, it decides **flat** vs **frequency selective** fading.

**C. Doppler spread (BD)**
Width of the spectrum when a pure tone is sent; BD ≈ fm = v/λ. It measures how fast the channel changes.

**D. Coherence time (Tc)**
Time over which the channel is constant; Tc ≈ 0.423/fm. Compared with symbol period, it decides **fast** vs **slow** fading.

| Parameter | Classifies | Rule |
|---|---|---|
| στ, Bc | Flat / frequency selective | Bs < Bc → flat |
| BD, Tc | Fast / slow | Ts < Tc → slow |

## Check yourself

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

1. What is path loss?
   A. The increase in noise at the receiver
   B. The reduction in signal power between transmitter and receiver, PL = 10 log(Pt/Pr)
   C. The time delay between the first and last multipath component
   D. The loss of signal caused only by cable attenuation
   Answer: B. Path loss is the dB difference between transmitted and received power as the wave travels. (Key point: Definition of path loss)

2. In PL(d) proportional to d^n, what is the value of n in free space?
   A. 4
   B. 3
   C. 2
   D. 1
   Answer: C. n = 2 in free space; urban areas have 2.7 to 5. (Key point: Path loss exponent)

3. How is the RMS delay spread defined?
   A. The first moment of the power delay profile
   B. The time at which energy falls X dB below the maximum
   C. The square root of the second central moment of the power delay profile
   D. The inverse of the Doppler spread
   Answer: C. στ = sqrt(mean of τ^2 − (mean τ)^2); it is the main measure of time dispersion. (Key point: RMS delay spread)

4. Which pair of parameters decides fast versus slow fading?
   A. Doppler spread BD and coherence time Tc
   B. RMS delay spread and coherence bandwidth
   C. Mean excess delay and path loss exponent
   D. Maximum excess delay and Bc
   Answer: A. Doppler spread and coherence time compared with the symbol period give fast or slow fading; στ and Bc give flat or frequency selective. (Key point: Classifying fading)

## Related in Wireless Communication

- [Parameters of mobile multipath channel](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Amobile-multipath-channel-parameters) — Wireless Communication, chapter 3
- [Large scale path loss and small scale fading](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Alarge-scale-loss-small-fading) — Wireless Communication, chapter 3
- [Path loss vs fading; time dispersion fading](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Apath-loss-vs-fading-time-dispersion) — Wireless Communication, chapter 3
- [Doppler effect; cordless phone path loss](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Adoppler-effect-cordless-phone) — Wireless Communication, chapter 3, asked 2×
- [Rayleigh versus Rician fading channel](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Arayleigh-vs-rician-fading) — Wireless Communication, chapter 3
- [Numerical: two-ray vs free-space path loss at 800 MHz](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atwo-ray-path-loss-800mhz) — Wireless Communication, chapter 3

## Related in other subjects

- [Friis equation and free space path loss](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afriis-path-loss-free-space) — Propagation and Antenna, chapter 5, asked 2×
- [Path loss in radio wave propagation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Apath-loss-derivation) — Propagation and Antenna, chapter 5, asked 2×
- [Friis, path loss and LOS distance](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afriis-path-loss-los-distance) — Propagation and Antenna, chapter 5
- [Free space path loss: 6 GHz, 10,500 km](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Afspl-derivation-6ghz-10500km) — Propagation and Antenna, chapter 5