---
title: "Log-distance path loss model"
subject: "Wireless Communication"
chapter: 3
key: "semester_7/wireless_communication:log-distance-path-loss"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/log-distance-path-loss
site: BCT Engineering
---

# Log-distance path loss model

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

- Asked in: 2077 Chaitra, 4 marks
- Video: [Log Distance Model/ Log Normal Shadowing Model](https://www.youtube.com/watch?v=4rAnCdPz9cw) — Disha A. Shukla
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/log-distance-path-loss
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-3

## Question

Explain log-distance path loss model.

## Short answer

The log-distance model predicts average path loss as PL(d)=PL(d0)+10n log(d/d0), where n is path loss exponent. A log-normal shadowing term Xσ (zero-mean Gaussian) can be added for random variations. Common n: free space 2, urban cellular 2.7–3.5.

## Answer

The **log-distance path loss model** states that the average received power falls off logarithmically with distance, whether indoors or outdoors. The average path loss is proportional to the n-th power of distance:

```text
PL(d) ∝ (d/d0)ⁿ

PL(d) [dB] = PL(d0) + 10n log(d/d0)
```
- **n** = path loss exponent: how fast loss grows with distance.
- **d0** = close-in reference distance in the far field (1 km for large cells, 100 m or 1 m for micro/indoor cells).
- **PL(d0)** = loss at d0, measured or found from free space: 20 log(4πd0/λ).

| Environment | n |
|---|---|
| Free space | 2 |
| Urban cellular radio | 2.7 – 3.5 |
| Shadowed urban cellular | 3 – 5 |
| In building, LOS | 1.6 – 1.8 |
| Obstructed in building | 4 – 6 |
| Obstructed in factories | 2 – 3 |

**Log-normal shadowing:** at the same distance, actual loss varies with surroundings, so a zero-mean Gaussian term Xσ (in dB, σ typically 6–12 dB) is added:

```text
PL(d) = PL(d0) + 10n log(d/d0) + Xσ
```

**Example:** PL(d0 = 100 m) = 70 dB, n = 3.5 → at 1 km, PL = 70 + 35 × log 10 = 105 dB.

## Check yourself

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

1. The log-distance model is:
   A. PL(d) = PL(d0) + 10n log(d/d0)
   B. PL(d) = PL(d0) + 20 log(d0/d)
   C. PL(d) = PL(d0) · n · d
   D. PL(d) = PL(d0) − 10n log(d/d0)
   Answer: A. Average loss grows as the n-th power of distance, giving 10n log(d/d0) in dB. (Key point: Model equation)

2. Which exponent range is typical for urban cellular radio?
   A. 1.6 to 1.8
   B. 2.7 to 3.5
   C. 4 to 6
   D. Exactly 2
   Answer: B. Urban cellular is 2.7 to 3.5; free space is 2; shadowed urban is 3 to 5. (Key point: Path loss exponent)

3. What does Xσ represent?
   A. Fixed reference loss
   B. Free-space loss
   C. Zero-mean Gaussian variation (σ about 6 to 12 dB) at the same distance
   D. Doppler shift
   Answer: C. Surroundings differ at the same distance, so a zero-mean Gaussian term in dB is added. (Key point: Role of shadowing term)

4. PL(100 m) = 70 dB and n = 3.5. What is PL at 1 km?
   A. 70 dB
   B. 140 dB
   C. 87.5 dB
   D. 105 dB
   Answer: D. PL = 70 + 35 log 10 = 105 dB. (Key point: Example calculation)

## Related in Wireless Communication

- [Two-ray model advantages and path loss derivation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atwo-ray-advantages-path-loss) — Wireless Communication, chapter 3
- [Okumura model and median path loss numerical](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-model-median-path-loss) — Wireless Communication, chapter 3
- [Numerical: Okumura path loss, EIRP and received power](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-eirp-received-power) — Wireless Communication, chapter 3, asked 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
- [Numerical: Okumura median path loss (unity gain)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Aokumura-median-loss-unity-gain) — Wireless Communication, chapter 3, asked 3×
- [Two-ray model path and phase difference](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atwo-ray-path-phase-difference) — Wireless Communication, chapter 3, asked 2×

## Related in other subjects

- [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
- [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×
- [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