---
title: "Derivation of co-channel reuse ratio"
subject: "Wireless Communication"
chapter: 2
key: "semester_7/wireless_communication:derive-cochannel-reuse-ratio"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/derive-cochannel-reuse-ratio
site: BCT Engineering
---

# Derivation of co-channel reuse ratio

Wireless Communication · Chapter 2: Cellular Mobile Communication Concepts

- Asked in: 2080 Bhadra, 4 marks
- Video: [Hexagonal Cell Geometry | Proof of Frequency Reuse Ratio Cellular Concept](https://www.youtube.com/watch?v=_i2pe569j5E) — ECIT Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/derive-cochannel-reuse-ratio
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-2

## Question

Derive the formula for co-channel reuse ratio: Q = √(3N).

## Short answer

With hexagonal cells, moving i and j cells along axes and using cosine rule gives D² = 3R²(i²+ij+j²). Cluster size N = i²+ij+j² from area scaling, so D = R√(3N), and co‑channel reuse ratio Q = D/R = √(3N).

## Answer

The **co-channel reuse ratio** Q = D/R, where D is the distance between centres of the nearest co-channel cells and R is the cell radius.

**Assumptions:** hexagonal cells of radius R (centre to vertex); distance between centres of two adjacent cells = √3 R.

**Locating a co-channel cell:** in a hexagonal layout, the nearest co-channel cell is found by moving *i* cells along a chain of hexagons, turning 60° anticlockwise, and moving *j* cells.

```text
                     * Co-channel cell
                    /
                   /  j cells
                  /   (after a 60° turn)
 A *-------------*
      i cells
 D = straight-line distance from A to the co-channel cell
```

The two paths have lengths i√3R and j√3R, and the angle between them (inside the triangle) is 120°. By the cosine rule:
```text
D² = (i√3R)² + (j√3R)² − 2(i√3R)(j√3R) cos 120°
   = 3R²i² + 3R²j² + 3R²ij        (cos 120° = −1/2)
   = 3R² (i² + ij + j²)
```

**Relating to cluster size:** the area of a hexagon is proportional to the square of the distance between centres of neighbouring hexagons. A cluster of N cells repeats with centre spacing D, while single cells repeat with spacing √3R, so
```text
N = (D / √3R)² = i² + ij + j²
```
Substituting:
```text
D² = 3R² N
D  = R √(3N)
Q  = D / R = √(3N)
```

| N | i, j | Q = √(3N) |
|---|---|---|
| 3 | 1, 1 | 3.00 |
| 4 | 2, 0 | 3.46 |
| 7 | 2, 1 | 4.58 |
| 12 | 2, 2 | 6.00 |

A larger Q means more separation between co-channel cells, so less co-channel interference but fewer channels per cell (less capacity).

## Check yourself

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

1. The co-channel reuse ratio Q is defined as:
   A. R/D
   B. D/R, the distance between nearest co-channel cell centres divided by cell radius
   C. N/R
   D. D x R
   Answer: B. Q = D/R. (Key point: Definition of Q)

2. For hexagonal cells of radius R, the distance between centres of adjacent cells is:
   A. 2R
   B. R
   C. 3R
   D. sqrt(3) R
   Answer: D. This is the basic spacing used in the derivation. (Key point: Adjacent cell spacing)

3. What does the cosine rule with 120 degrees give for D^2?
   A. 3R^2 (i^2 + ij + j^2)
   B. 3R^2 (i^2 - ij + j^2)
   C. R^2 (i + j)^2
   D. 3R^2 (i^2 + j^2)
   Answer: A. -2ab cos120 = +ab, so the cross term is +3R^2 ij. (Key point: Cosine rule)

4. Combining D^2 = 3R^2 N, the result for Q is:
   A. Q = 3N
   B. Q = sqrt(N/3)
   C. Q = sqrt(3N)
   D. Q = N/3
   Answer: C. D = R sqrt(3N), so Q = D/R = sqrt(3N). (Key point: Final result)

## Related in Wireless Communication

- [CCI vs ACI and co-channel reuse ratio proof](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acci-aci-prove-reuse-ratio) — Wireless Communication, chapter 2
- [Frequency reuse and co-channel reuse ratio](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afrequency-reuse-cochannel-reuse-ratio) — Wireless Communication, chapter 2
- [Numerical: co-channel distance for 7-cell reuse](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acochannel-distance-seven-cell) — Wireless Communication, chapter 2
- [CCI/ACI, reuse ratio proof and channels per cell](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acci-aci-reuse-duplex-channels) — Wireless Communication, chapter 2
- [Prove Q = √3N; optimal N for 15 dB S/I (n = 4)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Areuse-ratio-optimal-n-n4) — Wireless Communication, chapter 2
- [Frequency reuse and channel assignment strategies](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afrequency-reuse-channel-assignment) — Wireless Communication, chapter 2

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