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The Geometry of Cellular Coverage

Cellular networks achieve their spectral efficiency through spatial reuse: the same frequencies are assigned to geographically separated cells so that co-channel interference remains tolerable. The hexagonal cell model, introduced by MacDonald (1979), provides the canonical deterministic framework for analysing this reuse. Although real deployments never produce perfect hexagons, the model yields closed-form expressions for the signal-to-interference ratio (SIR) and co-channel distance that have guided cellular planning for four decades. Understanding its assumptions — and its limitations — is essential before moving to the stochastic models of Section 21.2.

Definition:
Frequency Reuse Factor and Co-Channel Distance

In a hexagonal cellular layout, the frequency reuse factor (or cluster size) $N$ is the number of cells per reuse cluster. The total bandwidth $W$ is partitioned into $N$ disjoint groups, and each cell in a cluster is assigned one group of bandwidth $W/N$ . Valid cluster sizes satisfy $N = i^2 + ij + j^2$ for non-negative integers $i, j$ (yielding $N = 1, 3, 4, 7, 9, 12, \ldots$ ).

The co-channel reuse distance $D$ is the distance between the centres of two nearest cells using the same frequency group:

$D = R\sqrt{3N}$

where $R$ is the cell radius (centre to vertex of the hexagon). The co-channel reuse ratio is:

$q = \frac{D}{R} = \sqrt{3N}$

The constraint $N = i^2 + ij + j^2$ arises from the requirement that the hexagonal tiling be preserved under the reuse pattern. The values $N = 1, 3, 4, 7$ cover all practical cellular systems: GSM used $N = 4$ or $N = 7$ ; LTE and 5G NR typically use $N = 1$ (universal reuse) combined with ICIC or interference cancellation.

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Definition:
Signal-to-Interference Ratio in the Hexagonal Model

Consider a mobile at the cell edge (distance $R$ from its serving BS) in an interference-limited network with path-loss exponent $\alpha > 2$ . With $N_I$ first-tier co-channel interferers at approximate distance $D$ from the mobile, the worst-case downlink SIR is:

$\text{SIR}_{\text{edge}} = \frac{R^{-\alpha}}{N_I \cdot D^{-\alpha}} = \frac{1}{N_I}\left(\frac{D}{R}\right)^{\alpha} = \frac{(3N)^{\alpha/2}}{N_I}$

For a standard hexagonal layout, the number of first-tier co-channel interferers is $N_I = 6$ . Thus:

$\text{SIR}_{\text{edge}} = \frac{(3N)^{\alpha/2}}{6}$

This expression assumes all interferers are at exactly distance $D$ , which is an approximation — the actual distances range from $D - R$ to $D + R$ depending on the mobile's position. A more precise analysis accounts for the exact geometry of all six first-tier interferers, but the simplified formula captures the essential scaling behaviour.

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Theorem: Co-Channel SIR Scaling Law

In a hexagonal cellular network with reuse factor $N$ , cell radius $R$ , path-loss exponent $\alpha > 2$ , and 6 first-tier co-channel interferers, the worst-case cell-edge SIR satisfies:

$\text{SIR}_{\text{edge}} = \frac{(3N)^{\alpha/2}}{6}$

Equivalently, for a target SIR requirement $\text{SIR}_{\min}$ , the minimum reuse factor is:

$N_{\min} = \frac{1}{3}\left(6\,\text{SIR}_{\min}\right)^{2/\alpha}$

The cell-edge SIR increases as $N^{\alpha/2}$ — a power law in the cluster size.

Increasing $N$ pushes co-channel interferers farther away (since $D = R\sqrt{3N}$ ), and the path-loss exponent $\alpha$ amplifies this distance gain. Higher $\alpha$ means more aggressive signal attenuation with distance, so interference decays faster and a smaller $N$ suffices. In free space ( $\alpha = 2$ ), reuse is very costly because interference does not decay fast enough; in urban environments ( $\alpha = 4$ ), aggressive reuse ( $N = 1$ or $N = 3$ ) becomes feasible.

Proof

Path-loss model

Assume a simplified path-loss model $\ell(d) = d^{-\alpha}$ . The desired signal power from the serving BS at distance $R$ is $S = P \cdot R^{-\alpha}$ . Each co-channel interferer at distance $D$ contributes interference $I_i = P \cdot D^{-\alpha}$ (equal power assumed).

SIR computation

The total first-tier interference is:

$I_{\text{total}} = \sum_{i=1}^{6} P \cdot D_i^{-\alpha} \approx 6 \cdot P \cdot D^{-\alpha}$

where we approximate all six interferer distances as $D$ . The SIR is:

$\text{SIR} = \frac{P \cdot R^{-\alpha}}{6 \cdot P \cdot D^{-\alpha}} = \frac{1}{6}\left(\frac{D}{R}\right)^{\alpha}$

Substitution of co-channel distance

Substituting $D/R = \sqrt{3N}$ :

$\text{SIR}_{\text{edge}} = \frac{(\sqrt{3N})^{\alpha}}{6} = \frac{(3N)^{\alpha/2}}{6}$

Solving for $N$ given a target $\text{SIR}_{\min}$ :

$(3N)^{\alpha/2} = 6\,\text{SIR}_{\min} \quad\Rightarrow\quad N = \frac{1}{3}(6\,\text{SIR}_{\min})^{2/\alpha}$

which must be rounded up to the nearest valid cluster size. $\blacksquare$

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Hexagonal Cell Layout and Frequency Reuse

Animated construction of a hexagonal cell grid with

N = 7

frequency reuse. Shows the cluster colouring, co-channel distance

D = R\sqrt{3N}

, and the six first-tier co-channel interferers that determine the cell-edge SIR.

The hexagonal cell model with reuse factor

N = 7

. Co-channel cells (same colour) are separated by distance

D = R\sqrt{3N}

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Hexagonal Reuse and SIR Analysis

Explore how the frequency reuse factor $N$ and path-loss exponent $\alpha$ affect the cell-edge SIR in a hexagonal layout. The simulation computes the exact SIR by summing contributions from multiple tiers of co-channel interferers. Increase $N$ to observe how the SIR grows as $(3N)^{\alpha/2}/6$ . Higher path-loss exponents yield steeper SIR improvements with reuse, but also reduce the absolute received signal power.

Parameters

Reuse factor

N

3

Path-loss exponent

\alpha

3.5

Example: GSM Frequency Planning

A GSM operator has 25 MHz of bandwidth and requires a minimum cell-edge SIR of 12 dB (approximately 15.85 in linear scale) for acceptable voice quality. The urban environment has $\alpha = 3.5$ .

(a) Compute the minimum reuse factor $N$ . (b) Determine the bandwidth per cell. (c) If the cell radius is $R = 1$ km, what is the co-channel reuse distance? (d) Verify the cell-edge SIR with the chosen $N$ .

Solution

Minimum reuse factor

(a) $N_{\min} = \frac{1}{3}(6 \times 15.85)^{2/3.5} = \frac{1}{3}(95.1)^{0.571} = \frac{1}{3} \times 14.87 = 4.96$ .

Rounding up to the nearest valid cluster size: $N = 7$ (since $N = 4$ gives only 4.96 and would not satisfy the requirement with margin; $N = 7 = 2^2 + 2 \cdot 1 + 1^2$ is the next valid size).

Bandwidth per cell

(b) $W_{\text{cell}} = 25/7 \approx 3.57$ MHz per cell.

With 200 kHz GSM carriers: $\lfloor 3570/200 \rfloor = 17$ carriers per cell, supporting approximately 17 $\times$ 8 = 136 simultaneous voice calls (8 time slots per carrier).

Co-channel distance

(c) $D = R\sqrt{3 \times 7} = 1 \times \sqrt{21} = 4.58$ km.

Co-channel cells are separated by nearly 4.6 km.

SIR verification

(d) $\text{SIR}_{\text{edge}} = (3 \times 7)^{3.5/2}/6 = 21^{1.75}/6 = 195.3/6 = 32.6$ (15.1 dB).

This exceeds the 12 dB requirement by 3.1 dB, providing a comfortable margin for shadowing and fading. $\blacksquare$

Quick Check

In a hexagonal cellular network with reuse factor $N = 4$ and path-loss exponent $\alpha = 4$ , what is the worst-case cell-edge SIR (considering 6 first-tier co-channel interferers)?

$\text{SIR} = 12^2 / 6 = 24.0$ (13.8 dB)

$\text{SIR} = 4^2 / 6 = 2.67$ (4.3 dB)

$\text{SIR} = (3 \times 4)^{4} / 6 = 3456$ (35.4 dB)

$\text{SIR} = \sqrt{3 \times 4}/6 = 0.577$ ( $-2.4$ dB)

Correction:

\text{SIR} = 12^2 / 6 = 24.0

(13.8 dB)

$\text{SIR} = (3N)^{\alpha/2}/6 = (3 \times 4)^{4/2}/6 = 12^2/6 = 144/6 = 24$ (13.8 dB). The formula gives $(3N)^{\alpha/2}/6$ with $3N = 12$ and $\alpha/2 = 2$ .

Historical Note: MacDonald and the Hexagonal Cell

1979

The hexagonal cell model was formalised by V. H. MacDonald in his seminal 1979 Bell System Technical Journal paper "The Cellular Concept." MacDonald showed that regular hexagons provide the most efficient tessellation of the plane among shapes that approximate circular coverage areas (circles cannot tile the plane without gaps). The $D = R\sqrt{3N}$ relationship and the valid cluster sizes $N = i^2 + ij + j^2$ became the foundation of all first-generation (AMPS) and second-generation (GSM) cellular planning. While modern networks have moved beyond static frequency planning, the hexagonal model remains the standard pedagogical tool and continues to appear in 3GPP evaluation methodology documents.

Common Mistake: The Hexagonal Model Overestimates Real-World SIR

Mistake:

Using the hexagonal cell-edge SIR formula $\text{SIR} = (3N)^{\alpha/2}/6$ as a reliable predictor of deployed network performance.

Correction:

The hexagonal model assumes equal cell radii, uniform transmit power, omnidirectional path loss, and exact equidistant interferers. Real deployments have irregular site placement, terrain effects, building shadowing, and antenna tilt variations that can reduce cell-edge SIR by 5--10 dB compared to the hexagonal prediction. Use the hexagonal model for first-order dimensioning and reuse planning, but validate with system-level simulations or stochastic geometry analysis for deployment-level predictions.

⚠️Engineering Note

Practical Frequency Planning Constraints

The hexagonal model yields the minimum reuse factor $N$ for a target SIR, but practical frequency planning must also account for:

Adjacent channel interference (ACI): GSM carriers within 200 kHz spacing suffer 20--30 dB less isolation than co-channel interference. Frequency plans must ensure adjacent channels are assigned to non-adjacent cells.
Site constraints: Regulatory restrictions, lease costs, and physical obstructions prevent ideal hexagonal site placement. Real deployments have 20--50% site location variance from the ideal grid.
Terrain and propagation asymmetry: Cells in hilly terrain may have radii varying 2:1, breaking the equal- $R$ assumption. Propagation asymmetry creates unequal interference contributions from the six first-tier interferers.
Modern approach: LTE and 5G NR use $N = 1$ (universal reuse) with inter-cell interference coordination (ICIC) and successive interference cancellation (SIC) instead of static frequency planning. This eliminates the bandwidth loss from $N > 1$ reuse at the cost of requiring sophisticated interference management.

Practical Constraints

•
GSM: ACI requires 400+ kHz separation between adjacent cells
•
Site placement variance: 20-50% from ideal hexagonal grid
•
LTE/5G NR: universal reuse ( $N = 1$ ) with ICIC

Frequency Reuse Factor

The number of cells $N$ in a reuse cluster. The available bandwidth is divided into $N$ groups, each assigned to one cell in the cluster. Valid hexagonal cluster sizes are $N = i^2 + ij + j^2$ for non-negative integers $i, j$ . The co-channel reuse distance is $D = R\sqrt{3N}$ .

Co-Channel Reuse Distance

The minimum distance $D$ between the centres of two cells using the same frequency group. In a hexagonal layout with reuse factor $N$ and cell radius $R$ : $D = R\sqrt{3N}$ . Larger $D$ reduces co-channel interference but requires a larger cluster size, reducing per-cell bandwidth.

Related: Frequency Reuse Factor

Hexagonal Cell Model