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Directional Antennas and Capacity Gain

In a practical cellular deployment, each cell site uses directional antennas that divide the 360-degree coverage area into sectors (typically 3 sectors of 120 degrees each). Sectorisation provides a dual benefit: it increases the capacity of each site (more simultaneous transmissions per location) and reduces co-channel interference (each sector antenna illuminates only a fraction of the interference field). Understanding the SIR improvement from sectorisation — and why the gain is less than the factor-of- $S$ reduction in interferers that a naive analysis would predict — requires careful accounting of the antenna radiation pattern and the geometry of interference.

Definition:
Sectorisation

Sectorisation divides each cell site into $S$ sectors using directional antennas, each with beamwidth $\theta = 360°/S$ . Each sector operates as an independent cell with its own frequency resources. The key parameters are:

Number of sectors $S$ : typically 3 (tri-sector) or 6.
Antenna beamwidth $\theta_{\text{3dB}}$ : the half-power (3 dB) beamwidth, typically $65°$ for a 3-sector configuration.
Front-to-back ratio (FBR): the ratio of the main-lobe gain to the back-lobe gain, typically 20--30 dB.

With $S$ -sector sites and reuse factor $N$ , the system has $S \cdot N$ effective channels. Each sector has bandwidth $W/N$ (the reuse is at the site level, not the sector level in standard configurations).

The effective SIR with sectorisation is:

$\text{SIR}_{\text{sector}} = \frac{G_{\text{main}} \cdot R^{-\alpha}} {\sum_{i} G(\phi_i) \cdot D_i^{-\alpha}}$

where $G_{\text{main}}$ is the main-lobe antenna gain and $G(\phi_i)$ is the gain in the direction of the $i$ -th interferer.

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Definition:
Sectorisation Gain

The sectorisation gain $G_S$ is the ratio of the SIR with $S$ sectors to the SIR with omnidirectional antennas:

$G_S = \frac{\text{SIR}_{S\text{-sector}}}{\text{SIR}_{\text{omni}}}$

For an ideal sectorised antenna (perfect $\theta = 360°/S$ beamwidth, zero gain outside the main beam), the number of effective first-tier interferers seen by a sector is reduced from 6 to approximately $6/S$ , giving:

$G_S^{\text{ideal}} = S$

In practice, side lobes and back lobes reduce this gain. For realistic 3-sector antennas, the typical sectorisation gain is $G_3 \approx 2.0$ -- $2.5$ (rather than the ideal factor of 3).

Theorem: SIR Improvement from Sectorisation

In a hexagonal cellular network with reuse factor $N$ and $S$ -sector sites, the worst-case cell-edge SIR is:

$\text{SIR}_{\text{sector}} = \frac{(3N)^{\alpha/2}}{N_I^{\text{eff}}}$

where $N_I^{\text{eff}}$ is the effective number of co-channel interferers after accounting for antenna directivity:

$N_I^{\text{eff}} = \sum_{i=1}^{6} \frac{G(\phi_i)}{G_{\text{main}}} \cdot \left(\frac{D}{D_i}\right)^{\alpha}$

For an ideal 3-sector antenna with perfect 120-degree beamwidth and considering exact interferer geometry:

$N_I^{\text{eff}} = 2 \quad\text{(ideal tri-sector)}$

yielding $G_3 = 6/2 = 3$ . For a realistic antenna pattern with 20 dB front-to-back ratio:

$N_I^{\text{eff}} \approx 2 + 4 \times 10^{-2} = 2.04$

yielding $G_3 \approx 2.94$ (4.7 dB), reduced to $\approx 2.5$ (4.0 dB) when second-tier interferers and imperfect patterns are included.

A tri-sector antenna "sees" only 2 of the 6 first-tier co-channel interferers in its main lobe (the other 4 are attenuated by the front-to-back ratio). The ideal gain of 3 comes from the 6/2 ratio. In practice, the non-zero back-lobe and side-lobe levels add residual interference from the 4 "hidden" interferers, reducing the gain to about 2.5.

Proof

Interferer geometry with tri-sector antennas

In a hexagonal layout with reuse factor $N$ , the 6 first-tier co-channel interferers are at angular positions separated by 60 degrees. With a 120-degree sector:

2 interferers fall within the main beam ( $|\phi_i| < 60°$ )
4 interferers fall in the side/back lobes ( $|\phi_i| \geq 60°$ )

Effective interference with ideal antenna

For an ideal antenna with gain $G = G_{\text{main}}$ for $|\phi| < 60°$ and $G = 0$ outside:

$N_I^{\text{eff}} = 2 \cdot \frac{G_{\text{main}}}{G_{\text{main}}} + 4 \cdot 0 = 2$

SIR gain: $G_3 = 6/2 = 3.0$ (4.77 dB).

Realistic antenna correction

With front-to-back ratio FBR $= 20$ dB ( $G_{\text{back}}/G_{\text{main}} = 0.01$ ):

$N_I^{\text{eff}} = 2 + 4 \times 0.01 = 2.04$

SIR gain: $G_3 = 6/2.04 = 2.94$ (4.69 dB).

Including second-tier interferers and realistic antenna roll-off typically reduces this to $G_3 \approx 2.5$ (4.0 dB) in system-level simulations. $\blacksquare$

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Sectorisation SIR Gain

Compare the SIR performance of omnidirectional and sectorised cell sites. The simulation shows the SIR distribution for users in a hexagonal network with different reuse factors, comparing 1-sector (omni) and 3-sector configurations. Observe how sectorisation provides approximately 2--3 times SIR improvement, with the exact gain depending on the path-loss exponent and antenna pattern.

Parameters

Path-loss exponent

\alpha

4

Reuse factor

N

3

Example: Capacity Gain from Sectorisation

A GSM operator uses $N = 7$ reuse with omnidirectional antennas and $\alpha = 4$ . The operator considers upgrading to tri-sector sites with realistic antenna gain $G_3 = 2.5$ .

(a) Compute the SIR with omnidirectional antennas. (b) Compute the SIR with tri-sector antennas. (c) Can the operator reduce the reuse factor to $N = 4$ with tri-sector antennas and still meet 12 dB SIR? (d) Compute the capacity gain (in terms of channels per site).

Solution

Omni SIR

(a) $\text{SIR}_{\text{omni}} = (3 \times 7)^{4/2}/6 = 21^2/6 = 441/6 = 73.5$ (18.7 dB).

Tri-sector SIR

(b) $\text{SIR}_{\text{sector}} = G_3 \times \text{SIR}_{\text{omni}} = 2.5 \times 73.5 = 183.8$ (22.6 dB).

The 4.0 dB sectorisation gain raises SIR from 18.7 to 22.6 dB.

Reduced reuse feasibility

(c) With $N = 4$ and tri-sector: $\text{SIR} = 2.5 \times (12)^2/6 = 2.5 \times 24 = 60.0$ (17.8 dB).

17.8 dB $>$ 12 dB, so $N = 4$ with sectorisation is feasible with 5.8 dB margin.

Capacity gain

(d) With $N = 7$ omni: $W/7$ bandwidth per cell, 1 cell/site. Channels per site $\propto 1/7$ .

With $N = 4$ tri-sector: $W/4$ bandwidth per sector, 3 sectors/site. Channels per site $\propto 3/4$ .

Capacity gain: $(3/4)/(1/7) = 21/4 = 5.25\times$ .

Sectorisation plus reduced reuse yields a 5.25 $\times$ capacity improvement. $\blacksquare$

Quick Check

Why is the practical sectorisation gain for 3-sector cells typically 2.5 rather than the ideal value of 3?

Because the antenna beamwidth is wider than the ideal 120 degrees

Because the antenna has non-zero side lobes and back lobes that admit residual interference from out-of-beam interferers

Because only 2 of the 6 interferers are in the same sector

Because sectorisation reduces the reuse distance

Correction:

Because the antenna has non-zero side lobes and back lobes that admit residual interference from out-of-beam interferers

An ideal sector antenna perfectly blocks the 4 out-of-beam interferers, leaving only 2. Real antennas have finite front-to-back ratio (20--30 dB) and side lobes, so the 4 "blocked" interferers contribute residual interference that reduces the gain from 3 to approximately 2.5.

Sectorisation

The practice of dividing each cell site into $S$ sectors using directional antennas, each covering $360°/S$ of azimuth. Sectorisation reduces co-channel interference by attenuating out-of-beam interferers and increases site capacity by a factor approximately equal to $S$ (reduced by practical antenna imperfections to $\sim 0.8S$ in practice).

Signal-to-Interference Ratio (Hexagonal Model)

The ratio of desired signal power to co-channel interference power at the cell edge. In the hexagonal model with reuse factor $N$ and 6 first-tier interferers: $\text{SIR} = (3N)^{\alpha/2}/6$ . Sectorisation improves this by reducing the effective number of interferers.

Sectorisation

Directional Antennas and Capacity Gain

Definition: Sectorisation

Definition: Sectorisation Gain

Theorem: SIR Improvement from Sectorisation

Interferer geometry with tri-sector antennas

Effective interference with ideal antenna

Realistic antenna correction

Sectorisation SIR Gain

Parameters

Example: Capacity Gain from Sectorisation

Omni SIR

Tri-sector SIR

Reduced reuse feasibility

Capacity gain

Quick Check

Sectorisation

Signal-to-Interference Ratio (Hexagonal Model)

Definition:
Sectorisation

Definition:
Sectorisation Gain