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Capacity per Square Kilometre

The Shannon capacity $C = \log_2(1 + \text{SINR})$ measures spectral efficiency per link (bits/s/Hz). But a cellular operator cares about aggregate capacity per unit area — how many bits/s/Hz can be delivered over each km $^2$ of the service region. This area spectral efficiency (ASE) combines per-link spectral efficiency with the spatial density of cells. A dense network with many low-SINR links can deliver higher ASE than a sparse network with fewer high-SINR links, because the sum of many small rates exceeds a few large rates. Understanding ASE is essential for network planning and for evaluating the economic case for densification.

Definition:
Area Spectral Efficiency

The area spectral efficiency (ASE) of a cellular network is defined as:

$\mathcal{A} = \lambda \cdot \bar{R} \quad\text{(bits/s/Hz/km$^2$)}$

where $\lambda$ is the BS density (cells/km $^2$ ) and $\bar{R}$ is the average per-cell (or per-user) spectral efficiency:

$\bar{R} = \mathbb{E}[\log_2(1 + \text{SINR})] = \int_0^{\infty} \frac{p_c(2^t - 1)}{\ln 2}\,dt$

For the PPP model with $\alpha > 2$ , the ASE scales linearly with $\lambda$ because $p_c(\tau)$ — and hence $\bar{R}$ — is independent of $\lambda$ in the interference-limited regime.

The linear scaling $\mathcal{A} \propto \lambda$ is the fundamental promise of densification. However, it assumes the interference-limited regime. In practice, densification encounters diminishing returns when noise becomes non-negligible (ultra-dense regime), when backhaul becomes the bottleneck, or when propagation transitions from NLOS to LOS at short distances.

Definition:
SINR--Density Trade-Off

In an interference-limited network, increasing BS density $\lambda$ does not change the per-link SINR distribution (by the PPP invariance result), but increases the number of simultaneously active links. The trade-off is:

$\mathcal{A}(\lambda) = \lambda \cdot \bar{R}(\alpha)$

where $\bar{R}(\alpha)$ is determined solely by the propagation environment. For $\alpha = 4$ and Rayleigh fading:

$\bar{R} = \int_0^{\infty} \frac{1}{\ln 2}\cdot\frac{1}{1 + \sqrt{2^t - 1}\, \arctan(\sqrt{2^t - 1})}\,dt \approx 1.49 \;\text{bits/s/Hz}$

so the ASE is approximately $\mathcal{A} \approx 1.49\lambda$ bits/s/Hz/km $^2$ .

Theorem: Linear Scaling of ASE with Density

Under the PPP model with path-loss exponent $\alpha > 2$ , Rayleigh fading, and nearest-BS association in the interference-limited regime, the area spectral efficiency satisfies:

$\mathcal{A}(\lambda) = \lambda \cdot C(\alpha)$

where $C(\alpha) = \int_0^{\infty} \frac{p_c(2^t-1)}{\ln 2}\,dt$ is a constant that depends only on $\alpha$ . In particular:

$\alpha = 3$ : $C(3) \approx 1.05$ bits/s/Hz
$\alpha = 4$ : $C(4) \approx 1.49$ bits/s/Hz
$\alpha = 5$ : $C(5) \approx 1.78$ bits/s/Hz

The ASE increases without bound as $\lambda \to \infty$ , with each doubling of BS density doubling the area capacity.

Linear ASE scaling is the theoretical justification for ultra-dense networks. Each new BS adds an independent communication link with the same average rate $C(\alpha)$ , because the interference from all other BSs precisely offsets the distance gain. Higher path-loss exponents $\alpha$ yield larger $C(\alpha)$ because interference decays faster, improving per-link SINR. The practical limit to this scaling comes from LOS propagation transitions, backhaul constraints, and pilot contamination.

Proof

Per-link rate independence

From Theorem 21.3, the SINR distribution of a typical user is independent of $\lambda$ :

$\Pr[\text{SINR} > \tau] = p_c(\tau, \alpha)$

Therefore the ergodic rate per link is:

$\bar{R} = \mathbb{E}[\log_2(1+\text{SINR})] = C(\alpha)$

which is a function of $\alpha$ alone.

ASE computation

With $\lambda$ BSs per km $^2$ , each serving on average one user with ergodic rate $C(\alpha)$ , the aggregate rate per km $^2$ is:

$\mathcal{A} = \lambda \cdot C(\alpha)$

This is exact in the interference-limited regime where noise is negligible compared to aggregate PPP interference.

Behaviour with multiuser load

With $K$ users per cell, each user receives a fraction $1/K$ of the resources on average. The ASE becomes:

$\mathcal{A} = \lambda \cdot K \cdot \frac{C(\alpha)}{K} = \lambda \cdot C(\alpha)$

which is independent of $K$ in the interference-limited regime (assuming each active user faces the same SINR distribution). The ASE depends only on spatial density and propagation. $\blacksquare$

,

Area Spectral Efficiency vs. Density

Explore how the area spectral efficiency scales with network parameters. Adjust the path-loss exponent $\alpha$ to see how propagation conditions affect the per-link rate constant $C(\alpha)$ . Vary the number of users per cell $K$ to observe the impact of user load on per-user throughput (while ASE remains approximately constant). The plot shows both ASE and the per-user rate as functions of cell density.

Parameters

Path-loss exponent

\alpha

4

K

(users per cell)10

Example: Network Dimensioning with ASE

An operator must deliver 100 Gbps aggregate capacity over a 10 km $^2$ service area using 100 MHz bandwidth. The environment has $\alpha = 4$ .

(a) Compute the required ASE. (b) Using the PPP model result $C(4) \approx 1.49$ bits/s/Hz, determine the minimum BS density. (c) How many base stations are needed? (d) If the average ISD (inter-site distance) is $d = 1/\sqrt{\lambda}$ , what is the ISD?

Solution

Required ASE

(a) Total area capacity: $100 \times 10^9$ bits/s. Bandwidth: $100 \times 10^6$ Hz. Area: 10 km $^2$ .

$\mathcal{A}_{\text{req}} = \frac{100 \times 10^9}{100 \times 10^6 \times 10} = 100$ bits/s/Hz/km $^2$ .

Minimum BS density

(b) $\lambda_{\min} = \mathcal{A}_{\text{req}}/C(4) = 100/1.49 = 67.1$ BS/km $^2$ .

Total BSs

(c) $N_{\text{BS}} = \lambda \times A = 67.1 \times 10 = 671$ BSs.

Inter-site distance

(d) $d = 1/\sqrt{67.1} = 0.122$ km $= 122$ m.

This is an ultra-dense small-cell deployment with BSs approximately every 122 m — typical of 5G mmWave deployments in dense urban environments. $\blacksquare$

Quick Check

Under the PPP model in the interference-limited regime, what happens to the per-user throughput when the BS density $\lambda$ is doubled but the number of users per cell $K$ remains fixed?

Per-user throughput doubles

Per-user throughput stays approximately the same

Per-user throughput halves because of more interference

Per-user throughput increases by factor $\sqrt{2}$

Correction:

Per-user throughput stays approximately the same

The SINR distribution is invariant to $\lambda$ under the PPP model, so the ergodic rate per link $C(\alpha)$ is unchanged. With $K$ users sharing each cell, the per-user rate remains $C(\alpha)/K$ . The benefit of densification appears in the ASE (aggregate bits/s/Hz/km $^2$ ), not the individual user rate.

⚠️Engineering Note

Practical Limits to ASE Scaling in Ultra-Dense Networks

The linear ASE scaling $\mathcal{A} = \lambda C(\alpha)$ is a theoretical upper bound. Real ultra-dense deployments face several mechanisms that cause ASE saturation:

LOS/NLOS transition: At inter-site distances below the LOS breakpoint (50--200 m in urban), both serving and interfering links transition to LOS propagation with $\alpha_L \approx 2$ , severely degrading SINR.
Pilot contamination: The number of orthogonal pilot sequences is limited by the coherence time-bandwidth product. In dense networks, pilot reuse creates correlated interference that degrades channel estimation.
Synchronisation: Dense deployments with non-ideal backhaul have imperfect time synchronisation, creating inter-cell interference from non-aligned OFDM symbols.
Physical limitations: Below 10 m inter-site distance, near-field effects, coupling, and site acquisition costs make further densification impractical.
Empirical observation: 3GPP studies show ASE saturation at approximately $\lambda = 100$ -- $200$ BS/km $^2$ for sub-6 GHz bands in dense urban environments.

Practical Constraints

•
LOS breakpoint: 50-200 m (urban), 200-500 m (suburban)
•
ASE saturation: ~100-200 BS/km² for sub-6 GHz
•
Synchronisation tolerance: ±1.5 μs (LTE), ±3 μs (NR)

Key Takeaway

Network densification is the primary lever for increasing area capacity. The ASE scales linearly with BS density $\lambda$ in the interference-limited PPP model, because densification improves aggregate throughput without changing per-link SINR. However, practical limits — LOS propagation transitions, backhaul constraints, pilot contamination, and deployment costs — cause this scaling to saturate in ultra-dense regimes. The optimal densification strategy combines macro cells for coverage with small cells for capacity in a HetNet architecture.

Area Spectral Efficiency (ASE)

The aggregate spectral efficiency per unit area, $\mathcal{A} = \lambda \bar{R}$ (bits/s/Hz/km $^2$ ), where $\lambda$ is the BS density and $\bar{R}$ is the average per-link spectral efficiency. Under the PPP model, ASE scales linearly with $\lambda$ , providing the theoretical basis for network densification.

Area Spectral Efficiency