Reuse distance and spectral efficiency

Spectral efficiency of a cellular system is the traffic (or number of channels) carried per unit bandwidth per unit area, e.g. channels/MHz/km². It depends directly on how often the same frequency is reused in space.

Relation between reuse distance and cluster size:

D = R √(3N)     →     N = (D/R)² / 3

A smaller reuse distance D (for a given R) means a smaller cluster size N.

Effect on capacity: with S total duplex channels,

Channels per cell  k = S / N
Total capacity     C = M × S = (number of cells / N) × S

So C ∝ 1/N ∝ 1/D². If D is minimised:

1. Each cell gets more channels (S/N larger).
2. The same frequency is reused more times over the same area (more clusters M).
3. More users are served per MHz per km², so spectral efficiency is maximised.

Example: with S = 420 channels and 70 cells:

Limit: D cannot be reduced indefinitely, because co-channel interference rises:

S/I = (√(3N))ⁿ / i₀ = (D/R)ⁿ / i₀

Therefore the minimum reuse distance that still meets the required S/I (e.g. 18 dB for AMPS, giving N = 7 for n = 4) gives the maximum spectral efficiency. Sectoring, power control and better modulation allow a smaller D and hence higher efficiency.