Bits per Hertz
Spectrum is the one thing you cannot manufacture — it is licensed and auctioned for billions. So the number that matters is not raw rate but rate per hertz.
Spectral Efficiency
A link at 2 b/s/Hz gives 40 Mbps in a 20 MHz channel — or 2 Gbps in a gigahertz. One number, comparable across every system ever built.
Three Levers
Modulation order (log₂M), code rate Rc, and roll-off α. Denser modulation, lighter coding, tighter shaping — each buys η and each costs SNR.
Putting Numbers On It
Uncoded, with α = 0.25: QPSK gives 2/1.25 = 1.6. At α = 0.1, 64-QAM gives 6/1.1 = 5.45 — and a 3/4 code drops that to 4.09.
The Ceiling on η
Divide capacity by B and the ceiling depends on SNR alone: 3.46 b/s/Hz at 10 dB, 6.66 at 20 dB. Roughly one more b/s/Hz per 3 dB.
Mind the Gap
The curve is Shannon’s ceiling; the bars are real schemes. Slide the SNR and see which ones the channel can actually support — and how much headroom is left.
A Century of Progress
- AM broadcast — ~0.5 b/s/Hz, analog
- GSM (2G) — ~0.17 b/s/Hz
- LTE 2×2 MIMO — ~7.5 b/s/Hz peak
- WiFi 6 — ~8 b/s/Hz per stream (1024-QAM)
- 5G massive MIMO — 30+ b/s/Hz per cell
How 5G Gets 30+
Those figures come from spatial multiplexing: MIMO opens several parallel channels in the same band, and each obeys Shannon separately. The bound applies per channel — MIMO multiplies the channels.
Overheads Nobody Advertises
No real link delivers its theoretical η. The cyclic prefix (~7% of every OFDM symbol), guard subcarriers, pilot tones, and control headers together take 20–30% off the top. Also: always ask whether a quoted figure is peak or average.
What you learned
- η = Rb/B in b/s/Hz — the yardstick, because spectrum is finite
- η = Rc·log₂M/(1+α): modulation, coding, and roll-off
- Ceiling η ≤ log₂(1+SNR) — about +1 b/s/Hz per 3 dB
- MIMO multiplies channels; overheads cost 20–30%; peak ≠ average