Spectral Efficiency | WRL-101

Spectrum is the one resource in wireless that cannot be manufactured. It is licensed, auctioned for billions, and shared by everyone. So the number engineers care about most is not raw data rate but rate per hertz — how much traffic you extract from each slice of band you were granted. That figure is spectral efficiency, and it is the single yardstick that lets you compare a 1930s AM station with a WiFi 7 link.

The Definition

Spectral efficiency η is simply bit rate divided by the bandwidth it occupies:

Spectral Efficiency

\eta = \frac{R_b}{B} \quad \left[\text{bits/s/Hz}\right]

Units are bits per second per hertz (b/s/Hz) — and note they are dimensionless: bits per second divided by cycles per second. A link at 2 b/s/Hz delivers 40 Mbps in a 20 MHz channel, or 2 Gbps in a gigahertz.

What the Last Two Lessons Bought Us

We can now compute η from the choices of Module 5 and M6-L2. Nyquist and the roll-off fix the symbol rate (R_s = B/(1+α)); the modulation order fixes the bits per symbol (log₂M); and the error-correcting code spends a fraction R_c of them on redundancy. Together:

Practical Spectral Efficiency

\eta = \frac{R_c \, \log_2 M}{1 + \alpha}

R_c is the code rate (e.g. 3/4 means three data bits per four transmitted). Every factor is a design lever: denser modulation raises log₂M, tighter pulse shaping shrinks α, weaker coding raises R_c — each at a cost in required SNR.

Worked values

Modulation	log₂M	Roll-off α	Code rate	η (b/s/Hz)
BPSK	1	0.25	1 (uncoded)	0.8
QPSK	2	0.25	1	1.6
16-QAM	4	0.25	1	3.2
64-QAM	6	0.1	1	5.45
64-QAM	6	0.1	3/4	4.09
256-QAM	8	0.1	1	7.27

Shannon’s Ceiling, Restated

Divide Shannon’s capacity (M6-L1) by the bandwidth and the ceiling turns into a ceiling on spectral efficiency — a limit that depends only on SNR:

The Shannon Bound on η

\eta \;\le\; \log_2\!\left(1 + \mathrm{SNR}\right)

At 10 dB SNR the ceiling is log₂(11) = 3.46 b/s/Hz; at 20 dB it is log₂(101) = 6.66; at 30 dB, log₂(1001) = 9.97. Roughly, every 3 dB of SNR buys about one more b/s/Hz once you are well above the noise.

Compare that with the table above and the picture snaps into focus. 64-QAM at 5.45 b/s/Hz needs an SNR whose Shannon ceiling exceeds 5.45 — about 17 dB at minimum, and in practice several dB more, because a real constellation with a real code does not reach the bound. The gap between a scheme’s η and Shannon’s η at the same SNR is exactly the implementation gap that LDPC and turbo codes closed to a fraction of a dB.

Real Systems

Published figures need care, because they are quoted in two very different flavours. Peak spectral efficiency assumes the best modulation, the lightest coding, and every antenna stream working — you standing beside the access point. Average (or cell-edge) efficiency is what a real deployment achieves across all its users, and it is several times lower.

System	Typical peak η	How it gets there
AM broadcast	~0.5 b/s/Hz	Analog; 10 kHz for one voice channel
GSM (2G)	~0.17 b/s/Hz	GMSK, heavy coding, 200 kHz carriers
LTE (4G), 2×2 MIMO	~7.5 b/s/Hz	64-QAM × 2 spatial streams
WiFi 6 (802.11ax)	~7.2 b/s/Hz per stream delivered (8.3 on-subcarrier)	1024-QAM, code rate 5/6, OFDM — M11-L1 constructs both
5G NR, massive MIMO	30+ b/s/Hz (cell)	256-QAM × many spatial layers

The 5G row looks like it breaks Shannon — it does not. Those figures come from spatial multiplexing: MIMO (Module 10) opens several parallel channels in the same band, and each obeys Shannon separately. The bound applies per channel; MIMO multiplies the number of channels.

How to Improve It

Higher-order modulation — more bits per symbol, needing ~6 dB more SNR per extra 2 bits (M5-L3).
Tighter pulse shaping — a smaller α recovers the excess bandwidth, at the cost of filter complexity and timing sensitivity.
Stronger, more efficient codes — LDPC and turbo let you run a higher code rate at the same error rate, or the same rate at lower SNR.
MIMO / spatial multiplexing — the big one: reuse the same hertz N times over with N antenna streams.
Adaptive modulation and coding — track the channel and always run the densest setting it supports, so average efficiency approaches peak.
Denser spatial reuse — smaller cells reuse the same spectrum more often per square kilometre (bits/s/Hz/km² is the honest metric for a network).

The overheads nobody advertises. A real link never delivers its theoretical η. OFDM spends the cyclic prefix (typically 7% of every symbol), guard subcarriers at the band edges, pilot tones for channel estimation, and control channels and headers before any user data moves. Between them these routinely take 20–30% off the number the modulation and coding suggest. When a datasheet and a measurement disagree, this is usually why.

Key Takeaways

η = R_b/B in b/s/Hz — the yardstick that matters, because spectrum is the resource you cannot make more of.
In practice η = R_c · log₂M / (1 + α): modulation order, code rate, and roll-off are the three levers.
Shannon’s bound becomes η ≤ log₂(1 + SNR) — 3.46 b/s/Hz at 10 dB, 6.66 at 20 dB, ~1 more b/s/Hz per 3 dB.
Always ask whether a quoted figure is peak or average; they differ by several times.
Figures above the single-channel bound come from MIMO spatial multiplexing, not from beating Shannon — the bound holds per channel.
Real links lose 20–30% to cyclic prefix, guard bands, pilots, and control overhead.

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