The Hard Ceiling
Module 5’s ladder kept asking: how many bits can a channel really carry? In 1948, Claude Shannon answered exactly — a single equation that no modulation or coding can ever beat.
Capacity in One Line
C is the max error-free rate (bits/s), B the bandwidth (Hz), and SNR the linear power ratio. Use a dB figure only after converting: SNR = 10^(dB/10).
Dial-up to WiFi
A phone line — 3.1 kHz at 30 dB — gives 3100 × log₂(1001) ≈ 31 kbps, exactly where dial-up modems stalled. A 20 MHz WiFi channel at 25 dB reaches ~166 Mbps per stream.
Bandwidth Beats Power
Bandwidth is linear — double B, double C. SNR is logarithmic — once it is high, a whole 3 dB of extra power buys only about one more bit per symbol. That is why every generation chases wider channels, not just louder ones.
Which Are You Short Of?
Bandwidth-limited (lots of SNR, little spectrum): reach for higher-order QAM. Power-limited (lots of spectrum, faint signal, like a deep-space probe): spread over more bandwidth instead. The formula tells you which lever to pull.
Why You Can’t Beat It
Shannon proved both halves: below C a code exists that makes errors vanish; above C, errors are unavoidable by any code. C is a true wall. What he didn’t give was the codes themselves — only proof they exist.
Capacity Calculator
The curve is C against SNR for your bandwidth. Note how it climbs fast at first, then bends into a slow logarithmic crawl.
−1.59 dB
Spend the widest bandwidth imaginable and there is still a floor: below ln 2 ≈ −1.59 dB of energy per bit, no code on any bandwidth can communicate reliably.
Codes That Get Close
For 50 years the ceiling sat far above real systems. Then modern codes closed the gap to a fraction of a dB:
- Turbo (1993) — 3G, 4G LTE
- LDPC (1962, revived 1990s) — WiFi, DVB-S2, 5G data
- Polar (2009) — first proven-capacity code, 5G control
What you learned
- C = B log₂(1 + SNR) is the hard ceiling on error-free bit rate
- SNR is a linear ratio; bandwidth is linear in C, SNR only logarithmic
- Bandwidth-limited → higher QAM; power-limited → more bandwidth
- Floor of −1.59 dB Eb/N₀; LDPC/turbo/polar press within a fraction of a dB