Shannon Channel Capacity

Lesson notes

Overview

Maximum error-free data rate grows with bandwidth and the log of signal-to-noise ratio. This topic is rated advanced and is number 37 of 93 in Engineering. Deeper models, formal results and harder reasoning.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

Shannon’s theorem states that the maximum rate of error-free data on a noisy channel is C = B log₂(1 + S/N), where B is the bandwidth in hertz and S/N the signal-to-noise ratio. Doubling the bandwidth doubles the capacity, but to double it by raising power requires squaring the S/N. It tells engineers how close a modem or wireless link is to the theoretical limit, and good codes now approach it.

Reading the relationship

The expression "C = B log₂(1 + S/N)" is a compact summary of the idea. It uses the symbols C, B, S, N; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and a logarithm, which turns multiplication into addition and compresses large values. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.

Common pitfalls

Always ask what the safety margin is and which assumption would break the design first. Idealised models hide losses, noise and limits.

How it connects

It builds on “Pulse-Width Modulation”, the topic before it in Engineering. Once you are comfortable, continue with “The Diode I–V Curve”.

Why it matters

Engineers use this idea to design structures, machines and systems that are safe, efficient and reliable.