PYTHON · NUMPY

Channels, capacity, and the matched filter

Move the matching idea out of impedance and into information.

01Challenge

Build four receivers, measure the signal-to-noise ratio each achieves, and find out that the winner is the one shaped like the signal.

02Model

Shannon put a ceiling on every channel: C = B·log2(1 + SNR). Bandwidth and signal-to-noise are the only currencies, and you cannot print either. Given a channel, the job is to claw back every decibel toward that ceiling — and the tool is the matched filter, a receiver shaped like the signal it expects. It is optimal for a reason that should now sound familiar: among all receivers of a given energy, the one aligned with the signal extracts the most from it. That is conjugate matching, written in a different alphabet.

03Guided practice

1. The pulse. A shaped burst, sent through noise. 2. The receivers. A boxcar averager, a gate, a filter tuned to the wrong frequency, and one shaped exactly like the pulse. 3. The score. SNR for a unit-energy filter is (s·h)²/σ², and Cauchy–Schwarz says that is largest when h is parallel to s. The matched filter hits 2E/N0, the theoretical best, and nothing beats it.

04Feedback

PASS WHEN PASS when the matched filter beats every alternative and lands on 2E/N0.

If your run fails, check:

A cell errored — run it again.
The matched filter should win and hit the theoretical SNR.

05Retrieve & space

What two things set channel capacity? (bandwidth and signal-to-noise ratio).
What shape is the optimal receiver? (the shape of the signal it expects).
Why is the matched filter a matching problem? (it aligns the receiver with the signal — conjugate matching in the information domain).

06Mastery & project

You can explain why the matched filter is optimal and connect it to impedance matching.

The habit: to get the most out of a channel, shape yourself like what you are listening for.

← Impedance everywhere: the universal matching principle Bandwidth is a budget: Bode–Fano →

PYTHON · NUMPY

Channels, capacity, and the matched filter

Race four receivers against the same noisy pulse and see which one is optimal, and why.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you