PE1-3.3 The General Gaussian Problem
You can solve binary detection under Gaussian assumptions for the same-covariance and same-mean cases, including diagonal and non-diagonal covariance and equal and unequal variances.
Gaussian assumptions are made everywhere because they are often true by the central limit theorem and because they make the algebra close in a form you can read. The same-covariance case gives a linear detector — a dot product and a threshold — while the same-mean case gives a quadratic one, and knowing which structure your problem has tells you the receiver's shape before any computation. Work the diagonal-covariance case first; the non-diagonal one is the same argument after a whitening transformation.
Work through these
Introduction to the general Gaussian problem
The case where everything is normally distributed, which is both tractable and a good model for thermal noise. The next four items are its sub-cases.
Binary detection with the same covariance: diagonal covariance matrix
The simplest version, where the noise components are independent. The detector turns out to be a distance comparison, which is a satisfying result.
Binary detection with the same covariance: non-diagonal covariance matrix
The correlated version, which requires whitening first. The whitening step is the idea to carry away, because it recurs throughout signal processing.
Same mean: uncorrelated signal components with equal variances
The case where the signals differ in variance rather than in mean, with equal variances across components. The detector is now quadratic rather than linear.
Same mean: uncorrelated signal components with unequal variances
The same with unequal variances, which weights the components differently. Comparing the two shows what the structure is responding to.
Linear versus quadratic detector structure
The summary of the topic: when the detector is a weighted sum and when it must square. It answers the question the four cases above were exploring.
Sign in to keep your progress.
Free resources
We haven't checked most of these for screen reader use yet.
Links last checked 30 Aug 2026.
Stuck here?
Ask a mentor. A real person answers, and they can see exactly which topic you're on. Usually within a couple of working days.
Checking your session…
Topics shown in module order.