PE2-2.4 Communication Theory & Detection
You can represent low pass and band pass signals in signal spaces, analyse PAM, and apply maximum likelihood detection under AWGN with the union bound to compute symbol error rates and choose constellations.
Signal space is the abstraction that makes modulation geometry: each symbol is a point, noise is a cloud around it, and error probability is about distances between points. Once you see it that way, constellation design becomes an obvious optimisation — spread the points as far apart as the power budget allows. The union bound is the working engineer's tool here: an upper bound that is loose at low SNR and tight where it matters.
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Introduction to communications
The frame for the second half of the course: getting information across a channel that corrupts it. It reconnects to the communication subject with the simulation tools now available.
Low pass and band-pass signals
The two signal classes and the relationship between them. The equivalent baseband representation is what makes simulating a radio system practical.
Signal spaces
Treating signals as points in a space, which turns detection into geometry. It is the single most useful abstraction in this half of the course.
Pulse amplitude modulation (PAM)
The simplest digital scheme, used as the worked example throughout. Everything about detection is developed on it before generalising.
Detection and the effects of AWGN
What the channel does, and the standard noise model. Additive white Gaussian noise is an assumption, and knowing when it fails is part of understanding it.
Maximum likelihood detection
The optimal decision rule, which in the geometry above is nearest neighbour. It connects the probability from the first half to the signal spaces of this one.
The union bound
A bound on error probability that is loose and computable, which is why it is used. Exact expressions are rare and this is usually close enough.
Symbol error rates
Turning the bound into the number a system is specified by. Simulating it and comparing against the theory is the exercise this topic exists for.
Choosing constellations
How the constellation shape affects performance, which is the design decision the whole topic has been building towards.
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