PE2-2.4 Communication Theory & Detection

What this is and why it exists

The second half of the course returns to communication, now with simulation available. One abstraction organises all of it.

Treat each transmitted symbol as a point in a space. Noise becomes a cloud around each point, and the probability of error becomes a question about distances. Modulation turns into geometry.

The vocabulary

• Baseband signal — a signal occupying frequencies near zero.
• Bandpass signal — one occupying a band around a carrier.
• Equivalent baseband representation — a complex low frequency description of a bandpass signal.
• Signal space — a space in which each candidate signal is a point.
• Constellation — the arrangement of those points.
• PAM — pulse amplitude modulation. The amplitude of each pulse carries the symbol.
• AWGN — additive white Gaussian noise. The standard channel model.
• Maximum likelihood detection — choosing the candidate that makes the observation most probable.
• Union bound — an upper bound on error probability, computed from pairwise distances.
• Symbol error rate — the probability that a whole symbol is decided wrongly.

The mental model

The equivalent baseband representation is the practical starting point. Simulating a signal at its actual carrier frequency would need samples fast enough for that carrier, which is enormously wasteful. Instead the carrier is stripped away and the signal is described by a complex low frequency signal carrying the same information. Everything about the modulation survives; only the carrier is set aside. That is what makes simulating a radio system possible on an ordinary computer.

Then the central abstraction. Pick a set of basis signals and every candidate transmission becomes a list of coordinates, which is a point. The arrangement of those points is the constellation. Amplitude modulation of pulses gives points along a line, evenly spaced.

Noise becomes a displacement of the received point away from the transmitted one. Under the standard model the noise is added, its statistics are Gaussian, and its power is spread evenly across frequency. Those three properties make the displacement equally likely in every direction and more likely to be small than large.

Maximum likelihood detection then has an immediate geometric reading. Choose the candidate that makes what you received most probable. Because the noise cloud shrinks with distance in every direction equally, that is exactly the nearest candidate point. The optimal receiver is a nearest neighbour search.

Error probability follows. A mistake happens when the noise carries the received point across the boundary to another candidate's region. That is more likely for candidates that are close and less likely for those far away. So the error rate is governed by the distances between points, and mostly by the smallest of them.

The union bound is the working engineer's tool. Computing the exact error probability means integrating a Gaussian over an awkwardly shaped region, and for most constellations no closed form exists. Instead, note one useful inequality. The probability of confusing the transmitted symbol with any other is at most the sum of the pairwise probabilities. That sum is straightforward, because each term depends only on the distance between two points.

It is an upper bound, and it counts some regions more than once, so it is loose. It is loose where the signal-to-noise ratio is low and errors are common. It is tight where the ratio is high and errors are rare, and that is where systems operate, so it is usually close enough.

Constellation design becomes an obvious optimisation once the geometry is clear. Errors are set by the smallest distance between points, and the average power is set by how far the points are from the origin. So the design problem is to spread the points as far apart as possible for a given average power. That is why constellations look the way they do, and why some shapes beat others at the same rate.

The exercise the topic exists for is to simulate the error rate and plot it against the theoretical curve. Agreement means both are right. Disagreement means one of them is wrong, and you have two independent things to check against each other.

What you should now be able to explain or do

Say why an equivalent baseband representation makes simulation practical. Place a modulation scheme in signal space and read its constellation. Explain the three properties of the standard noise model. Show that maximum likelihood detection is nearest neighbour detection. Derive a union bound and say where it is loose, and state the constellation design problem.

Check yourself