Prerequisites

Before You Begin

• Linear algebra: eigenvalue decomposition, positive semidefinite matrices(Review ch01)

  Self-check: Can you state and prove the spectral theorem for Hermitian matrices, and show that the set of PSD matrices forms a convex cone?

• Probability: joint distributions, conditional expectations, Markov chains(Review ch02)

  Self-check: Can you state the data processing inequality $I(X;Z) \leq I$ for the Markov chain $X \to Y \to Z$ and prove it using the chain rule for mutual information?

• Optimisation: convex sets, KKT conditions, Lagrange duality(Review ch03)

  Self-check: Can you formulate a convex optimisation problem, derive the KKT conditions, and solve the water-filling problem $\max \sum_i \log(1 + p_i \lambda_i)$ subject to $\sum p_i \leq P$?

• Information theory: entropy, mutual information, channel capacity, AEP(Review ch11)

  Self-check: Can you state the channel coding theorem, define the capacity $C = \max_{p(x)} I$ of a DMC, and sketch the achievability proof using random coding and joint typicality?

• MIMO capacity and precoding(Review ch17)

  Self-check: Can you derive the MIMO capacity $C = \max_{\mathbf{K}_x: \mathrm{tr}(\mathbf{K}_x) \leq P} \log\det(\mathbf{I} + \mathbf{H}\mathbf{K}_x\mathbf{H}^{H}/\sigma^2)$ and show that the optimal $\mathbf{K}_x$ follows from water-filling on the channel singular values?