Prerequisites

Before You Begin

• Singular value decomposition (SVD) and matrix rank(Review ch01)

  Self-check: Can you decompose an $M \times N$ matrix $\mathbf{H}$ as $\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ and state the relationship between rank and the number of nonzero singular values?

• Maximum-likelihood detection and decision regions(Review ch09)

  Self-check: Can you write the ML decision rule for a linear model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ and explain why it minimises the probability of error?

• Diversity order and the Alamouti space-time code(Review ch10)

  Self-check: Can you construct the $2 \times 2$ Alamouti codeword matrix and compute its diversity order for a $2 \times N_r$ system?

• MIMO channel capacity and water-filling(Review ch15)

  Self-check: Can you state the MIMO capacity formula $C = \log_2 \det(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^{H})$ and explain the water-filling power allocation across eigenmodes?

• MMSE estimation and the matrix inversion lemma(Review ch01)

  Self-check: Can you derive the MMSE estimator $\hat{\mathbf{x}} = \mathbf{R}_{xy}\mathbf{R}_{yy}^{-1}\mathbf{y}$ and apply the Woodbury identity to simplify the resulting expression?