Prerequisites

Before You Begin

• Matrix inversions, pseudo-inverses, and the Woodbury identity(Review ch01)

  Self-check: Can you compute the Moore-Penrose pseudo-inverse $\mathbf{H}^\dagger = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ and apply the Woodbury identity $(\mathbf{A} + \mathbf{U}\mathbf{B}\mathbf{V})^{-1}$ to simplify expressions involving rank-one updates?

• Gaussian channel capacity and mutual information(Review ch11)

  Self-check: Can you derive the capacity of the AWGN channel $C = \log_2(1 + \text{SNR})$ and explain the role of the water-filling power allocation in parallel Gaussian channels?

• MIMO capacity, SVD decomposition, and eigenmode transmission(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 decompose the MIMO channel into parallel eigenmodes via the SVD?

• Linear precoding (MRT, ZF, MMSE) and MMSE-SIC capacity achievement(Review ch16)

  Self-check: Can you design a ZF precoder $\mathbf{W} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ and explain why MMSE-SIC achieves the MIMO channel capacity regardless of decoding order?

• Dirty paper coding (Costa theorem) and BC-MAC duality concepts(Review ch16)

  Self-check: Can you state Costa's theorem — that known interference at the transmitter does not reduce capacity — and explain its role in achieving the broadcast channel capacity?