Prerequisites

Before You Begin

• Matrix inverses, trace, and determinant identities(Review ch01)

  Self-check: Can you apply the matrix inversion lemma $(\mathbf{A} + \mathbf{u}\mathbf{v}^H)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^H\mathbf{A}^{-1}}{1+\mathbf{v}^H\mathbf{A}^{-1}\mathbf{u}}$ and state the relationship between $\text{tr}(\mathbf{A}\mathbf{B})$ and $\text{tr}(\mathbf{B}\mathbf{A})$?

• Antenna arrays and beamforming fundamentals(Review ch07)

  Self-check: Can you express the array response vector $\mathbf{a}(\theta) = [1, e^{j2\pi d\sin\theta/\lambda}, \ldots]^T$ for a uniform linear array and explain how the beamwidth narrows as the number of elements $M$ increases?

• MIMO channel capacity and spatial multiplexing(Review ch15)

  Self-check: Can you write the ergodic MIMO capacity $C = \mathbb{E}\!\left[\log_2\det\!\left(\mathbf{I} + \frac{\text{SNR}}{N_t}\mathbf{H}\mathbf{H}^{H}\right)\right]$ and explain why the capacity grows linearly with $\min(N_t, N_r)$ at high SNR?

• Multi-user MIMO precoding and detection(Review ch17)

  Self-check: Can you describe ZF precoding in the downlink $\mathbf{W} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ and explain why it eliminates inter-user interference when the base station has full CSI?

• Law of large numbers and concentration inequalities(Review ch01)

  Self-check: Can you state the strong law of large numbers for i.i.d. random variables and explain why $\frac{1}{M}\sum_{m=1}^{M}|h_m|^2 \to 1$ almost surely when $h_m \sim \mathcal{CN}(0,1)$?