Prerequisites & Notation

Before You Begin

• MIMO outage probability and ergodic capacity(Review ch10)

  Self-check: Can you write the outage probability $P_{\rm out}(R)$ for an $n_t \times n_r$ i.i.d. Rayleigh channel at target rate $R$, and distinguish it operationally from the ergodic capacity $\bar C = \mathbb{E}_{\mathbf{H}} \log\det(\mathbf{I} + \tfrac{\text{SNR}}{n_t}\mathbf{H}\mathbf{H}^{H})$?

• Wishart-eigenvalue distribution of $\mathbf{H}\mathbf{H}^{H}$(Review ch10)

  Self-check: Can you state the joint density of the nonzero eigenvalues of $\mathbf{H}\mathbf{H}^{H}$ when $\mathbf{H}$ is $n_r \times n_t$ with i.i.d. $\mathcal{CN}(0, 1)$ entries? And identify the Vandermonde factor $\prod_{i<j} (\lambda_i - \lambda_j)^2$ that drives the DMT exponent?

• Rank and determinant criteria for space-time codes(Review ch11)

  Self-check: Can you state the rank criterion ($\mathrm{rank}(\boldsymbol{\Delta}) \ge r$ implies diversity $r n_r$) and the determinant criterion ($\det(\boldsymbol{\Delta}\boldsymbol{\Delta}^H)$ controls coding gain) for the codeword error matrix $\boldsymbol{\Delta} = \ntn{X} - \hat{\ntn{X}}$?

• Alamouti scheme and V-BLAST (ZF and ML) receivers(Review ch11)

  Self-check: Can you compute the diversity order of Alamouti on an $n_r$-receive channel ($2 n_r$), and of zero-forcing V-BLAST on an $n_t \times n_r$ channel with $n_r \ge n_t$ ($n_r - n_t + 1$)? And explain why the latter is so much smaller than $n_t n_r$?

• Gallager random-coding exponent and exponential equality(Review ch13)

  Self-check: Can you state $f(\text{SNR}) \doteq \text{SNR}^{a}$ iff $\lim \log f / \log \text{SNR} = a$, and describe why additive $o(\log\text{SNR})$ terms — constants, polylog prefactors — are invisible to this notation?

• Outage exponent of a scalar Rayleigh channel(Review ch11)

  Self-check: Can you show that for a scalar Rayleigh channel at fixed rate $R$, $P_{\rm out}(R) \doteq \text{SNR}^{-1}$ (diversity order 1)? And extend this to Alamouti on an $n_r$-receive channel ($\doteq \text{SNR}^{-2 n_r}$)?