Chapter Summary

Chapter 4 Summary: Achievable Rate Analysis

Key Points

• 1.

  The use-and-then-forget (UatF) bound provides a tractable lower bound on the achievable rate by treating the channel estimate as deterministic and absorbing estimation error into worst-case Gaussian noise, yielding closed-form SINR expressions that become tight under channel hardening.

• 2.

  MRC combining achieves SINR that grows linearly with $N_t$ but is interference-limited: the denominator $\sum_j {P_t}_{j} \beta_{j} + \sigma^2$ is independent of $N_t$, and multi-user interference does not vanish even with infinitely many antennas.

• 3.

  ZF combining eliminates inter-user interference (from estimated channels) at the cost of $K$ spatial degrees of freedom, yielding an effective array gain of $N_t - K$ and a noise-limited (rather than interference-limited) rate expression.

• 4.

  MMSE combining optimally balances noise enhancement and interference suppression, achieving the highest rate among linear receivers; its large-system behavior is characterized via random matrix theory (Stieltjes transform of the Marchenko-Pastur distribution).

• 5.

  Sum rate scaling: MRC achieves $\Theta(N_t/\ln N_t)$, while ZF and MMSE achieve $\Theta(N_t)$ with optimal user loading, confirming that massive MIMO delivers spatial multiplexing gain proportional to the number of antennas.

• 6.

  Power scaling: all three schemes support transmit power reduction as $1/N_t$ (exponent $\alpha = 1$) with no rate loss in the single-cell case — the array gain perfectly compensates the reduced power, enabling green communication.

• 7.

  Pilot contamination from other cells creates a finite rate ceiling that does not vanish with $N_t$ and reduces the power scaling exponent to $\alpha = 1/2$, motivating the pilot decontamination techniques of Chapter 3.