Chapter Summary

Chapter 13 Summary: Fading Channels

Key Points

• 1.

  The fading channel model $Y = HX + Z$ introduces a random multiplicative gain $H$, and the capacity depends critically on two factors: the fading time scale (ergodic vs. quasi-static) and the availability of CSI (CSIR only vs. full CSI).

• 2.

  Ergodic capacity with CSIR only is $C_{\text{erg}} = \mathbb{E}[\frac{1}{2}\log(1 + |H|^2 \text{SNR})]$, achieved by constant-power Gaussian transmission. The transmitter does not need channel knowledge — the law of large numbers averages over fading states.

• 3.

  Fading always reduces ergodic capacity relative to the AWGN channel at the same average SNR (Jensen's inequality applied to the concave $\log$ function), but the penalty is modest: at 20 dB Rayleigh SNR, only about 8% capacity loss.

• 4.

  Water-filling over fading states with full CSI yields $P^*(h) = (\nu - N/|h|^2)^+$, the same structure as parallel Gaussian channels. The gain over CSIR-only is most significant at low SNR and vanishes at high SNR.

• 5.

  Outage capacity governs quasi-static fading, where a single channel realization persists for the entire codeword. The $\epsilon$-outage capacity is determined by the $\epsilon$-quantile of the instantaneous capacity $\frac{1}{2}\log(1 + |H|^2 \text{SNR})$ and can be dramatically lower than the ergodic capacity.

• 6.

  Diversity is the key weapon against outage: $L$ independent fading branches reduce $P_{\text{out}}$ from $\text{SNR}^{-1}$ to $\text{SNR}^{-L}$, each additional branch providing an order of magnitude improvement per 10 dB.

• 7.

  MIMO capacity $C = \log\det(\mathbf{I} + \frac{1}{N}\mathbf{H}\mathbf{K}_x\mathbf{H}^{H})$ decomposes via SVD into $\min(n_t, n_r)$ parallel sub-channels. Telatar's scaling law shows ergodic capacity grows as $\min(n_t, n_r) \cdot \log(\text{SNR})$ at high SNR — the spatial multiplexing gain.

• 8.

  Isotropic input $\mathbf{K}_x = (P/n_t)\mathbf{I}$ is optimal without CSIT for i.i.d. Rayleigh MIMO, while water-filling over singular values is optimal with CSIT. The CSIT gain is most pronounced when the channel is ill-conditioned.