Summary

Chapter 11 Summary: Information Theory for Wireless Channels

Key Points

• 1.

  Entropy and mutual information provide the mathematical foundation for quantifying information. Channel capacity $C = \max_{p(x)} I$ is the maximum rate for reliable communication, and Shannon's coding theorem guarantees that capacity-approaching codes exist.

• 2.

  AWGN capacity $C = B\log_2(1 + \text{SNR})$ reveals the fundamental bandwidth-power trade-off. The Shannon limit $E_b/N_0 = \ln 2 = -1.59$ dB is the absolute minimum energy per bit for reliable communication, approached in the infinite-bandwidth regime.

• 3.

  Flat-fading channels require distinguishing ergodic capacity (long codewords spanning many fading realisations, $C_{\text{erg}} = E[\log_2(1+\gamma)]$) from outage capacity (delay-limited systems where the channel is quasi-static). Water-filling in time with full CSI allocates more power to stronger channel states.

• 4.

  Frequency-selective channels decompose into parallel independent sub-channels via OFDM. Water-filling across frequency allocates power optimally: more to strong sub-carriers, none to deeply faded ones. The total capacity is the sum of individual sub-channel capacities.

• 5.

  Multiple antennas increase capacity through array gain. SIMO with $N_r$ antennas achieves $C = \log_2(1 + \|\mathbf{h}\|^2 \cdot \text{SNR})$ via MRC. MISO with CSIT achieves the same through beamforming, but without CSIT provides only diversity gain. MIMO (Chapter 15) offers a linear capacity scaling with $\min(N_t, N_r)$.

• 6.

  Practical coding (turbo, LDPC, polar) closes the gap to capacity from 8-10 dB (uncoded) to under 1 dB. Adaptive modulation and coding (AMC) provides a finite-rate approximation to water-filling, achieving 85-95% of ergodic capacity in 4G/5G systems.