Summary

Chapter 14 Summary: Orthogonal Frequency Division Multiplexing (OFDM)

Key Points

• 1.

  Parallel channel decomposition. OFDM converts a frequency-selective channel into $N$ independent flat-fading sub-channels by dividing the wideband signal into narrow orthogonal subcarriers spaced at $\Delta f = 1/T_{\text{sym}}$. Each subcarrier requires only a single-tap equaliser: $\hat{X}[k] = Y[k]/H[k]$.

• 2.

  DFT-based implementation with cyclic prefix. The OFDM transmitter is an IDFT and the receiver is a DFT, both efficiently computed via the FFT. The cyclic prefix (of length $N_{\text{cp}} \geq L - 1$) converts linear channel convolution into circular convolution, eliminating both ISI and ICI. The CP overhead $\eta_{\text{CP}} = N_{\text{cp}}/(N + N_{\text{cp}})$ is typically 5--7% in LTE/NR.

• 3.

  Channel estimation via pilots. Known pilot symbols inserted into the time-frequency grid enable channel estimation. The LS estimator $\hat{H}[k] = Y_p[k]/X_p[k]$ is simple but noise-enhanced; the MMSE estimator exploits frequency-domain correlation for lower MSE at the cost of requiring channel statistics. Pilot spacing must satisfy the Nyquist condition in both time (Doppler) and frequency (delay spread).

• 4.

  PAPR is the major transmitter challenge. The OFDM signal has a maximum PAPR of $N$ (= $10\log_{10} N$ dB) when subcarriers add constructively. Practical PAPR at $\text{CCDF} = 10^{-3}$ is typically 10--12 dB for $N = 1024$. Reduction techniques include clipping (with filtering), selected mapping (SLM), and tone reservation.

• 5.

  Synchronisation is critical. A normalised fractional CFO $\epsilon_F$ creates an ICI floor of approximately $3/(\pi^2 \epsilon_F^2)$, limiting achievable SINR regardless of transmit power. Both time and frequency synchronisation are essential, with preamble-based methods (Schmidl-Cox) for acquisition and pilot-based tracking for residual correction.

• 6.

  AMC maximises throughput. OFDM's per-subcarrier flat-fading model enables adaptive modulation and coding: subcarriers with high SNR use high-order modulation (64/256-QAM) and high code rates, while weak subcarriers use QPSK or are nulled. CQI-based link adaptation is fundamental to LTE and 5G NR.

• 7.

  Generalisations address OFDM limitations. SC-FDMA (DFT-spread OFDM) reduces PAPR for uplink via DFT precoding; FBMC improves spectral containment by replacing rectangular pulses with well-localised filters; OTFS operates in the delay-Doppler domain for high-mobility resilience. CP-OFDM remains dominant due to MIMO compatibility and implementation maturity.