Summary

Chapter 8 Summary: Digital Modulation

Key Points

• 1.

  Signal-space representation reduces optimal detection of $M$ waveforms in AWGN to a nearest-neighbour search in $\mathbb{R}^N$ ($N \leq M$). The Gram-Schmidt procedure constructs an orthonormal basis, and the sufficient statistics $\mathbf{r} = \mathbf{s}_m + \mathbf{w}$ capture all information relevant to detection.

• 2.

  QAM and PSK constellations are the workhorses of modern digital communications. QPSK achieves the same BER as BPSK at double the spectral efficiency. Gray mapping ensures that the most likely symbol errors cause only one bit error, giving $\text{BER} \approx \text{SER}/\log_2 M$ at high SNR.

• 3.

  Constant-envelope modulations (MSK, GMSK) encode information in frequency or phase and are robust to nonlinear amplification. MSK is the minimum-index ($h = 0.5$) continuous-phase FSK scheme with BPSK-equivalent BER; GMSK adds Gaussian pre-filtering for spectral compactness (used in GSM with $BT = 0.3$).

• 4.

  Nyquist pulse shaping prevents ISI: the first Nyquist criterion requires the folded spectrum $\sum_k P(f - k/T_s) = T_s$. The raised-cosine filter satisfies this with roll-off factor $\alpha$, occupying bandwidth $W = (1+\alpha)R_s/2$.

• 5.

  Root-raised-cosine (RRC) matched filtering splits the Nyquist pulse equally between transmitter and receiver: TX RRC $*$ RX RRC $=$ raised cosine, achieving both ISI-free detection and maximum output SNR of $2E_s/N_0$.

• 6.

  The bandwidth efficiency plane maps each modulation format to a point $(\eta, E_b/N_0)$. The Shannon limit defines the boundary below which reliable communication is impossible. Practical uncoded modulations operate 8-10 dB from the Shannon limit; modern coded modulation (LDPC, turbo, polar) closes the gap to 1-2 dB.