Chapter Summary

Chapter Summary

Key Points

• 1.

  The $M$-ary decision problem. Given $M$ hypotheses $\mathcal{H}_0,\ldots,\mathcal{H}_{M-1}$ with densities $f_m$, priors $\pi_m$, and observation $\mathbf{y}$, the MAP rule is $g(\mathbf{y}) = \arg\max_m \pi_m f_m(\mathbf{y})$, and the ML rule (equal priors) drops the $\pi_m$.

• 2.

  Signal-space detection. For $M$ finite-energy signals in AWGN, Gram--Schmidt orthogonalisation produces an orthonormal basis $\{\phi_1,\ldots,\phi_N\}$ with $N \leq M$. The vector of projections $\mathbf{y} \in \mathbb{R}^N$ is a sufficient statistic, and ML detection is the minimum-Euclidean-distance rule: pick the constellation point closest to $\mathbf{y}$.

• 3.

  Voronoi decision regions. The minimum-distance decoder partitions $\mathbb{R}^N$ into polytopes (Voronoi cells). The geometry of these cells --- in particular the distance to the nearest boundary --- controls performance.

• 4.

  Union and nearest-neighbor bounds. Symbol error probability obeys $P_e \leq \sum_{m'\neq m} P(m\to m') \leq (M-1)Q(d_{\min}/(2{\sigma^2}^{1/2}))$. The nearest-neighbor approximation $P_e \approx K_{\min} Q(d_{\min}\sqrt{\text{SNR}/2})$ is tight at high SNR, where $K_{\min}$ counts nearest neighbors.

• 5.

  Exact formulas for standard constellations. BPSK: $P_e = Q(\sqrt{2E_s/N_0})$. $M$-PSK: $P_e \approx 2Q(\sqrt{2\text{SNR}}\sin(\pi/M))$. Square $M$-QAM: $P_e = 1 - (1 - 2(1-1/\sqrt{M})Q(\sqrt{3\text{SNR}/(M-1)}))^2$.

• 6.

  MGF averaging and Craig's formula. The Craig integral $Q(x) = \frac{1}{\pi}\int_0^{\pi/2}\exp(-x^2/(2\sin^2\phi))\,d\phi$ converts SER in fading to a single integral of the SNR MGF --- a closed-form reduction that works for Rayleigh, Nakagami, and Rician channels.

• 7.

  Error exponents. For $n$ i.i.d. observations, the ML error probability decays as $P_e \doteq e^{-n D_{\min}}$, where $D_{\min} = \min_{m\neq m'} D(f_m \| f_{m'})$ (roughly; the precise exponent is the minimum Chernoff information). The same KL quantity that governs detection also governs channel capacity (ITA Ch. 4) --- this is the operational link between Books FSI and ITA.