Prerequisites & Notation

Before You Begin

Chapter 5 established the BICM capacity results — what rate a BICM transmitter can support. This chapter turns to error probability — what SNR is required to hit a target error rate, and how that SNR depends on the labelling, the binary code, and the interleaver. Reader requirements are the same as Ch. 5, plus a working understanding of binary linear codes (weight enumerators) and fading-channel PEP machinery (Craig's integral, Chernoff bound, diversity order).

BICM encoder/decoder, bit metric, and the Gray-labelling near-optimality story(Review ch05)
Self-check: Can you write down the BICM bit-metric LLR $\lambda_\ell(y)$ and explain why under Gray labelling on square QAM it is well-approximated by a sign-flipped projection onto the in-phase or quadrature axis?
Signal-space constellations, $d_{\min}$ , and the coding-gain criterion(Review ch01)
Self-check: Can you state the Gaussian PEP in terms of squared Euclidean distance $P(\mathbf{x} \to \hat{\mathbf{x}}) \le Q(d(\mathbf{x}, \hat{\mathbf{x}}) / (2\sigma))$ and identify which factor gives the diversity slope?
TCM pairwise error probability and the Chernoff/Bhattacharyya bound(Review ch02)
Self-check: Can you bound the Gaussian PEP by $Q(x) \le \frac{1}{2}\exp(-x^2/2)$ and use this to derive the TCM error-event exponent along a trellis path?
Binary linear codes and weight enumerators $W_d$ (Review ch11)
Self-check: Can you state the minimum Hamming distance $d_H$ of a binary code, write the weight enumerator $W(z) = \sum_d W_d z^d$ , and relate it to the union-bound bit error probability?
Rayleigh fading channel model and diversity order(Review ch10)
Self-check: Can you state the Rayleigh PEP $\mathbb{E}[Q(\sqrt{|h|^2 \gamma})] \sim (4\gamma)^{-1}$ at high SNR and define diversity order as the high-SNR slope of $\log P_e$ versus $\log \text{SNR}$ ?
Q-function, Craig's integral, and MGF-based PEP(Review ch03)
Self-check: Can you write Craig's representation $Q(x) = \frac{1}{\pi}\int_0^{\pi/2} e^{-x^2/(2\sin^2\theta)}\,d\theta$ and use it to average $Q$ over a Rayleigh-distributed amplitude in closed form?

Notation for This Chapter

Chapter 5's BICM notation (constellation $\mathcal{X}$ , labelling $\mu$ , bit-channel subsets $\mathcal{X}_\ell^{(b)}$ ) carries over. The symbols below are the additional ones introduced in this chapter for the pairwise-error and diversity analysis. See also the book-level notation table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{c}, \hat{\mathbf{c}}$	A pair of distinct BICM codewords (binary sequences before mapping)	s01
$d_H$	Hamming distance between two binary codewords; $d_{H,\min}$ for the code minimum	s01
$P(\mathbf{c} \to \hat{\mathbf{c}})$	Pairwise error probability that the decoder prefers $\hat{\mathbf{c}}$ over the transmitted $\mathbf{c}$	s01
$d^2_{\rm avg}(\mu, \ell)$	Average squared Euclidean distance between pairs $(s, \hat s)$ differing in label bit $\ell$ under labelling $\mu$	s01
$d^2_{\rm avg}(\mu)$	Average of $d^2_{\rm avg}(\mu, \ell)$ over bit positions $\ell \in \{0, \ldots, L-1\}$	s01
$\beta_\ell(\mu)$	Bhattacharyya factor for bit channel $\ell$ : $\beta_\ell = \mathbb{E}[\sqrt{p(y \mid b_\ell = 1)/p(y \mid b_\ell = 0)}]$	s01
$\ell(\mu, b, \hat b)$	Number of constellation points with label bit equal to $b$ whose partner (flipping that one label bit) takes value $\hat b \ne b$ ; depends on $\mu$	s02
$L_{\min}(\mu)$	Minimum over bit positions $\ell$ of the minimum number of DISTINCT constellation positions (in the sense of Euclidean geometry) that differ in bit $\ell$ between label pairs	s03
$d_{\rm div}$ or $d$	Diversity order: high-SNR slope of $-\log P_e / \log \text{SNR}$	s03
$d_{\rm BICM}(\mu)$	BICM diversity order under labelling $\mu$ ; equals $d_H \cdot L_{\min}(\mu)$ on fully-interleaved Rayleigh	s03
$\gamma_c$	Coding gain (linear scale or dB): the horizontal offset of the high-SNR BER curve at a given diversity order	s03
$\|h\|$	Fading amplitude (Rayleigh unless stated); $\|h\|^2 \sim \text{Exp}(1)$ in the unit-mean normalisation	s03
$W_d, c_d$	Weight enumerator coefficient (number of codewords at Hamming weight $d$ ) and input-weight- $d$ multiplicity of a convolutional code	s04
$P(d)$	Diversity- $d$ PEP: the probability that two codewords at Hamming distance $d$ are confused at the receiver	s04
$N$	Interleaver length in symbols (or coded bits, divided by $L$ )	s05
$T_c$	Coherence time of the fading channel, in symbol periods	s05
$N_{\rm eff}$	Effective number of independent fading samples seen by a codeword: $N_{\rm eff} \approx N / T_c$	s05

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