Prerequisites & Notation

Before You Begin

Chapter 20 sits at the interface of three strands. From Part II (BICM) we take the point-to-point coded-modulation framework — the separation of code and modulation and the capacity metric $C = \log_2(1 + \text{SNR})$ that every later chapter of the book is compared to. From Ch. 10 (MIMO capacity) we take the multi-antenna model $\ntn{Y} = \mathbf{H} \mathbf{X} + \mathbf{w}$ , the Rayleigh-fading assumption, and the spatial-multiplexing sum- capacity $\log_2 \det(\mathbf{I} + \text{SNR}\, \mathbf{H}\mathbf{H}^{H} / n_t)$ . From the MIMO book (Ch. 18, forward-referenced throughout) we borrow the massive-MIMO regime $n_r \gg 1$ , in which the Wishart $\tfrac{1}{n_r}\mathbf{H}^{H} \mathbf{H} \to \mathbf{I}$ (channel hardening) and the fading variance collapses. Channel hardening is the main reason the coded-modulation problem simplifies as the array grows: the fade becomes effectively deterministic, and the link budget is dominated not by diversity but by hardware constraints — specifically, the cost of high-resolution ADCs and RF chains at 64-256 antennas.

This chapter is about those hardware constraints and the coded- modulation design choices they force. Two constraints dominate. First, ADC resolution: a 12-bit ADC at a few GSamples/sec burns $\sim 1$ W per receive chain, which at $n_r = 128$ is intractable; the pragmatic answer has been 1-4 bit ADCs and careful coded- modulation design around the resulting quantisation loss. Second, RF chain count: activating $n_t = 64$ full RF chains is again prohibitive, motivating spatial modulation and related index- modulation schemes that embed information in which antennas are active, not in their amplitudes alone. These two ideas — low- resolution quantisation and index-based information transmission — are the operational content of this chapter.

BICM and the CM/BICM capacity gap (Ch. 5)(Review ch05)
Self-check: Can you write down the BICM achievable rate $R_{\text{BICM}} = \sum_{i=1}^m I(B_i; Y)$ , contrast it with the CM capacity $I(X; Y)$ , and identify the conditions under which the gap is small (high SNR, Gray labelling) or large (low SNR, non-Gray)?
MIMO channel model and capacity (Ch. 10)(Review ch10)
Self-check: Can you state the ergodic MIMO capacity with CSI at the receiver only, $\mathbb{E}[\log_2 \det(\mathbf{I}_{n_r} + (\text{SNR}/n_t)\, \mathbf{H}\mathbf{H}^{H})]$ , and explain why it grows as $\min(n_t, n_r) \log_2 \text{SNR}$ at high SNR?
Entropy and mutual information on a binary-symmetric channel (Ch. 4 / Book ITA)(Review ch01)
Self-check: Can you derive the BSC capacity $C_{\text{BSC}} = 1 - h_2(p)$ , where $h_2(p) = -p \log_2 p - (1-p)\log_2(1-p)$ ? This formula is the heart of the 1-bit-quantised capacity proof in §2.
Q-function and AWGN error probability
Self-check: Can you write $Q(x) = \tfrac{1}{\sqrt{2\pi}} \int_x^\infty e^{-t^2/2}\, dt$ , recall that $Q(0) = 1/2$ and $Q(x) \approx \tfrac{1}{2} e^{-x^2/2}$ at large $x$ , and distinguish this from the quantiser $Q(\cdot)$ used later in this chapter?
Basic ADC operation and sampling theorem (Book Telecom Ch. 3)
Self-check: Can you describe how an analogue-to-digital converter samples at rate $f_s$ and uniformly quantises each sample to one of $2^b$ levels, and explain why each additional bit of resolution costs roughly $2\times$ in power (a $6\,$ dB SNR improvement)?

Notation for This Chapter

Notation combines the MIMO symbols from Ch. 10 (channel matrix $\mathbf{H}$ , SNR $\text{SNR}$ , transmitted codeword $\mathbf{X}$ ) with chapter-specific symbols for quantisation (ADC bits $b$ , quantiser $Q(\cdot)$ , quantisation-SNR $\mathrm{SNR}_q$ ) and for index modulation (active antennas $n_a$ , antenna index $k$ ). Note the potential collision with the Q-function $Q(\cdot)$ : the quantiser $Q(\cdot)$ is a deterministic nonlinear function that maps real/complex numbers to a finite alphabet; $Q$ is the Gaussian tail probability. They are different objects and the text carefully distinguishes them.

Symbol	Meaning	Introduced
$n_t, n_r$	Number of transmit and receive antennas	s01
$n_a$	Number of active transmit antennas in Generalised Spatial Modulation ( $1 \le n_a \le n_t$ )	s04
$b$	Number of bits per sample at the ADC (resolution)	s01
$Q(\cdot)$	Quantiser: deterministic map from $\mathbb{R}$ (or $\mathbb{C}$ ) to a $2^b$ -level alphabet. NOT the Q-function.	s01
$\mathrm{SNR}_q$	Quantisation SNR: $\mathrm{SNR}_q = 6.02\, b + 1.76$ dB for a uniform $b$ -bit ADC on a full-scale sinusoid	s01
$\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$	MIMO channel matrix (i.i.d. $\mathcal{CN}(0,1)$ under Rayleigh)	s02
$\text{SNR}$	Receive signal-to-noise ratio per antenna	s02
$C$	Capacity (bits per channel use)	s02
$Q(\cdot)$	Gaussian tail probability (Q-function); distinct from the quantiser $Q(\cdot)$	s02
$h_2(p)$	Binary entropy function: $-p \log_2 p - (1-p) \log_2(1-p)$	s02
$M$	QAM constellation size (symbols per transmit antenna)	s03
$k$	Antenna index, $k \in \{1, 2, \ldots, n_t\}$	s03
$\mathbf{X}$	Transmitted codeword / signal vector in $\mathbb{C}^{n_t}$	s02
$\mathbf{w} \in \mathbb{C}^{n_r}$	Receiver noise, $\mathcal{CN}(0, \sigma^2 \mathbf{I})$	s02
$R_{\mathrm{SM}}$	Spatial Modulation rate: $R_{\mathrm{SM}} = \log_2 n_t + \log_2 M$ bits/ch.use	s03
$R_{\mathrm{GSM}}$	Generalised SM rate: $\lfloor \log_2 \binom{n_t}{n_a} \rfloor + n_a \log_2 M$ bits/ch.use	s04
$N_{\mathrm{RF}}$	Number of active RF chains (digital-to-RF conversions per Tx)	s05

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