Prerequisites & Notation

Before You Begin

Every chapter until now has treated the receiver front-end as ideal: the observation $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ is available to the digital baseband at infinite precision. This chapter addresses the single most energy-hungry component of a massive-MIMO receiver — the analog-to-digital converter — and asks what happens when its resolution is deliberately reduced, in the extreme case to a single sign bit per I/Q rail. The result is a hardware-aware rate theory that cannot be read off from the Shannon formula $C = \log_2(1 + \text{SNR})$ . We assume familiarity with the following prior material.

Linear MIMO receivers: MRC, ZF, MMSE and their achievable rates(Review ch09)
Self-check: Can you write the closed-form SINR of MMSE combining for user $k$ in a massive MIMO uplink with perfect CSI?
Use-and-then-forget (UatF) rate bound and massive MIMO rate scaling(Review ch04)
Self-check: Can you state the per-user UatF rate expression and explain why it lower-bounds the ergodic capacity under imperfect CSI?
Gaussian channel capacity and mutual information for memoryless channels(Review ch13)
Self-check: Can you write $C = \log_2(1 + \text{SNR})$ from AEP or differential-entropy arguments?
Bussgang's theorem for a Gaussian input passed through a memoryless nonlinearity
Self-check: Given $Y = g(X)$ with $X$ Gaussian, can you write $\mathbb{E}[XY]$ in terms of $g$ and the variance of $X$ ?
Constellation design, PSK, QAM, and minimum-distance receivers(Review ch15)
Self-check: Can you compare the symbol error probability of QPSK and 16-QAM at the same SNR?
ADC basics: sampling rate, resolution, SNDR, figure of merit
Self-check: Can you explain why ADC power scales as $2^b f_s$ for a Walden-type converter?

Notation for This Chapter

Symbols introduced or specialized in this chapter. We reserve $b$ for the per-antenna ADC resolution in bits and $f_s$ for the sampling rate, and we write $Q_b(\cdot)$ for a uniform mid-rise quantizer with $2^b$ output levels per real dimension. See NGlobal Notation Table for the master table of symbols that are shared across the book.

Symbol	Meaning	Introduced
$Q_b(\cdot)$	Uniform mid-rise scalar quantizer with $2^b$ levels. $b=1$ is the sign quantizer	s01
$Q_{1}(\cdot)$	Sign quantizer: $Q_1(y) = \operatorname{sign}(\operatorname{Re}(y)) + j\,\operatorname{sign}(\operatorname{Im}(y))$	s01
$b, b_n$	ADC resolution in bits (global, or per antenna $n$ )	s01
$f_s$	ADC sampling rate (samples per second per I/Q rail)	s01
$P_{\text{ADC}}(b)$	ADC power consumption: $P_{\text{ADC}}(b) = c_0\,2^b f_s$ (Walden model)	s01
$\mathbf{y}_q$	Quantized observation: $\mathbf{y}_q = Q_b(\mathbf{y})$	s01
$\mathbf{B}$	Bussgang linear gain matrix $\mathbf{B} = \mathbb{E}[\mathbf{y}_q \mathbf{y}^H](\mathbb{E}[\mathbf{y}\mathbf{y}^H])^{-1}$	s02
$\mathbf{d}$	Bussgang residual distortion, decorrelated from $\mathbf{y}$	s02
$\rho_b$	Bussgang distortion factor for $b$ -bit uniform quantizer (e.g., $\rho_1 = 1 - 2/\pi$ )	s02
$\alpha$	Fraction of antennas equipped with high-resolution ADCs in a mixed-ADC receiver	s03
$\mathcal{H}, \mathcal{L}$	Index sets of high-resolution and low-resolution antennas, $\|\mathcal{H}\| = \alpha N_r$	s03
$P_{\text{bud}}$	Total ADC power budget constraint	s04
$\eta$	Energy efficiency: bits per joule $\eta = R / P_{\text{rx}}$	s05
$\kappa_b$	Effective SNR penalty of $b$ -bit quantization, $\kappa_b = 1 - \rho_b$	s02

← Ch 18 1-bit ADC Receivers