Prerequisites & Notation

Before You Begin

Part V of this book takes the theory developed in Parts I-IV and asks a harder question: how does any of it survive a standardized air interface? Chapter 22 is the grounding chapter — it shows how 5G NR operationalizes massive MIMO within the constraint of a 1-ms slot, a fixed pilot budget, a quantized codebook, and the need to manage beams in mobility. The chapter assumes you are fluent in single-cell massive MIMO theory (Part I), multi-user precoding (Part II), and FDD feedback (Chapter 8). We also rely on OFDM fundamentals from the Telecom book. The emphasis here is on the information geometry of CSI compression and the scheduling geometry of beam management, not on the protocol stack.

Massive MIMO system model and channel hardening: $\frac{1}{N_t} \mathbf{H}_{k}^{H} \mathbf{H}_{k} \to \beta_{k}$ almost surely(Review ch01)
Self-check: Can you state why, in the massive regime, per-user SINR with MRC grows as $N_t$ times a constant, and why interference from favorable-propagation users vanishes?
Uplink-downlink duality in TDD, and the pilot overhead budget $\tau_p / \tau_c$ (Review ch03)
Self-check: Given coherence block length $\tau_c = 200$ and $K = 10$ orthogonal uplink pilots, can you compute the fraction of symbols used for training?
Linear precoding: MRT $\mathbf{v}_{k} = \mathbf{H}_{k} / \|\mathbf{H}_{k}\|$ , ZF precoding, regularized ZF(Review ch06)
Self-check: Can you derive the ZF precoder from the pseudoinverse of $\mathbf{H}$ and explain when regularization matters?
Codebook-based FDD CSI feedback: Grassmannian quantization, feedback overhead scaling(Review ch08)
Self-check: Can you state the rate loss of Lloyd-type quantization of a unit-norm Grassmannian beam with $B$ feedback bits in $N_t$ dimensions?
OFDM numerology: subcarrier spacing $\Delta f$ , useful symbol time $1/\Delta f$ , cyclic prefix, relation to $B_c$ and $\sigma_\tau$ (Review ch24)
Self-check: Can you compute the OFDM symbol duration and cyclic-prefix overhead for $\Delta f = 30$ kHz and explain why $\Delta f$ must exceed $f_D$ ?
Multi-user MIMO capacity region for the broadcast channel (downlink)(Review ch16)
Self-check: Can you sketch the MU-MIMO BC rate region and state why dirty-paper coding is optimal while ZF is within a bounded gap?

Notation for This Chapter

Chapter 22 adds 5G NR-specific symbols to the massive MIMO notation of Chapters 1-10. The chapter-local symbols $\mu$ , $N_p$ , $L$ , $P_{\text{CSI}}$ , and the procedure labels P1/P2/P3 are not global $\ntn{}$ tokens — they belong to this chapter and to the 3GPP specifications it mirrors. See NGlobal Notation Table for the master table of customizable symbols.

Symbol	Meaning	Introduced
$\mu$	NR numerology index, $\mu \in \{0,1,2,3,4\}$ ; subcarrier spacing $\Delta f = 15\,\text{kHz}\cdot 2^{\mu}$	s01
$T_{\text{slot}}$	NR slot duration $= 1\,\text{ms}/2^{\mu}$ (14 OFDM symbols per slot)	s01
$\Delta f$	Subcarrier spacing (15, 30, 60, 120, or 240 kHz)	s01
$N_p^{\text{CSI-RS}}$	Number of CSI-RS ports (up to 32 in Rel-15, up to 64 in later releases); equals the logical antenna count visible to the UE	s02
$T_{\text{CSI}}$	CSI-RS periodicity in slots (typically 5, 10, 20, 40, or 80 slots)	s02
$\mathbf{S}_{i,k}$	Reference signal matrix carrying CSI-RS or SRS samples on the time-frequency grid	s02
$L$	Number of DFT beams summed in the Type II linear combination codebook (Rel-15: $L \in \{2,3,4\}$ )	s03
$B_{\text{CSI}}$	CSI feedback payload in bits per reporting instance	s03
$N_{\text{SSB}}$	Number of Synchronization Signal Blocks transmitted in one SS burst (up to 8 in FR1, up to 64 in FR2)	s04
$T_{\text{sweep}}$	P1 beam-sweep duration; $T_{\text{sweep}} = N_{\text{SSB}} \cdot T_{\text{SSB}}$	s04
$R_k$	Per-user achievable rate for user $k$ (bits/s/Hz), raw symbol per the chapter's request	s01
$\mathbf{H}_{k,m}$	Downlink channel from TRP $m$ to user $k$ , $N_t \times 1$ per subcarrier	s05
$M_{\text{TRP}}$	Number of coordinated TRPs in multi-TRP operation	s05
$B_c$	Channel coherence bandwidth; determines pilot density along the frequency dimension	s02
$T_c$	Channel coherence time; determines CSI-RS periodicity and beam-tracking cadence	s04
$f_D$	Maximum Doppler frequency; bounds the allowable CSI ageing between feedback instances	s04
$\sigma_\tau$	RMS delay spread; bounds the minimum subcarrier spacing for a given cyclic prefix	s01

← Ch 21 The NR MIMO Framework: Numerology, FR1/FR2, TDD/FDD