Prerequisites & Notation

Before You Begin

Part V of this book turns from hardware and standards to the extreme deployment frontier: the sky. Low-Earth-orbit satellite constellations — Starlink, OneWeb, Kuiper, and the coming 6G NTN proposals — promise broadband coverage to the entire globe, including oceans, deserts, and regions where terrestrial cells will never exist. Delivering that promise requires re-doing massive-MIMO system design under two physical constraints that no terrestrial cell ever faces. First, LEO satellites orbit at $\approx 7.5$ km/s relative to the ground, generating Doppler shifts of $6$ – $40$ kHz at Ka band — two to three orders of magnitude larger than any terrestrial mobility scenario. Second, the one-way propagation delay is $5$ – $20$ ms, large enough that closed-loop schemes honed for $\mu$ s-scale terrestrial feedback must be rethought. This chapter adapts the cell-free and user-centric ideas of Part III to the orbital regime, with macro- diversity across simultaneously visible satellites as the central design lever and the choice of waveform (OFDM vs OTFS) as the secondary one. We assume familiarity with the following prior material.

Massive MIMO fundamentals, channel hardening and favorable propagation(Review ch01)
Self-check: Can you state why $\frac{1}{N_t} \mathbf{H}_{k}^{H} \mathbf{H}_{k} \to 1$ as $N_t \to \infty$ and what it implies for linear precoding?
Cell-free and user-centric massive MIMO — APs without cells(Review ch11)
Self-check: Can you state the difference between centralized cell-free and user-centric cluster operation, and name the serving-set selection rule?
Linear precoding — MRT, ZF, MMSE — in the multiuser downlink(Review ch06)
Self-check: Can you write the ZF precoder $\mathbf{W} = \mathbf{H}^{H} (\mathbf{H} \mathbf{H}^{H})^{-1}$ and explain when it leaves residual interference?
Distributed processing: local vs centralized combining, LSFD, fronthaul(Review ch13)
Self-check: Can you sketch level 1, 2, 3, 4 cooperation and state what each level sends over the fronthaul?
OFDM wideband signal model, CP length, subcarrier spacing, ICI(Review ch10)
Self-check: Can you state the condition on the CP length that eliminates inter-symbol interference in OFDM, and the condition on $f_D / \Delta f$ that bounds inter-carrier interference?
Friis free-space path loss, link budget, noise power from bandwidth and $G/T$ (Review ch15)
Self-check: Can you write $\beta_{\text{FSPL}} = (4\pi d / \lambda)^2$ and the receive SNR in terms of $P_t$ , antenna gains, and noise temperature?
Doppler shift for a moving terminal, coherence time $T_c \sim 1/f_D$ (Review ch18)
Self-check: Can you derive $f_D = v f_0 / c$ and explain why the LEO case breaks the usual 'slow time variation' assumption?

Notation for This Chapter

Symbols specific to this chapter. The altitude $h$ , the elevation angle $\theta_{\text{el}}$ , the satellite velocity $v_{\text{sat}}$ , and the number of simultaneously visible satellites $M$ are chapter-local and are not global $\ntn{}$ tokens; they are defined here. See NGlobal Notation Table for the master table.

Symbol	Meaning	Introduced
$h$	Satellite altitude above the Earth's surface (km). LEO: $500$ – $2000$ ; MEO: $\approx 8000$ – $20000$ ; GEO: $\approx 35786$	s01
$R_E$	Earth radius ( $\approx 6371$ km)	s01
$\theta_{\text{el}}$	Elevation angle of the satellite from the user terminal (rad or deg)	s01
$v_{\text{sat}}$	Satellite orbital velocity, $v_{\text{sat}} = \sqrt{G M_E / (R_E + h)}$ ( $\approx 7.5$ km/s for LEO)	s01
$d_{\text{slant}}$	Slant range from terminal to satellite as a function of $h$ and $\theta_{\text{el}}$	s01
$\tau_{\text{prop}}$	One-way propagation delay, $\tau_{\text{prop}} = d_{\text{slant}} / c$ (ms)	s01
$M$	Number of satellites simultaneously visible to a terminal under the macro-diversity architecture	s03
$\mathbf{H}_{m}$	Downlink channel vector from satellite $m$ to the user; $\mathbf{H}_{m} \in \mathbb{C}^{N_t}$ where $N_t$ is the satellite array size	s03
$f_D$	Peak one-way Doppler shift, $f_D = v_{\text{sat}} f_0 / c$ at zenith overhead pass	s02
$\Delta f_D(\theta_{\text{el}})$	Instantaneous Doppler shift as a function of elevation; varies from $+f_D$ at horizon-approach to $-f_D$ at horizon-departure	s02
$G/T$	Receiver figure of merit: antenna gain divided by system noise temperature (dB/K)	s02
$L_{\text{rain}}$	Rain fade attenuation (dB), dominant excess loss at Ka band	s02
$T_{\text{visible}}$	Duration a given LEO satellite remains above the minimum elevation for a fixed terminal	s01
$T_{\text{HO}}$	Handover interval — how often a terminal must switch its master satellite	s05
$T_c$	Channel coherence time $\approx 1 / f_D$ ; for LEO Ka band this is sub-millisecond	s02

← Ch 22 Non-Terrestrial Networks: Orbits, Coverage, Delays