Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1), signals and systems (Chapter 4), large-scale propagation (Chapter 5), and small-scale fading (Chapter 6).

  • Complex vectors, inner products, and matrix-vector products(Review ch01)

    Self-check: Can you compute the inner product aHb\mathbf{a}^H \mathbf{b} for complex vectors a,b∈CN\mathbf{a}, \mathbf{b} \in \mathbb{C}^N and describe its geometric meaning?

  • Fourier transforms and spatial frequency(Review ch04)

    Self-check: Do you understand how spatial sampling at spacing dd relates to the Nyquist condition in the spatial-frequency domain?

  • Free-space and large-scale path loss(Review ch05)

    Self-check: Can you write the Friis transmission equation and identify the role of antenna gains and effective area?

  • Multipath propagation and channel impulse response(Review ch06)

    Self-check: Can you express the received signal as a sum over multipath components, each with amplitude, delay, and phase?

Chapter 7 Notation

Key symbols introduced or heavily used in this chapter.

SymbolMeaningIntroduced
a(θ)\mathbf{a}(\theta)Array steering vector for direction θ\thetas02
AF(θ)\mathrm{AF}(\theta)Array factor of an antenna arrays02
DDAntenna directivity (dimensionless)s01
GGAntenna gain (directivity times radiation efficiency)s01
AeA_eEffective aperture area of an antennas01
NNNumber of antenna elements in an arrays02
ddInter-element spacing in the arrays02
θ0\theta_0Scan (steering) angle of the main beams02
w\mathbf{w}Beamforming weight vector ∈CN\in \mathbb{C}^Ns02
H\mathbf{H}MIMO channel matrix ∈CNr×Nt\in \mathbb{C}^{N_r \times N_t}s04
← Ch 6Antenna Parameters