Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1) and signals and systems (Chapter 4). The geometric signal-space viewpoint relies on inner products and orthogonal projections from Chapter 1, while pulse shaping and spectral analysis require Fourier transforms and LTI system theory from Chapter 4.

Inner products, norms, and orthogonal projections(Review ch01)
Self-check: Can you project a vector $\mathbf{v}$ onto a subspace spanned by an orthonormal set $\{\mathbf{e}_1, \ldots, \mathbf{e}_N\}$ and compute the projection error?
Gram-Schmidt orthogonalisation(Review ch01)
Self-check: Can you apply the Gram-Schmidt procedure to convert a set of linearly independent vectors into an orthonormal basis?
Fourier transforms and frequency-domain analysis(Review ch04)
Self-check: Can you compute the Fourier transform of a rectangular pulse and state Parseval's theorem?
Bandpass signals and complex baseband representation(Review ch04)
Self-check: Can you convert a passband signal $x(t)\cos(2\pi f_c t)$ to its complex baseband equivalent?
Matched filter and correlator receiver(Review ch04)
Self-check: Can you explain why the matched filter $h(t) = s^*(T-t)$ maximises the output SNR at time $T$ ?

Chapter 8 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$s(t)$	Transmitted signal waveform	s01
$\varphi_n(t)$	Orthonormal basis function ( $n$ -th)	s01
$\mathbf{s} = [s_1, \ldots, s_N]^T$	Signal vector (coordinates in signal space)	s01
$M$	Constellation size (number of signal points)	s01
$d_{\min}$	Minimum Euclidean distance between constellation points	s01
$R_s$	Symbol rate (symbols per second)	s02
$W$	Signal bandwidth (Hz)	s04
$\alpha$	Roll-off factor of raised-cosine filter ( $0 \leq \alpha \leq 1$ )	s04
$\eta$	Spectral efficiency (bits/s/Hz)	s05
$E_b/N_0$	Energy per bit to noise spectral density ratio	s05
$E_s$	Average energy per symbol	s01
$T_s$	Symbol duration	s02

← Ch 7 Signal-Space Concepts