Prerequisites & Notation

This chapter combines linear algebra (Ch. 1), probability (Ch. 2), and optimisation (Ch. 3). The following should be comfortable.

Inner products, norms, and orthogonal projections(Review ch01)
Self-check: Can you compute $\langle f, g \rangle = \int_{-\infty}^{\infty} f(t) g^*(t)\,dt$ ?
Complex exponentials and Euler's formula
Self-check: Can you express $\cos(\omega t)$ in terms of $e^{j\omega t}$ ?
Probability density functions, expectation, and autocorrelation(Review ch02)
Self-check: Can you compute $R_X(\tau) = \mathbb{E}[X(t)X^*(t-\tau)]$ ?
Basic convexity and optimisation concepts(Review ch03)
Self-check: Do you understand why maximising a concave function is a convex optimisation problem?
Integration, differentiation, and improper integrals
Self-check: Can you evaluate $\int_{-\infty}^{\infty} e^{-a|t|}\,dt$ for $a > 0$ ?

Symbols introduced in this chapter. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$x(t)$	Continuous-time signal	s01
$h(t)$	Impulse response of an LTI system	s01
$*$	Convolution operator	s01
$X(f)$	Fourier transform of $x(t)$	s02
$H(f)$	Transfer function (frequency response)	s02
$f_s$	Sampling frequency (samples/second)	s03
$x[n]$	Discrete-time signal	s03
$\tilde{x}(t)$	Complex baseband (envelope) signal	s04
$\hat{x}(t)$	Hilbert transform of $x(t)$	s04
$h(t, \tau)$	Time-variant impulse response	s05
$S_X(f)$	Power spectral density of $X(t)$	s06