Prerequisites & Notation

Before You Begin

This chapter assumes fluency with the material on linear time-varying (LTV) channels and multipath fading. Brush up on the items below if any feel unfamiliar. Everything else is built up from scratch.

Linear time-invariant (LTI) systems, convolution, and the impulse response(Review Telecom Ch. 4)
Self-check: Can you derive $y(t) = h(t) \star x(t)$ from the eigenfunction property of complex exponentials?
Multipath propagation: rays, delays, fading(Review Telecom Ch. 6)
Self-check: Can you write a multipath channel as $h(\tau, t) = \sum_i h_i(t)\,\delta(\tau - \tau_i(t))$ ?
Doppler effect and mobility-induced channel variation(Review Telecom Ch. 6)
Self-check: Given a reflector moving at speed $v$ toward the receiver, can you compute the Doppler shift $\nu = (v/c)\,f_0$ ?
Bello's system functions (at a qualitative level)(Review Telecom Ch. 6.6)
Self-check: Do you recall that a time-varying channel can be described by several equivalent functions related by Fourier transforms?
Fourier analysis and 1D / 2D convolution(Review Telecom Ch. 4)
Self-check: Can you state Parseval's theorem and the convolution theorem?
Circularly symmetric complex Gaussian random variables(Review FSP Ch. 8)
Self-check: Can you write the PDF of a zero-mean $\mathcal{CN}(0, \sigma^2)$ variable?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table and the book's notation preferences panel, which lets you customize the base symbols.

Symbol	Meaning	Introduced
$\tau$	Delay variable (seconds), continuous axis of the DD plane	s01
$\nu$	Doppler variable (hertz), continuous axis of the DD plane	s01
$P$	Number of resolvable propagation paths	s01
$h_i, \tau_i, \nu_i$	Complex gain, delay, and Doppler shift of the $i$ -th path	s01
$h(\tau, t)$	Time-variant impulse response (Bello's delay-time function)	s02
$H(f, t)$	Time-varying transfer function (Bello's frequency-time function)	s02
$h(\tau, \nu)$	Delay-Doppler spreading function (the central object of this book)	s03
$S_h(\tau, \nu)$	Scattering function: power spectrum of the spreading function	s04
$\tau_{\max}, \nu_{\max}$	Maximum delay spread, maximum Doppler spread	s04
$B_c, T_c$	Coherence bandwidth and coherence time	s04
$f_D$	Maximum Doppler frequency: $f_D = (v_{\max}/c)\,f_0$	s01

Back to chapter overview From Multipath and Mobility to the Delay-Doppler Representation