Prerequisites & Notation

Before You Begin

This chapter assumes familiarity with NumPy arrays, complex numbers, and basic linear algebra. If any of these items feel unfamiliar, review the linked material first.

NumPy array creation, slicing, and broadcasting (Chapter 5)(Review ch05)
Self-check: Can you create a 1D array and compute element-wise operations?
Complex exponentials and Euler's formula
Self-check: Do you know that $e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)$ ?
Matrix multiplication and linear systems (Chapter 6)(Review ch06)
Self-check: Can you express a matrix-vector product using the @ operator?
Basic calculus: integrals and summations
Self-check: Are you comfortable with $\int_{-\infty}^{\infty} f(t)\,dt$ and $\sum_{n=0}^{N-1}$ ?

Notation for This Chapter

Symbols and conventions introduced in this chapter. We use $j$ for the imaginary unit (engineering convention) and lowercase for signals.

Symbol	Meaning	Introduced
$x[n]$ , $x(t)$	Discrete-time and continuous-time signals	s01
$X[k]$ , $X(f)$	DFT coefficients and continuous Fourier transform	s01
$N$	Length of the DFT (number of samples)	s01
$f_s$	Sampling frequency in Hz	s01
$\\omega$	Angular frequency $2\pi f$ (rad/s)	s01
$h[n]$ , $H(f)$	Impulse response and frequency response of a filter	s02
$*$	Convolution operator: $(x * h)[n] = \sum_m x[m]\,h[n-m]$	s02
$w[n]$	Window function applied to a signal segment	s03
$S_{xx}(f)$	Power spectral density of signal $x$	s03
$\\psi(t)$ , $\\phi(t)$	Wavelet (mother) and scaling functions	s05

← Ch 6 Fourier Transforms