Prerequisites & Notation

Before You Begin

This chapter assumes comfort with NumPy array operations (Chapter 5), basic linear algebra in NumPy/SciPy (Chapter 6), and undergraduate-level probability theory. If any item feels completely foreign, review the linked material first.

NumPy arrays, broadcasting, and random number generation (Chapter 5)(Review ch05)
Self-check: Can you draw 10000 samples from a Gaussian using rng.standard_normal(10000)?
Linear algebra: Cholesky, eigendecomposition (Chapter 6)(Review ch06)
Self-check: Can you compute the Cholesky factor of a positive-definite matrix with np.linalg.cholesky?
Probability basics: PDF, CDF, expectation, variance
Self-check: Do you know that $\mathrm{Var}[X] = E[X^2] - (E[X])^2$ ?
Complex numbers and baseband signal representation
Self-check: Do you know that a complex Gaussian has independent real and imaginary parts?

Notation for This Chapter

Symbols and conventions introduced in this chapter. We use $j$ for the imaginary unit (engineering convention) and boldface for vectors/matrices.

Symbol	Meaning	Introduced
$f_X(x)$	Probability density function (PDF) of random variable $X$	s01
$F_X(x)$	Cumulative distribution function (CDF)	s01
$\\mathcal{N}(\\mu, \\sigma^2)$	Gaussian (normal) distribution with mean $\mu$ and variance $\sigma^2$	s01
$\\mathcal{CN}(\\mathbf{0}, \\mathbf{R})$	Circularly symmetric complex Gaussian with covariance $\mathbf{R}$	s01
$\\mathbf{R}$	Covariance or correlation matrix	s01
$\\mathbf{L}$	Lower-triangular Cholesky factor ( $\mathbf{R} = \mathbf{L}\mathbf{L}^H$ )	s01
$P_b$ , BER$	Bit error rate (probability of bit error)	s03
$E_b/N_0$	Bit energy to noise spectral density ratio	s03
$h$ , $\\mathbf{H}$	Fading channel coefficient (scalar) or MIMO channel matrix	s04
$K$	Ricean $K$ -factor (ratio of LOS to scattered power)	s04
$\\hat{\\theta}$	Estimator of parameter $\theta$	s02
$N_s$	Number of Monte Carlo samples	s03

← Ch 8 Distributions and Sampling