Prerequisites & Notation

Before You Begin

This chapter assumes familiarity with NumPy array operations (Chapter 5), basic signal processing concepts (Chapter 7), and statistics/Monte Carlo simulation (Chapter 9). We build on the Q-function and BER simulation framework from Chapter 9.

NumPy arrays, broadcasting, and vectorized computation (Chapter 5)(Review ch05)
Self-check: Can you create and manipulate complex-valued NumPy arrays?
FFT, convolution, and filtering (Chapter 7)(Review ch07)
Self-check: Can you compute the cross-correlation of two signals using np.correlate?
Monte Carlo simulation and BER estimation (Chapter 9)(Review ch09)
Self-check: Can you estimate BER with confidence intervals from a vectorized simulation?

Notation for This Chapter

Symbols and conventions used throughout the chapter. Boldface denotes vectors/matrices; $j = \sqrt{-1}$ is the imaginary unit.

Symbol	Meaning	Introduced
$M$	Constellation size (number of symbols)	s01
$s_m$	Complex constellation point for symbol index $m$	s01
$E_s$	Average symbol energy	s01
$E_b$	Average bit energy; $E_b = E_s / \log_2 M$	s01
$N_0$	One-sided noise power spectral density	s02
$\\sigma_n^2$	Noise variance per dimension; $\sigma_n^2 = N_0/2$	s02
$P_b$	Bit error rate (BER)	s02
$P_s$	Symbol error rate (SER)	s02
$Q(x)$	Gaussian tail probability $Q(x) = \frac{1}{2}\mathrm{erfc}(x/\sqrt{2})$	s02
$h(t)$	Matched filter impulse response	s03
$d_{\\min}$	Minimum Euclidean distance between constellation points	s01

← Ch 19 Constellation Design and Mapping