Readiness review

This course is genuinely applied: every technique it introduces exists to answer "what does this actually do to a real signal?" That means it leans hard on two courses that come before it — fourier-analysis for the transform machinery (Fourier series, the Fourier transform, convolution, sampling, the DFT), and probability, once chapter 12's random-signal material begins. This chapter teaches, in this course's own voice, the handful of facts and conventions from each that later chapters reach for constantly — not a survey of everything those two courses contain, and not a substitute for them either: fourier-analysis and probability are where to go for the full, rigorous treatment of anything covered here more briefly.

Section 1 draws the one distinction the rest of the course depends on: whether a signal's argument is a real number or an integer. Section 2 teaches complex numbers and Euler's formula in the j notation this course uses — the imaginary unit written j, not i, following the same convention its primary source and the wider signal-processing field both use. Section 3 makes the bracket-vs-parenthesis convention, x[n] vs. x(t), precise, and introduces the unit impulse and unit step every later sequence is built from. Section 4 derives the geometric series closed forms, the single computation the z-transform and the DFT both reduce to at a key step. Section 5 is genuinely new: how NumPy's 0-based array indexing maps onto this course's own x[n] notation, and the one place the two conventions quietly disagree. Section 6 teaches the random variables, expectation, and variance that chapter 12's random-signal material stands on.

If a fact in this chapter feels familiar, that's fine — read it anyway. This course's own notation (j for the imaginary unit, x[n] for a discrete-time signal, E[X[n]] for a random sample's mean) is fixed here and used unchanged from chapter 1 forward.