Prerequisite Theory

:::info Prerequisites

This section assumes familiarity with the below:

• Complex numbers (including Euler’s formula)
• Basic trigonometry and calculus
• Introductory concepts in signals
• Systems such as periodicity and sampling

No advanced math beyond these basics is required, but some intuition for time and frequency domains will be helpful.

:::

The list of prerequisites and helpful resources can be found below.

Quick study path (suggested)

Topic	Suggested Time on Topic (in hours)
Pythagorean theorem	1
Complex numbers & trigonometry	4-8
Basic calculus (integrals / limits)	4-8
Linear Algebra (vectors, basis)	4-8
Signals & sampling (Nyquist, aliasing)	6-10
Practical DFT/FFT & coding (NumPy/Scipy)	4-8

Total: = 23-43 hours

Helpful resources

Quick Recap — Pythagorean Theorem

• NASA — Pythagorean Theorem
  • Anything will do — NASA made it, so why not?
  • Pythagorean Theorem

Foundations — Complex numbers & trigonometry

• Khan Academy — Complex numbers & Trigonometry

  • Good interactive exercises for Euler’s formula, magnitude/phase, unit circle.
  • Euler's formula & Euler's identity | Series | AP Calculus BC | Khan Academy

• 3Blue1Brown — Visual intuition (unit circle / phasors)

  • Excellent visual intuition for complex exponentials and rotating phasors.
  • Complex number fundamentals | Ep. 3 Lockdown live math

Basic calculus / prerequisite math

• Khan Academy — Single variable calculus (integrals & limits)

  • Clear, bite-sized lessons on definite integrals and limits (essential for the CFT section).
  • Khan Academy's Calculus playlist

• Paul’s Online Math Notes — Calculus I / II

  • Concise reference notes and worked examples for integrals and series.
  • Calculus I
  • Calculus II

Linear algebra (optional but helpful)

• Gilbert Strang’s MIT OCW lectures / book — Introduction to Linear Algebra

  • Focus on vectors, inner products and basis functions (signals as vectors).
  • Downloadable lectures

• 3Blue1Brown — Essence of linear algebra (playlist)

  • Visual, intuitive take on basis and projections.
  • Essence of Linear Algebra

Signals & systems / sampling theory

• Intro Signals & Systems (textbook or course)

  • Read the introductory chapters on continuous vs discrete signals, linearity, and sampling.
  • Oppenheim & Willsky - Signals and Systems - 2nd Edition
  • MIT OCW - Signals and Systems

• Nyquist, aliasing, and sampling theorem

  • West & Jill's Data Communication and Computer Networks A Business Users Approach, Ninth Edition
    • Chapter 3: Transmission Fundamentals
    • Chapter 4: Analog and Digital Signals
    • Chapter 7: Data Transmission and Encoding

DFT / FFT practicals and coding

• NumPy / SciPy docs — numpy.fft and scipy.fft

  • Practical examples for computing FFTs, fftshift, real-input symmetry, and inverse transforms.
  • NumPy
  • SciPy

• Richard Lyons — Understanding Digital Signal Processing (book)

  • Very practical, engineering-focused explanations and examples (recommended chapters on DFT/FFT).
  • Understanding Digital Signal Processing

• Small hands-on project:
  • Load a simple 1-D signal (sine + noise), compute np.fft.fft, plot magnitude/phase, then ifft to verify reconstruction.
  • Try zero-padding and fftshift to see visualization effects.

Visual / interactive tools

• Desmos or GeoGebra — quickly plot complex exponentials, sines/cosines, and see effects of phase/amplitude changes.
• Online FFT visualizers — interactively play with windowing, zero-padding and see spectral leakage (search for “FFT visualizer” or “spectral leakage demo”).