Prerequisite Theory

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”).