What is a Discrete Fourier Transform? | Week 14 | MIT Fall 2020 | Grant Sanderson

An overview with Julia of what the Discrete Fourier Transform (DFT) does, by applying it to analyze sounds, including how it is defined, along with a comparison between the runtime of a naively- [Click “↓ ↓ ↓ Show More “ for the Outline:] implemented DFT and one using the Fast Fourier Transform (FFT) algorithm. Course homepage: Pluto notebook: 0:00 Introduction 2:00 Time series data from sound recordings 2:37 Julia notebook: Playing with sound - WAV files 4:02 Drawing waveforms 5:45 Effect of frequency 6:00 Combining (superposing) different frequencies 6:45 Julia: FFT function 6:59 Discrete Fourier Transform (DFT) vs Fast Fourier Transform (FFT) 8:18 Plotting an FFT 8:58 Musical overtones: Magnitude of the FFT 12:13 Analyzing a sound file using the FFT 12:53 Defining the DFT mathematically 13:35 First term of the DFT 14:08 Visualizing the DFT in the complex plane 14:20 Equally-spaced points on unit circle in the complex plane 15:12 Idea of Fourier transform of a signal: walking around a circle 16:13 Adding complex numbers as adding vectors 17:10 Magnitude of DFT gives information about frequency 17:35 Angle of DFT gives information about phase 18:39 Interpreting the second term of the DFT 21:06 General formula for DFT 22:10 Implementing the DFT in Julia 23:01 Julia: Writing “i“ as im 23:33 Julia: Array comprehension 25:35 Comparison of DFT with FFT results 26:49 Julia: isapprox for testing approximate equality 26:56 Efficiency of the implementation 28:23 Pre-computing an array of powers 28:57 Julia: Modulo (%) 29:14 Julia: OffsetArray for zero-based indexing 32:02 Computational complexity of DFT vs FFT 34:12 DFT as polynomials Video on FFT from Reducible: