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Discrete Fourier Transform (DFT) vs Fast Fourier Transform (FFT)
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Classroom Contents
Understanding the Discrete Fourier Transform - Week 14
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- 1 Introduction
- 2 Time series data from sound recordings
- 3 Julia notebook: Playing with sound - WAV files
- 4 Drawing waveforms
- 5 Effect of frequency
- 6 Combining (superposing) different frequencies
- 7 Julia: FFT function
- 8 Discrete Fourier Transform (DFT) vs Fast Fourier Transform (FFT)
- 9 Plotting an FFT
- 10 Musical overtones: Magnitude of the FFT
- 11 Analyzing a sound file using the FFT
- 12 Defining the DFT mathematically
- 13 First term of the DFT
- 14 Visualizing the DFT in the complex plane
- 15 Equally-spaced points on unit circle in the complex plane
- 16 Idea of Fourier transform of a signal: walking around a circle
- 17 Adding complex numbers as adding vectors
- 18 Magnitude of DFT gives information about frequency
- 19 Angle of DFT gives information about phase
- 20 Interpreting the second term of the DFT
- 21 General formula for DFT
- 22 Implementing the DFT in Julia
- 23 Julia: Writing "i" as im
- 24 Julia: Array comprehension
- 25 Comparison of DFT with FFT results
- 26 Julia: isapprox for testing approximate equality
- 27 Efficiency of the implementation
- 28 Pre-computing an array of powers
- 29 Julia: Modulo (%)
- 30 Julia: OffsetArray for zero-based indexing
- 31 Computational complexity of DFT vs FFT
- 32 DFT as polynomials
