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MATHEMATICS OF THE
DISCRETE FOURIER TRANSFORM (DFT)

JULIUS O. SMITH III

Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University
Stanford, California 94305 USA

Preface
- Chapter Outline
- Acknowledgments
- Errata
Introduction to the DFT
- DFT Definition
- Inverse DFT
- Mathematics of the DFT
- DFT Math Outline
Complex Numbers
- Factoring a Polynomial
- The Quadratic Formula
- Complex Roots
- Fundamental Theorem of Algebra
- Complex Basics
- The Complex Plane
- More Notation and Terminology
- Elementary Relationships
- Euler's Identity
- De Moivre's Theorem
- Conclusion
- Complex_Number Problems
Proof of Euler's Identity
- Euler's Identity
- Positive Integer Exponents
- Properties of Exponents
- The Exponent Zero
- Negative Exponents
- Rational Exponents
- Real Exponents
- A First Look at Taylor Series
- Imaginary Exponents
- Derivatives of f(x)=a^x
- Back to e
- e^(j theta)
- Back to Mth Roots
- Roots of Unity
- Euler_Identity Problems
Sinusoids and Exponentials
- Sinusoids
- Exponentials
  - Why Exponentials are Important
  - Audio Decay Time (T60)
- Complex Sinusoids
- Sinusoid Problems
Geometric Signal Theory
- The DFT
- Signals as Vectors
  - An Example Vector View:
- Vector Addition
- Vector Subtraction
- Signal Metrics
- The Inner Product
- Signal Reconstruction from Projections
- Signal Projection Problems
The DFT Derived
- Geometric Series
- Orthogonality of Sinusoids
  - Nth Roots of Unity
  - DFT Sinusoids
- Orthogonality of the DFT Sinusoids
- Norm of the DFT Sinusoids
- An Orthonormal Sinusoidal Set
- The Discrete Fourier Transform (DFT)
- Frequencies in the ``Cracks''
- Normalized DFT
- The Length 2 DFT
- Matrix Formulation of the DFT
- DFT Problems
Fourier Theorems for the DFT
- The DFT and its Inverse Restated
  - Notation and Terminology
  - Modulo Indexing, Periodic Extension
- Signal Operators
- Even and Odd Functions
- The Fourier Theorems
- DFT Theorems Problems
DFT Applications
- Spectrum Analysis of a Sinusoid
- Spectrograms
  - Spectrogram of Speech
- Filters and Convolution
- Correlation Analysis
- Power Spectral Density
- Coherence
- Recommended Further Reading
Taylor Series Expansions
- Informal Derivation of Taylor Series
- Taylor Series with Remainder
- Formal Statement of Taylor's Theorem
- Weierstrass Approximation Theorem
- Points of Infinite Flatness
- Differentiability of Audio Signals
Logarithms and Decibels
- Logarithms
  - Changing the Base
  - Logarithms of Negative and Imaginary Numbers
- Decibels
- Electrical Engineering 101
- Exercises
Digital Audio Number Systems
- Linear Number Systems
- Logarithmic Number Systems for Audio
- Round-Off Error Variance
Matrices
- Matrix Multiplication
- Solving Linear Equations Using Matrices
Continuous/Discrete Fourier
- Discrete Time Fourier Transform (DTFT)
- Fourier Transform (FT) and Inverse
  - Existence of the Fourier Transform
  - The Continuous-Time Impulse
- Fourier Series (FS)
  - Relation of the DFT to Fourier Series
Continuous Fourier Theorems
- Differentiation Theorem
- Scaling Theorem
- The Uncertainty Principle
Sampling Theory
- Introduction to Sampling
  - Reconstruction from Samples--Pictorial Version
  - Reconstruction from Samples--The Math
- Aliasing of Sampled Signals
  - Continuous-Time Aliasing Theorem
- Sampling Theorem
- Geometric Sequence Frequencies
Fast Fourier Transform (FFT)
- Decimation in Time
- The Radix 2 FFT
  - Radix 2 FFT Complexity is N Log N
- Other FFT Algorithms
- FFT Software
Matlab/Octave Examples
- Complex Numbers in Matlab and Octave
  - Complex Number Manipulation
- Factoring Polynomials in Matlab
- Geometric Signal Theory
- The DFT
- Spectrogram Computation
Figures and Tables
- Figures
- Tables
Bibliography
Index
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``Mathematics of the Discrete Fourier Transform (DFT)'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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Copyright © 2003-10-09 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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