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MATHEMATICS OF THE
DISCRETE FOURIER TRANSFORM (DFT)
WITH MUSIC AND AUDIO APPLICATIONS

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 Function
  - Coherence Function in Matlab
- Recommended Further Reading
Fast Fourier Transform (FFT)
- Decimation in Time
- The Radix 2 FFT
  - Radix 2 FFT Complexity is N Log N
- Other FFT Algorithms
- FFT Software
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
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
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
About this document ...

Preface

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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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MATHEMATICS OF THE DISCRETE FOURIER TRANSFORM (DFT) WITH MUSIC AND AUDIO APPLICATIONS

MATHEMATICS OF THE
DISCRETE FOURIER TRANSFORM (DFT)
WITH MUSIC AND AUDIO APPLICATIONS