**Search Mathematics of the DFT**

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DISCRETE FOURIER TRANSFORM (DFT)

WITH AUDIO APPLICATIONS

SECOND EDITION

**JULIUS O. SMITH III
**

- Preface

- Introduction to the DFT

- 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
- Direct Proof of De Moivre's Theorem
- Euler_Identity Problems

- Sinusoids and Exponentials
- Sinusoids
- Exponentials
- Complex Sinusoids
- Circular Motion
- Projection of Circular Motion
- Positive and Negative Frequencies
- Plotting Complex Sinusoids versus Frequency
- Sinusoidal Amplitude Modulation (AM)
- Sinusoidal Frequency Modulation (FM)
- Analytic Signals and Hilbert Transform Filters
- Generalized Complex Sinusoids
- Sampled Sinusoids
- Powers of
*z* - Phasors and Carriers
- Importance of Generalized Complex Sinusoids
- Comparing Analog and Digital Complex Planes

- Sinusoid Problems

- Geometric Signal Theory
- The DFT
- Signals as Vectors
- Vector Addition
- Vector Subtraction
- Scalar Multiplication
- Linear Combination of Vectors
- Linear Vector Space
- Signal Metrics
- The Inner Product
- Signal Reconstruction from Projections
- Signal Projection Problems

- The DFT
Derived
- Geometric Series
- Orthogonality of Sinusoids
- Orthogonality of the DFT Sinusoids
- Norm of the DFT Sinusoids
- An Orthonormal Sinusoidal Set
- The Discrete Fourier Transform (DFT)
- Frequencies in the ``Cracks''
- Spectral Bin Numbers
- Fourier Series Special Case
- Normalized DFT
- The Length 2 DFT
- Matrix Formulation of the DFT
- DFT Problems

- Fourier Theorems for the DFT
- The DFT and its Inverse Restated
- Signal Operators
- Even and Odd Functions
- Fourier Theorems
- Linearity
- Conjugation and Reversal
- Symmetry
- Shift Theorem
- Convolution Theorem
- Dual of the Convolution Theorem
- Correlation Theorem
- Power Theorem
- Rayleigh Energy Theorem (Parseval's Theorem)
- Stretch Theorem (Repeat Theorem)
- Downsampling Theorem (Aliasing Theorem)
- Zero Padding Theorem (Spectral Interpolation)
- Interpolation Theorems

- DFT Theorems Problems

- DFT Applications
- Why a DFT is usually called an FFT in practice
- Spectrum Analysis of a Sinusoid
- Spectrograms
- Filters and Convolution
- Correlation Analysis
- Power Spectral Density
- Coherence Function
- Recommended Further Reading

- Fast Fourier Transforms (FFT)
- Mixed-Radix Cooley-Tukey FFT
- Prime Factor Algorithm (PFA)
- Rader's FFT Algorithm for Prime Lengths
- Bluestein's FFT Algorithm
- Fast Transforms in Audio DSP
- Related Transforms
- FFT Software

- Continuous/Discrete Transforms

- Continuous Fourier Theorems

- Sampling Theory
- Introduction to Sampling
- Aliasing of Sampled Signals
- 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

- Digital Audio Number Systems

- Matrices

- Matlab/Octave Examples
- Complex Numbers in Matlab and Octave
- Factoring Polynomials in Matlab
- Geometric Signal Theory
- The DFT
- Spectrogram Computation

- Bibliography
- Index for this Document
- About this document ...

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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https://ccrma.stanford.edu/~jos/mdft/