NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
- 00. Table Of Content
- A Basic Tutorial on Sampling Theory
- A First Look at Taylor Series
- About this document
- Acknowledgement
- Alias Operator
- Aliasing of Sampled Continuous-Time Signals
- An Example of Changing Coordinates in 2D
- An Example Vector View
- An Orthonormal Sinusoidal Set
- Appendix A: Linear Time-Invariant Filters and Convolution
- Appendix A: Round-Off Error Variance
- Appendix B: Electrical Engineering 101
- Appendix B: Introductory Statistical Signal Processing
- Appendix C: Mathematica/Matlab Examples
- Appendix D: The Similarity Theorem
- Appendix: Matlab Examples
- Application of the Shift Theorem to FFT Windows
- Applications of Cross-Correlation
- Audio Decay Time
- Autocorrelation
- Back to e
- Back to e^(j theta)
- Bandlimited Interpolation in Time
- Bibliography
- Binary Integer Fixed-Point Numbers
- Cauchy-Schwarz Inequality
- Causal FIR Filters
- Changing the Base
- Circular Motion
- Coherence
- Comparing Analog and Digital Complex Planes
- Complex Numbers
- Complex Roots
- Complex Sinusoids
- Conclusions
- Conjugation and Reversal
- Constructive and Destructive Interference of Sinusoids
- Convolution
- Convolution Representation
- Convolution Representation of LTI Filters
- Convolution Theorem
- Correlation
- Correlation Theorem
- Cross-Correlation
- DB for Display
- DB SPL
- DBm Scale
- DBV Scale
- De Moivre's Theorem
- Decibels
- Decimation Operator
- Decimation Theorem (Aliasing Theorem)
- Derivation of Taylor Series Expansion with Remainder
- Derivatives of f(x)=a^x
- DFT Math Outline
- Difference Equation
- Digital Filter Theory Summary
- Does it Work?
- Dual of the Convolution Theorem
- Dynamic Range
- Elementary Relationships
- Euler's Formula
- Euler's Theorem
- Even and Odd Functions
- Example 1: FFT of a Simple Sinusoid
- Example 2: FFT of a Not-So-Simple Sinusoid
- Example 3: FFT of a Zero-Padded Sinusoid
- Example 4: Blackman Window
- Example 5: Use of the Blackman Window
- Example 6: Hanning-Windowed Complex Sinusoid
- Example Applications of the DFT
- Example Sinusoids
- Exponentials
- Factoring a Polynomial with Real Roots
- Figure 7.2
- Figure 7.3
- Figuring Out Sampling Theory by Playing Around with Complex Sinusoids
- Finiteness
- FIR Filters
- Flip Operator
- Floating-Point Numbers
- Footnotes
- Formal Statement of Taylor's Theorem
- Fourier Theorems for the DFT
- Fractional Binary Fixed-Point Numbers
- Frequencies in the \
- Frequency Response
- Further Notes on Differentiability of Audio Signals
- General Conditions
- General Formula for Two's-Complement, Integer Fixed-Point Numbers
- Generalized Complex Sinusoids
- Geometric Series
- Geometric Signal Theory
- Gram-Schmidt Orthogonalization
- Graphical Convolution
- How Many Bits are Enough for Digital Audio?
- Imaginary Exponents
- In-Phase and Quadrature Sinusoidal Components
- Index
- Informal Derivation of Taylor Series Expansion
- Introduction
- Introduction to Complex Numbers
- Introduction to Digital Filter Analysis
- Introduction to the DFT
- Law Companding
- Linear Number Systems for Digital Audio
- Linear Phase Terms
- Linearity
- Linearity and Time-Invariance
- Linearity of the Inner Product
- Little Endian Formula
- Logarithmic Fixed-Point Numbers
- Logarithmic Number Systems for Digital Audio
- Logarithms
- Logarithms of Negative and Imaginary Numbers
- Logarithms, Decibels, and Number Systems
- LTI Filters and the Convolution Theorem
- Mathematica for Selected Plots
- Mathematics of the DFT
- Matlab Examples
- Matrices
- Matrix Formulation of the DFT
- Matrix Multiplication
- Method 1: Additive Synthesis
- Modulo Indexing, Periodic Extension
- More Notation and Terminology
- Motivating Example and Overview
- Multiplication of Decimal Numbers
- Music 320 Background Reader Mathematics of the Discrete Fourier Transform (DFT)
- Negative Exponents
- Norm Induced by the Inner Product
- Norm of the DFT Sinusoids
- Normalized DFT
- Notation and Terminology
- Numerical Tools in Mathematica
- Numerical Tools in Matlab
- One's Complement Fixed-Point Format
- Order
- Orthogonality
- Orthogonality of Sinusoids
- Orthogonality of the DFT Sinusoids
- Other Norms
- Outline
- Phase Delay and Group Delay
- Polynomial Multiplication
- Positive and Negative Frequencies
- Positive Integer Exponents
- Power Theorem
- Powers of
- Preface
- Projection
- Projection of Circular Motion
- Proof of Euler's Identity
- Properties of DB Scales
- Properties of Exponents
- Pulse Code Modulation (PCM)
- Rational Exponents
- Rayleigh Energy Theorem (Parseval's Theorem)
- Real Exponents
- Reconstruction from Samples--Pictorial Version
- Reconstruction from Samples--The Math
- Recovering a Continuous-Time Signal from its Samples
- Repeat Operator
- Sampled Sinusoids
- Shannon's Sampling Theorem
- Shift Operator
- Shift Theorem
- Sidebar on Mathematica
- Signal Metrics
- Signal Operators
- Signal Reconstruction from Projections
- Signals as Vectors
- Sinusoids
- Sinusoids and Exponentials
- Sinusoids at the Same Frequency
- Special Case: The Mth Roots of Unity
- Specific DB Scales
- Spectral Phase
- Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and the FFT
- Stretch Operator
- Stretch Theorem (Repeat Theorem)
- Symmetry
- The Analytic Signal and Hilbert Transform Filters
- The BiQuad Section
- The Complex Plane
- The DFT
- The DFT and its Inverse
- The DFT Derived
- The DFT Matrix
- The Discrete Fourier Transform (DFT) Derived
- The Exponent Zero
- The Fourier Theorems
- The Fundamental Theorem of Algebra
- The Inner Product
- The Length 2 DFT
- The Pythagorean Theorem in N-Space
- The Quadratic Formula
- The Weierstrass (Polynomial) Approximation Theorem
- Transfer Function
- Triangle Difference Inequality
- Triangle Inequality
- Two's Complement Fixed-Point Format
- Vector Addition
- Vector Cosine
- Vector Subtraction
- What frequencies are representable by a geometric sequence?
- When Do We Have to Swap Bytes When Changing Computers?
- Why (Generalized) Complex Sinusoids are Important
- Why Exponentials are Important
- Why Sinusoids are Important
- Zero Padding
- Zero Padding Theorem