NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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
- Music 320 Background Reader Mathematics of the Discrete Fourier Transform (DFT)
- 00. Table Of Content
- Index
- Preface
- Outline
- Introduction to the DFT
- The DFT
- Mathematics of the DFT
- DFT Math Outline
- Introduction to Complex Numbers
- Factoring a Polynomial with Real Roots
- The Quadratic Formula
- Complex Roots
- The Fundamental Theorem of Algebra
- Complex Numbers
- The Complex Plane
- More Notation and Terminology
- Elementary Relationships
- Euler's Formula
- De Moivre's Theorem
- Numerical Tools in Matlab
- Numerical Tools in Mathematica
- Proof of Euler's Identity
- Euler's Theorem
- Positive Integer Exponents
- Properties of Exponents
- The Exponent Zero
- Negative Exponents
- Rational Exponents
- Special Case: The Mth Roots of Unity
- Real Exponents
- A First Look at Taylor Series
- Imaginary Exponents
- Derivatives of f(x)=a^x
- Back to e
- Sidebar on Mathematica
- Back to e^(j theta)
- Informal Derivation of Taylor Series Expansion
- Derivation of Taylor Series Expansion with Remainder
- Formal Statement of Taylor's Theorem
- The Weierstrass (Polynomial) Approximation Theorem
- Further Notes on Differentiability of Audio Signals
- Logarithms, Decibels, and Number Systems
- Logarithms
- Changing the Base
- Logarithms of Negative and Imaginary Numbers
- Decibels
- Properties of DB Scales
- Specific DB Scales
- DBm Scale
- DBV Scale
- DB SPL
- DB for Display
- Dynamic Range
- Linear Number Systems for Digital Audio
- Pulse Code Modulation (PCM)
- Binary Integer Fixed-Point Numbers
- One's Complement Fixed-Point Format
- Two's Complement Fixed-Point Format
- General Formula for Two's-Complement, Integer Fixed-Point Numbers
- Little Endian Formula
- Fractional Binary Fixed-Point Numbers
- How Many Bits are Enough for Digital Audio?
- When Do We Have to Swap Bytes When Changing Computers?
- Logarithmic Number Systems for Digital Audio
- Floating-Point Numbers
- Logarithmic Fixed-Point Numbers
- Law Companding
- Appendix A: Round-Off Error Variance
- Appendix B: Electrical Engineering 101
- Sinusoids and Exponentials
- Sinusoids
- Example Sinusoids
- Why Sinusoids are Important
- In-Phase and Quadrature Sinusoidal Components
- Sinusoids at the Same Frequency
- Constructive and Destructive Interference of Sinusoids
- Exponentials
- Why Exponentials are Important
- Audio Decay Time
- Complex Sinusoids
- Circular Motion
- Projection of Circular Motion
- Positive and Negative Frequencies
- The Analytic Signal and Hilbert Transform Filters
- Generalized Complex Sinusoids
- Sampled Sinusoids
- Powers of
- Why (Generalized) Complex Sinusoids are Important
- Comparing Analog and Digital Complex Planes
- Mathematica for Selected Plots
- Acknowledgement
- Geometric Signal Theory
- Signals as Vectors
- An Example Vector View
- Vector Addition
- Vector Subtraction
- Signal Metrics
- Other Norms
- The Inner Product
- Linearity of the Inner Product
- Norm Induced by the Inner Product
- Cauchy-Schwarz Inequality
- Triangle Inequality
- Triangle Difference Inequality
- Vector Cosine
- Orthogonality
- The Pythagorean Theorem in N-Space
- Projection
- Signal Reconstruction from Projections
- An Example of Changing Coordinates in 2D
- General Conditions
- Gram-Schmidt Orthogonalization
- Appendix: Matlab Examples
- The Discrete Fourier Transform (DFT) Derived
- The DFT Derived
- Geometric Series
- Orthogonality of Sinusoids
- Orthogonality of the DFT Sinusoids
- Norm of the DFT Sinusoids
- An Orthonormal Sinusoidal Set
- Frequencies in the \
- Normalized DFT
- The Length 2 DFT
- Matrix Formulation of the DFT
- Matrices
- Matrix Multiplication
- The DFT Matrix
- Matlab Examples
- Figure 7.2
- Figure 7.3
- Fourier Theorems for the DFT
- The DFT and its Inverse
- Notation and Terminology
- Modulo Indexing, Periodic Extension
- Signal Operators
- Flip Operator
- Shift Operator
- Convolution
- Graphical Convolution
- Polynomial Multiplication
- Multiplication of Decimal Numbers
- Correlation
- Stretch Operator
- Zero Padding
- Repeat Operator
- Decimation Operator
- Alias Operator
- Even and Odd Functions
- The Fourier Theorems
- Linearity
- Conjugation and Reversal
- Symmetry
- Shift Theorem
- Linear Phase Terms
- Application of the Shift Theorem to FFT Windows
- Convolution Theorem
- Dual of the Convolution Theorem
- Correlation Theorem
- Power Theorem
- Rayleigh Energy Theorem (Parseval's Theorem)
- Stretch Theorem (Repeat Theorem)
- Decimation Theorem (Aliasing Theorem)
- Zero Padding Theorem
- Bandlimited Interpolation in Time
- Conclusions
- Appendix A: Linear Time-Invariant Filters and Convolution
- LTI Filters and the Convolution Theorem
- Appendix B: Introductory Statistical Signal Processing
- Cross-Correlation
- Applications of Cross-Correlation
- Autocorrelation
- Coherence
- Appendix C: Mathematica/Matlab Examples
- Appendix D: The Similarity Theorem
- Example Applications of the DFT
- Spectrum Analysis of a Sinusoid: Windowing, Zero-Padding, and the FFT
- 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
- Spectral Phase
- A Basic Tutorial on Sampling Theory
- Introduction
- Reconstruction from Samples--Pictorial Version
- Reconstruction from Samples--The Math
- Aliasing of Sampled Continuous-Time Signals
- Shannon's Sampling Theorem
- Figuring Out Sampling Theory by Playing Around with Complex Sinusoids
- What frequencies are representable by a geometric sequence?
- Recovering a Continuous-Time Signal from its Samples
- Method 1: Additive Synthesis
- Does it Work?
- Introduction to Digital Filter Analysis
- Motivating Example and Overview
- FIR Filters
- Convolution Representation
- Finiteness
- Causal FIR Filters
- Transfer Function
- Order
- The BiQuad Section
- Digital Filter Theory Summary
- Linearity and Time-Invariance
- Difference Equation
- Convolution Representation of LTI Filters
- Frequency Response
- Phase Delay and Group Delay
- Bibliography
- About this document
- Footnotes