NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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
<< Previous page TOC INDEX Next page >>
Index
- 20 dB boost
- Properties of DB Scales
- 3 dB boost
- Properties of DB Scales
- convolution
- Convolution Representation of LTI
- Aliasing
- Alias Operator
- aliasing operator
- Alias Operator
- amplitude response
- LTI Filters and the | Frequency Response | Frequency Response
- anti-aliasing lowpass filter
- Alias Operator
- antilog
- Logarithms
- antilogarithm
- Logarithms
- antisymmetric functions
- Even and Odd Functions
- average power
- Appendix A: Round-Off Error
- base
- Logarithms
- base
logarithm
- Logarithms
- base
logarithm
- Logarithms
- bel
- Decibels
- bin number
- Frequencies in the ''Cracks''
- binary
- Binary Integer Fixed-Point Numbers
- binary digits
- Binary Integer Fixed-Point Numbers
- bits
- Binary Integer Fixed-Point Numbers
- canonical with respect to delay
- Order
- causal
- Causal FIR Filters | Difference Equation
- characteristic of the logarithm
- Logarithms
- coefficient of projection
- The Discrete Fourier Transform
- column-vector
- Matrices
- comb-filter
- Constructive and Destructive Interference
- common logarithm
- Logarithms
- commutativity of convolution
- Convolution
- companding
- Dynamic Range
- complex matrix
- Matrices
- complex matrix transpose
- Matrices
- convolution
- Convolution
- convolution operator ''
''
- Convolution Representation
- convolution representation
- Appendix A: Linear Time-Invariant
- correlation operator
- Correlation
- cross-talk
- Frequencies in the ''Cracks''
- cyclic convolution
- Convolution
- dB scale
- Decibels
- decibel
- Decibels
- decimal numbers
- Binary Integer Fixed-Point Numbers
- DFT as a digital filter
- Frequencies in the ''Cracks''
- DFT matrix
- The DFT Matrix
- difference equation
- Difference Equation
- difference equation coefficients
- Difference Equation
- digit
- Binary Integer Fixed-Point Numbers
- digital filter
- Appendix A: Linear Time-Invariant
- Discrete Fourier Transform
- The DFT and its
- Discrete Fourier Transform (DFT)
- The DFT
- dynamic range
- Dynamic Range
- dynamic range of magnetic tape
- Dynamic Range
- energy
- Decibels
- Euler's Formula
- Euler's Formula
- Euler's Theorem
- Euler's Theorem
- even functions
- Even and Odd Functions
- expected value
- Appendix A: Round-Off Error
- fast convolution
- Convolution Theorem
- filter
- Digital Filter Theory Summary
- filter coefficients
- Difference Equation
- filter delay for amplitude envelopes
- Phase Delay and Group
- filter order
- Difference Equation
- filter power response
- Frequency Response
- filter time delay
- Phase Delay and Group
- finite-impulse-response filter
- FIR Filters
- FIR filter
- FIR Filters
- flip operator
- Flip Operator
- Fourier Dual
- Dual of the Convolution
- frequency bin
- Frequencies in the ''Cracks''
- frequency response
- LTI Filters and the | Frequency Response
- frequency-domain aliasing
- Alias Operator | Alias Operator
- geometric sequence
- Geometric Series
- geometric series
- Geometric Series
- group delay
- Phase Delay and Group
- Hermitian spectra
- Symmetry
- Hermitian symmetry
- Conjugation and Reversal
- Hermitian transpose
- Matrices
- hex
- Binary Integer Fixed-Point Numbers
- hexadecimal
- Binary Integer Fixed-Point Numbers
- ideal bandlimited interpolation
- Zero Padding Theorem | Bandlimited Interpolation in Time
- ideal lowpass filtering operation in the frequency domain
- Bandlimited Interpolation in Time
- identity matrix
- Matrix Multiplication
- IDFT
- The DFT and its
- impulse response
- Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
- impulse signal
- Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
- indicator function
- Linear Phase Terms
- Intensity
- Decibels
- intensity level
- DB SPL
- interpolation operator
- Zero Padding Theorem
- inverse DFT
- The DFT | The DFT and its
- inverse DFT matrix
- The DFT Matrix
- irrational number
- Real Exponents
- JND
- Decibels
- just-noticeable difference
- Decibels
- lag
- Correlation
- lagged product
- Correlation
- length
even rectangular windowing operation
- Bandlimited Interpolation in Time
- linear
- Linearity and Time-Invariance
- linear filter
- Linearity and Time-Invariance
- linear phase FFT windows
- Application of the Shift
- linear phase signal
- Linear Phase Terms
- linear phase term
- Shift Theorem | Linear Phase Terms
- linear transformation
- Matrix Multiplication
- logarithm
- Logarithms
- loudness
- DB SPL
- LTI filters
- Linearity and Time-Invariance
- main lobe
- Frequencies in the ''Cracks''
- mantissa
- Logarithms
- matrix
- Matrices
- matrix columns
- Matrices
- matrix multiplication
- Matrix Multiplication
- matrix rows
- Matrices
- matrix transpose
- Matrices
- mean
- Appendix A: Round-Off Error
- mean of a random variable
- Appendix A: Round-Off Error
- mean of a signal
- Appendix A: Round-Off Error
- mean square
- Appendix A: Round-Off Error
- modulo
- Modulo Indexing, Periodic Extension
- moments
- Appendix A: Round-Off Error
- multiplication of large numbers
- Logarithms
- multiplying two numbers convolves their digits
- Multiplication of Decimal Numbers
- natural logarithm
- Logarithms
- non-commutativity of matrix multiplication
- Matrix Multiplication
- normalized inverse DFT matrix
- The DFT Matrix
- normalized DFT matrix
- The DFT Matrix
- normalized DFT sinusoids
- An Orthonormal Sinusoidal Set | Normalized DFT
- normalized frequency
- The DFT and its
- octal
- Binary Integer Fixed-Point Numbers
- odd functions
- Even and Odd Functions
- order
- Order
- orthogonal
- The DFT Matrix
- orthogonality of sinusoids
- Orthogonality of Sinusoids
- orthonormal
- The DFT Matrix
- PCM
- Pulse Code Modulation (PCM)
- periodic
- Modulo Indexing, Periodic Extension
- periodic extension
- Frequencies in the ''Cracks'' | Modulo Indexing, Periodic Extension
- phase delay
- Phase Delay and Group
- phase response
- LTI Filters and the | Frequency Response | Frequency Response
- phase response at zero frequency
- Phase Delay and Group
- phon amplitude scale
- DB SPL
- polar form
- Rational Exponents
- polynomial approximation
- Derivation of Taylor Series
- power
- Decibels
- pressure
- Decibels
- primitive
th root of unity
- Orthogonality of Sinusoids
- rational
- Rational Exponents
- real filter
- Difference Equation
- rectangular window
- Frequencies in the ''Cracks''
- rms level
- Appendix A: Round-Off Error
- roots of unity
- Orthogonality of Sinusoids
- row-vector
- Matrices
- sample mean
- Appendix A: Round-Off Error
- sample variance
- Appendix A: Round-Off Error
- second central moment
- Appendix A: Round-Off Error
- sensation level
- DB SPL
- shift operator
- Shift Operator
- sidelobes
- Frequencies in the ''Cracks''
- signal
- Digital Filter Theory Summary
- signal dynamic range
- Dynamic Range
- similarity theorem
- Appendix D: The Similarity
- sinc function
- Frequencies in the ''Cracks''
- skew-Hermitian
- Conjugation and Reversal
- sone amplitude scale
- DB SPL
- Sound Pressure Level
- DB SPL
- spectral leakage
- Frequencies in the ''Cracks''
- spectrum
- The Discrete Fourier Transform | The DFT and its
- SPL
- DB SPL
- square matrix
- Matrices
- stable
- Difference Equation
- standard deviation
- Appendix A: Round-Off Error
- Stretch Operator
- no title | Stretch Operator
- symmetric functions
- Even and Odd Functions
- Taylor Series Expansion
- no title | Informal Derivation of Taylor
- time-domain aliasing
- Alias Operator
- time-invariant
- Linearity and Time-Invariance
- Toeplitz
- Matrix Multiplication
- transfer function
- Transfer Function | Convolution Representation of LTI
- transform pair
- Notation and Terminology
- transpose of a matrix product
- Matrix Multiplication
- unilateral
transform
- Convolution Representation of LTI
- unit pulse signal
- Appendix A: Linear Time-Invariant
- unitary
- The DFT Matrix
- variance
- Appendix A: Round-Off Error
- window
- Frequencies in the ''Cracks''
- Zero padding
- Zero Padding
- zero phase signal
- Symmetry