**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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

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## 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
normalizedinverseDFT 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
Taylortime-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