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20 dB boost : 15.2.1
3 dB boost : 15.2.1
coherenceml : 9.6.1
A-weighted dB scale : 15.2.2.5
absolutely integrable : 11.2.1
ADSR envelope : 8.2.4.4
alias operator : 8.2.15
aliased sinc function : 7.7 | 8.4.13
aliasing : 8.2.15 | 13.2
aliasing operator : 8.2.15
aliasing theorem : 8.4.11
continuous time : 13.2.1
AM index : 5.3.5
amplitude of a sinusoid : 5.1
amplitude response : 9.3.2
analytic signal : 5.3.7
anti-aliasing lowpass filter : 8.2.15
anti-Hermitian : 8.4.2
antilinear : 6.9.1
antilogarithm, antilog : 15.1
antisymmetric functions : 8.3
Argand diagram : 3.6
attack level : 8.2.4.4
autocorrelation : 9.4.3
average power : 6.8 | 16.3
Banach space : 6.8.3
bandlimited : 4.8
bandlimited downsampling : 8.2.14
bandlimited interpolation : 8.4.13
of spectra : 8.2.11
time or frequency domain : 8.2.11
bandlimited signals cannot be time limited : 12.3
base (of a logarithm) : 15.1
beats : 5.3.5
bel : 15.2
Bessel function : 5.3.6.1
Bessel generating function : 5.3.6.1
bin number (DFT) : 7.8
bin numbers : 7.8
bits (binary digits) : 16.1.2
Blackman window : 9.1.4
Bluestein FFT : 10.4
cardinal sine : 13.1.2
carrier wave : 5.3.5 | 5.3.11.1
Cartesian coordinates : 3.6
Cauchy-Schwarz inequality : 6.9.3
causal : 5.3.12 | 8.2.4.3
causal signal : 8.2.8
causal signals : 8.2.8
causal signals, periodic : 8.2.8
causal zero padding : 8.2.9
causalperiodicsignals : 8.2.8
characteristic of a logarithm : 15.1
chirp signals : 10.4
chirp z transform algorithm : 10.4
circular convolution : 8.2.4
circular cross-correlation : 9.4.1
click removal : 9.4.2
CODEC : 16.2.3
coefficient of projection : 7.6
coherence function : 9.6 | 9.6
column vector : 17.1
comb filter : ^ | 5.1.5 | 5.1.5
common logarithm : 15.1
commutativity of convolution : 8.2.4.1
companding : 15.2.3
completing the square : 3.2
complex amplitude : 5.3.11.1
complex conjugate : 3.7
complex matrix : 17
complex matrix transpose : 17
complex multiplication : 3.5
complex numbers : 3 | 3.3 | 3.5 | 3.7
complex numbers in matlab : 18.1
complex plane : 3.6
complex roots of a polynomial : 3.3
complex vector space : 6.10.4
complexity of FFT : 10.1.2.1
conjugation and reversal symmetries (DFT) : 8.4.2
constant modulus : 5.3
continuous-time aliasing : 13.2.1
convolution : 8.2.4 | 8.2.4
ADSR example : 8.2.4.4
filter interpretation : 8.2.4.2
filter representation : 9.3
frequency domain : 8.4.6
graphical : 8.2.4.6
matched filter example : 8.2.4.5
smoother example : 8.2.4.3
convolution as a filter : 8.2.4.2
convolution commutativity : 8.2.4.1
convolution theorem : 8.4.5
convolution theorem dual : 8.4.6
correlation : 8.2.5
correlation analysis : 9.4
correlation operator : 8.2.5
correlation theorem : 8.4.7
cosine, two vectors : 6.9.6
cps : 5.1
critical bandwidth of hearing : 5.3.5
cross-correlation : 9.4.1
cross-correlation, circular : 9.4.1
cross-correlation, unbiased : 9.4.2
cross-covariance : 9.4.3
cross-spectral density : 9.4.1
cross-talk : 7.7
cubic spline : 14.5
cycles per second : 5.1
cyclic convolution : 8.2.4
dB for display : 15.2.2.6
dB per decade : 15.2.1
dB per octave : 15.2.1
dB properties : 15.2.1
dB relative to full scale : 15.2.2
dB scale : 15.2
dB SPL : 15.2.2.4
dBA : 15.2.2.5
dBFS : 15.2.2
dBm scale : 15.2.2.1
dBu scale : 15.2.2.2
dBV scale : 15.2.2.3
DCT : 10.6.1
de Moivre's theorem : 3.10
de Moivre's theorem, proof : 4.15
decibel : 15.2
decimal numbers : 16.1.2
decimation : 8.2.14
decimation in frequency : 10.1.1
decimation in time : 10.1.1 | 10.1.1
decimation theorem : 8.4.11
delta function : 11.2.2
DFT : 7
applications : 9
as a digital filter : 7.7
bin amplitude response : 18.4.2
definition : 2.1
math outline : 2.4
normalized : 7.10
DFT mathematics overview : 2.3
DFT matrix : 7.12 | 7.12
DFT matrix in matlab : 18.4.3
DFT sinusoids : 7.2.2 | 18.4.1
differentiability of audio signals : 14.6
differentiation theorem : 12.1
digit : 16.1.2
digital filter : 9.3
Discrete Cosine Transform (DCT) : 10.6.1
Discrete Fourier Transform (DFT) : 2.1 | 7 | 8.1
Discrete Time Fourier Transform (DTFT) : 11.1
downsampling operator : 8.2.14
downsampling theorem : 8.4.11
DTFT : 11.1
duality (Fourier) : 8.4.6
dynamic range : 15.2.3
dynamic range of magnetic tape : 15.2.3
energy : 15.2
energy of a signal : 6.8
energy theorem : 8.4.9 | 8.4.9
entire function : 5.3.6.1
essential singularity : 14.5
Euclidean norm : 6.8
Euler's Identity : 3.9 | 3.9 | 4 | 15.1.2
even and odd functions : 8.3
even functions : 8.3
exp(j theta) : 4.12
expected value : 16.3
exponent : 15.1
exponentials : 5.2
exponents
properties of : 4.3
rational : 4.6
factored form of a polynomial : 3.1
factoring a polynomial : 3.1
fast convolution : 8.4.5
Fast Fourier Transform (FFT) : 10
feedback FM : 5.3.6
FFT : 10
audio signal processing : 10.5
Bluestein FFT : 10.4
complexity : 10.1.2.1
decimation in time : 10.1.1
mixed-radix Cooley-Tukey : 10.1
number theory transform : 10.6.2
Rader FFT : 10.3
radix 2 : 10.1.2
software : 10.7
FFT notation : 8.1.1
FFT window : 7.7 | 9.1.4
filter : 8.2.4.2
fixed-point FFTs : 10.1.3
flip operator : 8.2.2 | 8.2.2
FM index : 5.3.6.2
FM modulation frequency : 5.3.6.1
FM synthesis spectrum : 5.3.6.2
folding frequency : 8.4.13
formants : 9.2.1
Fourier duality : 8.4.6
Fourier series : 7.9
Fourier series and the DFT : 11.3
Fourier series coefficient : 11.3
Fourier symmetries : 8.4.3
Fourier theorems : 8 | 8.4
Fourier theorems (DFT) : 8 | 8.4
aliasing theorem : 8.4.11
convolution theorem : 8.4.5
convolution theorem dual : 8.4.6
correlation theorem : 8.4.7
downsampling theorem : 8.4.11
energy theorem (Rayleigh) : 8.4.9
Parseval's theorem : 8.4.8
periodic interpolation (in time) : 8.4.13
power theorem : 8.4.8
shift theorem : 8.4.4
stretch (repeat) theorem : 8.4.10
zero-padding (spectral interpolation) theorem : 8.4.12
Fourier transform : 11.2
Fourier transform cases : 11
Fourier transform existence : 11.2.1
Fourier Transform theorems : 12
continuous-time aliasing : 13.2.1
differentiation : 12.1
scaling or similarity : 12.2
uncertainty principle : 12.3
frame : 8.2.10
frequency bin : 7.8
frequency domain : 5.1.6
frequency modulation : 5.3.6 | 5.3.6
frequency resolution : 5.3.5
frequency response : 9.3.1
frequency-domain aliasing : 8.2.15 | 8.2.15
FS (Fourier Series) : 11.3
FT (Fourier Transform) : 11.2
fundamental theorem of algebra : 3.4
Gaussian function : 12.3.1
generalized function : 11.2.2
generating function : 5.3.6.1
geometric sequence : 7.1
geometric sequence frequencies : 13.4
geometric series : 7.1 | 7.1
geometric signal theory : 6
Gibb's phenomenon : 7.7
Good-Thomas FFT algorithm : 10.2
Gram-Schmidt orthogonalization : 6.10.6
graphical convolution : 8.2.4.6
half-open interval : 8.1
Hann window : 9.1.5
Hanning window : 9.1.5
Heisenberg uncertainty principle : 12.3.1
Hermitian spectra : 8.4.3
Hermitian symmetry : 8.4.2
Hermitian transpose : 6.9 | 7.12 | 17
Hertz : 5.1
hexadecimal : 16.1.2
Hilbert transform : 5.3.7
Hz : 5.1
ideal lowpass filter : 8.4.13.1
idempotent : 18.3.5
identity matrix : 17.1
IDFT : 2.2 | 8.1
imaginary exponents : 4.9
imaginary part : 3.5
impulse response : 8.2.4.2 | 9.3
impulse signal : 8.2.4.2 | 9.3
impulse train : 11.3.1
impulse, continuous time : 11.2.2
impulse-train signal : 8.2.4.2
indicator function : 8.4.4.2
inner product : 6.9
inner product in matlab : 18.3.3
integrable function : 11.2.1
intensity : 15.2
intensity level : 15.2.2.4
interpolation kernel : 13.1.2
interpolation operator : 8.2.12 | 8.2.12
inverse DFT : 2.2 | 8.1
inverse DFT matrix : 7.12
irrational number : 4.7
ITU-R 468 noise weighting : 15.2.2.5
just-noticeable difference (JND) : 15.2
lag : 8.2.5
lagged product : 8.2.5
linear algebra : 6.10.6
linear combination : 5.3.11.2 | 6.6
linear number systems for digital audio : 16.1
linear phase : 8.4.4.2
linear phase FFT windows : 8.4.4.4
linear phase signals : 8.4.4.2
linear phase term : 8.4.4 | 8.4.4.1 | 8.4.4.1
linear transformation : 17.1
linear vector space : 6.7
linear, time-invariant filters and convolution : 9.3
linearity of the DFT : 8.4.1
linearly dependent : 6.10.4
linearly independent : 6.10.2
logarithm : 15.1
logarithmic number systems for audio : 16.2
logarithms
changing the base : 15.1.1
of imaginary numbers : 15.1.2
loudness : 15.2.2.4
lowpass filter (ideal) : 8.4.13.1
Lp norms : 6.8.1
machine epsilon : 18.3.5.2
Maclaurin series : 14.3
magnitude of a sinusoid : 5.1
magnitude spectrum : 5.1.6
main lobe : 7.7
mantissa : 15.1
matched filter : 8.2.4.5 | 8.2.4.5
matlab listings
coherence function : 9.6.1
complex numbers : 18.1
DFT bin response : 18.4.2
DFT matrix : 18.4.3
factoring polynomials : 18.2
inner product : 18.3.3
orthogonalization : 18.3.6
signal energy, power : 18.3.2.1
signal metrics : 18.3.2
spectrogram : 18.5
subspace projection : 18.3.5
vector cosine : 18.3.4
Matlab/Octave examples : 18
matrix : 17
matrix multiplication : 17.1
matrix transpose : 17
maximally flat : 14.2
mean of a random variable : 16.3
mean of a signal : 6.8 | 16.3
mean square : 6.8 | 16.3
mean value : 16.3
mixed radix : 10.1
mixed-radix FFT : 10.1
modulation index : 5.3.5
modulo : 8.1.2
modulo indexing : 8.1.2
moments of a function : 16.3
monic polynomial : 3.1
Mth roots of unity : 4.13
mu-law coding : 16.2.3
multiplication in the time domain is convolution in the frequency domain : 8.4.6
multiplication of large numbers : 15.1
multiplying two numbers convolves their digits : 8.2.4.8
natural logarithm : 15.1
NDFT : 7.10
non-removable singularity : 14.5
nonlinear system of equations : 3.1
norm of DFT Sinusoids : 7.4
norm properties : 6.8.2
normalized inverse DFT matrix : 7.12
normalized DFT : 7.10 | 8.4.9
normalized DFT matrix : 7.12
normalized DFT sinusoids : 7.5 | 7.5 | 7.10 | 8.4.8.1
normalized frequency : 8.1
normalized radian frequency : 11.1
normed : 6.8.3
normed linear vector space : 6.8.3
Nth roots of unity : 7.2.1
number systems for digital audio : 16
byte swapping : 16.1.5
fixed point
one's complement : 16.1.2.1
two's complement : 16.1.2.2
floating point : 16.2.1
fractional fixed point : 16.1.3
how many bits are enough : 16.1.4
logarithmic : 16.2
logarithmic fixed point : 16.2.2
mu law : 16.2.3
PCM : 16.1.1
number theoretic transform : 10.6.2
Nyquist limit : 8.4.13 | 13
Nyquist rate : 8.4.13 | 13
Nyquist sampling theorem : 13
octal : 16.1.2
Octave : 18
Octave Symbolic Manipulation Toolbox : 4.7 | 4.11
odd and even functions : 8.3
Ohm's law : 15.3
operator notation : 8.2.1
operators
alias : 8.2.15
downsampling : 8.2.14
flip : 8.2.2
interpolation : 8.2.12
repeat : 8.2.13
shift : 8.2.3
stretch : 8.2.6
orthogonal basis computation in matlab : 18.3.6
orthogonal complement : 18.3.5
orthogonal projection : 6.9.9
orthogonality : 6.9.7 | 7.12
orthogonality of DFT sinusoids : 7.3
orthogonality of sinusoids : 7.2
orthonormal : 7.12
overlap-add : 8.4.13.2
Padé approximation : 14.2
parabola : 3.2
Parseval's theorem : 8.4.8
PCM : 16.1.1
peak amplitude : 5.1
peak meter : 15.2.2.2
periodic : 8.1.2 | 11.3
periodic extension : 7.7 | 8.1.2
periodic interpolation : 8.4.13
periodogram method for power spectrum estimation : 9.5
phase : 5.1
phase modulation : 5.3.6
phase negation : 8.4.2
phase response : 9.3.3
phasor : 5.3.11.1 | 5.3.11.1
phasor analysis : 5.3.6.2
phon amplitude scale : 15.2.2.4
phoneme : 9.2.1
piecewise constant-phase spectra : 8.4.3
pitch shifting : 9.4.2
polar coordinates : 3.6
polar form : 3.9
polar form of a complex number : 4.13
polynomial
factoring : 3.1
roots : 3.3
polynomial approximation : 14.2
polynomial multiplication : 8.2.4.7
positive and negative frequencies : 5.3.3
positive-frequency sinusoid : 5.3.1
power : 15.2
power spectral density : 9.4.3
power spectral density estimation : 9.5
power spectrum : 9.4.3
power theorem : 8.4.8
power theorem, normalized DFT : 8.4.8.1
pressure : 15.2
prime factor algorithm (PFA) : 10.2
primitive root of unity : 4.14 | 7.2.1
probability density function : 16.3
probability distribution : 16.3
projection error : 6.9.9
projection in matlab : 18.3.5
projection matrix : 18.3.5
projection of signals : 6.9.9
projection sum : 6.10
Pythagorean theorem in N-Space : 6.9.8
quadratic formula : 3.2 | 3.2
Rader FFT : 10.3
radian frequency : 5.1
radix 2 FFT : 10.1.2 | 10.1.2
random variable : 16.3
rational number : 4.6
Rayleigh's energy theorem : 8.4.9
re-index : 10.2
real part : 3.5
real vector space : 6.10.4
rectangular form : 3.9
rectangular window : 7.7 | 8.4.13.1
rectilinear coordinates : 3.6
remainder term : 14.1 | 14.3
removable singularity : 14.5
repeat (stretch) theorem : 8.4.10
repeat operator : 8.2.13
resolution of spectral peaks : 5.3.5
ring modulator : 5.3.5
rms level : 16.3
root mean square : 6.8
roots of a polynomial : 3.1 | 3.3
roots of unity : 4.14 | 4.14 | 7.2.1
round-off error variance : 16.3
row vector : 17.1
sample circular cross-covariance : 9.4.3
sample coherence function : 9.6
sample mean : 6.8 | 16.3
sample variance : 6.8 | 16.3
sampling rate : 8.4.13
sampling theorem : 13 | 13.3
scalar : 6.5
scalar multiplication : 6.5
scale factor : 6.5
scaling theorem : 12.2
Schwarz inequality : 6.9.3
second central moment : 16.3
second moments of a signal : 12.3.1
sensation level : 15.2.2.4
Shannon sampling theorem : 13
shift operator : 8.2.3 | 8.2.3
shift theorem : 8.4.4
shift theorem and FFT windows : 8.4.4.4
side band : 5.3.5
sidelobes : 7.7
sifting property : 11.2.2
signal dynamic range : 15.2.3
signal energy : 6.8
signal metrics : 6.8
signal mix : 6.6
signal operators : 8.2
signal projection : 6.9.9
similarity theorem : 12.2
sinc function : 7.7 | 13.1.2
sinc function, aliased : 7.7
sinusoidal amplitude modulation : 5.3.5
sinusoids and exponentials : 5
sinusoids at the same frequency : 5.1.4
sinusoids, importance of : 5.1.2
skew-Hermitian : 8.4.2
smoothing example : 8.2.4.3 | 8.2.4.4
smoothing, frequency domain : 8.4.6
sone amplitude scale : 15.2.2.4
Sound Pressure Level (SPL) : 15.2.2.4
spectral graphs : 5.3.4
spectral interpolation : 7.7 | 8.2.11 | 8.4.12
spectral leakage : 7.7
spectral magnitude representation : 5.1.6
spectral plot : 5.3.4
spectral power : 8.4.8
spectral representation : 5.1.6 | 5.3.4
spectrogram : 9.2
spectrogram in matlab : 18.5
spectrum : 7.6 | 8.1 | 8.2.11
spectrum complex conjugate : 8.4.2
speech spectrogram : 9.2.1
SPL : 15.2.2.4
split radix : 10.1.2
square integrable : 11.2.1
square matrix : 17
standard deviation : 16.3
statistical signal processing : 16.3
Stone-Weierstrass polynomial approximation theorem : 14.4
stretch (repeat) theorem : 8.4.10
stretch operator : 8.2.6
subspace : 6.10.4
subspace projection : 18.3.5
sum and difference frequencies : 5.3.5
sustain level : 8.2.4.4
symmetric functions : 8.3
system identification : 9.4.5 | 9.6
Taylor series : 4.8 | 14
formal statement : 14.3
remainder bound : 14.2
remainder term : 14.1
simplified derivation : 14.1
temporal energy density : 6.8
theorems for the DFT : 8.4
time constant : 5.2
time scale modification : 9.4.2
time-bandwidth product : 12.3.3
time-domain aliasing : 8.2.15
time-limited signals : 12.3.2
Toeplitz matrix : 17.1
transcendental number : 4.11
transform pair : 8.1.1
transpose of a complex matrix : 17
transpose of a matrix product : 17.1
tremolo effect : 5.3.5
twiddle factors : 10.1.1
unbiased cross-correlation : 9.4.2
uncertainty principle : 12.3
unit pulse signal : 9.3
unitary : 7.12
variance : 6.8
variance of a random variable : 16.3
vector addition : 6.3
vector cosine : 6.9.6
vector cosine in matlab : 18.3.4
vector representation of signals : 6.2
vector space : 6.7
vector subtraction : 6.4
virtual analog : 8.2.4.4
voltage transfer : 15.2.2.2
Weierstrass polynomial approximation theorem : 14.4
Welch's method : 9.5
window : 7.7
windowing in the time domain equals smoothing in the frequency domain : 8.4.6
Winograd transform : 10.2
zero padding : 8.2.7 | 8.4.12 | 9.1
zero padding example : 9.1.3
zero padding in the time domain equals ideal interpolation in the frequency domain : 8.2.11
zero padding, causal : 8.2.9
zero padding, spectral : 8.4.13
zero phase signal : 8.4.3
zero phase signals : 8.4.4.3
zero-padding theorem : 8.4.12
zero-phase signal : 8.4.4.3
zeros of a polynomial : 3.1


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``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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA