Index

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20 dB boost : 15.2.1

3 dB boost : 15.2.1

coherenceml : 9.6.1

A-weighted dB scale : 15.2.2.4

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 : 15.2.2.5

dB per decade : 15.2.1

dB per octave : 15.2.1

dB properties : 15.2.1

dB scale : 15.2

dB SPL : 15.2.2.3

dBA : 15.2.2.4

dBm scale : 15.2.2.1

dBV scale : 15.2.2.2

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

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 : 15.2.2.3

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 : 15.2.2.4

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 : 15.2.2.3

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

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 : 15.2.2.3

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 : 15.2.2.3

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 : 15.2.2.3

Sound Pressure Level (SPL) : 15.2.2.3

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 : 15.2.2.3

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

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.