Index for this Document

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 frequency : 5.3.5

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 Full Scale (dBFS) : 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 | 6.8.3

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 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