Index

20 dB boost : 12.2.1

3 dB boost : 12.2.1

absolutely integrable : 16.2.1

alias operator : 8.2.11

aliased sinc function : 7.7

Aliasing : 8.2.11 | 18.2

aliasing operator : 8.2.11

amplitude of a sinusoid : 5.1

amplitude response : 15.1

anti-aliasing lowpass filter : 8.2.11

antilog : 12.1

antilogarithm : 12.1

antisymmetric functions : 8.3

Argand diagram : 3.6

average power : 6.5 | 13.3

Banach spaces : 6.5.3

bandlimited interpolation : 8.4.13

bandlimited interpolation in the frequency domain : 8.2.6

bandlimited interpolation in the time domain : 8.2.6

bandlimited interpolation of spectra : 8.2.7

bandlimited signals cannot be time limited : 17.3

base : 12.1

bel : 12.2

bin (discrete Fourier transform) : 7.7

bin number : 7.7

binary : 13.1.2

binary digits : 13.1.2

bits : 13.1.2

carrier : 5.3.8.1

Cartesian coordinates : 3.6

characteristic of the logarithm : 12.1

circular cross-correlation : 15.2.1

CODEC : 13.2.3

coefficient of projection : 7.6

coherence function : 15.4

column-vector : 14

comb-filter : 5.1.5

common logarithm : 12.1

commutativity of convolution : 8.2.3

companding : 12.2.3 | 13.2.3

complex amplitude : 5.3.8.1

complex conjugate : 3.7

complex matrix : 14

complex matrix transpose : 14

complex multiplication : 3.5

complex number : 3.5

complex numbers : 3.3

notation and terminology : 3.7

complex numbers in matlab : 20.1

complex plane : 3.6

conjugation and reversal symmetries : 8.4.2

constant modulus : 5.3

convolution : 8.2.3 | 8.2.3

graphical : 8.2.3.1

convolution in the frequency domain : 8.4.6

convolution representation : 15.1

convolution theorem : 8.4.5

convolution theorem dual : 8.4.6

correlation : 8.2.4

correlation analysis : 15.2

correlation operator : 8.2.4

correlation theorem : 8.4.7

cross-correlation, circular : 15.2.1

cross-covariance : 15.3

cross-spectral density : 15.2.1

cross-talk : 7.7

cubic spline : 11.5

cyclic convolution : 8.2.3

dB per decade : 12.2.1

dB per octave : 12.2.1

dB scale : 12.2

De Moivre's theorem : 3.10

decibel : 12.2

decibels : 12.2

decimal numbers : 13.1.2

decimation in frequency : 19.1

decimation in time : 19.1 | 19.1

decimation theorem : 8.4.11

delta function : 16.2.2

DFT

normalized : 7.8

DFT applications : 9

DFT as a digital filter : 7.7

DFT bin amplitude response : 20.4.2

DFT matrix : 7.10 | 7.10

DFT matrix in matlab : 20.4.3

dft sinusoids : 7.2.2 | 20.4.1

differentiability of audio signals : 11.6

differentiation theorem : 17.1

digit : 13.1.2

digital filter : 15.1

discrete cosine transform (DCT) : 19.3.1

Discrete Fourier Transform : 8.1

Discrete Fourier Transform (DFT) : 2.1

Discrete Fourier Transform (DFT) derived : 7

discrete time Fourier transform (DTFT) : 16.1

downsampling operator : 8.2.10

downsampling theorem (aliasing theorem) : 8.4.11

dynamic range : 12.2.3

dynamic range of magnetic tape : 12.2.3

energy : 12.2

energy of a signal : 6.5

energy theorem : 8.4.9 | 8.4.9

essential singularity : 11.5

Euler's Formula : 3.9

Euler's identity : 4

Euler's Theorem : 4.1

even functions : 8.3

exp(j theta) : 4.12

expected value : 13.3

exponents

properties of : 4.3

rational : 4.6

factored form of a polynomial : 3.1

fast convolution : 8.4.5

fast Fourier transform : 19

Fast Fourier Transform (FFT) : 19.2

FFT : 19.2

FFT notation : 8.1.1

FFT software : 19.4

flip operator : 8.2.1 | 8.2.1

Fourier series : 16.3 | 16.3

Fourier series and the DFT : 16.3.1

Fourier series coefficient : 16.3

Fourier symmetries : 8.4.3

Fourier theorems : 8 | 8.4

differentiation : 17.1

scaling or similarity : 17.2

uncertainty principle : 17.3

Fourier theorems (DFT) : 8.4

convolution theorem : 8.4.5

convolution theorem dual : 8.4.6

correlation theorem : 8.4.7

downsampling (aliasing) theorem : 8.4.11

energy theorem (Rayleigh) : 8.4.9

interpolation theorem (in time) : 8.4.13

Parseval's theorem : 8.4.8

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 theorems for the DFT : 8

Fourier theorems in continuous time : 17

Fourier transform : 16.2

Fourier transform existence : 16.2.1

Fourier transforms for continuous/discrete time/frequency : 16

frequency bin : 7.7

frequency response : 15.1

frequency-domain aliasing : 8.2.11 | 8.2.11

fundamental theorem of algebra : 3.4

Gaussian function : 17.3.1

generalized function : 16.2.2

geometric sequence : 7.1

geometric series : 7.1 | 7.1

geometric signal theory : 6

graphical convolution : 8.2.3.1

Heisenberg uncertainty principle : 17.3.1

Hermitian spectra : 8.4.3

Hermitian symmetry : 8.4.2

Hermitian transpose : 7.10 | 14

hex : 13.1.2

hexadecimal : 13.1.2

ideal bandlimited interpolation : 8.2.7 | 8.4.13

ideal bandlimited interpolation in the time domain : 8.4.13

ideal lowpass filtering operation in the frequency domain : 8.4.13

identity matrix : 14.1

IDFT : 2.2 | 8.1

imaginary part : 3.5

impulse response : 15.1

impulse signal : 15.1

impulse train : 16.3.1

impulse, continuous time : 16.2.2

indicator function : 8.4.4.1

instantaneous frequency : 5.1

instantaneous phase : 5.1

Intensity : 12.2

intensity level : 12.2.2.3

interpolation operator : 8.2.8 | 8.2.8

interpolation theorem : 8.4.13

inverse DFT : 2.2 | 8.1

inverse DFT matrix : 7.10

irrational number : 4.7

JND : 12.2

just-noticeable difference : 12.2

lag : 8.2.4

lagged product : 8.2.4

linear combination : 5.3.8.2

linear number systems for digital audio : 13.1

linear phase FFT windows : 8.4.4.2

linear phase signal : 8.4.4.1

linear phase term : 8.4.4 | 8.4.4.1

linear phase terms : 8.4.4.1

linear transformation : 14.1

linear, time-invariant filters and convolution : 15.1

linearity of the DFT : 8.4.1

logarithm : 12.1

logarithmic number systems for audio : 13.2

logarithms : 12.1

changing the base : 12.1.1

loudness : 12.2.2.3

Lp norms : 6.5.1

magnitude of a sinusoid : 5.1

main lobe : 7.7

mantissa : 12.1

matched filter : 8.2.3.1

matlab

complex numbers : 20.1

DFT matrix : 20.4.3

Matlab examples : 20

matrices : 14

matrix : 14

matrix columns : 14

matrix multiplication : 14.1

matrix rows : 14

matrix transpose : 14

mean : 13.3

mean of a random variable : 13.3

mean of a signal : 6.5 | 13.3

mean square : 13.3

mean value : 13.3

modulo : 8.1.2

modulo indexing : 8.1.2

moments : 13.3

monic : 3.1

Mth roots : 4.13

Mth roots of unity : 4.14

mu-law companding : 13.2.3

multiplication in the time domain : 8.4.6

multiplication of large numbers : 12.1

multiplying two numbers convolves their digits : 8.2.3.3

natural logarithm : 12.1

non-commutativity of matrix multiplication : 14.1

non-removable singularity : 11.5

nonlinear system of equations : 3.1

norm properties : 6.5.2

normalized inverse DFT matrix : 7.10

normalized DFT : 7.8 | 8.4.9

normalized DFT matrix : 7.10

normalized DFT sinusoid : 8.4.8

normalized DFT sinusoids : 7.5 | 7.8

normalized frequency : 8.1 | 16.1

number systems for digital audio : 13

floating point : 13.2.1

fractional fixed point : 13.1.3

how many bits are enough : 13.1.4

logarithmic : 13.2

logarithmic fixed point : 13.2.2

mu law : 13.2.3

one's complement fixed point : 13.1.2.1

PCM : 13.1.1

two's complement fixed point : 13.1.2.2

when byte swapping is needed : 13.1.5

number theoretic transform : 19.3.2

Nyquist sampling theorem : 18

octal : 13.1.2

odd functions : 8.3

ohm's law : 12.3

operators : 8.2

alias : 8.2.11

downsampling : 8.2.10

flip : 8.2.1

interpolation : 8.2.8

repeat : 8.2.9

shift : 8.2.2

stretch : 8.2.5

orthogonal : 7.10

orthogonality : 6.6.7

orthogonality of sinusoids : 7.2

orthogonality of the DFT sinusoids : 7.3

orthonormal : 7.10

parabola : 3.2

Parseval's theorem : 8.4.8

PCM : 13.1.1

peak amplitude : 5.1

periodic : 8.1.2 | 16.3

periodic extension : 7.7 | 8.1.2

periodogram method : 15.3

periodogram method for power spectrum estimation : 15.3

phase : 5.1

phase negation : 8.4.2

phase response : 15.1

phasor : 5.3.8.1 | 5.3.8.1

phasor angle : 5.3.8.1

phasor magnitude : 5.3.8.1

phon amplitude scale : 12.2.2.3

polar coordinates : 3.6

polar form : 4.13

polynomial approximation : 11.2

polynomial multiplication : 8.2.3.2

power : 12.2

power spectral density : 15.2.4

power spectrum : 15.2.4

power theorem : 8.4.8

pressure : 12.2

prime factor algorithm : 19.3.3

prime factor algorithm (PFA) : 19.3.3

primitive root of unity : 7.2.1

Pythagorean theorem in N-Space : 6.6.8

quadratic formula : 3.2

radian frequency : 5.1

radix 2 FFT : 19.2 | 19.2

rational : 4.6

Rayleigh's energy theorem : 8.4.9

real part : 3.5

rectangular window : 7.7

rectilinear coordinates : 3.6

remainder term : 11.1

repeat operator : 8.2.9

repeat theorem (stretch theorem) : 8.4.10

rms level : 13.3

root mean square : 6.5

roots : 3.1

roots of unity : 4.14 | 4.14 | 7.2 | 7.2.1

round-off error variance : 13.3

row-vector : 14

sample coherence function : 15.4

sample mean : 6.5 | 13.3

sample variance : 6.5 | 13.3

sampling theorem : 18 | 18.3

sampling theory : 18

scaling theorem : 17.2

second central moment : 13.3

second moments : 17.3.1

second moments of a signal : 17.3.1

sensation level : 12.2.2.3

Shannon sampling theorem : 18

shift operator : 8.2.2 | 8.2.2

shift theorem : 8.4.4

shift theorem and FFT Windows : 8.4.4.2 | 8.4.4.2

sidelobes : 7.7

sifting property : 16.2.2

signal dynamic range : 12.2.3

signal energy : 6.5

signal metrics : 6.5

signal operators : 8.2

signal processing analysis : 15

similarity theorem : 17.2

sinc function : 7.7 | 8.4.13

sinc function, aliased : 7.7

sinusoid : 5.1

sinusoids and exponentials : 5

sinusoids at the same frequency : 5.1.4

skew-Hermitian : 8.4.2

smoothing in the frequency domain : 8.4.6

sone amplitude scale : 12.2.2.3

Sound Pressure Level : 12.2.2.3

spectral interpolation : 8.2.7

spectral interpolation (ideal) : 8.4.12

spectral leakage : 7.7

spectrum : 7.6 | 8.1

spectrum complex conjugate : 8.4.2

SPL : 12.2.2.3

split radix : 19.2

square integrable : 16.2.1

square matrix : 14

standard deviation : 13.3

Stone-Weierstrass polynomial approximation theorem : 11.4

stretch operator : 8.2.5

stretch theorem (repeat theorem) : 8.4.10

symmetric functions : 8.3

system identification : 15.2.3 | 15.4

Taylor series expansion : 4.8 | 4.8 | 11 | 11.1

Taylor series remainder term : 11.1

Taylor Series with remainder : 11.2

theorems for the DFT : 8.4

time constant : 5.2

time-bandwidth product : 17.3.3

time-domain aliasing : 8.2.11 | 8.2.11

time-limited signals : 17.3.2

Toeplitz : 14.1

transcendental number : 4.11

transform pair : 8.1.1

transpose of a matrix product : 14.1

twiddle factors : 19.1

unbiased cross-correlation : 15.2.1

uncertainty principle for Fourier pairs : 17.3

unit pulse signal : 15.1

unitary : 7.10

variance : 6.5 | 13.3

vector addition : 6.3

vector representation of signals : 6.2

vector subtraction : 6.4

Weierstrass polynomial approximation theorem : 11.4

Welch's method : 15.3

window : 7.7

windowing in the time domain : 8.4.6

zero padding : 8.2.6 | 8.4.12 | 9.1

zero padding in the frequency domain : 8.2.6 | 8.4.13

zero padding in the time domain : 8.2.6

zero phase signal : 8.4.3

zero-padding theorem : 8.4.12

zero-phase signal : 8.4.4.1

zeros : 3.1