Index

20 dB boost : 11.2.1

3 dB boost : 11.2.1

absolutely integrable : 14.2.1

alias operator : 8.2.11

aliased sinc function : 7.7

aliasing : 8.2.11 | 16.2

aliasing operator : 8.2.11

aliasing theorem

continuous time : 16.2.1

amplitude of a sinusoid : 5.1

amplitude response : 9.3.2

anti-aliasing lowpass filter : 8.2.11

anti-Hermitian : 8.4.2

antilogarithm, antilog : 11.1

antisymmetric functions : 8.3

Argand diagram : 3.6

average power : 6.5 | 12.3

Banach spaces : 6.5.3

bandlimited : 4.8

bandlimited interpolation : 8.4.13

of spectra : 8.2.7

time or frequency domain : 8.2.7

bandlimited signals cannot be time limited : 15.3

base (of a logarithm) : 11.1

bel : 11.2

bin number (DFT) : 7.7

bits (binary digits) : 12.1.2

Blackman window : 9.1.4

carrier wave : 5.3.8.1

Cartesian coordinates : 3.6

causal : 8.2.6

characteristic of a logarithm : 11.1

circular convolution : 8.2.3

circular cross-correlation : 9.4.1

CODEC : 12.2.3

coefficient of projection : 7.6

coherence function : 9.6

column vector : 13

comb filter : 5.1.5

common logarithm : 11.1

commutativity of convolution : 8.2.3.1

companding : 11.2.3 | 12.2.3

complex amplitude : 5.3.8.1

complex conjugate : 3.7

complex matrix : 13

complex matrix transpose : 13

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

complexity of FFT : 17.2.1

conjugation and reversal symmetries (DFT) : 8.4.2

constant modulus : 5.3

continuous-time aliasing : 16.2.1

convolution : 8.2.3 | 8.2.3

filter representation : 9.3

frequency domain : 8.4.6

graphical : 8.2.3.2

convolution commutativity : 8.2.3.1

convolution theorem : 8.4.5

convolution theorem dual : 8.4.6

correlation : 8.2.4

correlation analysis : 9.4

correlation operator : 8.2.4

correlation theorem : 8.4.7

cross-correlation, circular : 9.4.1

cross-covariance : 9.4.4

cross-spectral density : 9.4.1

cross-talk : 7.7

cubic spline : 10.5

cyclic convolution : 8.2.3

dB for display : 11.2.2.4

dB per decade : 11.2.1

dB per octave : 11.2.1

dB properties : 11.2.1

dB scale : 11.2

dB SPL : 11.2.2.3

dBm scale : 11.2.2.1

dBV scale : 11.2.2.2

de Moivre's theorem : 3.10

decibel : 11.2

decimal numbers : 12.1.2

decimation in frequency : 17.1

decimation in time : 17.1 | 17.1

decimation theorem : 8.4.11

delta function : 14.2.2

DFT

applications : 9

as a digital filter : 7.7

bin amplitude response : 18.4.2

definition : 2.1

math outline : 2.4

normalized : 7.8

DFT matrix : 7.10 | 7.10

DFT matrix in matlab : 18.4.3

DFT sinusoids : 7.2.2 | 18.4.1

differentiability of audio signals : 10.6

differentiation theorem : 15.1

digit : 12.1.2

digital filter : 9.3

discrete time Fourier transform (DTFT) : 14.1

discrete cosine transform (DCT) : 17.3.1

Discrete Fourier Transform (DFT) : 2.1 | 7 | 8.1

downsampling operator : 8.2.10

downsampling theorem (aliasing theorem) : 8.4.11

DTFT : 14.1

duality (Fourier) : 8.4.6

dynamic range : 11.2.3

dynamic range of magnetic tape : 11.2.3

energy : 11.2

energy of a signal : 6.5

energy theorem : 8.4.9 | 8.4.9

essential singularity : 10.5

Euler's Identity : 3.9 | 3.9 | 4 | 11.1.2

even functions : 8.3

exp(j theta) : 4.12

expected value : 12.3

exponent : 11.1

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) : 17

FFT : 17

FFT notation : 8.1.1

FFT software : 17.4

FFT window : 7.7 | 9.1.4

flip operator : 8.2.1 | 8.2.1

folding frequency : 8.4.13

formants : 9.2.1

Fourier duality : 8.4.6

Fourier series and the DFT : 14.3

Fourier series coefficient : 14.3

Fourier symmetries : 8.4.3

Fourier theorems : 8 | 8.4

Fourier theorems (continuous time) : 15

continuous time aliasing : 16.2.1

differentiation : 15.1

scaling or similarity : 15.2

uncertainty principle : 15.3

Fourier theorems (DFT) : 8 | 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

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 : 14.2

Fourier transform existence : 14.2.1

Fourier transforms for continuous/discrete time/frequency : 14

frequency bin : 7.7

frequency response : 9.3.1

frequency-domain aliasing : 8.2.11 | 8.2.11

FS (Fourier Series) : 14.3

FT (Fourier Transform) : 14.2

fundamental theorem of algebra : 3.4

Gaussian function : 15.3.1

generalized function : 14.2.2

geometric sequence : 7.1

geometric sequence frequencies : 16.4

geometric series : 7.1 | 7.1

geometric signal theory : 6

graphical convolution : 8.2.3.2

Hann window : 9.1.5

Hanning window : 9.1.5

Heisenberg uncertainty principle : 15.3.1

Hermitian spectra : 8.4.3

Hermitian symmetry : 8.4.2

Hermitian transpose : 7.10 | 13

hexadecimal : 12.1.2

ideal lowpass filter : 8.4.13.1

identity matrix : 13.1

IDFT : 2.2 | 8.1

imaginary part : 3.5

impulse response : 9.3

impulse signal : 9.3

impulse train : 14.3.1

impulse, continuous time : 14.2.2

indicator function : 8.4.4.1

instantaneous frequency : 5.1

instantaneous phase : 5.1

integrable function : 14.2.1

intensity : 11.2

intensity level : 11.2.2.3

interpolation operator : 8.2.8 | 8.2.8

inverse DFT : 2.2 | 8.1

inverse DFT matrix : 7.10

irrational number : 4.7

just-noticeable difference (JND) : 11.2

lag : 8.2.4

lagged product : 8.2.4

linear combination : 5.3.8.2

linear number systems for digital audio : 12.1

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 | 8.4.4.1

linear transformation : 13.1

linear, time-invariant filters and convolution : 9.3

linearity of the DFT : 8.4.1

logarithm : 11.1

logarithmic number systems for audio : 12.2

logarithms

changing the base : 11.1.1

of imaginary numbers : 11.1.2

loudness : 11.2.2.3

lowpass filter (ideal) : 8.4.13.1

Lp norms : 6.5.1

Maclaurin series : 10.3

magnitude of a sinusoid : 5.1

main lobe : 7.7

mantissa : 11.1

matched filter : 8.2.3.2

Matlab : 18

Matlab/Octave examples : 18

matrix : 13

matrix multiplication : 13.1

matrix transpose : 13

maximally flat : 10.2

mean of a random variable : 12.3

mean of a signal : 6.5 | 12.3

mean square : 12.3

mean value : 12.3

modulo : 8.1.2

modulo indexing : 8.1.2

moments of a function : 12.3

monic polynomial : 3.1

Mth roots of unity : 4.13

mu-law companding : 12.2.3

multiplication in the time domain is convolution in the frequency domain : 8.4.6

multiplication of large numbers : 11.1

multiplying two numbers convolves their digits : 8.2.3.4

natural logarithm : 11.1

non-removable singularity : 10.5

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 : 7.5 | 8.4.8

normalized DFT sinusoids : 7.8

normalized frequency : 8.1 | 14.1

Nth roots of unity : 7.2.1

number systems for digital audio : 12

floating point : 12.2.1

fractional fixed point : 12.1.3

how many bits are enough : 12.1.4

logarithmic : 12.2

logarithmic fixed point : 12.2.2

mu law : 12.2.3

one's complement fixed point : 12.1.2.1

PCM : 12.1.1

two's complement fixed point : 12.1.2.2

when byte swapping is needed : 12.1.5

number theoretic transform : 17.3.2

Nyquist limit : 8.4.13

Nyquist rate : 8.4.13

Nyquist sampling theorem : 16

octal : 12.1.2

Octave : 18

odd functions : 8.3

Ohm's law : 11.3

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

orthogonality : 6.6.7 | 7.10

orthogonality of DFT sinusoids : 7.3

orthogonality of sinusoids : 7.2

orthonormal : 7.10

overlap-add : 8.4.13.2

Padé approximation : 10.2

parabola : 3.2

Parseval's theorem : 8.4.8

PCM : 12.1.1

peak amplitude : 5.1

periodic : 8.1.2 | 14.3

periodic extension : 7.7 | 8.1.2

periodic interpolation : 8.4.13

periodogram method for power spectrum estimation : 9.5

phase : 5.1

phase negation : 8.4.2

phase response : 9.3.3

phasor : 5.3.8.1 | 5.3.8.1

phon amplitude scale : 11.2.2.3

polar coordinates : 3.6

polar form of a complex number : 4.13

polynomial

factoring : 3.1

roots : 3.3

polynomial approximation : 10.2

polynomial multiplication : 8.2.3.3

positive and negative frequencies : 5.3.3

power : 11.2

power spectral density : 9.4.4

power spectral density estimation : 9.5

power spectrum : 9.4.4

power theorem : 8.4.8

pressure : 11.2

prime factor algorithm (PFA) : 17.3.3

primitive root of unity : 4.14 | 7.2.1

probability density function : 12.3

probability distribution : 12.3

projection : 6.6.9

Pythagorean theorem in N-Space : 6.6.8

quadratic formula : 3.2 | 3.2

radian frequency : 5.1

radix 2 FFT : 17.2 | 17.2

random variable : 12.3

rational number : 4.6

Rayleigh's energy theorem : 8.4.9

real part : 3.5

rectangular window : 7.7 | 8.4.13.1

rectilinear coordinates : 3.6

remainder term : 10.1 | 10.3

removable singularity : 10.5

repeat (stretch) theorem : 8.4.10

repeat operator : 8.2.9

rms level : 12.3

root mean square : 6.5

roots of a polynomial : 3.1 | 3.3

roots of unity : 4.14 | 4.14 | 7.2.1

round-off error variance : 12.3

row vector : 13

sample coherence function : 9.6

sample mean : 6.5 | 12.3

sample variance : 6.5 | 12.3

sampling rate : 8.4.13

sampling theorem : 16 | 16.3

scaling theorem : 15.2

second central moment : 12.3

second moments of a signal : 15.3.1

sensation level : 11.2.2.3

Shannon sampling theorem : 16

shift operator : 8.2.2 | 8.2.2

shift theorem : 8.4.4

shift theorem and FFT windows : 8.4.4.2

sidelobes : 7.7

sifting property : 14.2.2

signal dynamic range : 11.2.3

signal energy : 6.5

signal metrics : 6.5

signal operators : 8.2

similarity theorem : 15.2

sinc function : 7.7

sinc function, aliased : 7.7

sinusoids and exponentials : 5

sinusoids at the same frequency : 5.1.4

skew-Hermitian : 8.4.2

smoothing, frequency domain : 8.4.6

sone amplitude scale : 11.2.2.3

Sound Pressure Level (SPL) : 11.2.2.3

spectral interpolation : 8.2.7 | 8.4.12

spectral leakage : 7.7

spectrogram : 9.2

spectrogram in matlab : 18.5

spectrum : 7.6 | 8.1

spectrum complex conjugate : 8.4.2

speech spectrogram : 9.2.1

SPL : 11.2.2.3

split radix : 17.2

square integrable : 14.2.1

square matrix : 13

standard deviation : 12.3

Stone-Weierstrass polynomial approximation theorem : 10.4

stretch (repeat) theorem : 8.4.10

stretch operator : 8.2.5

symmetric functions : 8.3

system identification : 9.4.3 | 9.6

Taylor series : 10

remainder bound : 10.2

remainder term : 10.1

Taylor series expansion : 4.8 | 4.8

theorems for the DFT : 8.4

time constant : 5.2

time-bandwidth product : 15.3.3

time-domain aliasing : 8.2.11

time-limited signals : 15.3.2

Toeplitz matrix : 13.1

transcendental number : 4.11

transform pair : 8.1.1

transpose of a complex matrix : 13

transpose of a matrix product : 13.1

twiddle factors : 17.1

unbiased sample cross-correlation : 9.4.1

uncertainty principle : 15.3

unit pulse signal : 9.3

unitary : 7.10

variance : 6.5

variance of a random variable : 12.3

vector addition : 6.3

vector representation of signals : 6.2

vector subtraction : 6.4

Weierstrass polynomial approximation theorem : 10.4

Welch's method : 9.5

window : 7.7

windowing in the time domain equals smoothing in the frequency domain : 8.4.6

zero padding : 8.2.6 | 8.4.12 | 9.1

zero padding example : 9.1.3

zero padding in the time domain equals ideal interpolation in the frequency domain : 8.2.7

zero padding, spectral : 8.4.13

zero phase signal : 8.4.3

zero-padding theorem : 8.4.12

zero-phase signal : 8.4.4.1

zeros of a polynomial : 3.1