Index

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FIR filter design

optimal least-squares impulse response : 15.2

absolutely integrable : 3.2.1

acyclic convolution : 3.3.5

acyclic FFT convolution : 9.1.2

additive synthesis : 2 | 8 | 8.1.4 | 11.4.6

alias component matrix : 12.3.8

aliased sinc function : 2.5

aliasing components : 3.3.12

aliasing theorem for the DTFT : 3.3.12

aliasing, time domain : 9.1.4.3

allpass filter : 12.5.2

amplitude envelope : 8

analysis modulation matrix : 12.3.8

analytic signal : 2.1 | 15.5

asinc function : 2.5

associate peaks : 8.3.2

audio spectrogram : 7.3

audio spectrogram hop size : 7.3.2

auditory filter : 7.3.3.3

auditory filter bank : 7.3.3.2

auditory filter banks : 7.3.1

autocorrelation : 3.3.7

autocorrelation computation : 6.9

autocorrelation function : 17.2.3

average : 17.1.8

bandlimited signals cannot be time limited : 3.4.16

bandpass filter : 15.5

Bark frequency scale : 18.5

Bark warping : 18.7

Bartlett window : 4.4

baseband signal : 10.1.2

bias : 5.7

bias of parabolic interpolation : 5.7

biased autocorrelation : 6

biased sample autocorrelation : 6.6

bibliography : 20

bilinear transform : 18.6

bilinear transform frequency warping : 18.2

bin number : 7.1.3

Blackman window : 4.3.3

Blackman window matlab example : 19.1.1

Blackman-Harris window : 4.3.6

Blackman-Harris window family : 4.3

Blackman-Harris window, frequency-domain implementation : 4.3.7

bounded variation : 3.5

brown noise : 6.14

cepstral smoothing : 11.2.1

cepstrum : 15.7

characteristic function : 16.13.4

Chebyshev polynomials : 4.9.4.1

chirp signal : 10.2.1

chirp, Gaussian-windowed : 16.5

chirplet : 16.5

chirplet modeling : 8.6.2

chirplets : 4.10 | 16

circular convolution : 9.1

coherent addition of signals : 6.15

COLA (constant overlap-add) : 7.1.1

COLA constraint : 9.2.1

COLA constraint, frequency domain : 9.3.2

COLA dual : 9.3

colored noise : 6.14

complex demodulation : 10.3.2

complex Gaussian integral : 16.3

compression : 12

Conjugate Quadrature Filters : 12.3.7

constant overlap-add (COLA) : 11.4.1

constant overlap-add (COLA) property : 7.1.1

constant overlap-add property : 9

constant-overlap-add : 9.2.1

continuous probability distribution : 17.1.3

convolution : 3.3.5 | 9.1

acyclic : 9.1.2

acyclic in matlab : 9.1.2.1

cyclic : 9.1.1

cyclic, or circular : 9.1

FFT overlap-add in matlab : 9.2.5

FFT, overlap-add : 9.2

fractional : 15.10.7

in Matlab or Octave : 9.1.3

short signals : 9.1

convolution theorem : 3.3.5 | 3.3.5 | 3.4.6

convolution, continuous time : 3.4.6

correlation : 3.3.6

correlation analysis : 17.2

correlation theorem : 3.3.6 | 3.3.7

covariance : 6.4

critical band of hearing : 7.3.2

cross synthesis : 11.1

cross-correlation : 17.2.1

cross-power spectral density : 17.2.2 | 17.2.2

cubic polynomial phase interpolation : 8.6.1

cut-off frequency : 15.1

cycles per second : 3.4.1

cyclic autocorrelation : 6.8

cyclic convolution : 9.1

cyclic FFT convolution : 9.1.1

dc sampling filter : 10.3.1

decimation operator : 12.1.2

deconvolution : 9.1.2

delta function : 3.4.9

demos : 11.7

denoising : 6.1.1

deterministic : 5.8.2

deterministic part : 8.4.1

detrend : 6.9

DFT filter bank : 10.3 | 10.3.4.2

differentiation theorem : 3.4.2 | 3.5

differentiation theorem dual, DTFT : 3.3.13

differentiation theorem dual, FT : 3.4.3

digital filter design, FIR : 15

digital prolate spheroidal sequence (DPSS) : 4.7

Dirichlet function : 2.5

discrete probability distribution : 16.10

Discrete Prolate Spheroidal Sequences (DPSS) : 4.7

discrete time Fourier transform (DTFT) : 3.1

Dolph window : 4.9

Dolph-Chebyshev and Hamming windows compared : 4.9.3

Dolph-Chebyshev window : 4.9 | 4.9

Dolph-Chebyshev window length computation : 4.9.4.4

Dolph-Chebyshev window, theory : 4.9.4

downsampling : 3.3.12

downsampling (decimation) operator : 12.1.2

DPSS window : 4.7

DTFT

aliasing theorem : 3.3.12

convolution theorem : 3.3.5

correlation theorem : 3.3.6

downsampling theorem : 3.3.12

energy theorem : 3.3.8

even symmetry : 3.3.3.1

linearity : 3.3.1

power theorem : 3.3.8

repeat operator : 3.3.10

repeat theorem : 3.3.11

scaling operator : 3.3.10

scaling theorem : 3.3.11

shift theorem : 3.3.4

stretch operator : 3.3.9

stretch theorem : 3.3.11

symmetry : 3.3.3

time reversal : 3.3.2

DTFT Fourier theorems : 3.3

effective length of a window : 2.7.1

energy theorem : 3.3.8

ensemble average : 17.1.6

entropy : 16.11.1 | 16.11.1

envelope break-points : 8.6.1

envelope follower : 7.3.3.6

equivalent rectangular bandwidth : 18.8

excitation pattern : 7.3.1 | 7.3.2 | 7.3.3.2

expected value : 17.1.6 | 17.1.6 | 17.3

exponential window : 4.5

extended lapped transforms : 12.7.2

FBS modifications : 10.8.2.1

FFT convolution speed : 9.1.4

FFT input buffer : 11.4.2

fftshift utility in matlab : 5.4.1

filter

overlap-add FFT convolution : 9.2

filter bank summation interpretation of the STFT : 10

filter bank, perfect reconstruction : 12.3

filter banks : 12

paraunitary : 12.5

filter design : 18

example of window method : 15.4.2

Hilbert transform filter : 15.5

least-squares, linear-phase FIR : 15.11.6

filter design, FIR

frequency-sampling method : 15.3

window method : 15.4

filter-bank interpretation of the STFT : 10.1.2

Filter-Bank Summation (FBS) : 10.3.4

filtered white noise : 6.14 | 6.14

filters

audio, FIR : 9.1.4.1

lossless : 12.5.2

lossless examples : 12.5.3

finite support : 6.6

finite-impulse-response : 15.4

FIR digital filter design

frequency-sampling method : 15.3

window method : 15.4

FIR filter design

by linear programming : 15.12

least-squares, linear phase : 15.11.6

optimal methods : 15.11

FIR fractional delay filter : 15.10

first-order moment : 16.13.1

flip operator : 3.4.7

formants : 7.2.1

Fourier dual : 5 | 10.5

Fourier theorems

continuous time : 3.4

discrete time : 3.3

DTFT

differentiation dual : 3.3.13

FT

differentiation dual : 3.4.3

Fourier theorems (continuous time)

convolution theorem : 3.4.6

differentiation : 3.4.2

flip theorem : 3.4.7

gaussian pulse : 3.4.10

impulse train : 3.4.13

power theorem : 3.4.8

rectangular pulse : 3.4.11

sampling theorem : 3.4.15

scaling or similarity : 3.4.4

shift theorem : 3.4.5

uncertainty principle : 3.4.16

Fourier transform : 3.2

Fourier transform existence : 3.2.1

Fourier transforms for continuous/discrete time/frequency : 3

fractional delay filter : 15.10

fractionally iterated convolution : 15.10 | 15.10.7

fractionally iterated multiplication : 15.10.1

frame : 7.1.3

frequency resolution : 2.5.2 | 2.7

frequency sampling for FIR filter design : 15.3

frequency shifting : 11.5

frequency trajectories : 8.3.2

frequency warping

allpass : 18

bilinear transform : 18.2

non-parametric : 19.3.5

frequency-shifting : 8.1.8

Gaussian chirp : 4.10 | 16

Gaussian distributed : 5.8.2

Gaussian distribution

maximum entropy property : 16.11

Gaussian function : 3.4.16.1 | 16

Gaussian integral : 16.2.1

gaussian pulse : 3.4.10

Gaussian random variable, closed under addition : 16.14

Gaussian window : 16.1

Gaussian window function : 4.10

Gaussian, Fourier transform of : 16.4

Gaussian-windowed chirp : 16.5

generalized function : 3.4.9

generalized Hamming window family : 4.2 | 4.2.6

Gibbs phenomenon : 2.5.1

glossary of notation : 14

graphic equalizer : 15.6

graphical convolution : 9.1

graphical equalizers : 9.3.3

Group-Additive Synthesis : 8.6.3.2

Haar filter bank : 12.3.3

Hamming and Dolph-Chebyshev windows compared : 4.9.3

Hamming window : 4.2.4

Hann window : 4.2.1 | 4.2.1

Hann-Poisson window : 4.6

hanning window : 4.2.1

harmonic : 2.7.1

Heisenberg uncertainty principle : 3.4.16.1

Hermitian : 3.3.3

Hermitian spectrum : 15.5

Hilbert transform : 15.9

Hilbert transform filter design : 15.5

hop size : 6.12 | 7.1.3 | 9.2.1

ideal lowpass filter : 15.4

impulse train : 3.4.13

impulse, continuous time : 3.4.9

impulse, sinc : 3.4.12

independent events : 17.1.2 | 17.3.1

independent random variables : 17.3.1

inner product : 3.3.8 | 3.4.8

instantaneous loudness : 7.3.2 | 7.3.3.6

interpolation kernel : 7.3.3.3

interpolation kernel, spectral, ideal : 19.3.5.1

interpolation of a DFT : 5.2

inverse FFT synthesis : 8.6.2

iterated convolution : 15.10.7

Kaiser window : 4.8

Kaiser window beta parameter : 4.8.3

Kaiser-Bessel window : 4.8

lagged product : 6.4

Laurent expansion : 15.8 | 15.8

least squares estimation : 5.8.1

least squares sinusoidal parameter estimation : 5.8.1

likelihood function : 5.8.3

linear least squares : 5.8.1.1

linear phase : 9.1.4.2

linear phase term : 3.3.4

linear prediction spectral envelope : 11.2.2

linearity of the DTFT : 3.3.1

long-term loudness : 7.3.3.6

lossless filter : 12.5.2

lossless filter examples : 12.5.3

lossless filters : 12.5.2

lossless transfer function matrix : 12.5.2

loudness : 7.3 | 7.3.1

loudness spectrogram : 7.3.2 | 7.3.2

loudness spectrogram, examples : 7.3.3

loudness versus time : 7.3.3.6

loudness versus time and frequency : 7.3.2

low-pass filtering by FFT : 9.1.4.2

lowpass filter, ideal : 15.1

magnitude-only analysis/synthesis : 11.4.7

main-lobe width : 2.6

masking : 11.6

matlab

discrete prolate spheroidal window : 19.1.2

DPSS window : 4.7.1

minimum zero-padding factor : 19.2.4

peak finder : 19.2

phase unwrapping : 19.3.4

spectrogram : 19.3

spectrum analysis windows : 19.1

window method for FIR filter design : 15.4.1

matlab examples : 19

matlab listing

dpssw : 19.1.2

findpeaks : 19.2.1

maxr : 19.2.2

npwarp : 19.3.5

oboeanal : 19.2.5

qint : 19.2.3

spectrogram : 19.3.1

testspectrogram : 19.3.2 | 19.3.3

unwrap : 19.3.4

zero-phase blackman : 19.1.1

zpfmin : 19.2.4

maximum likelihood estimator : 5.8.2

maximum likelihood sinusoidal parameter estimation : 5.8.2

mean of a distribution : 16.13.1

mean of a random process : 17.1.7

minimum phase : 15.7

minimum phase filters : 15.7

minimum phase means a causal cepstrum : 15.8

modulated lapped transform : 4.2.13

MPEG filter banks : 12.7

multi-resolution STFT : 7.3.2

multirate filter banks : 12

multirate noble identities : 12.2.5

multiresolution sinusoidal modeling : 8.1.7

multiresolution STFT : 7.3.3.1 | 7.3.3.1

munchkinization : 11.5

noble identities : 12.2.5

noise : 6.1.2

mean : 17.1.7

synthesis example : 6.14.2

white : 17.3

noise process : 17.1.4

noise spectral analysis

periodogram : 6.11

Welch's method : 6.12

noise spectrum analysis : 6

pink noise example : 6.14.3

noise, filtered : 6.14

non-coherent addition of signals : 6.15

non-parametric : 15.10

nonlinear spectral phase modification : 15.10

nonuniform resampling : 7.3.3.3

normal distribution : 5.8.2

normalized frequency : 3.1

normalized radian frequency : 2.2

notation glossary : 14

oboe spectrum analysis : 5.5

oddly-stacked Princen-Bradley filter bank : 12.7.2

OLA modifications : 10.8.2.1

optimized windows : 4.11

orthogonal two-channel filter banks : 12.3.8

orthogonality principle : 5.8.1.2

overlap-add convolution in matlab : 9.2.5

overlap-add decomposition : 9.2.1

overlap-add FFT convolution : 9.2

overlap-add FFT processor : 9

overlap-add interpretation of the STFT : 9 | 10.1.1

overlap-add method : 7.1.4

overlap-add, with modifications : 9.5

overtone : 8

panning : 6.16

paraconjugate : 12.3.8

paraconjugation : 12.5.1

parametric filter design : 15.10

paraunitary filter bank : 12.5.5

paraunitary filter banks : 12.5

Parseval's theorem : 3.3.8

PARSHL : 11.4

partial overtone : 8

partition of unity property : 9.2.1

PDF : 17.1.3

peak detection : 11.4.3

peak matching : 11.4.4

peak-finding : 5.8

perfect reconstruction : 10.1.3

perfect reconstruction filter bank, conditions for : 12.4.5

perfect reconstruction filter banks : 12.4

perfect reconstruction filter banks, critically sampled : 12.3

periodic sinc function : 2.5

periodogram : 6.11

periodogram method : 6.12 | 6.12

periodogram method for power spectrum estimation : 6.12

phase unwrapping : 15.10 | 19.3.4

phase vocoder : 8.1.3

phons : 7.3.3.6

pink noise : 6.14 | 6.14.2

pitch detection : 11.6

Poisson summation formula : 9.3.1

Poisson summation formula, continuous time : 3.4.14

Poisson window : 4.5

polyphase component filters : 12.2.1

polyphase components : 12.2

polyphase decomposition : 12.1.3 | 12.2.1 | 12.2.2

polyphase filter bank : 12.1.3

polyphase matrix : 12.4

polyphase signals : 12.1.3

Portnoff window : 10.7

power spectral density : 17.2.5

smoothed : 6.7

power spectrum : 17.2.5

power theorem : 3.3.8 | 3.4.8

pre-emphasis : 11.4.8

preemphasis : 11.6

preprocessing : 11.6

probability density function : 17.1.3

probability distribution : 16.10 | 17.1.1 | 17.1.1

processing gain : 6.15

prolate spheroidal wave function : 4.7

prolate spheroidal window : 4.7

Pseudo-QMF filter bank : 12.7.1

quadratic interpolation : 5.6

quadratically interpolated FFT (QIFFT) method : 5

quadrature mirror filterbanks (QMF) : 12.3.5

radians per second : 3.4.1

raised-cosine window : 4.2.1

random process : 17.1.4

random variable : 17.1.3

random variables : 17.1

Rayleigh's energy theorem : 3.3.8

rectangular pulse : 3.4.11

rectangular window : 2.3 | 2.5 | 4.1

rectangular window side-lobes : 2.5.1

references by topic : 20

Remez multiple exchange algorithm : 15.4.2.4

repeat operator : 3.3.10

repeat theorem : 3.3.11

residual signal : 8.4.1

resolution bandwidth : 2.6

resolution of frequencies : 2.7

resolution window length : 2.7

resolving sinusoids : 2.6

Riemann Lemma : 3.5

roll-off rate : 3.5

running-sum lowpass filter : 10.3.1

sample autocorrelation : 6 | 6.4

sample autocorrelation function : 6.9

sample mean : 17.1.8

sample mean of a random process : 17.1.8

sample power spectral density : 6.5

sample PSD : 6

sample variance : 6.4 | 17.1.10 | 17.1.10

sampled rectangular pulse : 3.4.13

sampling synthesis : 8.6.3.1

sampling theory : 3.4.15

scaling theorem : 3.4.4

second central moment : 16.13.2 | 17.1.9

second moments of a signal : 3.4.16.1

shah symbol : 3.4.13

shift operator : 3.3.4

shift theorem : 3.3.4 | 3.3.4 | 3.4.5

short time Fourier transform : 7

downsampling : 10.8

modifications : 10.9

short-term loudness : 7.3.3.6

short-time Fourier transform (STFT) : 7.1

side-lobe width : 2.6

sifting property : 2.1 | 3.4.9

signal model : 5.8.1

similarity theorem : 3.4.4

sinc function : 2.5 | 15.4

sinc function, aliased (periodic) : 2.5

sine window : 4.2.13 | 4.2.13

sines + noise + transients model : 11.4.13

sines + noise spectral modeling : 8.4

sines+noise+transients : 8

sinusoidal amplitude estimation : 5.8.1.1

sinusoidal modeling : 8 | 8

sinusoidal parameter estimation

general case : 5.8.1.3

known frequency : 5.8.1.2

known frequency and phase : 5.8.1.1

least squares : 5.8.1

sinusoidal spectrum analysis : 2

Slepian window : 4.7 | 4.7

sliding DFT : 10.3.4.2

sones : 7.3.3.6

specific loudness : 7.3.1 | 7.3.2 | 7.3.3.4

spectral display : 7.1

spectral envelope : 11.2

cepstral smoothing : 11.2.1

linear prediction : 11.2.2

spectral interpolation : 5

spectral interpolation, ideal : 5.1 | 19.3.5.1

spectral modeling : 8

history : 8.1

spectral modeling applications : 11.4.9

spectral modeling synthesis : 8

spectral modifications : 9 | 9.2

spectral transformations : 11.4.5

spectrogram : 7.2 | 19.3.1

spectrogram parameters : 7.2

spectrogram, for audio display : 7.3

spectrum : 2.1

spectrum analysis : 4

noise : 6

oboe data : 5.5

sinusoids or spectral peaks : 2

statistical formulation : 17

time varying : 7

speech spectrogram : 7.2.1

square integrable : 3.2.1

stationary : 6.1.1 | 17.1.6

stationary stochastic process : 17.1.5

statistical signal processing : 17

step size : 7.1.3

stereo panning : 6.16

STFT : see short-time Fourier transformtextbf

filter-bank interpretation : 10.1.2

overlap-add interpretation : 10.1.1

weighted overlap-add : 9.7

STFT filter bank, downsampled : 10.8.1

stochastic part : 8.4.1

stochastic process : 6 | 17.1.4

stop-band attenuation : 15.4.2.3

stretch operator : 3.3.9 | 3.3.9 | 12.1.1

stretch theorem : 3.3.11

strong COLA constraint : 9.3.2.1 | 9.3.2.1

subtractive synthesis : 8.4

symmetric Toeplitz operator : 4.7

symmetry of the DTFT for real signals : 3.3.3

third-octave filter bank : 7.3.1

time aliasing : 15.10.6

time compression/expansion : 11.5

time domain aliasing : 9.1.2.2

time limited : 15.4

time reversal and the DTFT : 3.3.2

time scale modification : 8.5.3 | 11.5

time-bandwidth product : 3.4.16.3

time-domain aliasing : 9.1.4.3

time-frequency displays : 7

time-frequency distributions : 7.1

time-frequency reassignment : 8.1.7.2

time-limited interpolation : 5.2

time-limited signals : 3.4.16.2

time-scale modification : 8.1.8

time-varying OLA modifications : 9.5

total variation : 3.5

transform coders : 7.1.4

transient detector : 8.5.2

transition band : 15.10.6

transpose, filter bank : 12.3.4 | 12.4.7

triangular window : 4.4

twiddle factor : 12.1.2

two-sided Taylor expansion : 15.8

type II polyphase decomposition : 12.2.3

unbiased estimator : 17.1.8 | 17.1.10

uncertainty principle : 3.4.16

unimodular polynomial matrix : 12.5.5

unwrapped phase : 15.10

unwrapping phase : 19.3.4

upsampling (stretch) operator : 12.1.1

variance : 17.1.9 | 17.1.9

variance of a distribution : 16.13.2

vocoder : 11.3

wah-wah pedal : 10.12

wavelet filter banks : 13

wavelets : 13.2

weak COLA constraint : 9.3.2

weighted overlap add : 9.7

weighted overlap-add : 9.7

Welch autocorrelation : 6.12.1 | 6.12.2

Welch's method for spectrum analysis : 6.12

Welch's method, windowed : 6.13

white noise : 6.1.1 | 6.1.2 | 6.3 | 6.3.1 | 6.4 | 6.4 | 6.5 | 6.5 | 6.7 | 6.10 | 6.11 | 6.11.1 | 6.14 | 6.14 | 6.14 | 6.14.2 | 17.3

window function : 4

window method, FIR filter design : 15.4 | 15.6

windowing effect : 2.4

windows

Bartlett : 4.4

Blackman : 4.3.3 | 19.1.1

Chebyshev : 4.9

Dolph-Chebyshev : 4.9

Dolph-Chebyshev theory : 4.9.4

DPSS : 4.7

exponential : 4.5

frequency resolution : 4.8.5

generalized Hamming : 4.2 | 4.2.6

Hann-Poisson : 4.6

Kaiser : 4.8

Kaiser-Bessel : 4.8

no side-lobes case : 4.6

optimized : 4.11

Poisson : 4.5

Prolate Spheroidal : 4.7

rectangular : 2.5 | 4.1

sine : 4.2.13

Slepian : 4.7

triangular : 4.4

windows for spectrum analysis : 4

zero padding : 5.3

zero padding, minimum : 19.2.4

zero padding, zero-phase form : 5.4

zero-centered : 2.3

zero-padding factor : 7.1.3

zero-phase windows : 2.5

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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.