Index

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