Index

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Index

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Index

20 dB boost
Properties of DB Scales
3 dB boost
Properties of DB Scales
convolution
Convolution Representation of LTI
Aliasing
Alias Operator
aliasing operator
Alias Operator
amplitude response
LTI Filters and the | Frequency Response | Frequency Response
anti-aliasing lowpass filter
Alias Operator
antilog
Logarithms
antilogarithm
Logarithms
antisymmetric functions
Even and Odd Functions
average power
Appendix A: Round-Off Error
base
Logarithms
base $2$ logarithm
Logarithms
base $10$ logarithm
Logarithms
bel
Decibels
bin number
Frequencies in the “Cracks”
binary
Binary Integer Fixed-Point Numbers
binary digits
Binary Integer Fixed-Point Numbers
bits
Binary Integer Fixed-Point Numbers
canonical with respect to delay
Order
causal
Causal FIR Filters | Difference Equation
characteristic of the logarithm
Logarithms
coefficient of projection
The Discrete Fourier Transform
column-vector
Matrices
comb-filter
Constructive and Destructive Interference
common logarithm
Logarithms
commutativity of convolution
Convolution
companding
Dynamic Range
complex matrix
Matrices
complex matrix transpose
Matrices
convolution
Convolution
convolution operator “$\
Convolution Representation
convolution representation
Appendix A: Linear Time-Invariant
correlation operator
Correlation
cross-talk
Frequencies in the “Cracks”
cyclic convolution
Convolution
dB scale
Decibels
decibel
Decibels
decimal numbers
Binary Integer Fixed-Point Numbers
DFT as a digital filter
Frequencies in the “Cracks”
DFT matrix
The DFT Matrix
difference equation
Difference Equation
difference equation coefficients
Difference Equation
digit
Binary Integer Fixed-Point Numbers
digital filter
Appendix A: Linear Time-Invariant
Discrete Fourier Transform
The DFT and its
Discrete Fourier Transform (DFT)
The DFT
dynamic range
Dynamic Range
dynamic range of magnetic tape
Dynamic Range
energy
Decibels
Euler’s Formula
Euler’s Formula
Euler’s Theorem
Euler’s Theorem
even functions
Even and Odd Functions
expected value
Appendix A: Round-Off Error
fast convolution
Convolution Theorem
filter
Digital Filter Theory Summary
filter coefficients
Difference Equation
filter delay for amplitude envelopes
Phase Delay and Group
filter order
Difference Equation
filter power response
Frequency Response
filter time delay
Phase Delay and Group
finite-impulse-response filter
FIR Filters
FIR filter
FIR Filters
flip operator
Flip Operator
Fourier Dual
Dual of the Convolution
frequency bin
Frequencies in the “Cracks”
frequency response
LTI Filters and the | Frequency Response
frequency-domain aliasing
Alias Operator | Alias Operator
geometric sequence
Geometric Series
geometric series
Geometric Series
group delay
Phase Delay and Group
Hermitian spectra
Symmetry
Hermitian symmetry
Conjugation and Reversal
Hermitian transpose
Matrices
hex
Binary Integer Fixed-Point Numbers
hexadecimal
Binary Integer Fixed-Point Numbers
ideal bandlimited interpolation
Zero Padding Theorem | Bandlimited Interpolation in Time
ideal lowpass filtering operation in the frequency domain
Bandlimited Interpolation in Time
identity matrix
Matrix Multiplication
IDFT
The DFT and its
impulse response
Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
impulse signal
Appendix A: Linear Time-Invariant | FIR Filters | Difference Equation
indicator function
Linear Phase Terms
Intensity
Decibels
intensity level
DB SPL
interpolation operator
Zero Padding Theorem
inverse DFT
The DFT | The DFT and its
inverse DFT matrix
The DFT Matrix
irrational number
Real Exponents
JND
Decibels
just-noticeable difference
Decibels
lag
Correlation
lagged product
Correlation
length $M$ even rectangular windowing operation
Bandlimited Interpolation in Time
linear
Linearity and Time-Invariance
linear filter
Linearity and Time-Invariance
linear phase FFT windows
Application of the Shift
linear phase signal
Linear Phase Terms
linear phase term
Shift Theorem | Linear Phase Terms
linear transformation
Matrix Multiplication
logarithm
Logarithms
loudness
DB SPL
LTI filters
Linearity and Time-Invariance
main lobe
Frequencies in the “Cracks”
mantissa
Logarithms
matrix
Matrices
matrix columns
Matrices
matrix multiplication
Matrix Multiplication
matrix rows
Matrices
matrix transpose
Matrices
mean
Appendix A: Round-Off Error
mean of a random variable
Appendix A: Round-Off Error
mean of a signal
Appendix A: Round-Off Error
mean square
Appendix A: Round-Off Error
modulo
Modulo Indexing, Periodic Extension
moments
Appendix A: Round-Off Error
multiplication of large numbers
Logarithms
multiplying two numbers convolves their digits
Multiplication of Decimal Numbers
natural logarithm
Logarithms
non-commutativity of matrix multiplication
Matrix Multiplication
normalized inverse DFT matrix
The DFT Matrix
normalized DFT matrix
The DFT Matrix
normalized DFT sinusoids
An Orthonormal Sinusoidal Set | Normalized DFT
normalized frequency
The DFT and its
octal
Binary Integer Fixed-Point Numbers
odd functions
Even and Odd Functions
order
Order
orthogonal
The DFT Matrix
orthogonality of sinusoids
Orthogonality of Sinusoids
orthonormal
The DFT Matrix
PCM
Pulse Code Modulation (PCM)
periodic
Modulo Indexing, Periodic Extension
periodic extension
Frequencies in the “Cracks” | Modulo Indexing, Periodic Extension
phase delay
Phase Delay and Group
phase response
LTI Filters and the | Frequency Response | Frequency Response
phase response at zero frequency
Phase Delay and Group
phon amplitude scale
DB SPL
polar form
Rational Exponents
polynomial approximation
Derivation of Taylor Series
power
Decibels
pressure
Decibels
primitive $N$th root of unity
Orthogonality of Sinusoids
rational
Rational Exponents
real filter
Difference Equation
rectangular window
Frequencies in the “Cracks”
rms level
Appendix A: Round-Off Error
roots of unity
Orthogonality of Sinusoids
row-vector
Matrices
sample mean
Appendix A: Round-Off Error
sample variance
Appendix A: Round-Off Error
second central moment
Appendix A: Round-Off Error
sensation level
DB SPL
shift operator
Shift Operator
sidelobes
Frequencies in the “Cracks”
signal
Digital Filter Theory Summary
signal dynamic range
Dynamic Range
similarity theorem
Appendix D: The Similarity
sinc function
Frequencies in the “Cracks”
skew-Hermitian
Conjugation and Reversal
sone amplitude scale
DB SPL
Sound Pressure Level
DB SPL
spectral leakage
Frequencies in the “Cracks”
spectrum
The Discrete Fourier Transform | The DFT and its
SPL
DB SPL
square matrix
Matrices
stable
Difference Equation
standard deviation
Appendix A: Round-Off Error
Stretch Operator
no title | Stretch Operator
symmetric functions
Even and Odd Functions
Taylor Series Expansion
no title | Informal Derivation of Taylor
time-domain aliasing
Alias Operator
time-invariant
Linearity and Time-Invariance
Toeplitz
Matrix Multiplication
transfer function
Transfer Function | Convolution Representation of LTI
transform pair
Notation and Terminology
transpose of a matrix product
Matrix Multiplication
unilateral $z$ transform
Convolution Representation of LTI
unit pulse signal
Appendix A: Linear Time-Invariant
unitary
The DFT Matrix
variance
Appendix A: Round-Off Error
window
Frequencies in the “Cracks”
Zero padding
Zero Padding
zero phase signal
Symmetry

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