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Index: Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications -- Second Edition
Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications -- Second Edition
Fourier Theorems
Rayleigh Energy Theorem (Parseval's Theorem)
Downsampling Theorem (Aliasing Theorem)
Click for https://ccrma.stanford.edu/~jos/mdft/Modulo_Indexing_Periodic_Extension.html
Click for https://ccrma.stanford.edu/~jos/mdft/Stretch_Operator.html
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Index: Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications -- Second Edition
Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications -- Second Edition
Fourier Theorems
Rayleigh Energy Theorem (Parseval's Theorem)
Downsampling Theorem (Aliasing Theorem)
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Stretch Theorem (Repeat Theorem)
Theorem:
For all
,
Proof:
Recall the
stretch operator
:
Let
, where
,
. Also define the new denser frequency grid associated with length
by
, and define
as usual. Then
But
Thus,
, and by the
modulo indexing
of
,
copies of
are generated as
goes from 0 to
.
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``
Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition
'', by
Julius O. Smith III
,
W3K Publishing
, 2007,
ISBN
978-0-9745607-4-8
Copyright ©
2024-04-02
by
Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),
Stanford University