12 ms·

Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications

91 points by mindcrime 4y ago

texaslonghorn5 4y ago
https://ccrma.stanford.edu/~jos/mdft/Chapter_Outline.html https://ccrma.stanford.edu/~jos/mdft/Chapter_Outline.html interesting looking page. compared to ctcf, there is a lot of cool math that pops out in the dtdf case (sampling, aliasing, etc). also it's fundamentally how we deal with audio in computer systems so this is highly practical info. it is also fun to look at the various other discrete flavors of signal processing. https://ccrma.stanford.edu/~jos/mdft/Fourier_Transforms_Continuous_Discrete_Time_Frequency.html https://ccrma.stanford.edu/~jos/mdft/Fourier_Transforms_Cont... you can even extend this to finite algebraic structures. I think most fourier analysis references will have some information as there is relevance to Dirichlet theorem on primes.
jmwilson 4y ago
Having first been exposed to the DFT in a very similar matter to this text, I think there is a significant advantage in first deriving it from the discrete time Fourier transform (DTFT, not to be confused with DFT, and notwithstanding the name is actually a continuous function) instead of trying to introduce it independently on its own. This is the approach taken in Oppenheim and Schafer textbook, but requires the reader have a bit more background in mathematics. In this way, the DFT is seen as a sampling of the DTFT when the signal is convolved with a window function, and explains why the spectrum of a signal is smeared when its period is not a multiple of the transform size. This textbook says "there is no leakage when the signal being analyzed is truly periodic and we can choose N to be exactly a period, or some multiple of a period" -- actually there still is, it's just that the DFT happens to sample precisely at the nulls of those sidelobes. The sidelobes are further seen as a consequence of the window function, and explains why certain window choices have better sidelobe attenuation at the tradeoff of wider main lobe/lower frequency resolution.
- a-dub 4y ago
  richard lyons' textbook "understanding digital signal processing" has a nice treatment of spectral leakage issues and choice of window functions. i think it may even discuss the integer multiple issue you speak of with nice illustrations of the "wrapping" effect.
- deleted 4y ago
  [deleted]
rerdavies 4y ago
Just part of a vast trove of mathematics for audio, by Julius O. Smith III: https://ccrma.stanford.edu/~jos/sitemap.html https://ccrma.stanford.edu/~jos/sitemap.html
blueprint 4y ago
Not to be confused with density-functional theory, which is also DFT. https://en.wikipedia.org/wiki/Density_functional_theory https://en.wikipedia.org/wiki/Density_functional_theory
khiner 4y ago
Throwing a shameless plug in here for a set of Jupiter notebooks I made that follow each chapter of this book: https://karlhiner.com/jupyter_notebooks/mathematics_of_the_dft https://karlhiner.com/jupyter_notebooks/mathematics_of_the_d... Reproduces many of the interesting results in Python, providing charts and animations along the way. Hope it’s of use to someone! (I also have a similar series for two of the other three books in this series - Digital Filters and Physical Modeling.)
cozzyd 4y ago
Just wanted to say that Julius is one of my favorite professors from undergrad. (I randomly took a bunch of clases at ccrma for fun ... But now it turns out I do a bunch of signal processing )
vermarish 4y ago
Tangientially related: I finished my B.S. eight months ago and have since been disillusioned with the prospects of ever going to grad school. This submission is so interesting that after skipping through it, I was inspired to draft a roadmap toward grad school over the last two hours -- mostly reexamining project ideas and considering which PhDs in my network might find those interesting once developed. Thank you for submitting this, it is absolutely fascinating.

91 points by mindcrime 4y ago

https://ccrma.stanford.edu/~jos/mdft/Chapter_Outline.html https://ccrma.stanford.edu/~jos/mdft/Chapter_Outline.html interesting looking page. compared to ctcf, there is a lot of cool math that pops out in the dtdf case (sampling, aliasing, etc). also it's fundamentally how we deal with audio in computer systems so this is highly practical info. it is also fun to look at the various other discrete flavors of signal processing. https://ccrma.stanford.edu/~jos/mdft/Fourier_Transforms_Continuous_Discrete_Time_Frequency.html https://ccrma.stanford.edu/~jos/mdft/Fourier_Transforms_Cont... you can even extend this to finite algebraic structures. I think most fourier analysis references will have some information as there is relevance to Dirichlet theorem on primes.

jmwilson 4y ago

Having first been exposed to the DFT in a very similar matter to this text, I think there is a significant advantage in first deriving it from the discrete time Fourier transform (DTFT, not to be confused with DFT, and notwithstanding the name is actually a continuous function) instead of trying to introduce it independently on its own. This is the approach taken in Oppenheim and Schafer textbook, but requires the reader have a bit more background in mathematics. In this way, the DFT is seen as a sampling of the DTFT when the signal is convolved with a window function, and explains why the spectrum of a signal is smeared when its period is not a multiple of the transform size. This textbook says "there is no leakage when the signal being analyzed is truly periodic and we can choose N to be exactly a period, or some multiple of a period" -- actually there still is, it's just that the DFT happens to sample precisely at the nulls of those sidelobes. The sidelobes are further seen as a consequence of the window function, and explains why certain window choices have better sidelobe attenuation at the tradeoff of wider main lobe/lower frequency resolution.

a-dub 4y ago
richard lyons' textbook "understanding digital signal processing" has a nice treatment of spectral leakage issues and choice of window functions. i think it may even discuss the integer multiple issue you speak of with nice illustrations of the "wrapping" effect.
deleted 4y ago
[deleted]

rerdavies 4y ago

Just part of a vast trove of mathematics for audio, by Julius O. Smith III: https://ccrma.stanford.edu/~jos/sitemap.html https://ccrma.stanford.edu/~jos/sitemap.html

blueprint 4y ago

Not to be confused with density-functional theory, which is also DFT. https://en.wikipedia.org/wiki/Density_functional_theory https://en.wikipedia.org/wiki/Density_functional_theory

khiner 4y ago

Throwing a shameless plug in here for a set of Jupiter notebooks I made that follow each chapter of this book: https://karlhiner.com/jupyter_notebooks/mathematics_of_the_dft https://karlhiner.com/jupyter_notebooks/mathematics_of_the_d... Reproduces many of the interesting results in Python, providing charts and animations along the way. Hope it’s of use to someone! (I also have a similar series for two of the other three books in this series - Digital Filters and Physical Modeling.)

cozzyd 4y ago

Just wanted to say that Julius is one of my favorite professors from undergrad. (I randomly took a bunch of clases at ccrma for fun ... But now it turns out I do a bunch of signal processing )

vermarish 4y ago

Tangientially related: I finished my B.S. eight months ago and have since been disillusioned with the prospects of ever going to grad school. This submission is so interesting that after skipping through it, I was inspired to draft a roadmap toward grad school over the last two hours -- mostly reexamining project ideas and considering which PhDs in my network might find those interesting once developed. Thank you for submitting this, it is absolutely fascinating.