[HN Gopher] Mathematics of the Discrete Fourier Transform (DFT) ...
       ___________________________________________________________________
        
       Mathematics of the Discrete Fourier Transform (DFT) with Audio
       Applications
        
       Author : mindcrime
       Score  : 44 points
       Date   : 2022-11-19 07:45 UTC (2 days ago)
        
 (HTM) web link (ccrma.stanford.edu)
 (TXT) w3m dump (ccrma.stanford.edu)
        
       | jmwilson wrote:
       | 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.
        
       | texaslonghorn5 wrote:
       | 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_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.
        
       ___________________________________________________________________
       (page generated 2022-11-21 23:01 UTC)