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Can anyone explain to me what is the point of using complex numbers to get the Discrete Fourier Transform when the Discrete Cosine Transform and Discrete Sine Transform exist and both use only real numbers. Is there not another way of having a Discrete "Cosine and Sine" Transform all in one that does not use complex numbers? After all frequencies are suppose to be physical cosines and sines, not complex numbers.

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You are a bit confused; DFT uses complex exponentials as a basis set; DCT uses cosines only as a basis set; DST uses sines only as a basis set. Quite related, they are, but completely different. –  J. M. Feb 7 '12 at 11:05
    
My question is not asking "How are complex exponentials different from cosines and sines?". My question is why are complex exponentials needed at all for the DFT? For comparison, a Fourier series can be represented using complex exponentials but its not mathematically absolutely necessary since it can be recasted using only sines and cosines. –  user782220 Feb 7 '12 at 22:20
    
Notational convenience comes to mind, among other things (there is for instance that nasty factor of $1/2$ in the constant term that you have to keep track of when you insist on working entirely in $\mathbb R$). Have you by any chance seen Van Loan and Briggs/Henson? Or Bracewell and Brigham, for that matter? I think they explain it better than I can... –  J. M. Feb 7 '12 at 22:56
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