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Unformatted text preview: Chapter 5 Finite Length Discrete Fourier Transfom Discrete Fourier Transform (DFT) Usually, we do not have an infinite amount of data which is required by the DTFT. Instead, we have 1 image, a segment of speech, etc. Also, most real world data are not of the convenient form a n u [ n ]. Finally, on a computer, we can not calculate an uncountably infinite (continuum) of frequencies as required by the DTFT. ACTUAL DATA ANALYSIS on a computer Use a DFT to look at a finite segment of data. In our development in the previous section of x [ n ] periodic with x [ n ] the part of the signal that was repeated, we could have assumed that our finite segment of data came from windowing an infinite length sequence x [ n ] x [ n ] = x [ n ] w R [ n ] where w R [ n ] is a rectangular window: 1 w R [ n ] = 1 , n = 0 , 1 , , N 1 , otherwise x [ n ] = x [ n ] w R [ n ] is just the samples of x [ n ] between n = 0 and n = N 1 . x [ n ] is 0 everywhere else. Therefore, it is defined n , and we can take its DTFT: X () = DTFT( x [ n ]) = X n = x [ n ] e j n = X n = x [ n ] w R [ n ] e j n = N 1 X n =0 x [ n ] e j n So, X () = N 1 X n =0 x [ n ] e j n = N 1 X n =0 x [ n ] e j n as we saw before. Lets say now that we want to sample X () (which is continuous and periodic with period 2 ) so we store it on a computer. Sample X (): Assume we want 8 points in frequency then sample X () at 8 uniformly spaced points on the unit circle: Values of frequency are 0 , / 4 , / 2 , , 7 / 4 or 2 k/ 8 , k = 0 , 1 , , 7 . 2 If we let k = N , what happens? If k = N , we get repetition of the points we sampled so only N samples are unique. Define Discrete Fourier Transform (DFT) as X [ k ] = X ( 2 k N ) for = 2 k N , k = 0 , 1 , . . . , N 1, i.e. only look at the N distinct sampled frequencies of X ()....
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 Spring '07
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