NOTES_DFT

# NOTES_DFT - Chapter 5 Finite Length Discrete Fourier...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 (Ω) = DTFT( x [ n ]) = ∞ X n =-∞ x [ n ] e- j Ω n = ∞ X n =-∞ x [ n ] w R [ n ] e- j Ω n = N- 1 X n =0 x [ n ] e- j Ω n So, X (Ω) = N- 1 X n =0 x [ n ] e- j Ω n = N- 1 X n =0 x [ n ] e- j Ω n as we saw before. Let’s say now that we want to sample X (Ω) (which is continuous and periodic with period 2 π ) so we store it on a computer. Sample X (Ω): Assume we want 8 points in frequency – then sample X (Ω) at 8 uniformly spaced points on the unit circle: Values of frequency are 0 , π/ 4 , π/ 2 , ··· , 7 π/ 4 or 2 πk/ 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 πk N ) for Ω = 2 πk N , k = 0 , 1 , . . . , N- 1, i.e. only look at the N distinct sampled frequencies of X (Ω)....
View Full Document

{[ snackBarMessage ]}

### Page1 / 21

NOTES_DFT - Chapter 5 Finite Length Discrete Fourier...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online