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 .
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 periodic with
the part of the signal that was repeated, we could have assumed
that our finite segment of data came from ``windowing" an infinite
Let's say now that we want to sample (which is continuous and periodic with period ) so we store it on a computer.
Assume we want 8 points in frequency - then sample at 8 uniformly spaced points on the unit circle:
Values of frequency are
If we let , what happens? If , we get repetition of the points we sampled so only samples are unique.
Define Discrete Fourier Transform (DFT) as
Note: The resolution of the samples of the frequency spectrum is since we sample the spectrum at points that are spaced apart in frequency, that is, .
Note: If we looked at the samples of for all to for frequencies , we would get the closely related Discrete Fourier Series (DFS) which is of course periodic with period since is periodic.
The DFT is just one period of the DFS.
DFT is more useful because who wants to store all the samples of a periodic signal?!! The DFT is just one period of the DFS.
Shorthand Notation for the DFT:
Let root of unity since . You may also write simply as .
Given and , find .
is an integer with and (as usual), find its DFT.
Synthesis: INVERSE DFT
How can we recover from ?
Synthesis formula is
Prove this gives back :
ORTHOGONALITY OF EXPONENTIALS AGAIN!
Ex. Find the IDFT of .
and , find .
Given and , find .
Selected DFT Properties
If is real, then
If is even, then is real. (For a finite length function to be even, )