FREQUENCY DOMAIN REPRESENTATION - Digital Video Processing for Engineers - page 86

So we get exactly the same magnitude at the frequency
component X 2 . The difference is the phase of X 2 . Therefore we can
see that the DFT does not just pick out the frequency components
of a signal, but is sensitive to the phase of those components. The
phase, as well as amplitude of the frequency components X k , can
be represented because the DFT output is complex.
The process of the DFT is to correlate the N sample input data
stream x i against N equally spaced complex frequencies. If the
input data stream is one of these N complex frequencies, then we
will get a perfect match, and get zero in the other N 1 frequencies
which don't match. But what happens if we have an input data
stream with a frequency in between one of the N frequencies?
To review, we have looked at three simple examples. The first
was a constant level signal, so the DFT output was just the zero
frequency or DC component. The second example was a complex
frequency which matched exactly to one of the frequency bins,
X k , of the DFT. The third was the same complex frequency, but
with a phase offset. The fourth will be a complex frequency not
matched to one of the N frequencies used by the DFT.

12.1.4 Fourth DFT Example
We will look at an input signal of frequency e þ j2.1 p i/8 . This is
rather close to e þ j2 p i/8 , so we would expect a rather strong output
at X 1 . Let's see what the N ¼ 8 DFT result is

hopefully the
arithmetic is all correct. Slogging through this arithmetic is purely
optional

e

e the details are shown to provide a complete example.

Generic DFT equation for N ¼ 8:X k ¼ P i ¼ 0 x i e j2 p ki = 8
X 0 ¼ X 7 1 ¼ X 7

i ¼ 0 e þ j2 : 1 p i = 8 i ¼ 0 e þ j2 : 1 p i = 8

¼½ 1 þ j0 þ½ 0 : 6788 þ j0 : 7343 þ½ 0 : 0785 þ j0 : 9969

þ½ 0 : 7853 þ j0 : 6191 þ½ 0 : 9877 j0 : 1564 þ½ 0 : 5556

j0 : 8315 þ½ 0 : 2334 j0 : 9724 þ½ 0 : 8725 j0 : 4886

¼ 0 : 3777 j0 : 0986

X 1 ¼ X 7

i ¼ 0 e þ j2 : 1 p i = 8

e j2 p i = 8

¼ X 7

i ¼ 0 e þ j0 : 1 p i = 8

¼½ 1 þ j0 þ½ 0 : 9992 þ j0 : 0393 þ½ 0 : 9969

þ j0 : 0785 þ½ 0 : 9931 þ j0 : 1175 þ½ 0 : 9877 þ j0 : 1564

þ½ 0 : 9808 þ j0 : 1951 þ½ 0 : 9724 þ j0 : 2334 þ½ 0 : 9625

þ j0 :

2714

¼ 7 :

8925 þ j1

:

0917