Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 95

for n ¼ 1

X ð0Þe j0 þ X ð1Þe j 2 þ X ð2Þe jp þ X ð3Þe j 3 2

X 3

xð1Þ¼ 1
4

X ðkÞe j k 2 ¼ 1
4

k ¼0

¼ 1

4 ðX ð0ÞþjX ð1ÞX ð2ÞjX ð3ÞÞ
¼ 1

4 ð10 þ jð2 þ j2Þð2Þjð2 j2ÞÞ ¼ 2

for n ¼ 2

X ð0Þe j0 þ X ð1Þe jp þ X ð2Þe j2p þ X ð3Þe j3p

X 3

xð2Þ¼ 1
4

X ðkÞe jkp ¼ 1
4

k ¼0

¼ 1

4 ðX ð0ÞX ð1ÞþX ð2ÞX ð3ÞÞ
¼ 1

4 ð10 ð2 þ j2Þþð2Þð2 j2ÞÞ ¼ 3

and for n ¼ 3

X ð0Þe j0 þ X ð1Þe j 3 2 þ X ð2Þe j3p þ X ð3Þe j 9 2

X 3

xð3Þ¼ 1
4

¼ 1
4

X ðkÞe j kp 2

k ¼0

¼ 1

4 ðX ð0ÞjX ð1ÞX ð2ÞþjX ð3ÞÞ
¼ 1

4 ð10 jð2 þ j2Þð2Þþjð2 j2ÞÞ ¼ 4

This example actually verifies the inverse DFT. Applying the MATLAB function ifft() we obtain
>> x ¼ ifft([10 2þ2j 2 2 2j])
x ¼ 1234

Now we explore the relationship between the frequency bin k and its associated frequency.
Omitting the proof, the calculated N DFT coefficients XðkÞ represent the frequency components
ranging from 0 Hz (or radians/second) to f s Hz (or u s radians/second), hence we can map the frequency
bin k to its corresponding frequency as follows:

u ¼ ku s
N

ð radians per second Þ (4.12)

or in terms of Hz,

f ¼ kf s
N

ð Hz Þ (4.13)

where u s ¼ 2 pf s .