Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 112

Applying DFT Equation (4.8) to x w ðnÞ for k ¼ 0; 1; 2; 3, respectively,

X ðkÞ¼x w ð0ÞW k 4 þ xð1ÞW k 4 þ xð2ÞW k 4 þ xð3ÞW k3

4

We obtain the following results:
X ð0Þ¼3:3334
X ð1Þ¼2 j1:3334
X ð2Þ¼0:6666
X ð3Þ¼2 þ j1:3334

NT ¼ 1

4$0:01 ¼ 25 Hz

D f ¼

Applying Equations (4.19) , (4.22), and (4.23) leads to

4 jX ð0Þj ¼ 0:8334; f 0 ¼ tan 1

A 0 ¼ 1

0
3:3334

P 0 ¼ 1

¼ 0 0 ;

4 2 jX ð0Þj 2 ¼ 0:6954

4 jX ð1Þj ¼ 0:6009; f 1 ¼ tan 1 1:3334

A 1 ¼ 1 P 1 ¼ 1

4 2 jX ð1Þj 2

¼146:31 0 ;

2

¼ 0:3611

4 jX ð2Þj ¼ 0:1667; f 2 ¼ tan 1

A 2 ¼ 1

0
0:6666

P 1 ¼ 1

4 2 jX ð2Þj 2 ¼ 0:0278

¼ 0 0 ;

Similarly,

4 jX ð3Þj ¼ 0:6009; f 3 ¼ tan 1 1:3334

A 3 ¼ 1 P 3 ¼ 1 ¼ 146:31 0 ;

4 2 jX ð3Þj 2 ¼ 0:3611 2

b. Since N ¼ 4, from the Hamming window function, we have
w hm ð0Þ¼0:54 0:46 cos 2p 0
4 1

¼ 0:08

w hm ðnÞ¼0:54 0:46 cos 2p 1
4 1

¼ 0:77

Similarly,w hm ð2Þ¼0:77, w hm ð3Þ¼0:08. Next, the windowed sequence is computed as
x w ð0Þ¼xð0Þw hm ð0Þ¼1 0:08 ¼ 0:08
x w ð1Þ¼xð1Þw hm ð1Þ¼2 0:77 ¼ 1:54
x w ð2Þ¼xð2Þw hm ð2Þ¼3 0:77 ¼ 2:31
x w ð0Þ¼xð3Þw hm ð3Þ¼4 0:08 ¼ 0:32

Applying DFT Equation (4.8) to x w ðnÞ for k ¼ 0; 1; 2; 3, respectively,

X ðkÞ¼x w ð0ÞW k 4 þ xð1ÞW k 4 þ xð2ÞW k 4 þ xð3ÞW k3

4