Page 520 - Advanced engineering mathematics

500 CHAPTER 14 The Fourier Integral and Transforms

For this example, choose N = 2 = 128 and M = 10, so M ≤ N/8. Sample the partial sum at
7
points jp/N = j/64 for j = 0,1,··· ,127. Thus define the finite sequence
127

j
127
u =[ f ( jp/N)] = .
j=0
64
j=0
The 128-point DFT of u has kth term
127
j

U k = e −πijk/64 .
64
j0
Fill in the missing places by defining
⎧
for k = 0,1,··· ,10
⎪U k
⎨
V k = 0 for k = 11,··· ,117
⎪
⎩
U k for k = 117,··· ,127.
Then
10
i πijk/64
S 10 ( jp/N) = S 10 ( j/64) = 1 + e
πk
k=−10,k =0
127
1
≈ V k e πijk/64 .
128
k=0
For comparison, we will compute S 10 (1/2), and then the approximate value from the last term in
this equation. First,
10
i
S 10 (1/2) = 1 + e πik/2 = 0.45847··· .
πk
k=−10,k =0
For the approximation, we first need the numbers U k :
127 127
j j
−πij/64
U 0 = = 127,U 1 = e =−1.0 + 40.735i,
64 64
j=0 j=0
127
j
U 2 = e −πij/32 =−1.0 + 20.355i,U 3 =−1.0 + 13.557i,
64
j=0
U 4 =−1.0 + 10.153i,U 5 =−1.0 + 8.1078i
U 6 =−1.0 + 6.7415i,U 7 =−1.0 + 5.7631i
U 8 =−1.0 + 5.0273i,U 9 =−1.0 + 4,4532i
U 10 =−1.0 + 3.9922i,U 118 =−1.0 − 3.9922i
U 119 =−1.0 − 4.4532i,U 120 =−1.0 − 5.0273i
U 121 =−1.0 − 5.7631i,U 122 =−1.0 − 6.7415i
U 123 =−1.0 − 8.1078i,U 124 =−1.0 − 10.153i
U 125 =−1.0 − 13.557i,U 126 =−1.0 − 20.355i
U 127 =−1.0 − 40.735i.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 16:43 THM/NEIL Page-500 27410_14_ch14_p465-504