Page 864 - Advanced_Engineering_Mathematics o'neil

844 Answers to Selected Problems

For ω
=±1,
ω 1 cos(K(1 + ω)) cos(K(1 − ω))

ˆ f S (ω) = − − ,
ω − 1 2 1 + ω 1 − ω
2
1
ˆ f S (1) = (1 − cos(2K)) =− ˆ f S (−1)
4
1
1
5. ˆ f C (ω) = 1 + ;
2 1+(1+ω 2 ) 1 + (1 − ω )
2
1
1 + ω 1 − ω
ˆ f S (ω) = −
2
2
2 1 + (1 + ω ) 1 + (1 − ω )
Section 14.5 The Discrete Fourier Transform
1. Approximate values are given in Table A.7.
3. Approximate values are given in Table A.8.
5. Approximate values are given in Table A.9.
1 5 k 2πijk/6
7. u j = (1 + i) e
6 k=0
u 0 =−1.33333 + 0.166667i,u 1 =−0.427030 + 0.549038i
u 2 =−0.016346 + 0.561004i,u 3 = 0.33333 + 0.500000i
u 4 = 0.849679 + 0.272329i,u 5 = 1.593696 − 2.049038i

TABLE A.7 Approximate Discrete Fourier Transform Values in Problem 1,
Section 14.5

k D[u](k)
–4 0.13292 – 0.01658i
−9
–3 0.09624 + 0.72830(10 )i
–2 0.13292 + 0.01658i
–1 2.93687 + 0.42794i
0 0.1.82396
1 2.93687 – 0.42794i
2 0.13292 – 0.01658i
−9
3 0.09624 – 0.72830(10 )i
4 0.13292 + 0.01658i

TABLE A.8 Approximate Discrete Fourier Transform Values in Problem 3,
Section 14.5

k D[u](k)
–4 0.65000 – 0.17321i
−9
–3 0.61667 – 0.25346(10 )i
–2 0.65000 + 0.17321i
–1 0.81667 + 0.40415i
0 2.45000
1 0.81667 – 0.40415i
2 0.65000 – 0.17321i
−9
3 0.61667 + 0.25346(10 )i
4 0.65000 + 0.17321i

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October 14, 2010 17:50 THM/NEIL Page-844 27410_25_Ans_p801-866