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14.7 DFT Approximation of the Fourier Transform 503

We can improve the accuracy by choosing N larger. With N = 2 = 512 we obtain
9
511
2π
e
ˆ
f (3) ≈ e −2π j/512 −6πij/512
512
j=0
= 0.10595 − 0.2994i.

EXAMPLE 14.18
We will continue the preceding example, but now carry out the approximation at more points.
8
Use L = 4 and N = 2 = 256. The approximations are obtained by
255
π
ˆ
e
f (k/4) ≈ e −π j/32 −πijk/128 .
32
j=0
To have |k|≤ N/8 = 32, we will only use this approximation for f (k/4) for k = 1,2,··· ,13.
ˆ
Table 14.1 gives the approximate values along with the actual values of f (ω).
ˆ
The real part of f (ω) is consistently approximated in this scheme with an error of about 0.05,
ˆ
while the imaginary part is approximated in many cases with an error of about 0.002. Improved
accuracy is achieved by choosing N larger.
This approximation was based on the assumption that f (ω) could be approximated by
ˆ
ˆ
an integral oer an interval [0,2π L]. We can extend these ideas to the case that f (ω) can be
π L −iωξ
approximated by an integral f (ξ)e dξ:
−π L
π L
ˆ
f (ω) ≈ f (ξ)e −iωξ dξ.
−π L
Then
0 L
ˆ
f (k/L) ≈ f (ξ)e −ikξ/L dξ + f (ξ)e −ikξ/L dξ.
−π L 0

ˆ
TABL E 14.1 DFT Approximation of f (ω) in Example 14.18
k DFT approximation of ˆ f (ω) ˆ f (ω)
(k = 1) ˆ f (1/4) 0.99107 - 0.23509i 0.94118 - 0.23529i
(k = 2) ˆ f (1/2) 0.84989 - 0.3996i .8 - 0.4i
(k = 3) ˆ f (3/4) 0.68989 - 0.4794i 0.64 - 0.48i
(k = 4) ˆ f (1) 0.54989 - 0.4992i 0.5 - 0.5i
(k = 5) ˆ f (5/4) 0.44013 - 0.4868i 0.39024 - 0.4878i
(k = 6) ˆ f (3/2) 0.35758 - 0.46033i 0.3077 - 0.4615i
(k = 7) ˆ f (7/4) 0.29605 0.42936i 0.24615 - 0.43077i
(k = 8) ˆ f (2) 0.24989 - 0.39839i 0.2 - 0.4i
(k = 9) ˆ f (9/4) 0.21484 - 0.36933i 0.16495 - 0.37113i
(k = 10) ˆ f (5/2) 0.18782 - 0.34282i 0.13793 - 0.34483i
(k = 11) ˆ f (11/4) 0.16668 - 0.31896i 0.11679 - 0.32117i
(k = 12) ˆ f (3) 0.14989 - 0.29759i 0.1 - 0.3i
(k = 13) ˆ f (13/4) 0.13638 - 0.27847i 0.086486 - 0.28108i

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October 14, 2010 16:43 THM/NEIL Page-503 27410_14_ch14_p465-504