Page 517 - Advanced engineering mathematics

14.5 The Discrete Fourier Transform 497

1 1 1 1 − e 4i 1 1 − e −4i
U k = − .
N N 2i 1 − e 4i/N−2πik/N 2i 1 − e −4i/N−2πik/N
Now approximate the exponential terms in the denominator by using the first two terms of the
x
power series of e about 0. This gives us
x
e ≈ 1 + x
for |x| << 1. Then
1 1 1 − e 4i 1 1 − e −4i
U k = −
N N 1 −[1 + (4i/N − 2πik/N)] 2i 1 −[1 + (−4i/N − 2πik/N)]
1 1 − e 4i 1 − e −4i
=− −
4 −2 + kπ 2 + kπ
1 1
4i
4i
=− [4 − πi(e − e −4i ) − 2(e + e −4i )]
4 π k − 4
2 2
1 1
=− [4 − 2πki sin(4) − 4cos(4)]
2 2
4 π k − 4
cos(4) − 1 1 πk sin(4)
= + i .
2 2
2 2
π k − 4 2 π k − 4
x
The approximation e ≈ 1 + x has led to an approximate expression for (1/N)U k that is exactly
equal to d k . This approximation cannot be valid for all k, however. First, the approximate value
x
used for e assumed that x is much less than 1 in magnitude. Further, the N-point DFT is periodic

of period N,so U k+N = U k , while there is no such periodicity in the d k s.
In general it would be difficult to derive an estimate on relative sizes of |k| and N that would
result in (1/N)U k approximating d k to within a given tolerance, and which would hold for a
reasonably broad class of functions. However, for many applications encountered in practice, the
empirical rule
N
|k|≤
8
has proved to be effective.

SECTION 14.5 PROBLEMS

−k
In each of Problems 1 through 6, compute D[u](k) for 8. U k = i , N = 5
k = 0,±1,··· ,±4. 9. U k = e −ik , N = 7
2
10. U k = k , N = 5
5
1. u =[cos( j)] j=0 11. U k = cos(k), N = 5
ij 5 12. U k = ln(k + 1), N = 6
2. u =[e ]
j=0
5
3. [1/( j + 1)]
j=0
2 5
4. [1/( j + 1) ] In each of Problems 13 through 16, compute the first seven
j=0
2 5
5. [ j ] complex Fourier coefficients d 0 ,d ±1 , d ±2 and d ±3 of f (t).
j=0
5
6. [cos( j) − sin( j)] Then use the DFT to approximate these coefficients, with
j=0
N = 128.
N
In each of Problems 7 through 12, a sequence [U k ]
k=0
is given. Determine the N-point inverse discrete Fourier 13. f (t) = cos(t) for 0 ≤ t ≤ 2, f has period 2
transform of this sequence. 14. f (t) = e −t for 0 ≤ t < 3, f has period 3
2
15. f (t) = t for 0 ≤ t < 1, f has period 1
2t
k
7. U k = (1 + i) , N = 6 16. f (t) = te for 0 ≤ t < 4, f has period 4
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October 14, 2010 16:43 THM/NEIL Page-497 27410_14_ch14_p465-504