Page 514 - Advanced engineering mathematics

494 CHAPTER 14 The Fourier Integral and Transforms

Again recognizing geometric series in the last two terms,
1 1 − (e ia−2πijk/N N 1 1 − (e −ia−2πijk/N N
)
)
U k = −
2i 1 − e ia−2πijk/N 2i 1 − e −ia−2πijk/N
1 1 − e iaN −2πik 1 1 − e −iaN −2πik
e
e
= −
2i 1 − e e 2i 1 − e −ia −2πik/N
ia −2πik/N
e
1 1 − e iaN 1 1 − e −iaN
= − ,
2i 1 − e ia−2πik/N 2i 1 − e −ia−2πik/N
in which we have used the fact that e −2πik = 1.
√
To be specific, let N = 5 and a = 2. Then
√ √ √ √
u 0 = 0,u 1 = sin( 2),u 2 = sin(2 2),u 3 = sin(3 2),u 4 = sin(4 2).
The 5-point DFT of u has kth term
√ √
1 1 − e 5i 2 1 1 − e −5 2
U k = √ − √ .
2i 1 − e i 2−2πik/5 2i 1 − e −i 2−2πik/5
For example,
√ √
1 1 − e 5i 2 1 1 − e −5i 2
U 0 = √ − √
2i 1 − e i 2 2i 1 − e −i 2
√ √ √
sin(4 2) + sin( 2) − sin(5 2)
= √ ≈−0.1820207591
2 − 2cos( 2)
and
√ √
1 1 − 5i 2 1 1 − e −5i 2
U 1 = √ − √
2i 1 − e i 2−2πi/5 2i 1 − e −i 2−2πi/5
≈ 0.4162488825 − 2.202105642i.

14.5.1 Linearity and Periodicity of the DFT

Linearity of the DFT is obvious because it is defined as a sum. For any numbers α and β or u
and v which are N-point sequences, then
D[αu + βv](k) = αU k + βV k .

We claim next that the N-point DFT is periodic of period N. This follows from the fact that
e −2πijk = 1, since ijk is an integer. Specifically,
N−1
−2πij(k+N)/N
U k+N = u j e
j=0
N−1 N−1

e
= u j e −2πijk/N −2πijk = u j e −2πijk/N = U k .
j=0 j=0
14.5.2 The Inverse N-Point DFT
Suppose we know the numbers U k ,the N-point DFT of some u. We would like to retrieve u.
We know that, whatever each u j is,

N−1

U k = u j e −2πijk/N .
j=0

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October 14, 2010 16:43 THM/NEIL Page-494 27410_14_ch14_p465-504