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492 CHAPTER 14 The Fourier Integral and Transforms

⎧
⎪1 for 0 ≤ t < K 7. Show that, under appropriate conditions on f and its
⎨
4. f (t) = −1 for K ≤ t < 2K derivatives,
⎪
0 for t ≥ 2K F S [ f (4) (t)](ω) = ω ˆ f S (ω) = ω f (0) + ωf (0).
3
⎩
4

−t
5. f (t) = e cos(t) 8. Show that, under appropriate conditions on f and its

sinh(t) for K ≤ t < 2K derivatives,
6. f (t) =
0 for 0 ≤ t < K and for t ≥ 2K F C [ f (4) (t)](ω) = ω ˆ f C (ω) + ω f (0) − f (3) (0).
2
4

14.5 The Discrete Fourier Transform
If f has fundamental period p, its complex Fourier series is
∞

d k e 2πikt/p ,
k=−∞
in which
1 p
d k = f (t)e −2πikt/p dt
p 0
for k = 0,±1,±2,···.
Under certain conditions on f , this series converges at t to to ( f (t+) + f (t−))/2.
Our objective is to define the discrete Fourier transform. To understand why this definition
will take the form that it does, consider the problem of approximating the coefficients d k in the
complex Fourier series. One way is to begin by subdividing [0, p] into N subintervals of equal
length p/N and choosing a point in each interval of the subdivision, say
jp ( j + 1)p

t j in ,
N N
for j = 0,1,2,··· , N − 1. Approximate d k by the Riemann sum
N−1
1 p
d k ≈ f (t j )e −2πikt j /p .
p N
j=0
This suggests the definition of the discrete Fourier transform, which acts on a sequence of
N given complex numbers and produces an infinite sequence of complex numbers, one for each
integer k, as follows.

N−1
Let N be a positive integer and let u =[u j ] be a sequence of N complex numbers. Then
j=0
the N-point discrete Fourier transform of u is the sequence D[u] defined by
N−1

D[u](k) = u j e −2πijk/N
j=0
for k = 0,±1,±2,···.

To simplify the notation, we will denote the N-point discrete Fourier transform of u by
U, with lower case for the input sequence and upper case for its discrete transform. In this
notation,

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October 14, 2010 16:43 THM/NEIL Page-492 27410_14_ch14_p465-504