Page 519 - Advanced engineering mathematics

14.6 Sampled Fourier Series 499

In these summations we use the 2M + 1 numbers
U N−M ,··· ,U N−1 ,U 0 ,··· ,U M .

Since M < N/8 we must fill in the missing values to obtain an N-point sequence. One way to do
this is to fill in these places with zeros to form
⎧
for k = 0,1,··· , M
⎪U k
⎨
V k = 0 for k = M + 1,··· , N − M − 1
⎪
for k = N − M,··· , N − 1.
⎩
U k
Then the Mth partial sum of the Fourier series of f (t), sampled at t = jp/N, is approximated by
N−1
1
S M ( jp/N) ≈ V k e 2πijk/N .
N
k=0
EXAMPLE 14.16
Let f (t) = t for 0 ≤ t < 2 and suppose f (t) has period 2. Part of the graph of f (t) is shown in
Figure 14.8.
The Fourier coefficients are

2

1 −2πikt/2 i/πk for k = 0
d k = te dt =
2 0 1 for k = 0.
The complex Fourier series of f (t) is
∞
i
πikt
1 + e .
πk
k=−∞,k =0
This converges to t on (0,2).The Mth partial sum is
M
i πikt
S M (t) = 1 + e
πk
k=−M,k =0
2

1.5

1

0.5

0
–2 –1 0 1 2 3 4 5
t

FIGURE 14.8 f (t) = tfor 0 ≤ t < 2, period 2.

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October 14, 2010 16:43 THM/NEIL Page-499 27410_14_ch14_p465-504