Page 701 - The Mechatronics Handbook

0066_Frame_C23 Page 9 Wednesday, January 9, 2002 1:52 PM

∗
where φ k (t) stands for the complex conjugate of the signal. If E k is equal to unity for all values of k, then
the φ(t) is an orthonormal set. It is relatively easy to approximate a given signal by an appropriate set of
orthonormal functions as this leads to minimum error between the actual signal and its approximation.
Thus, a given signal x(t) with finite energy over the interval t 1 < t < t 2 can be expressed as
∞
xt() = ∑ c k φ k t() (23.14)
k=∞
–
where

c k = ∫ t 2 xt()φ k t() t, k = 0, 1, 2,…
±
±
∗
d
t 1
This equation is referred to as the generalized Fourier series of x(t), and the constants c k , k = 0, ±1,
±2,…, are called the Fourier series coefficients with respect to the orthogonal set {φ(t)}.
x ˆ
Denoting the first M terms in Eq. (23.14) as (t), the resulting error function is
M
e M t() = xt() – ∑ c k φ k t() (23.15)
∗
k=0
By computing the average power of this error function and setting its derivatives with respect to c k to
zero, yields an optimal set of {c k } that minimizes the error energy. Also, if lim x ˆ (t) = x(t), then the
M→∞
basis functions are said to be complete, that is, the error energy is equal to zero. When dealing with
periodic signals, the time interval (t 1 , t 2 ) is equal to the period, T, of the signal. In addition, φ n (t) =
exp{jnω 0 } is often selected as the set of basis functions, for n = 0, ±1, ±2, …, and ω 0 = 2π/T. The methods
of computing the Fourier series coefficients are subsequently discussed.
The Complex Exponential Fourier Series
Let the signal x(t) be such that x(t) = x(t + T) and that it satisfies the Dirichlet conditions: 3
1. x(t) is absolutely integrable over its period, that is,

t +T
1
∫ xt() t < ∞
d
t
1
2. the number of maxima and minima of x(t) in each period is finite,
3. the number of discontinuities of x(t) in each period is finite,
then x(t) can be expanded as
∞
xt() = ∑ c k e jkω t , ω 0 = 2πf 0 (23.16)
0
k=∞
–
where

1 t +T – jkω t
1
c k = --- ∫ xt()e 0 dt (23.17)
T t
1
for any arbitrary value of t 1 .
The coefficients c k are called the complex Fourier series coefficients for the signal x(t), which, in general,
may be complex numbers.

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