Fourier Series Representation of an Arbitrary Periodic Function Consider now an arbitrary periodic function f (t) with period T, i.e., f (t) satisfies the property
f (t + nT ) = f (t), ∀n=I (2–352)
where I is the set of integers. Examples of arbitrary periodic functions include a square-wave (see Fig. 2–16) and a sawtooth (see Fig. 2–17). It is known that any arbitrary periodic function
Figure 2–16 Square-Wave Function
can be expressed as an infinite series of sines and cosines. This infinite series is called a Fourier series. Suppose now that we consider a function f (t) that is periodic with period T on the interval from zero to T. Then, in terms of a Fourier series expansion, the periodic function f (t) can be written as
where Ω = 2π/T is the fundamental frequency. It is known that the functions e ikΩt , (k = 0, ±1, ±2, . . .) are orthogonal over the time interval t ∈ [0, T ], i.e.,
The coefficients ck, (k = 0, ±1, ±2, . . .) are obtained as follows. Suppose we multiply both sides of Eq. (2–353) by e −ilΩt (where l ∈ I) and integrate over the period of the function (i.e., from zero to T). We then obtain
where
Noting that Ω = 2π/T, we have
Suppose now that we let m = k − l (we note that, because k and l are integers, m is also an integer). Then when m ≠ 0 we have
Furthermore, for the case that m = 0, we need to take the limit as m → 0 as
Because both the numerator and denominator approach zero as m → 0, we can use L’Hopital’s rule to obtain
Noting that the condition m = 0 is equivalent to the condition that k = l, we have
which implies
The expression for ck from Eq. (2–362) can then be used in Eq. (2–353) to obtain the Fourier series expansion of the periodic function f (t). It is noted that a Fourier series can be written in real form as follows. First, we note that
Then, we have
Now we have
from which we obtain
Similarly,
from which we obtain
Then we can define
Solving for ck and c−k in terms of ak and bk, we obtain
We can then write
Substituting the expressions for ck and c−k into this last equation, we obtain
Rearranging, we have
Then, using Eqs. (2–365) and (2–368), we obtain
Eq. (2–377) is a real form of a Fourier series for an arbitrary periodic function f (t).
1. What is a Fourier series representation? | ![]() |
2. How does a Fourier series represent an arbitrary periodic function? | ![]() |
3. Why is a Fourier series representation useful in mechanical engineering? | ![]() |
4. Can any periodic function be accurately represented by a Fourier series? | ![]() |
5. How can Fourier series be applied to practical mechanical engineering problems? | ![]() |