Fourier Series

Nearly any periodic function can be represented as a linear combination of simple, sinusoidal waves.  By applying a process known as a Fourier transform, one can represent a complicated periodic function as an approximation displaying each individual component in superposition.

The Fourier transformation process consists of rewriting the original F(t) function in the form of

where n is an integer from 1 to infinity, ω represents angular frequency, t is time, and a₀, a_n, and b_n are coefficients.  To determine the a₀, a_n, and b_n coefficients, it is necessary to evaluate three integrals:

and

After determining each coefficient, the process is completed by plugging each in to its respective place within the Fourier series form of F(t).

Example:

(Problem 3-28 from Classical Dynamics of Particles and Systems 5^th Edition by Thornton and Marion)

Obtain the Fourier expansion for the function

in the interval –π/ω < t < π/ω where ω = 1 rad/s.

Solution:

After plotting the function, we determine that the function is odd as F(-t) = -F(t), or the values on the left and right side of the y-axis are of equal absolute value but opposite in sign.  This means that a₀ must equal 0 as the integral of an odd function of period 2π over the interval –π/ω < t < π/ω is always 0.

So,

Similarly, when determining a_n, we have an odd function F(t) times an even function cos(nωt).  This results in an overall odd function F(t)cos(nωt).  Thus, for the same reasons as a₀,

When determining b_n, we have an odd function F(t) times an odd function sin(nωt) resulting in an overall even function F(t)sin(nωt).  In this case, we must take the integral.

                        

Thus,

or

By plotting the original function and the Fourier expansion including various numbers of terms, it is clear that as the number of terms included in the expansion increases, the Fourier series plot approaches that of the original function.  Below are plots of the original function, the Fourier series including two terms, the Fourier series including three terms, and the Fourier series including four terms, respectively.