[A Fourier series approximation of a square wave.]

Fourier Sin/Cos Series Demonstration

A Fourier series is a sum of sinusoidal (sine and cosine) functions with harmonic frequencies {ω_m= mω₀ } where m in an integer and ω₀ = 2πf₀ is the fundamental frequency. It is a useful mathematical tool because it can be used to analyze periodic functions with period T = 1/f₀ by decomposing such a function into sinusoidal components with constant (Fourier) coefficients a_m and b_m.

We refer to each term as having a certain (angular) frequency {ω_m= mω₀ } because the independent variable is often time. However, Fourier analysis is very general and the independent variable can, in fact, be anything including position so we write the unknown function in the generic form f(x).

Note: In order to simplify the model, the domain of the independent variable has been set to [0, 2π]. The domain can be changed in the Ejs model. Also, this EJS model is inefficient because the sine and cosine functions are evaluated repeatedly with the same argument. A more efficient implementation would create an array of sin/cos values after the number of sample points has been set. A much more efficient implementation would use the Fast Fourier Transformation (FFT) in the OSP numerics library.

Questions and Exercises:

• Set the function to zero and drag a sample point (red circle in the function plot).  What happens to the Fourier coefficients?  Can you drag a single point so that only sine coefficients appear?  Cosine coefficients?
• Set the function to zero and explain how you must arrange the sample points so that only sine coefficients appear,  How must the points be arranged so that only cosine coefficients appear?
• Set the function to zero and enter the following sine coefficients into the table:  {1, 0, 1/3, 0, 1/5, 0, 1/7, 0, 1/9}.  What function are you approximating? What function are you approximating if these coefficients are entered as cosine coefficients?  What if {1, 0, -1/3, 0, 1/5, 0, -1/7, 0, 1/9} is entered for the cosine coefficients?
• Select a square wave and observe the value of the Fourier series at the discontinuity.  What is the value of the series at the discontinuity?  Select other discontinuous functions and make a general observation about the value of the series at a discontinuity.
• Select a square wave and observe the value of the Fourier series at the discontinuity.  Note the Gibbs overshoot.  Does this overshoot have a maximum value?  Do all discontinuous functions have this overshoot?

One final and important note.  Although the Fourier coefficients are computed using a relatively simple and naive box integration, the resulting Fourier series passes through all the sample points.  This happy mathematical accident suggests that a Fourier series can be used as an interpolating function to compute f(x) between known data points.  When might this use of a Fourier series be a good idea?  Is a Fourier series or a power series more efficient for approximating functions?  Discuss.