Fourier, part 5

Fourier Approximations and Music

For this module, there are two computer algebra system files for each system. If you have your CAS open, save your file and close the program. When you click on the button below corresponding to your CAS, this will launch the CAS and will load a file corresponding to Parts 5-7 of this module.

Part 5: Fourier Approximations for General Functions of Period **2*pi**

In Part 4 we discovered integral formulas for the coefficients of trigonometric polynomials. In this part we will use those formulas to approximate general functions of period 2*pi.

Now suppose we have a 2*pi-periodic function that is not a trigonometric polynomial. For example, consider the function obtained by extending the absolute value function

f(t) = |t| for t between -pi and pi

to be periodic of period 2*pi. Because of the shape of the graph, this function is often referred to as a "triangular wave (function)."

Graph of Triangular Wave Function

Our approach here will be to use the same formulas as above to calculate the coefficients a_k and b_k. Then we will see how closely the trigonometric polynomials

approximate f. Our hope is that, as n increases, the approximations will be increasingly like the function f itself.

Let f be the "triangular wave" function.

(a) Use the formulas above to calculate the values of a₀, a₁, and a₂.
(b) Find a general formula for a_k where k is an odd positive integer.
(c) Find a general formula for a_k where k is an even positive integer.
(d) Notice that f is an even function. Use that fact to predict the values of the b_k.
Calculate the values of b_k where k is a positive integer.
Compare the graph of the "triangular wave" function with the n^th-order approximations for values of n = 1, 3, 4, and 5. Write down your observations. Explain why the approximations for n = 3 and n = 4 are the same. Enlarge your graphs as necessary to make the comparisons clear.
Compare the "triangular wave function" to your approximation with n = 5 by evaluating them at various values of t between 0 and pi.

For our second example, we will approximate the " saw-tooth wave" function obtained by extending the function

g(t) = t for t between -pi and pi
to be 2*pi-periodic.

Graph of Saw-Tooth Function

Using Steps 1 - 3 as a model, find and compare the Fourier approximations to the saw-tooth wave function. Your worksheet should include a discussion of odd and even functions and their relationship to the general formulas for a_k or b_k. Your worksheet should also include graphs of the saw-tooth wave function together with your approximations for various values of n. In addition, compare the saw-tooth wave function with your approximations by evaluating both g and the approximation at appropriate points. Describe what happens to the graph of your approximations near the points of discontinuity of g. What is the value of your approximation at the points of discontinuity of g?
For our third example, we will approximate a function that is discontinuous and, also, neither odd nor even. This "square wave" or "pulse function" is defined by extending the function

.
to be 2-pi periodic.

Graph of Pulse Function
Using the same procedures as for the functions f and g, examine the approximations to h. The general formulas for a_k and b_k are difficult to analyze. Look for values of k for which a_k or b_k are equal to 0.

In our last example in this part, we will approximate the "trapezoid wave" defined by extending the function

to be 2-pi periodic.

Graph of Trapezoid Wave Function

Follow the same steps as for the previous examples. Again, the general formulas for a_k and b_k are difficult to analyze. Look for values of k for which a_k or b_k are equal to 0.

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Fourier Approximations and Music

Part 5: Fourier Approximations for General Functions of Period 2*pi

Part 5: Fourier Approximations for General Functions of Period **2*pi**