We are surrounded by periodic or nearly periodic phenomena. For example, the graph below shows a particularly nice function of period 2 -- that is, the same basic pattern of length 2 repeats itself over-and-over again.
The TI-92 screen below shows a recording made of someone humming.
Notice that the same basic pattern, shown in salmon in the picture below, repeats over and over again.
This particular example is a real recording of a real person humming and the repetition is not perfect. This is often the case with real data. In theory, for example, a spring-and-mass system should oscillate like a perfect sine function but in practice imperfections in the experimental set up often change the results a bit.
The four graphs below show several additional examples. All four of these examples have period 3. The first two are particularly relevant for the study of Nancy's blown speaker. The left graph shows a very pure tone of period 3. The right graph shows the same signal after clipping. Notice it still has period 3. Clipping a signal doesn't change the period but it does change the signal in other ways. It is these other changes that caused Nancy's speaker to blow out. The tools that we develop in this module will enable us to understand exactly why Nancy's speaker blew out.
This module is about analyzing periodic phenomena. The simplest periodic functions of period lambda are functions of the form
This form is the easiest to understand physically -- the constant A is the amplitude and the constant delta is the phase shift and essentially comes from the time at which the function was "turned on." However, it is easier mathematically to work with the form
and as usual it is easy to switch back-and-forth between these two forms. So we will work with the second form.
As a consequence of our work in the previous module we can show that the component of a function f(t) on the interval [0, lambda] of the form
can be found by computing
-1, if 0 <= t < 1
f(t) = { on the interval [0, 2]
1, if 1 <= t < 2
of the form
Graph the function f(t) and this component. Think of the function f(t) as being periodic with period 2 and the same basic pattern above repeated over-and-over again.
Any function that is made by taking a basic pattern of length lambda (the length is measured along the x-axis) and repeating the same pattern over-and-over again has period lambda. All of the functions below have period 3.
sin 2 pi t/3
sin 4 pi t/3
sin 6 pi t/3
sin 8 pi t/3
as shown in the graphs below. The first graph is like our other examples of period 3 functions. A simple basic pattern of length 3 is repeated over-and-over again. Normally, we would say that the second graph has period 1.5 but it also has period 3. This time a more complicated pattern of length 3 (actually made up of two simple patterns of length 1.5) is repeated over-and-over again. The other two graphs are similar with progressively more complicated patterns of length 3 repeated over-and-over again. Notice that any time a function has period lambda it also has period n * lambda for any positive integer n.
Any constant function f(t) = c also has period lambda for any lambda at all!!
Our work in this module is based on using the following functions (thought of as vectors in the vector space C[0, lambda]) as "building blocks."
u0(t) = 1 (The constant function)
u1(t) = cos 2 pi t/lambda
v1(t) = sin 2 pi t/lambda
u2(t) = cos 2 pi 2 t/lambda
v2(t) = sin 2 pi 2 t/lambda
u3(t) = cos 2 pi 3 t/lambda
v3(t) = sin 2 pi 3 t/lambda
.
.
.
un(t) = cos 2 pi n t/lambda
vn(t) = sin 2 pi n t/lambda
.
.
.
Using the Finite Dimensional Projection Theorem (see the exercises below) we see that we can write any function, f(t), of period lambda in the form
where the coefficents are determined by
As it is stated, the Finite Dimensional Projection Theorem applies to vectors like u0, u1, v1, ... un, vn that are perpendicular to each other and are unit vectors.
We now have the necessary machinery to take a function f(t) that has period lambda and write it in the form
The first part
is called the Fourier polynomial of degree n and the second part R(t) is called the residual. For most functions, high degree Fourier polynomials are very good approximations.
The coefficents
are called to Fourier coefficients and they tell us the higher frequency components of f(t). These higher frequency components are very important. They give different musical instruments playing the same note their characteristic sounds. As we saw above the Fourier coefficients are determined by
Click here to open a new window with a Java applet you can use to investigate Fourier polynomials. Arrange your two windows so that they overlap and you can move back-and-forthe between the them by clicking on the exposed portion of the inactive window to make it active. When you are done with the applet, close its window.
This applet shows two things. First it shows one period of a periodic function using a heavy black line. Then it shows the zero-th Fourier polynomial (just a constant function) approximating this function using a heavy red line.
Click on the pale red bar at the bottom of the applet to approximate the function using the next Fourier polynomial. Each time you click on this bar, the graph of the old approximation is replaced by a thin pale red line and the new approximation (with one more term) is drawn with a heavy red line.
Click on the pale blue bar at the bottom to start over with the first Fourier polynomial.
You can change the original function (shown by the heavy black line) by clicking on the graph near one of the vertical pale blue grid lines. If you click right on top of one of these lines then the graph will change so that the value of the function at the indicated x-coordinate changes to the height at which you clicked.
You can even graph indicate functions with discontinuities. If you click just to the right of one of the pale blue vertical grid lines the values of the function coming in to the right of this linme will change but the values of the function on the left will not change. Similarly, If you click just to the left of one of the pale blue vertical grid lines the values of the function coming in to the left of this line will change but the values of the function on the right will not change.
Your CAS window has the machinery to compute and graph Fourier polynomials approximating functions of period lambda. Use this machinery to look at the Fourier polynomials of degree n = 0, 1, 2, 3, and 4 for each of the functions below.
Explain why Nancy's speaker blew.
If your workstation has a TI-CBL with one of the TI graphing calculators (the TI-82, 83, 85, or 92) with the microphone sensor then you can collect real sounds -- using, for example, a musical instrument or a person singing. Deep tones work the best.
The following links lead to instructions and programs for collecting sound using one of the TI graphing calculators with the TI-CBL.
The following links lead to instructions for uploading data from the TI graphing calculators to the computer.
Use the ideas developed in this module to analyze a real sound. You might even experiment with voiceprints. Voiceprints are often used to identify the person speaking but in order to be used in this way we must have a consistent way of describing different voices. One way of describing a voice is by finding its Fourier coefficients. You might experiment with different people singing the same note to see if their Fourier coefficients provide a useful way of identifying the singer.
