Looking for Periodicity
The Fourier Transform

In this module and the next module we develop techniques that can be used to analyze situations in which there is some periodicity. The techniques we develop are named after Jean Baptiste Joseph Fourier. A brief biographical sketch of Fourier is available at the MacTutor History of Mathematics Archive. Because periodicity is so common and so important these techniques are used in a surprizing number of situations. We begin by looking at three examples. You should look at all three before going on.

• Playing music on a CD player or casette tape recorder.
• Sunspots.
• Fox and Lynx in the Mackenzie River District.

The three examples above all have one common kernel -- periodicity. Most of the sounds we hear are made up of a combination of different periodic components. We need to know something about these different components to understand why Nancy's speakers blew even though the signal going to the speakers was weaker than signals that in the past had not blown the speakers out. Looking at the Mackenzie River District lynx data and at the sunspot data we see a rough periodicity. Analyzing this periodicity can help us understand our sun and the earth on which we live.

In order to look for periodicity we will use very selective metaphorical ears -- ears that pick-up signals of the form

for a fixed frequency w. Functions of this form can also be written in the form

The module Two Forms for Some Trigonometric Functions discusses converting back-and-forth between these two forms. The form

is often easier to work with mathematically than the first form, although the first form is easier to interpret physically. For this reason we will work with the second form.

This situation is identical mathematically to the situation we looked at in the preceding module. We are looking at this signal from the viewpoint or perspective of the frequency w. From this viewpoint we "see" just the part

That is, we are "measuring" the original function against the two "yardsticks"

If we use the notation

u = sin 2 pi w t
v = cos 2 pi w t

Then from this perspective we "hear" or "see"

In the preceding module the key mathematical facts that enabled us to "see" things in this way were

• The two vectors u and v were perpendicular.

• The two vectors u and v were unit vectors.

We will work with the vector space C[c, d] whose elements are continuous functions on the interval [c, d]. The interval [c, d] must satisfy one important criterion --

1. Show that the two vectors

   
   u = sin 2 pi w t
   v = cos 2 pi w t
   

   are perpendicular in the vector space C[c, d]. Recall that in this inner product space the inner product is given by

   

2. Note that these two vectors are not unit vectors. Find the magnitude of each of these vectors. answer

3. Find a formula for the coefficients a and b that tell us the "appearance"
   a sin 2 pi w t + b cos 2 pi w t

   of a signal f(t) as "viewed" from the perspective of the frequency w. answer

The magnitude of the frequency w component

of the signal f(t) is given by

This gives us a function A(w) that tells us the magnitude of the frequency w component of the signal f(t) for each frequency w. This function A(w) is called the Fourier (power) Transform of the signal f(t). It is important because it tells us how much of the signal f(t) is due to simple tones of various frequencies.

We often write a(w) and b(w) because the coefficients a and b depend on the frequency w. The coefficients a(w) and b(w) are often called the Fourier coefficients of the function f(t) at frequency w.

• Use your CAS window to compute a(w), b(w), and A(w) for each of the indicated functions f(t) and frequencies w below.

• Compare the original function f(t) to the function
  g(t) = a(w) cos 2 pi w t + b(w) sin 2 pi w t

• Discuss what you see.

1. 
    f(t) = 3 sin 2 pi (440 t + 0.15),  0 <= t <= 1/440
    w = 440
   
   

2. 
    f(t) = 3 sin 2 pi (440 t + 0.15),  0 <= t <= 1/450
    w = 450
   
   

3. 
    f(t) = 3 sin 2 pi (440 t + 0.15),  0 <= t <= 1/460
    w = 460
   
   

4. 
    f(t) = max(-0.5, min(0.5, sin 2 pi 440 t))),  0 <= t <= 1/440
    w = 440
   
   

5. Why did Nancy's speakers blow? You may be able to see the answer now. We will examine this question further in the next module.

The details of computing the Fourier power transform are fairly complex and the computations themselves can be quite time-consuming. In the next module we look at a simpler variation of this situation. We close this module by looking at the Fourier power transform for the Mackenzie River lynx data and at the Fourier power transform for the sunspot data we looked at earlier.

The graph below shows the Fourier power transform for the Mackenzie River lynx data. The x-axis is labeled with the period rather than the frequency because for this kind of data the period is more meaningful than the frequency. The y-axis is labeled from 0 to 1. The value 1 would mean that 100% of the signal was at that period. Notice that the Fourier power transform for the lynx data has a very pronounced spike at about nine years. This, to me, was very surprizing. This is data drawn from a real population out in the wild and although I had heard that this data was periodic, I didn't expect the periodicity to be so dramatic -- after all, populations in the wild are subject to all sorts of factors -- for example, weather or hunting.

The graph below shows the Fourier power transform for the sunspot data. The x-axis is labeled with the period rather than the frequency because for this kind of data the period is more meaningful than the frequency. The y-axis is labeled from 0 to 1. The value 1 would mean that 100% of the signal was at that period. Notice this graph is nowhere near as dramatic as the graph for the lynx data. This was even more surprizing to me. I expected a very pronounced periodicity in part because I had heard of the "sunspot cycle" and in part because astronomical data is filled with periodicity.