The Calculus Diaries - Jennifer Ouellette [94]
MIXING AND MATCHING
The culprit was a single equation describing how heat traveled through certain materials as a wave. Fourier concluded that every wavelike “signal,” no matter how complex, could be rebuilt from scratch by adding together many different waves mixed together according to a specific “recipe.” In other words, complicated periodic functions can be written as the sum of simple waves mathematically represented by sines and cosines (now known as the Fourier series). We can figure out which waves are present in a complex signal by taking an integral over all possible waves. That is the Fourier transform.
Fourier transforms are difficult for a beginning calculus student to grasp, and more complex signals require powerful computers to crunch the numbers, but the overall concept is straightforward enough. You just take apart the original signal to determine the “ingredients,” and then figure out how to rebuild that signal with a mixture of the same component sinusoid waves.
It’s a bit like trying to re-create at home your favorite restaurant’s spécialité de la maison, except you have to guess at the ingredients. The more sinusoids we use, the more accurately the resulting reconstructed waveform resembles the original—much as estimating the area underneath a curve gives a more accurate result if you use more and more rectangles in the method of exhaustion. Anytime we are adding together many different smaller pieces that add up over time, we are taking an integral.
There is a neat trick to determining whether a given waveform is an ingredient in our original signal. Earlier we saw two simple sine waveforms, representing the functions sin(x) and sin(2x). If we multiply sin(x) by itself, we get a wave that looks like this:
Note that it oscillates entirely above the x axis, unlike the original sine wave, which oscillated equally above and below the x axis. If we integrated it, the total area would gradually accumulate; it would just go up and up, with no subtractions. This is how we know that sin(x) is a component of our original signal—indeed, it is the only component wave of our original signal. In contrast, if we multiply sin(x) by sin(2x), we get a resulting wave that looks like the graph at the top of page 244.
This time, it oscillates fairly equally above and below the x axis. If we integrated it, the total area would oscillate around 0, because sometimes the area adds to the total, and sometimes it subtracts. This tells us that sin(2x) is not a component of our original signal. We would get a similar result if we multiplied sin(x) by sin(1.1x), sin(3x), or any other wave, because our original signal was not a complex waveform, but consisted of one simple wave as the sole ingredient.
Let’s see what happens when we have a signal that adds two waves together: the function sin(x) + sin(2x), which looks like this:
Now we perform the exact same process for each possible sine wave that could be a component. For instance, multiply the above wave by sin(x), and we get this:
Since most of the oscillation occurs above the x axis, we know that if we integrated it, the total area would accumulate, indicating that sin(x) is one of the components of our original signal. But if we multiply our original signal by sin(3x), we get something that looks like the illustration on the illustration below.
This tells us that sin(3x) is not a component of our original signal, because it oscillates equally above and below the x axis. We would get a similar result for every other possible wave we tried—a variable that is commonly designated in