The amplitude of each term satisfies a relationship with the original function called the Parseval equality (Wikipedia).
wikipedia), basically a method of summing little chunks of 'area' under a smoothly varying function. The funky-looking 'E' on the right side of the equation is the Greek letter Sigma, and represents summation of a series of values (wikipedia). On a personal note, it was curiosity about these symbols when I was in the eight-grade that motivated me to learn calculus on my own. It's value extends far beyond science and mathematics.
Physically, we can think of this as an expression of conservation of energy, that the total energy in the signal, the square of the amplitude of the function, f(x), must be equal to the total energy in each frequency composing the signal. If we plot the square of amplitude of each frequency (an^2+bn^2), we obtain a plot which is the equivalent of the power spectral density (PSD).
We'll examine a few more cases, such as the sawtooth wave, were the Fourier coefficients are computed to be
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Let's also look at the half-sine (rectifier) wave with Fourier coefficients
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Are the frequencies real?
One should be careful to define what one means by 'real'. I'd be inclined to define 'real' in this case as if the frequencies found are representative of the underlying physical mechanism for forming the original signal as opposed to an artifact of the sampling. And since all non-zero signals exhibit some sampling, there will always be sampling artifacts in the resulting PSD.
The Fourier series demonstrates that there is an alternative way to represent almost any finite function. In many ways, it is as indistinguishable as saying that you could represent the number 10 by 3+2+5 or 4+6. Think about the way you do pen-and-paper arithmetic, especially subtraction and division, and you'll realize these alternative representations are done all the time. The Fourier series becomes useful by providing an additional set of mathematical 'tricks' which may facilitate solving some other problem.
For example, in the case of the half-sine (rectifier) example above, such as signal is usually generating by running a sine wave signal (such as A/C) through a non-linear device like a rectifier, which chops off the negative (reverse) part of the current. In this case, the signal does not have to be formed by a summation of the frequencies in the Fourier series (though that would be an alternative way to form it). In this case, I would be inclined to say the frequency components of the signal are not 'real', but simply an alternative representation.
But suppose you're examining a process where you expect to have some regular structures or variation? Then the peaks might tell you more about the structure under examination, but one must exercise extreme care.
The major point of this section is to introduce the fact that non-zero power in the frequency bin of a given signal profile is NOT evidence of a real frequency indicating some intrinsic structure. I will demonstrate more examples of this in the coming sections.
More online Fourier resources
- Signals Systems Control - Java Applets for Fourier Series & Transforms
- Fourier Series
- Fourier Series Examples
Next Weekend: The Fourier Transform, FFT and PSD, Oh My!