Essential Facts About Fourier Series

Essential Facts About Fourier Series (Otherwise: Everything you wanted to know about Fourier Series but were afraid to ask)

When Fourier discovered/invented the Series named after him he did two things - things that you need to remember. •

Fourier discovered that a periodic signal could be expressed mathematically as a sum of sines and cosines. Each sine or cosine is multiplied by a coefficient, and then everything is added together.

•

Fourier not only discovered that the signal could be expressed mathematically as a sum of sines and cosines, he also discovered formulas that let you get the coefficients of the sines and cosines.

If you know what those two facts mean, then you are well on your way to understanding Fourier Series. ADVERTISEMENT

The Fourier Sum Here is the expression that Fourier found for a periodic signal.

In this expression note the following. •

This expression can be used to represent any periodic signal. ○ A periodic signal repeats. Say the time for a repetition is T seconds. Then, if the periodic signal is f(t), we would have: 

f(t+T) = f(t)

•

The sum could have an infinite number of terms.

•

All terms are at an integral multiple of a fundamental frequency: ○

fo = 1/T = fundamental frequency (Hertz)

○

ω o= 2π fo = fundamental angular frequency (radians/second)

○ The multiples of the fundamental frequency are call the harmonics.

The Fourier Coefficients Fourier also figured out a way to compute the coefficients in the series.

There is one exception to the rule:

And that is it. That is not to say that doing the integrals will be easy. It might not be, and you might need to do the integration numerically, especially if you have numerical data. That is another topic.

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A Note About

A Note About

Fourier Coefficients

Fourier Coefficients - 2

In a Fourier Series, there are two terms in the nth harmonic - a cosine term and a sine term. Together they give you the components of the signal at that frequency, i.e. the nth we have: ○

harmonic. Writing them out

ancos(nω ot) + bnsin(nω ot) = total component at the nth harmonic.

Now, the a's and b's can be computed starting with the definitions.

And, it is possible to compute the a's and b's by approximating those integrals. However, that isn't necessarily the way it is done. In most cases a different representation is used. Consider the following. ADVERTISEMENT

In the integral above, the function, f(t), is multiplied by a complex exponential. However, the complex exponential can be represented as a complex sum of the cosine and the sine. ejn2π

t/T

= cos(n2π t/T) + jsin(n2π t/T)

Using that representation gives the two integrals above. (And, note the presence of "j" multiplying the sine in the integral above, and in the expression for the complex exponential. That is what makes it complex.) Now, the neat result from this is that you can do one integral and get both the a's and b's simultaneously. In the fft functions in any analysis program that is what happens most of the time.