Derivation of Complex Fourier Series Coefficients

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On this page, we'll look at deriving where the formula for the Fourier Series coefficients come from. Namely, we look at the derivation of this equation:

optimal complex Fourier coefficients
[Equation 1]

The proof is fairly simple, assuming the Fourier Series g(t) does in fact converge to the original periodic function f(t). Assuming the periodic function f(t), with fundamnetal period T, has a Fourier Series representation (it always does for any real world periodic waveform). Then the following equation is true:

the function has a fourier series representation
[Equation 2]

The goal is to determine the values for the cn in equation [2]. This can be done by multiplying both sides of [2] with exp(-i*2*pi*m*t/T), and then integrating the product from t=[0,T]. The result will be the optimal solution for cm, where m represent any integer.

To see this in action, let m be any integer, and multiply equation [2] by the complex exponential:

first step in coefficient proof
[Equation 3]

Now integrate both sides of equation [3] with respect to t from [0, T]:

second step in coefficient proof
[Equation 4]

In equation [4], the order of summation and integration was swapped, which is proper because both operators are linear. The formal proof that infinite sums and integration can change their order is the life's work of some dead mathematician somewhere. Finally, in the last step of [4], the constant cn can be pulled outside the integral, since it is just a constant.

Continuing, the integral of [4] can be easily evaluated. When n does not equal m, the integral is always zero. When n=m, the integral will equal T:
third step in coefficient proof
[Equation 5]

Key point: As a result, the infinite sum only yields one non-zero term. Using equation [5] in equation [4], we get:

fourth step in coefficient proof
[Equation 6]

The integration gives zero for all terms except the mth term. As a result, equation [6] yields the solution for cm: which is equivalent to equation [1].

In the next section, we'll look at applying the Fourier Series math to solving electric circuits.


Next: An Application: Electric Circuits

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