Fourier Series - Complex Coefficients

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The previous page on Fourier Series used only real numbers. In engineering, physics and many applied fields, using complex numbers makes things easier to understand and more mathematically elegant. On this page, we'll redo the previous analysis using the complex form of the Fourier Series.

Again, we want to rewrite a periodic function f(t) with period T with the infinite sum of sinusoidal functions. In this case, we will use the complex exponential function as the basis. That is, we want to find the coefficients cn in the following formula:

complex form of Fourier Series
[Equation 1]

Note that the complex exponential is the sum of sinusoidal functions, since the following identity can be proven:

euler's identity
[Equation 2]

If you are unfamiliar with the rules of complex math (a neccessity for understanding the Fourier Transform), review the complex math tutorial page.

From Equation [1], the unknown Fourier coefficients are now the cn, where n is an integer between negative infinity and positive infinity. The optimal value for cn are:

optimal complex Fourier coefficients
[Equation 3]

Recall the square function:

Fourier series example

Figure 1. A periodic square waveform.

If you work through the math, you can find the optimal values for cn using equation [3]:

complex Fourier Coefficients for square wave
[Equation 4]

Note that the Fourier coefficients are complex numbers, even though the series in Equation [1], evaluated with the coefficients in Equation [4], result in a real function. If the following condition (equation [5]) is true, then the resultant function g(t) will be entirely real:

criteria for a fourier series to be real
[Equation 5]

In equation [5], the * represents the complex conjugate.

As an exercise, it would be enlightening to show that the results from the previous page on real Fourier coefficients, with the am, bn values, yield the exact same function as the complex version with the cn values given on this page. This can be shown with the help of the following identities, which should be memorized (along with Equation [2]):

cosine and sine complex identities
[Equation 6]

On the next page, we'll look at some more examples.

Next: Fourier Series Representation of the Cosine Function

Previous: Real (non-complex) Fourier Coefficients

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