# Engineering Analysis/Fourier Series

Before reading this chapter, students should be familiar with the fourier series decomposition method. Information about this can be found in Signals and Systems. |

The L_{2} space is an infinite function space, and therefore a linear combination of any infinite set of orthogonal functions can be used to represent any single member of the L_{2} space. The decomposition of an L_{2} function in terms of an infinite basis set is a technique known as the **Fourier Decomposition** of the function, and produces a result called the **Fourier Series**.

## Fourier Basis[edit | edit source]

Let's consider a set of L_{2} functions, , as follows:

We can prove that over a range , all of these functions are orthogonal:

Because is as an infinite orthogonal set in L_{2}, is also a valid basis set in the L_{2} space. Therefore, we can decompose any function in L_{2} as the following sum:

[Classical Fourier Series]

However, the difficulty occurs when we need to calculate the a and b coefficients. We will show the method to do this below:

## a_{0}: The Constant Term[edit | edit source]

Calculation of a_{0} is the easiest, and therefore we will show how to calculate it first. We use the value of a_{0} which minimizes the error in approximating by the Fourier series.

First, define an error function, E, that is equal to the squared norm of the difference between the function f(x) and the infinite sum above:

For ease, we will write all the basis functions as the set φ, described above:

Combining the last two functions together, and writing the norm as an integral, we can say:

We attempt to minimize this error function with respect to the constant term. To do this, we differentiate both sides with respect to a_{0}, and set the result to zero:

The φ_{0} term comes out of the sum because of the chain rule: it is the only term in the entire sum dependent on a_{0}. We can separate out the integral above as follows:

All the other terms drop out of the infinite sum because they are all orthogonal to φ_{0}. Again, we can rewrite the above equation in terms of the scalar product:

And solving for a_{0}, we get our final result:

## Sin Coefficients[edit | edit source]

Using the above method, we can solve for the a_{n} coefficients of the sin terms:

## Cos Coefficients[edit | edit source]

Also using the above method, we can solve for the b_{n} terms of the cos term.