Fourier Analysis I

I would like to start the engineering section of this blog with an introduction to Fourier Analysis, in particular Fourier Series. Fourier Series is quite a simple concept. Given you have a periodic function f(x), how can you best approximate it using sine and cosine functions. In more advanced cases, the function can be approximated using any function with a property called orthogonality. Two functions g(x) and h(x) are orthogonal in a closed interval [a, b] if they satisfy the following:

It is not that hard to prove that sine and cosine functions are orthogonal to each other and to themselves in the closed interval [0, 2π]:

Since sin(x) and cos(x) is symmetric at x=0, and sin(x) is odd, we need not work any further, because the integrand becomes odd across x=0, hence the integral is zero.

For the sine and cosine, it is more tedious:

Now, we are ready to approximate a function f(x) in terms of our sines and cosines. Fourier Theorem states that:

Where the coefficients are given by:

To understand how those coefficients are determined using those formulas, we need to make use of the orthogonality that we discussed just now. First for a0, integrate the entire expression with respect to x:

Using our sine and cosine orthogonality,

Therefore,

For the other two coefficients, multiply by cos(nx) and sin(nx) respectively and integrate term wise.

Fourier Series is very useful in solving linear homogenous differential equations due to the principle of superposition. In the coming posts, I’ll illustrate some of these applications, along with an interesting application to infinite series.