User:Eric.clay

Eric Clay's Signals and Systems Homepage

Hi Everyone,

I'm enrolled in Signals and Systems for Fall 2008. I'm a senior this year (not graduating) in electrical engineering.

Orthogonal Functions & Fourier Series

Orthogonal Functions

If we think of functions as vectors, then the concept of orthogonality between functions and vectors should be the same. Mathematically vectors are orthogonal if the inner ("dot") product is 0 and this can be extended to functions using $\int _{a}^{b}u^{*}(x)v(x)dx=0$.

Another way to think of this is that vectors are orthogonal if they have no component in each other. For functions, this can be roughly translated to mean that there is no overlap between the functions.

Fourier Series

When thinking in terms of orthogonal functions, it is helpful to take the idea of basis vectors and apply them to functions. The Fourier series can be used as a set of basis functions since it is infinitely repeating with n and allows us to use Fourier transforms to simplify calculations.