Question

Transform (1,i,0) , (1,2,1-i) to orthonormal basis

Answer 1

Main Idea

When working with signals many times it is helpful to break up a signal into smaller, more manageable parts. Hopefully by now you have been exposed to the concept of eigenvectors and there use in decomposing a signal into one of its possible basis. By doing this we are able to simplify our calculations of signals and systems through eigenfunctions of LTI systems.

Now we would like to look at an alternative way to represent signals, through the use of orthonormal basis. We can think of orthonormal basis as a set of building blocks we use to construct functions. We will build up the signal/vector as a weighted sum of basis elements.

Example 1

The complex sinusoids 1T√eiω0nt for all −∞<n<∞ form an orthonormal basis for L2([0,T]).

In our Fourier series equation, f(t)=∑∞n=−∞cneiω0nt, the {cn} are just another representation of f(t).

Note:For signals/vectors in a Hilbert Space, the expansion coefficients are easy to find.