1.4 Orthonormal basis expansions

Main idea

When working with signals many times it is helpful to break up a signal into smaller, more manageable parts. Hopefully bynow 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 blockswe use to construct functions. We will build up the signal/vector as a weighted sum of basis elements.

Alternate representation

Recall our definition of a basis : A set of vectors $\{b_i\}$ in a vector space $S$ is a basis if

1. The $b_i$ are linearly independent.
2. The $b_i$ span $S$ . That is, we can find $\{{\alpha }_i\}$ , where ${\alpha }_i\in \mathbb{C}$ (scalars) such that
   $\forall x, x\in S\colon x=\sum {\alpha }_i{b}_i$
   where $x$ is a vector in $S$ , $\alpha$ is a scalar in $\mathbb{C}$ , and $b$ is a vector in $S$ .

Condition 2 in the above definition says we can decompose any vector in terms of the $\{b_i\}$ . Condition 1 ensures that the decomposition is unique (think about this at home).

In general, given a basis $\{b_0, b_1\}()$ and a vector $x\in {ℝ}^2$ , how do we find the ${\alpha }_0$ and ${\alpha }_1$ such that

Finding the coefficients

Now let us address the question posed above about finding ${\alpha }_i$ 's in general for ${ℝ}^2$ . We start by rewriting [link] so that we can stack our $b_i$ 's as columns in a 2×2 matrix.

Simplifying our equation

To make notation simpler, we define the following two itemsfrom the above equations:

• Basis Matrix : $$B=\begin{pmatrix}⋮ & ⋮\\ b_0 & b_1\\ ⋮ & ⋮\\ \end{pmatrix}()$$
• Coefficient Vector : $$\alpha =\begin{pmatrix}{\alpha }_0\\ {\alpha }_1\\ \end{pmatrix}()$$

To get the ${\alpha }_i$ 's, we solve for the coefficient vector in [link]

Examples

Extending the dimension and space

We can also extend all these ideas past just ${ℝ}^2$ and look at them in ${ℝ}^n$ and ${ℂ}^n$ . This procedure extends naturally to higher (>2) dimensions. Given a basis $\{b_0, b_1, \dots , b_{n-1}\}$ for ${ℝ}^n$ , we want to find $\{{\alpha }_0, {\alpha }_1, \dots , {\alpha }_{n-1}\}$ such that