orthonormal basis

A subset $\{v_1,...,v_k\}$ of a vector space $\mathbf{V}$ , with the inner product $\langle,\rangle$ , is called orthonormal if $\langle v_i,v_j \rangle=0$ when $i \neq j$ . That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: $\langle v_i,v_i \rangle=1$ .

Representation of 1D Signal Using An Orthonormal Basis

Orthonormal basis function

\int_{-\infty}^\infty \phi(x,u_1) \phi^*(x,u_2) dx = \begin{cases}1, u_1 = u_2 \\ 0, u_1 \neq u_2\end{cases}

Each basis has norm 1
Different bases are orthogonal to each other

Inverse transform

Representing $f(x)$ as integral (limit of sum) of $φ(x,u)$ for all $u$ , with weight $F(u)$ $\int_{-\infty}^\infty F(u) \phi(x,u) \, du$

Forward transform

determining the weight through inner product $F(u) = \langle f(x), \phi(x,u)\rangle = \int_{-\infty}^\infty f(x) \phi^*(x,u) \, dx$

Orthonormal basis vectors

$\{\mathbf{h}_k,k=0,...,N-1\}$ are OBV if

\langle \mathbf{h}_k,\mathbf{h}_l \rangle =\delta_{k,l} =\begin{cases} 1 & k=l\\ 0 & k \neq l\end{cases}

with OBV

\begin{aligned} \langle \mathbf{h}_l, \mathbf{f} \rangle = \langle \mathbf{h}_l, \sum_{k=0}^{N-1} t(k) \mathbf{h}_k \rangle=t(l)=\mathbf{h}_l^H \mathbf{f} \\ \mathbf{t}=\begin{bmatrix} \mathbf{h}_0^H \\ \mathbf{h}_1^H\\ \vdots \\ \mathbf{h}_{N-1}^H \end{bmatrix} \\ \mathbf{B}^{-1}=\mathbf{B}^H \textrm{ or } \mathbf{B} = \mathbf{B} \mathbf{B}^H = \mathbf{I} \end{aligned}

where $\mathbf{B}$ is unitary.

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