0.4 Bases, orthogonal bases, biorthogonal bases, frames, tight  (Page 2/5)

Because this type of “orthogonality" requires two sets of vectors, the expansion set and the dual set, the system is called biorthogonal . Using [link] with the expansion in [link] gives

Although a biorthogonal system is more complicated in that it requires, not only the original expansion set, but the finding, calculating, andstorage of a dual set of vectors, it is very general and allows a larger class of expansions. There may, however, be greater numerical problemswith a biorthogonal system if some of the basis vectors are strongly correlated.

The calculation of the expansion coefficients using an inner product in [link] is called the analysis part of the complete process, and the calculation of the signal from the coefficients and expansion vectorsin [link] is called the synthesis part.

In finite dimensions, analysis and synthesis operations are simply matrix–vector multiplications. If the expansion vectors in [link] are a basis, the synthesis matrix has these basis vectors as columns and the matrix is square and non singular. If the matrix is orthogonal, itsrows and columns are orthogonal, its inverse is its transpose, and the identity operator is simply the matrix multiplied by its transpose. If itis not orthogonal, then the identity is the matrix multiplied by its inverse and the dual basis consists of the rows of the inverse. If thematrix is singular, then its columns are not independent and, therefore, do not form a basis.

Matrix examples

Using a four dimensional space with matrices to illustrate the ideas of this chapter, the synthesis formula $g\left(t\right)={\sum }_k{a}_k{f}_k\left(t\right)$ becomes

which can be compactly written in matrix form as

The synthesis or expansion [link] or [link] becomes

with the left-hand column vector $\mathbf{g}$ being the signal vector, the matrix $\mathbf{F}$ formed with the basis vectors ${\mathbf{f}}_{\mathbf{k}}$ as columns, and the right-hand vector $\mathbf{a}$ containing the four expansion coefficients, $a_k$ .

The equation for calculating the $k^{th}$ expansion coefficient in [link] is

which can be written in vector form as

where each $a_k$ is an inner product of the $k^{th}$ row of ${\tilde{\mathbf{F}}}^{\mathbf{T}}$ with $\mathbf{g}$ and analysis or coefficient [link] or [link] becomes

which together are [link] or

Therefore,

is how the dual basis in [link] is found.

If the columns of $\mathbf{F}$ are orthogonal and normalized, then

This means the basis and dual basis are the same, and [link] and [link] become

and

which are both simpler and more numerically stable than [link] .

The discrete Fourier transform (DFT) is an interesting example of a finite dimensional Fourier transform with orthogonal basis vectors where matrixand vector techniques can be informative as to the DFT's characteristics and properties. That can be found developed in several signal processingbooks.

Fourier series example

The Fourier Series is an excellent example of an infinite dimensional composition (synthesis) and decomposition (analysis). The expansionformula for an even function $g\left(t\right)$ over $0<x<2\pi$ is