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The dual frame vectors are also not unique but a set can be found such that [link] and, therefore, [link] hold (but [link] does not). A set of dual frame vectors could be found by adding a set of arbitrary but independent rows to F until it is square, inverting it, then taking the first $N$ columns to form $\tilde{\mathbf{F}}$ whose rows will be a set of dual frame vectors. This method of construction shows the non-uniqueness of the dual framevectors. This non-uniqueness is often resolved by optimizing some other parameter of the system [link] .
If the matrix operations are implementing a frame decomposition and the rows of F are orthonormal, then $\tilde{\mathbf{F}}={\mathbf{F}}^{\mathbf{T}}$ and the vector set is a tight frame [link] , [link] . If the frame vectors are normalized to $\left|\right|{\mathbf{x}}_{\mathbf{k}}\left|\right|=1$ , the decomposition in [link] becomes
where the constant $A$ is a measure of the redundancy of the expansion which has more expansion vectors than necessary [link] .
The matrix form is
where $\mathbf{F}$ has more columns than rows. Examples can be found in [link] .
The Shannon sampling theorem [link] can be viewied as an infinite dimensional signal expansion where the sinc functions are an orthogonal basis. The sampling theorem with critical sampling, i.e. at the Nyquist rate, is the expansion:
where the expansion coefficients are the samples and where the sinc functions are easily shown to be orthogonal.
Over sampling is an example of an infinite-dimensional tight frame [link] , [link] . If a function is over-sampled but the sinc functions remains consistentwith the upper spectral limit $W$ , using $A$ as the amount of over-sampling, the sampling theorem becomes:
and we have
where the sinc functions are no longer orthogonal. In fact, they are no longer a basis as they are not independent. They are, however, a tightframe and, therefore, have some of the characteristics of an orthogonal basis but with a “redundancy" factor $A$ as a multiplier in the formula [link] and a generalized Parseval's theorem.Here, moving from a basis to a frame (actually from an orthogonal basis to a tight frame) is almost invisible.
The discrete-time Fourier transform (DTFT) of the impulse response of an FIR digital filter $h\left(n\right)$ is its frequency response. The discrete Fourier transform (DFT) of $h\left(n\right)$ gives samples of the frequency response [link] . This is a powerful analysis tool in digital signal processing (DSP) and suggests that an inverse (or pseudoinverse)method could be useful for design [link] .
Frames tend to be more robust than bases in tolerating errors and missing terms. They allow flexibility is designing wavelet systems [link] where frame expansions are often chosen.
In an infinite dimensional vector space, if basis vectors are chosen such that all expansions converge very rapidly, the basis is called an unconditional basis and is near optimal for a wide class of signal representation and processing problems. This is discussed by Donoho in [link] .
Still another view of a matrix operator being a change of basis can be developed using the eigenvectors of an operator asthe basis vectors. Then a signal can decomposed into its eigenvector components which are then simply multiplied by the scalar eigenvalues toaccomplish the same task as a general matrix multiplication. This is an interesting idea but will not be developed here.
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