# 0.1 Signal dictionaries and representations

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This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

For a wide variety of signal processing applications (including analysis, compression, noise removal, and so on) it is useful toconsider the representation of a signal in terms of some dictionary  [link] . In general, a dictionary $\phantom{\rule{2pt}{0ex}}\Psi \phantom{\rule{2pt}{0ex}}$ issimply a collection of elements drawn from the signal space whose linear combinations can be used to represent or approximatesignals.

Considering, for example, signals in ${\mathbb{R}}^{N}$ , we may collect and represent the elements of the dictionary $\Psi$ as an $N×Z$ matrix, which we also denote as $\Psi$ . From this dictionary, a signal $x\in {\mathbb{R}}^{N}$ can be constructed as a linear combination of the elements (columns) of $\Psi$ . We write

$x=\Psi \alpha$
for some $\alpha \in {\mathbb{R}}^{Z}$ . (For much of our notation in this section, we concentrate on signals in ${\mathbb{R}}^{N}$ , though the basic concepts translate to other vector spaces.)

Dictionaries appear in a variety of settings. The most common may be the basis, in which case $\Psi$ has exactly $N$ linearly independent columns, and each signal $x$ has a unique set of expansion coefficients $\alpha ={\Psi }^{-1}x$ . The orthonormal basis (where the columns are normalized and orthogonal) is also ofparticular interest, as the unique set of expansion coefficients $\alpha ={\Psi }^{-1}x={\Psi }^{{}^{T}}x$ can be obtained as the inner products of $x$ against the columns of $\Psi$ . That is, $\alpha \left(i\right)=〈x,,,{\psi }_{i}〉,i=1,2,\cdots ,N$ , which gives us the expansion

$x=\sum _{i=1}^{N}〈x,,,{\psi }_{i}〉{\psi }_{i}.$

We also have that ${∥x∥}_{2}^{2}={\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}$ .

Frames are another special type of dictionary  [link] . A dictionary $\Psi$ is a frame if there exist numbers $A$ and $B$ , $0 such that, for any signal $x$

$A{∥x∥}_{2}^{2}\le \sum _{z}{〈x,,,{\psi }_{z}〉}^{2}\le B{∥x∥}_{2}^{2}.$
The elements of a frame may be linearly dependent in general (see [link] ), and so there may exist many ways to express a particular signal among the dictionary elements.However, frames do have a useful analysis/synthesis duality: for any frame $\Psi$ there exists a dual frame $\stackrel{˜}{\Psi }$ such that
$x=\sum _{z}〈x,,,{\psi }_{z}〉{\stackrel{˜}{\psi }}_{z}=\sum _{z}〈x,,,{\stackrel{˜}{\psi }}_{z}〉{\psi }_{z}.$
In the case where the frame vectors are represented as columns of the $N$ x $Z$ matrix $\Psi$ , the matrix $\stackrel{˜}{\Psi }$ containing the dual frame elements is simply the transpose of the pseudoinverse of $\Psi$ . A frame is called tight if the frame bounds $A$ and $B$ are equal. Tight frames have the special properties of (i) being theirown dual frames (after a rescaling by $1/A$ ) and (ii) preserving norms, i.e., ${\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}=A{∥x∥}_{2}^{2}$ . The remainder of this section discusses several importantdictionaries.

## The canonical basis

The standard basis for representing a signal is the canonical (or “spike”) basis. In ${\mathbb{R}}^{N}$ , this corresponds to a dictionary $\Psi ={I}_{N}$ (the $N×N$ identity matrix). When expressed in the canonical basis, signals are often said tobe in the “time domain.”

## Fourier dictionaries

The frequency domain provides one alternative representation to the time domain. The Fourier series and discrete Fourier transformare obtained by letting $\Psi$ contain complex exponentials and allowing the expansion coefficients $\alpha$ to be complex as well. (Such a dictionary can be used to represent real or complexsignals.) A related “harmonic” transform to express signals in ${\mathbb{R}}^{N}$ is the discrete cosine transform (DCT), in which $\Psi$ contains real-valued, approximately sinusoidal functions and the coefficients $\alpha$ are real-valued as well.

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
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