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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 Ψ 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 R N , we may collect and represent the elements of the dictionary Ψ as an N × Z matrix, which we also denote as Ψ . From this dictionary, a signal x R N can be constructed as a linear combination of the elements (columns) of Ψ . We write

x = Ψ α
for some α R Z . (For much of our notation in this section, we concentrate on signals in 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 Ψ has exactly N linearly independent columns, and each signal x has a unique set of expansion coefficients α = Ψ - 1 x . The orthonormal basis (where the columns are normalized and orthogonal) is also ofparticular interest, as the unique set of expansion coefficients α = Ψ - 1 x = Ψ T x can be obtained as the inner products of x against the columns of Ψ . That is, α ( i ) = x , ψ i , i = 1 , 2 , , N , which gives us the expansion

x = i = 1 N x , ψ i ψ i .

We also have that x 2 2 = i = 1 N x , ψ i 2 .

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

A x 2 2 z x , ψ z 2 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 Ψ there exists a dual frame Ψ ˜ such that
x = z x , ψ z ψ ˜ z = z x , ψ ˜ z ψ z .
In the case where the frame vectors are represented as columns of the N x Z matrix Ψ , the matrix Ψ ˜ containing the dual frame elements is simply the transpose of the pseudoinverse of Ψ . 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., i = 1 N x , ψ i 2 = A x 2 2 . The remainder of this section discusses several importantdictionaries.

A simple, redundant frame Ψ containing three vectors that span R 2 .

The canonical basis

The standard basis for representing a signal is the canonical (or “spike”) basis. In R N , this corresponds to a dictionary Ψ = 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 Ψ contain complex exponentials and allowing the expansion coefficients α to be complex as well. (Such a dictionary can be used to represent real or complexsignals.) A related “harmonic” transform to express signals in R N is the discrete cosine transform (DCT), in which Ψ contains real-valued, approximately sinusoidal functions and the coefficients α are real-valued as well.

Questions & Answers

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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
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
on number 2 question How did you got 2x +2
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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use the y -intercept and slope to sketch the graph of the equation y=6x
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Need help solving this problem (2/7)^-2
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4
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