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Preliminaries

Module by: Marco Duarte, Ronald DeVore

We previously described Shannon's Theorem plus encoding: the Nyquist sampling rate is the minimal required sampling rate to recover the entire class of bandlimited signals. We have seen that this sampling rate may be prohibitively large for broadband signals. We see a way to improve upon this situation: we will pose a different model for the signals which is more restrictive than the assumption that the signals are bandlimited. Fortunately, there are several real world scenarios in which one knows much more information about the signals of interest. For example, they may be written in terms of very few fundamental building blocks (such as sine waves or chirps). This leads us to define new signal classes based on notions of sparsity and seek to determine if we can improve on sampling and encoding in this new setting.

Let us define the general setting for this section. Let XX be a Banach space of functions. The typical examples are X=Lp⁢(R),Lp⁢(Rd),Lp⁢(-T,T)X=Lp⁢(R),Lp⁢(Rd),Lp⁢(-T,T), 1≤p≤∞1≤p≤∞. We denote the norm on XX by ∥⋅∥X∥⋅∥X. We define a dictionary DD as any collection of functions D⊆XD⊆X such that ∥g∥X=1∥g∥X=1 for all g∈Dg∈D, i.e. all the elements of the dictionary are normalized. While the definition is very broad, in practice dictionaries usually have more structure. Some examples include D=BD=B, a basis for XX, such as (i) the Fourier basis on [-π,π][-π,π], (ii) a wavelet basis,1 (iii) redundant families of waveforms of the form ψa,b,σ=e-a⁢(t-b)2⁢ei⁢σ⁢xψa,b,σ=e-a⁢(t-b)2⁢ei⁢σ⁢x, i.e. D={ψa,b,σ}a,b,σD={ψa,b,σ}a,b,σ, and (iv) wavelet packets.

Definition 1

We define the class of nn-sparse signals as Σn:=Σn(D)={s=∑g∈Λcgg,Λ⊆D,♯Λ≤n}Σn:=Σn(D)={s=∑g∈Λcgg,Λ⊆D,♯Λ≤n}. We also say that ss has sparsity nn in DD if s∈Σn⁢(D)s∈Σn⁢(D), i.e. if it can be written as the linear combination of nn functions from DD. We note that ΣnΣn is not a linear space; we instead have Σn+Σn⊆Σ2⁢nΣn+Σn⊆Σ2⁢n.

Footnotes

  1. Wavelet basis form orthonormal systems for L2⁢(I)L2⁢(I).

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