System Approximations and Generalized Measurements in Modern Sampling Theory

Abstract

This chapter studies several aspects of signal reconstruction of sampled data in spaces of bandlimited functions. In the first part, signal spaces are characterized in which the classical sampling series uniformly converge, and we investigate whether adaptive recovery algorithms can yield uniform convergence in spaces where non-adaptive sampling series does not. In particular, it is shown that the investigation of adaptive signal recovery algorithms needs completely new analytic tools since the methods used for nonadaptive reconstruction procedures, which are based on the celebrated Banach–Steinhaus theorem, are not applicable in the adaptive case.The second part analyzes the approximation of the output of stable linear time-invariant (LTI) systems based on samples of the input signal, and where the input is assumed to belong to the Paley–Wiener space of bandlimited functions with absolute integrable Fourier transform. If the samples are acquired by point evaluations of the input signal f, then there exist stable LTI systems H such that the approximation process does not converge to the desired output Hf even if the oversampling factor is arbitrarily large. If one allows generalized measurements of the input signal, then the output of every stable LTI system can be uniformly approximated in terms of generalized measurements of the input signal.The last section studies the situation where only the amplitudes of the signal samples are known. It is shown that one can find specific measurement functionals such that signal recovery of bandlimited signals from amplitude measurement is possible, with an overall sampling rate of four times the Nyquist rate.

Appendix

This appendix provides a short proof of Theorem 8 in Section 7.4.2

Proof (Theorem 8).

• 1. First, we prove the statement for the sets (7.18). To this end, let $g \in \mathcal{X}$ and ε > 0 be arbitrary. For all $M,N_{0} \in \mathbb{N}$, we have to show that there exists a functions f _∗ in the set (7.18) such that $\|g - f_{{\ast}}\|_{\mathcal{X}} <\epsilon$. Since $\mathcal{X}_{0}$ is a dense subset of $\mathcal{X}$, there exists a $q \in \mathcal{X}_{0}$ such that $\|g - q\|_{\mathcal{X}} <\epsilon /2$. Therewith, we define $f_{{\ast}}:= q + \tfrac{\epsilon } {2}\,f_{0}$ with $f_{0} \in \mathcal{D}_{weak}$ and with $\|f_{0}\|_{\mathcal{X}} = 1$. Then we get

  $$\displaystyle{ \|g - f_{{\ast}}\|_{\mathcal{X}}\leq \| g - q\|_{\mathcal{X}} + \tfrac{\epsilon } {2}\|f_{0}\|_{\mathcal{X}} <\tfrac{\epsilon } {2} + \tfrac{\epsilon } {2} =\epsilon \;. }$$

  Let $M,N_{0} \in \mathbb{N}$ be arbitrary. We still have to show that f _∗ is contained in set (7.18). To this end, we observe that for every $N \in \mathbb{N}$

  $$\displaystyle{ \|\mathrm{T}_{N}f_{{\ast}}\|_{\mathcal{Y}} =\|\mathrm{ T}_{N}q + \tfrac{\epsilon } {2}\mathrm{T}_{N}f_{0}\|_{\mathcal{Y}}\geq \tfrac{\epsilon } {2}\|\mathrm{T}_{N}f_{0}\|_{\mathcal{Y}}-\|\mathrm{ T}_{N}q\|_{\mathcal{Y}}\;. }$$

  Since $q \in \mathcal{X}_{0}$, (7.16) implies that there is an $N_{1} \geq N_{0}$ such that $1 \geq \|\mathrm{ T}_{N}q - q\|_{\mathcal{Y}}\geq \|\mathrm{ T}_{N}q\|_{\mathcal{Y}}-\| q\|_{\mathcal{Y}}$ for all N ≥ N ₁. Consequently

  $$\displaystyle{ \|\mathrm{T}_{N}q\|_{\mathcal{Y}}\leq 1 +\| q\|_{\mathcal{Y}}\leq 1 + C_{0}\|q\|_{\mathcal{X}}\quad \mathit{for\ \ all}\ N \geq N_{1}\;, }$$

  using for the last inequality that $\mathcal{X}$ is continuously embedded in $\mathcal{Y}$ with a certain constant $C_{0} <\infty$. Combining the last two inequalities, we get $\|\mathrm{T}_{N}f_{{\ast}}\|_{\mathcal{Y}}\geq \tfrac{\epsilon } {2}\|\mathrm{T}_{N}f_{0}\|_{\mathcal{Y}}- 1 - C_{0}\|q\|_{\mathcal{X}}$ for all N ≥ N ₁. Since $f_{0} \in \mathcal{D}_{\mathrm{weak}}$ there exits an $N_{2} \geq N_{1}$ such that

  $$\displaystyle{ \|\mathrm{T}_{N_{2}}f_{{\ast}}\|_{\mathcal{Y}}\geq \tfrac{\epsilon } {2}\|\mathrm{T}_{N_{2}}f_{0}\| - 1 - C_{0}\|q\|_{\mathcal{X}}> M }$$

  which shows that $f_{{\ast}}\in D(M,N_{2}) \subset \bigcup _{N\geq N_{0}}D(M,N)$. Thus the sets (7.18) are dense in $\mathcal{X}$ and it remains to show that these sets are open. To this end, let $M,N \in \mathbb{N}$ and f _∗ ∈ D(M, N) be arbitrary, i.e., $\|\mathrm{T}_{N}f_{{\ast}}\|_{\mathcal{Y}}> M$. Since T _N is a continuous linear operator $\mathcal{X} \rightarrow \mathcal{Y}$, there exists a δ > 0 and a neighborhood

  $$\displaystyle{ U_{\delta }(f_{{\ast}}) =\{ f \in \mathcal{X}:\| f - f_{{\ast}}\|_{\mathcal{X}} <\delta \} }$$

  of f _∗ such that $\|\mathrm{T}_{N}f\|_{\mathcal{Y}}> M$ for all f ∈ U _δ . Thus D(M, N) is open for all $M,N \in \mathbb{N}$ and since the union of (countable many) open sets is again open, the sets (7.18) are also open.

• 2. We prove (7.19). By the definition of the $\limsup$ operation, the set $\mathcal{D}_{\mathrm{weak}}$ can be written as

  $$\displaystyle{ \mathcal{D}_{\mathrm{weak}} =\Big\{ f \in \mathcal{X}\:\ \lim _{N_{0}\rightarrow \infty }\sup _{N\geq N_{0}}\|\mathrm{T}_{N}f\|_{\mathcal{Y}} = \infty \Big\} }$$

  and we note that for every fixed $f \in \mathcal{X}$ the sequence $\{\sup _{N\geq N_{0}}\|\mathrm{T}_{N}f\|_{\mathcal{Y}}\}_{N_{0}=1}^{\infty }$ is monotone decreasing. Assume that $f \in \mathcal{D}_{\mathrm{weak}}$ and choose $M \in \mathbb{N}$ arbitrary. Then, by the above definition of $\mathcal{D}_{\mathrm{weak}}$, it follows that for arbitrary N ₀ there exists an N ≥ N ₀ such that $\|\mathrm{T}_{N}f\|_{\mathcal{Y}}> M$, i.e., $f \in \bigcup _{N\geq N_{0}}D(M,N)$, and since this holds for all $M,N \in \mathbb{N}$ we have

  $$\displaystyle{ f \in \bigcap _{M=1}^{\infty }\ \bigcap _{ N_{0}=1}^{\infty }\ \bigcup _{ N=N_{0}}^{\infty }\ D(M,N) }$$

  which shows that $\mathcal{D}_{\mathrm{weak}} \subset \bigcap _{M=1}^{\infty }\ \bigcap _{N_{0}=1}^{\infty }\ \bigcup _{N=N_{0}}^{\infty }\ D(M,N)$. Conversely, assume that $f \in \bigcap _{M=1}^{\infty }\ \bigcap _{N_{0}=1}^{\infty }\ \bigcup _{N=N_{0}}^{\infty }\ D(M,N)$. Then to every arbitrary $M \in \mathbb{N}$ and $N_{0} \in \mathbb{N}$ there exists an N > N ₀ such that f ∈ D(M, N), i.e., that $\|\mathrm{T}_{N}f\|_{\mathcal{Y}}> M$. Thus $f \in \mathcal{D}_{\mathrm{weak}}$.

  Finally, we prove (7.20). Assume first that $f \in \mathcal{D}_{\mathrm{strong}}$. Then to every $M \in \mathbb{N}$ there exists an N ₀ = N ₀(M) such that $\|\mathrm{T}_{N}f\|_{\mathcal{Y}}> M$ for all N ≥ N ₀. In other words

  $$\displaystyle{ f \in \bigcap _{N=N_{0}(M)}^{\infty }D(M,N) \subset \bigcup _{ N_{0}=1}^{\infty }\bigcap _{ N=N_{0}}^{\infty }D(M,N)\qquad \text{for every}\ M \in \mathbb{N}\;, }$$

  which shows that $f \in \bigcap _{M=1}^{\infty }\ \bigcup _{N_{0}=1}^{\infty }\ \bigcap _{N=N_{0}}^{\infty }\ D(M,N)$. Conversely, assume that $f \in \bigcap _{M=1}^{\infty }\ \bigcup _{N_{0}=1}^{\infty }\ \bigcap _{N=N_{0}}^{\infty }\ D(M,N)$. This means that for any arbitrary $M \in \mathbb{N}$ the function f belongs to $\bigcup _{N_{0}=1}^{\infty }\ \bigcap _{N=N_{0}}^{\infty }\ D(M,N)$, i.e., there exists an N ₀ such that

  $$\displaystyle{ f \in \bigcap _{N=N_{0}}^{\infty }D(M,N),\qquad \text{i.e.,}\qquad \|\mathrm{T}_{ N}f\|_{\mathcal{Y}}> M\quad \mathrm{for\ \,all}\ N \geq N_{0}\;. }$$

  Thus $\lim _{N\rightarrow \infty }\|\mathrm{T}_{N}f\|_{\mathcal{Y}} = \infty$, i.e., $f \in \mathcal{D}_{\mathrm{strong}}$. □