Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation

Abstract

A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovation (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure, establish the equivalence between its graph connectivity and its phase retrievability by point evaluation measurements on the whole domain, apply the graph connected component decomposition to find its unique landscape decomposition and the set of FRI signals that have the same magnitude measurements. We construct discrete sets with finite density so that magnitude measurements of an FRI signal on the whole domain are determined by its phaseless samples taken on those discrete subsets, and we show that the corresponding phaseless sampling procedure has bi-Lipschitz property with respect to a new induced metric on the signal space and the standard $\ell ^{p}$-metric on the sampling data set. In this paper, we also propose an algorithm with linear complexity to reconstruct an FRI signal from its (un)corrupted phaseless samples on the above sampling set without restriction on the noise level and apriori information whether the original FRI signal is phase retrievable. The algorithm is theoretically guaranteed to be stable, and numerically demonstrated to approximate the original FRI signal in magnitude measurements.

Appendix A: Density of Phaseless Sampling Sets

In the appendix, we introduce a necessary condition on a discrete set $\Gamma$ such that ${\mathcal{M}}_{f, \Gamma }={\mathcal{M}}_{f}$ for all $f\in V(\Phi )$. We show that the density of such a discrete set $\Gamma$ is no less than the innovative rate of signals in $V(\Phi )$, see Theorem A.1 and Corollary A.2.

Theorem A.1

Let the domain $D$, the generator $\Phi : = (\phi _{\lambda })_{\lambda \in \Lambda }$, the family ${\mathcal{T}}=\{T_{\theta }, \theta \in \Theta \}$ of open sets and the linear space $V(\Phi )$ be as in Theorem 5.3, and let $\Gamma \subset D$. If ${\mathcal{M}}_{f, \Gamma }={\mathcal{M}}_{f}$ for all $f\in V(\Phi )$ with ${\mathcal{M}}_{f}=\{\pm f\}$, then

$$ D_{+}(\Gamma )\ge D_{+}(\Lambda ). $$

(A.1)

Proof

Take $x_{0}\in D$ and $r\ge r_{0}$. By (2.2) and (2.3), it suffices to prove that

$$ \#(\Gamma \cap B(x_{0}, r))\ge \# (\Lambda \cap B(x_{0}, r-r_{0})). $$

(A.2)

Assume, on the contrary, that (A.2) does not hold. Then we can find a nonzero vector $(d_{\lambda })_{\lambda \in \Lambda \cap B(x_{0}, r-r_{0})}$ such that

$$ \sum _{\lambda \in \Lambda \cap B(x_{0}, r-r_{0})} d_{\lambda }\phi _{\lambda }(\gamma )=0, \ \gamma \in \Gamma \cap B(x_{0}, r). $$

(A.3)

Recall that $\phi _{\lambda }, \lambda \in \Lambda$, are supported in $B(\lambda , r_{0})$ by Assumption 2.2. Hence

$$ \sum _{\lambda \in \Lambda \cap B(x_{0}, r-r_{0})} d_{\lambda }\phi _{\lambda }(\gamma )=0, \ \gamma \in \Gamma \backslash B(x_{0}, r). $$

(A.4)

Therefore the set

$$ W=\Big\{ f:=\sum _{\lambda \in \Lambda \cap B(x_{0}, r-r_{0})} c_{\lambda }\phi _{\lambda }: \ f(\gamma )=0, \ \gamma \in \Gamma \Big\} \subset V(\Phi ) $$

contains nonzero signals. Take a nonzero signal $f\in W$. By Theorem 4.4, $f=\sum _{i\in I} f_{i}$ for some nonzero signals $f_{i}\in V(\Phi ), i\in I$, such that ${\mathcal{M}}_{f_{i}}=\{\pm f_{i}\}, i\in I$, and $f_{i}f_{i}'=0$ for all distinct $i, i'\in I$. This together with $f\in W$ implies that $f_{i}(\gamma )=0$ for all $\gamma \in \Gamma$ and $i\in I$. Hence $0\in {\mathcal{M}}_{f_{i}, \Gamma }, i\in I$, which contradicts with ${\mathcal{M}}_{f_{i}, \Gamma }= {\mathcal{M}}_{f_{i}}=\{\pm f_{i}\}, i \in I$. □

From the above argument, we have the following result without the assumption on the family ${\mathcal{T}}$ of open sets in Theorem A.1.

Corollary A.2

Let the domain $D$ and the generator $\Phi =(\phi _{\lambda })_{\lambda \in \Lambda }$ satisfy Assumptions 2.1and 2.2respectively, and define the linear space $V(\Phi )$ by (1.1). If $\Gamma$ is a discrete set with ${\mathcal{M}}_{f, \Gamma }={\mathcal{M}}_{f}$ for all $f\in V(\Phi )$, then $D_{+}(\Gamma )\ge D_{+}(\Lambda )$.

We finish this appendix with a remark that the lower bound in (A.1) can be reached when the generator $\Phi =(\phi _{\lambda })_{\lambda \in \Lambda }$ satisfies that

$$ S_{\Phi }(\lambda , \lambda ')=\emptyset \ {\mathrm{for \ all\ distinct}} \ \lambda , \lambda '\in \Lambda . $$

(A.5)

As in this case, a signal $f\in V(\Phi )$ is nonseparable if and only if $f=c_{\lambda }\phi _{\lambda }$ for some $\lambda \in \Lambda$. Thus the set $\Gamma =\{ a(\lambda ), \lambda \in \Lambda \}$ is a phaseless sampling set whose upper density is the same as the rate of innovation, where $a(\lambda ), \lambda \in \Lambda$, are chosen so that $\phi _{\lambda }(a(\lambda ))\ne 0$.