Phase Retrieval of RealValued Signals in a ShiftInvariant Space
Abstract
Phase retrieval arises in various fields of science and engineering and it is well studied in a finitedimensional setting. In this paper, we consider an infinitedimensional phase retrieval problem to reconstruct realvalued signals living in a shiftinvariant space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order , we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate . We propose an algorithm to reconstruct nonseparable signals in a shiftinvariant space generated by a compactly supported continuous function . The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.
I Introduction
Phase retrieval plays important roles in signal/image/speech processing ([1]–[9]). It reconstructs a signal of interest from its magnitude measurements. The underlying recovery problem is possible to be solved only if we have additional information about the signal.
The phase retrieval problem of finitedimensional signals has received considerable attention in recent years ([10]–[14]). In the finitedimensional setting, a fundamental problem in phase retrieval is whether and how a vector (or ) can be reconstructed from its magnitude measurements , where is a measurement matrix. The phase retrievability has been fully characterized via the measurement matrix ([10, 14, 15]), and many algorithms have been proposed to reconstruct the vector from its magnitude measurements ([1, 8, 12, 13, 16, 17, 18]).
The phase retrieval problem in an infinitedimensional space is fundamentally different from a finitedimensional setting. There are several papers devoted to that topic ([19]–[26]). Thakur proved in [19] that realvalued bandlimited signals could be reconstructed from their phaseless samples taken at more than twice the Nyquist rate. The above result was extended to complexvalued bandlimited signals by Pohl, Yang and Boche in [22] with samples taken at more than four times the Nyquist rate. Recently, the phase retrievability of signals living in a principal shiftinvariant space was studied by Shenoy, Mulleti and Seelamantula in [24] when only magnitude measurements of their frequency are available.
Shiftinvariant spaces have been widely used in sampling theory, wavelet theory, approximation theory and signal processing, see [27]–[31] and references therein. In this paper, we consider the phase retrieval problem for realvalued signals in a principal shiftinvariant space
(I.1) 
where the generator is a realvalued continuous function with compact support. Our model of the generator is the Bspline of order , which is obtained by convoluting the indicator function on the unit interval times,
(I.2) 
Ia Contribution
In this paper, we show in Theorem II.2 that a realvalued signal is determined, up to a sign, from its magnitude , if and only if is nonseparable, i.e., it is not the sum of two nonzero signals in with their supports being essentially disjoint. As an application of Theorem II.2, we conclude that for any shiftinvariant space with continuous generator having compact support, not all signals in could be determined, up to a sign, from its magnitude , cf. [19, Theorem 1] for the shiftinvariant space generated by the sinc function .
Phase retrieval in a shiftinvariant space is a nonlinear sampling and reconstruction problem ([32, 33, 34]). In this paper, we show in Theorem II.6 and Corollary II.7 that a nonseparable spline signal in is determined, up to a sign, from its phaseless samples taken on the shiftinvariant set
(I.3) 
where and contains distinct points in .
The set in (I.3) has sampling rate , which is larger than the sampling rate needed for the phase retrievability of bandlimited signals [19, Theorem 1]. Let
(I.4) 
be the support length of the generator , which is the same as the order for the Bspline generator . A natural question is whether any nonseparable signal in the shiftinvariant space can be reconstructed from its phaseless samples taken on a set with sampling rate less than . From Example III.3 we see that any nonseparable linear spline signal in can be determined, up to a sign, from its phaseless samples taken on the set with sampling rate , where are three distinct points in . In Theorem III.4, we consider the phase retrieval problem of a nonseparable signal in the shiftinvariant space from its phaseless samples taken on a nonuniform set
(I.5) 
with sampling rate , where is the set of all positive/negative integers, and the sets and are contained in .
Stability of phase retrieval is of central importance. The reader may refer to [15, 35, 36, 37] for phase retrieval in finitedimensional setting and [38] for nonlinear frames. In this paper, we consider the scenario that phaseless samples taken on a sampling set
(I.6) 
are corrupted,
where is an odd integer, is a nonseparable signal in , and additive noises have the noise level . In Theorem IV.1, we establish the stability of phase retrieval in the above scenario.
The set in (I.6) has sampling rate . It becomes the shiftinvariant set in (I.3) for . Then as an application of Theorem IV.1, any nonseparable spline signal in can be reconstructed, up to a sign, approximately from its noisy phaseless samples on . The nonuniform sampling set in (I.5) can be interpreted as the limit of the sets as tends to infinity. Due to the exponential decay requirement (IV.29) about on the noise level, we cannot obtain from Theorem IV.1 that any nonseparable signal in the shiftinvariant space could be well approximated, up to a sign, when only its noisy phaseless samples on the nonuniform set are available.
Many algorithms have been proposed to solve the phase retrieval problem in finitedimensional setting ([1, 8, 12, 13, 16, 17, 18]). In this paper, we propose the MEPS algorithm to find an approximation of a nonseparable signal when its noisy phaseless samples are available. The MEPS algorithm contains four steps: minimization, extension, phase adjustment and sewing. Our numerical simulations indicate that the MEPS algorithm is robust against bounded additive noises , and the error between the reconstructed signal and the original signal is .
IB Organization
The paper is organized as follows. In Section II, we characterize the phase retrievability of a realvalued signal in a linear space from its magnitude . We also provide several equivalent statements for the phase retrievability of a signal in the shiftinvariant space when its phaseless samples on the shiftinvariant set in (I.3) are available only. In Section III, we present an illustrative example of the phase retrieval problem for linear spline signals, and we prove that any nonseparable signal in the shiftinvariant space could be determined, up to a sign, from its phaseless samples taken on the nonuniform sampling set in (I.5). In Section IV, we propose the MEPS algorithm to reconstruct a nonseparable signal in from its noisy phaseless samples on in (I.6), and we use it to establish the stability of the phase retrieval problem. In Section V, we present some simulations to demonstrate the stability of the proposed MEPS algorithm. Even though the stability requirement (IV.29) in Theorem IV.1 is not met for large , the MEPS algorithm still has high success rate to save phases of nonseparable signals in . All proofs are included in appendices.
Ii Phase retrievability and nonseparability
In this section, we consider the problem when a signal in a shiftinvariant space can be recovered, up to a sign, from its magnitude measurements , where is either the whole line or a shiftinvariant set .
Definition II.1.
Let be a linear space of realvalued continuous signals on the real line . A signal is said to be separable if there exist nonzero signals and in such that
(II.1) 
The set of all nonseparable signals in contains the zero signal. It is a cone of but not a convex set in general. A separable signal is the sum of two nonzero signals and with their supports being essentially disjoint. Then it cannot be recovered, up to a sign, from its magnitude measurements , since and . In the following theorem, we show that the converse is true.
Theorem II.2.
Let be a linear space of realvalued continuous signals on the real line . Then a signal is determined, up to a sign, by its magnitude measurements , if and only if is nonseparable.
Observe that all bandlimited signals are nonseparable, as they are analytic on the real line. Therefore, by Theorem II.2, we have the following result about bandlimited signals, cf. [19, Theorem 1].
Corollary II.3.
Any realvalued bandlimited signal is determined, up to a sign, by its magnitude measurements on the real line.
Let be a realvalued generator of the shiftinvariant space , and be its support length given in (I.4). Without loss of generality, we assume that
(II.2) 
otherwise replacing by for some . Clearly, is a separable signal in . Then from Theorem II.2 we obtain
Corollary II.4.
Let be a continuous function with compact support. Then not all signals in can be determined, up to a sign, by their magnitude measurements .
Next, we discuss the nonseparability of signals in a shiftinvariant space . For the case that (i.e., the generator is supported on ), one may verify that a signal is nonseparable if and only if there exists an integer such that
(II.3) 
This implies that any nonseparable signal in can be recovered, up to a sign, from its phaseless samples taken on the shiftinvariant set , where is so chosen that . So, from now on, we consider the phase retrieval problem only for signals in with the support length of the generator satisfying
(II.4) 
Before characterizing the nonseparability (and hence phase retrievability by Theorem II.2) of signals in a shiftinvariant space, let us consider nonseparability of piecewise linear signals.
Example II.5.
Due to the interpolation property of the Bspline of order , piecewise linear signals have the following expansion,
Therefore is separable if and only if there exist integers such that and . Thus the separable signal
is the sum of two nonzero signals supported in and respectively.
In the following theorem, we extend the support separation property in Example II.5 to separable signals in a shiftinvariant space.
Theorem II.6.
Let be a realvalued continuous function satisfying (II.2) and (II.4), , and let be a nonzero realvalued signal in . If all submatrices of
(II.5) 
are nonsingular, then the following statements are equivalent.

The signal is nonseparable.

for all , where and .

The signal is determined, up to a sign, from its phaseless samples , taken on the shiftinvariant set .
The nonsingularity of submatrices of the matrix in (II.5), i.e., , is also known as its full sparkness ([39, 40]), where
(II.6)  
and for a matrix .
Consider the matrix with its generating function being the continuous solution of a refinement equation,
(II.7) 
where . Under the assumption that
(II.8) 
for some polynomial having positive coefficients, it is known that the matrix in (II.5) is of full spark whenever , are distinct ([41, 42]). It is well known that the Bspline of order satisfies the refinement equation (II.7) with in (II.8) given by . This together with Theorem II.6 implies the following result for spline signals.
Corollary II.7.
Let contain distinct points in . Then any nonseparable spline signal in is determined, up to a sign, from its phaseless samples taken on the shiftinvariant set .
The full sparkness of the matrix in (II.5) implies that has linearly independent shifts, i.e., the linear map from sequences to signals is onetoone ([27, 43, 44]). Conversely, if is the continuous solution of a refinement equation (II.7) with linearly independent shifts, then in (II.5) is of full spark for almost all , see [44, Theorem A.2].
Iii Phaseless oversampling
A discrete set is said to have sampling rate if
(III.1) 
where is the cardinality of a set . Let be the continuous function satisfying (II.2), (II.4) and (II.5). It follows immediately from Theorem II.6 that nonseparable signals in can be fully recovered, up to a sign, from their phaseless samples taken on the shiftinvariant set with sampling rate , which is larger than the sampling rate required for recovering bandlimited signals [19, Theorem 1]. A natural question is to find necessary/sufficient conditions on a set such that any nonseparable signal in can be reconstructed from its phaseless samples taken on .
In this section, we first introduce a necessary condition on the sets .
Theorem III.1.
The lower bound estimate (III.2) is smaller than the sampling rate required for recovering bandlimited signals [19, Theorem 1]. So one may think that it can be improved. However as indicated in the example below, the lower bound estimate (III.2) is optimal if the requirement (II.5) on the generator is dropped.
Example III.2.
Let be a continuous function supported in and set . Similar to (II.3), one may verify that a signal in is nonseparable if and only if there exists such that
Hence given any with , all nonseparable signals in can be reconstructed, up to a sign, from their phaseless samples taken on the set with sampling rate one.
In this section, we next show that nonseparable signals in are determined, up to a sign, from their phaseless samples taken on a set with sampling rate . Before stating the result, let us briefly discuss an example of phaseless oversampling.
Example III.3.
(Continuation of Example II.5) Let and be a nonseparable piecewise linear signal. One may verify that distinct points are enough to determine and (hence ), up to a phase, from phaseless samples and . Particularly, solving
gives
and
For the case that at lease one of two evaluations and is nonzero,
(III.3) 
where the first equality follows from nonseparability of the signal , the second one is obtained by solving the equations
(III.4) 
and
From (III.3) we see that two distinct points could sufficiently determine .
For the case that , solving (III.4) yields
Then either for all or the phase of the signal on is determined up to the sign of nonzero evaluation . Therefore, we can continue the above procedure to determine the signal on if there are two distinct points in intervals for every .
Using the similar argument, we can prove by induction on that the signal , can be determined, up to a sign, by its phaseless samples taken on and , . By now, we conclude that a nonseparable signal in could be determined, up to a sign, by its phaseless samples on , where are distinct and . We remark that the additional point in the above phase retrievability is necessary in general. For instance, signals and in have the same magnitude measurements on , but .
Finally, we state the result on the phase retrieval of nonseparable signals in a shiftinvariant space with sampling rate .
Theorem III.4.
By the nonsingularity of any submatrices of the matrix in (II.5), there are at least distinct elements contained in such that (III.5) holds. Similarly there are at least distinct elements satisfying (III.6). Therefore (III.5) and (III.6) hold for some .
The requirements (III.5) and (III.6) in Theorem III.4 are met for any subsets , provided that is a refinable function with its symbol satisfying (II.8) [41]. Therefore as an application of Theorem III.4, we have the following result for spline signals.
Corollary III.5.
Let contain distinct points in , and be a subset of of size . Then any nonseparable spline signal in is determined, up to a sign, from its phaseless samples taken on .
Iv Stability of phase retrieval
Stability of phase retrieval is of central importance, as phaseless samples in lots of engineering applications are often corrupted. In this section, we establish the stability of phase retrieval in a shiftinvariant space, when its phaseless samples taken on the set are corrupted by additive noises ,
(IV.1) 
where is given in (I.6) for an odd integer , and has the noise level
For , it follows from Theorem II.6 that nonseparable signals in can be recovered, up to a sign, from their exact phaseless samples on . By Theorem III.4, nonseparable signals with finite duration are determined, up to a sign, from their exact phaseless samples on with sufficiently large .
To present an algorithm for phase retrieval in a noisy environment, we introduce four auxiliary functions. Let , and be as in Theorem III.4. Define
(IV.2) 
and
(IV.3) 
where and . Define
(IV.4) 
and
(IV.5) 
where , .
Take a threshold , we propose the following algorithm to construct an approximation
(IV.6) 
of the original signal
(IV.7) 
when only its noisy phaseless samples in (IV.1) are available.

For any , define and
(IV.10) recursively by the following:

If , we obtain from with replacing its th component by
(IV.11) where , and
for .

If , set
(IV.12) where is defined by
For , let
(IV.13) and
(IV.14) where is the symbol of a real number . Now we obtain from by updating its th terms by the unique solution , of the linear system,
(IV.15) where . One may verify that the support of is contained in for any and .
Finally define
(IV.16)


Set . Define and , recursively:

If , then we update from by replacing its th term with
(IV.17) where , is given by

If , set
(IV.18) where is defined by
