^{1}Helmholtz Zentrum München, Scientific Computing Research Unit, Neuherberg, Germany^{2}Department of Mathematics, Technische Universität München, Garching bei München, Germany^{3}Faculty of Mathematics, Universität Wien, Wien, Austria

In this study, we are concerned with the effect of certain linear transformations of a signal *f* on its phase. We are, in particular, interested in phase distortions caused by band-limiting operations. The band-limiting operators serve as a motivation for studying the class of phase-preserving operators. This class will be completely characterized.

## Introduction

In science and engineering, the problem of recuperating a square-integrable signal *f* from its transformed version *T* is given by the Fourier transform is of fundamental importance in signal analysis. A particular example which also served as a motivation for our studies arises in the field of optics. In diffraction imaging, the so-called diffraction pattern of the object *f* is measured usually (but not necessarily) in the far-field regime. In this regime, the diffraction pattern is given by the Fourier transform of the object. If we have full access to the Fourier transform of *f* the reconstruction of the signal is naturally no problem as the transform is one to one on the signal space. Unfortunately, this is almost never the case in practice. Not only is the signal corrupted with noise, experimental restrictions and more importantly physical limitations of the devices used to perform the measurement usually make it impossible to have full access to the Fourier transform. For example, in diffraction imaging, the sensor is usually a Charged-Coupled Device camera, which is, by construction, only able to measure the intensity of the incoming signal. As mentioned before, in the far-field, the signal is nothing but the Fourier transform of the object. Hence, the output of the measurement device is the squared modulus of the Fourier transform of the object, which means the phase information is completely lost. This leads to the problem of phase retrieval which is a notoriously difficult task. Even if we could solve this severe problem, there is yet another problem coming from the signal recording process. No sensor can cover the full range in frequency. Every recorder comes with a specific bandwidth characteristic, which means that the measured signal becomes artificially a band-limited signal and this creates another source of distortion in the signal recovery process. It is exactly this problem on which we are going to concentrate in the present study. To make this more clear, let us describe the problem in a more rigorous form.

Suppose the complex-valued signal *t* for which the signal *f*, defined on

and on *via* an usual extension argument, cannot have compact support. However, due to sensor characteristics, the Fourier transform is restricted to a certain bandwidth. Assume that the sensor bandwidth is given by *f* is compactly supported, its Fourier transform *f* by applying the inverse Fourier transform since the extension to the whole real line would be numerically impractical. The band-limiting operator which sends *f* to *f* by band-limited functions of bandwidth *f*, it is relatively easy to obtain a bound for *f* gets distorted by the band-limiting operation. One problem which we will address in this study is to find an estimate for the phase difference

The organization of the study is as follows. In Section 2, we introduce our notations and provide some auxiliary results. Section 3 is devoted to the study of the band-limiting operation and its effect on the phase of a signal. Finally, in Section 4, we will present a characterization of those operators which will preserve the phase of the signal.

## Preliminaries

We start by introducing some notations used throughout our presentation. Moreover, we will give some simple but nevertheless important facts on complex numbers.

The set of complex numbers different from zero will be denoted by *ϑ* congruent to *θ* modulo *θ* as the phase of *z*. We define a metric on *via*

The following simple result will be of some significance later.

Lemma 2.1. Let

Proof. According to the definition of

which implies

With

Combining the previous estimate with Eq. 1 and the fact that

The previous Lemma shows the intuitive fact that if the distance between *z* and *w* is small and if one of the two complex numbers stays sufficiently far away from zero (relative to

Let *f* be a complex-valued function defined on a subset

where *f*. The phase function

Lemma 2.2. Let

holds pointwise for every *x* in some subset

for every *x* in

Throughout this work, the *via*

## Phase Distance and Truncated Measurements

In this section, we want to compare the phase function of a compactly supported function *f* by band limiting the Fourier transform of *f*. We begin our consideration with some facts about localization of functions. For an introduction to the basic notions of Fourier analysis and operator theory see, for instance, [7,9].

Let

where *f* as time-limited or band-limited, respectively. The time- and band-limiting operator

We now introduce the concept of time-frequency localization of a function

Definition 3.1. Let *f* is called

The function *f* is called

In what follows we are mainly interested in the case where both *f* and the phase function

where

Theorem 3.2. Assume that

*Proof*. Since

where we used the elementary inequality

Since *f* and

for all

for all

on

An immediate consequence of this result shows that, for a certain class of

Corollary 3.3. Let

for every

*Proof*. The assumptions on *f* imply that

with

Corollary 3.3 shows, in particular, that if

then *n*. To see this, we take, for instance, the function

Let us briefly discuss the case of real-valued functions. First, we observe that, according to identity 2 the operator *f*, an

Theorem 3.4. Let

for every

*Proof.* Note that *f* and

Since

Now, Lemma 2.1 gives *f* is real-valued, this implies

In a similar fashion as before, we could add in Theorem 3.4 a certain regularity assumption (*f* is localized in the frequency domain. However,

Theorem 3.5 (2, Theorem 7). Let *f* is

where

In the setting of Theorem 3.4, inequality 4 leads to

for every

Remark 3.6. Consider the band-limiting operator *n*. Suppose that the corresponding real eigenvalues *f* (and therefore, of the phase

Observing that

**FIGURE 1**. Perturbation of a signal *f* by a prolate spheroidal wave function

## Phase-Preserving Operators

In Section 3, it has been shown under which conditions the phase of *f* is invariant under the action of the band-limiting operator, i.e.,

Which operators

In this section, we will give a precise characterization of operators on

Definition 4.1. We call a bounded linear operator

for every

The following statement is an immediate consequence of the previous definition.

Lemma 4.2. If

Proof. Let

which implies

Since

For

The function *ϕ* is called the symbol of the multiplication operator *ϕ* is real-valued with *T*, there exists a

In other words, the function *f*. To show this statement, we start by investigating the map

Definition 4.3. Let *f* and *g* pointwise linear-independent if

Lemma 4.4. Let

Moreover, if *f*, then

Proof. If

Therefore, *f*. Then, *T*, we obtain

Linear independence implies

Suppose that *T* would be a multiplication operator by a simple extension argument. In the following, we will construct a system which satisfies exactly the conditions above. To do so, we consider the system of exponentials

Lemma 4.5. Let *n* exponentials:

Then,

Proof. Without loss of generality, we may assume that

and

We define the matrix

Then, **Eqs 6 and7** into

with *j*, since

is a sum of terms of the form

with some

for every *R*. In case

which is a nonzero periodic function. Hence,

as

as

We now give a characterization of PPO.

Theorem 4.6. Let *T* is phase-preserving if and only if there exists

for every

Proof. The fact that every multiplication operator

is dense in

and

for some *f* is positive everywhere, Lemma 4.5 implies that *g* and *h*. Lemma 4.4 implies that *g* and *h* are arbitrary, we obtain a function

for every *ϕ* is not essentially bounded. Then, for every

has a positive Lebesgue measure. We now choose

which is possible due to density of *Q* in

With this choice of

Because

where *T*. Consequently,

on *Q*. Since *Q* is a dense subset of *Q*, i.e., on

The characterization of PPOs in Theorem 4.6 was stated in the space *f* was a Gaussian) which was based on Wiener’s Tauberian theorem. Equivalently, this means that the set of exponentials

is complete in *µ* on

Theorem 4.7. Let *µ* be a positive, finite Borel measure on

Proof. Assume by contradiction that

By the Riesz representation theorem, the functional

for some *q* is the Hölder conjugate exponent of *p*. The identities in Eq. 12 imply that

for every

By the uniqueness theorem of the Fourier–Stieltjes transform of measures, we have

Corollary 4.8. Let *T* is phase-preserving if and only if *T* is a multiplication operator with nonnegative symbol

Proof. It is clear that a multiplication operator on *f* is a Schwartz function, it lies in

where *µ* is a finite, positive Borel measure on

is dense in

We finish this section by establishing a relation between the previous characterization theorems and Section 3, where we obtained bounds on the phase of a compactly supported function *f* and the phase of its band-limited version

Assume that

is well-defined and bounded. It follows that

Similar to Definition 3.1, we say that a function

We define *ϕ* in such a way that

The existence of such a *ϕ* easily follows from the fact that the Gauss kernel is an approximate identity. Combining the two previous inequalities, we obtain the bound

This proves the following statement.

Corollary 4.9. Let

Note that an *f* can be achieved in an analogous manner to Section 3. Suppose that

We conclude that if *f* satisfies the assumptions derived in Section 3 (regularity and control over the total variation norm) which yielded a bound on the phase distance

## Conclusion

In this article, we were concerned with phase distortions caused by band limiting a compactly supported signal. This problem naturally arises in the field of optics such as diffraction imaging. Precise localization and regularity conditions were derived for which a bound on the phase of the input signal and the band-limited signal is achievable. The class of operators which leave the phase of any input signal invariant was characterized as multiplication operators with nonnegative symbol.

## Data Availability Statement

The original contributions presented in the study are included in the article, and further inquiries can be directed to the corresponding author.

## Author Contributions

Both authors listed have made a substantial contribution to the work and approved it for publication.

## Funding

The work was partially funded by the Helmholtz Association under the project Ptychography4.0. The open-access publication fees are payed by the University of Vienna *via* the open-access publishing agreement between Frontiers and the University of Vienna. The corresponding author (Lukas Liehr) is affiliated to the University of Vienna.

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

The authors thank the reviewers for their valuable comments which were very helpful to improve the paper.

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Keywords: signal analysis, Fourier transform, phase-preserving operator, phase distortion, band-limiting operator

Citation: Filbir F and Liehr L (2020) Phase Distortion by Linear Signal Transforms. *Front. Appl. Math. Stat.* 6:556585. doi: 10.3389/fams.2020.556585

Received: 28 April 2020; Accepted: 26 August 2020;

Published: 12 November 2020.

Edited by:

Juergen Prestin, University of Lübeck, GermanyReviewed by:

Robert Beinert, Technical University of Berlin, GermanyElijah Liflyand, Bar-Ilan University, Israel

Copyright © 2020 Liehr and Filbir. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Lukas Liehr, lukas.liehr@univie.ac.at