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Original Research ARTICLE

Front. Appl. Math. Stat., 12 November 2020 | https://doi.org/10.3389/fams.2020.556585

Phase Distortion by Linear Signal Transforms

  • 1Helmholtz Zentrum München, Scientific Computing Research Unit, Neuherberg, Germany
  • 2Department of Mathematics, Technische Universität München, Garching bei München, Germany
  • 3Faculty 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(f) is ubiquitous. The case where 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 fL2() is compactly supported and consider the decomposition f(t)=|f(t)|eiθ(t). The function θ(t) is called phase function, and it is well-defined only for those t for which the signal f(t) is different from zero. Clearly, the Fourier transform of f, defined on L1()L2() by

f(ξ)=f^(ξ)=f(t)e2πiξtdt

and on L2()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 [W,W]W>0. Although a restriction to different sets is mathematically possible, the restriction to a symmetric interval is the most relevant case in practice, where sensors usually act as low-pass filters. This will be the case we will concentrate on in the present study. As a result, not f^ but fW=χ[W,W]f is recorded during the measurement process. Applying the inverse Fourier transform gives fW=(χ[W,W]f^) which has the representation fW(t)=|fW(t)|eiθW(t). Note that since f is compactly supported, its Fourier transform f^ is an entire function and thus uniquely determined by f^W. The before mentioned approach seeks to recover 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 fW is actually an orthogonal projection of L2() onto the Paley–Wiener space PW2W, and hence fW is the best approximation of f by band-limited functions of bandwidth 2W with respect to the L2-norm. If we impose some moderate assumptions on the smoothness of f, it is relatively easy to obtain a bound for ||f(t)||fW(t)||. An interesting question now is how much the phase of 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 |θ(t)θW(t)|. It will turn out that again smoothness requirements are sufficient to get bounds for the phase differences. Moreover, we will also demonstrate under what conditions the phase is preserved by band-limiting operations. This raises the question what type of operators will leave the phase unchanged. We will present a characterization of these phase-preserving operators (PPO). It is obvious that every multiplication operator Tϕf=ϕf with ϕL() and ϕ0 almost everywhere is phase-preserving. However, it is not obvious whether or not these are the only linear operators with this property. We will demonstrate that this is indeed the case.

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 ×, and by T, we denote the set of complex numbers of modulus one. Any z× has a unique representation z=|z|eiθ with θ[0,2π), and clearly, any ϑ congruent to θ modulo 2π results into another representation z=|z|eiϑ. We will refer to θ as the phase of z. We define a metric on via

|θφ|2π:=mink|ξη+2πk|,θ,φ.

The following simple result will be of some significance later.

Lemma 2.1. Let z,w× with z=eiθ|z|,w=eiφ|w|. Then,

|θφ|2π2arcsin(min{1,|zw||z|})π|zw||z|.

Proof. According to the definition of |θφ|2π, we obtain

sin(12|θφ|2π)=12|eiθeiφ|,

which implies

|θφ|2π=2arcsin(12|eiθeiφ|)=2arcsin(min{1,12|eiθeiφ|}).(1)

With eiθ=z/|z| and eiφ=w/|w|, we obtain

|eiθeiφ|=|z|z|w|w||=|(zw)|w|+(|w||z|)w|z||w|||zw||z|+||w||z|||z|2|zw||z|.

Combining the previous estimate with Eq. 1 and the fact that arcsin(x)π2x for x[0,1], it yields the following statement.

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 |zw|), then the corresponding phase distance |θφ|2π is small.

Let f be a complex-valued function defined on a subset X. Then,

f(t)=|f(t)|eiθf(t),

where θf is a real-valued function which we will call in accordance with our previous terminology the phase function of f. The phase function θf is well-defined for all tX for which f(t) is different from zero. Henceforth, we will denote this set by D(f), i.e., D(f):={tX|f(t)0}. If g:Y is a second function with g(t)=|g(t)|eiθg(t) defined on Y, then we define θg and θf to be equal if θg(t)=θf(t), for all tD(f)D(g). Note that this relation is well-defined as for all tD(f)(g), the phase functions can be considered to be equal anyway. With these terminologies, we immediately get the following sequence from Lemma 2.1 the following consequence.

Lemma 2.2. Let X and f,g:X be complex-valued functions with f=eiθ|f| and g=eiφ|g|. Let 0ε1 be fixed and suppose that the inequality

|f(x)g(x)|ε|f(x)|

holds pointwise for every x in some subset JD(f)D(g). Then,

|θ(x)φ(x)|2π2arcsin(ε)

for every x in J.

Throughout this work, the Lp-norm of a measurable function f:E defined on some measurable set E is defined as usual via

||f||p={(E|f(x)|pdx)1p,1p<esssupxE|f(x)|,p=.

Phase Distance and Truncated Measurements

In this section, we want to compare the phase function of a compactly supported function fL2() with the phase function of fW:=(χ[W,W]f^), i.e., fW originates from 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 T,W be measurable sets. We define the following two operators acting on functions fL2()

RTf:=χTf,BWf=1(χWf),

where denotes the Fourier operator on L2(). In case T=[T,T], W=[W,W], T,W>0 we will write RT and BW instead of RT and BW, respectively. Both operators are orthogonal projections on L2(), and their range is L2(T) respective to those functions in fL2() with Fourier transform supported in W. In case T=[T,T] and W=[W,W], we speak f as time-limited or band-limited, respectively. The time- and band-limiting operator BWRT was studied by Slepian and Pollak [5,6] in some detail, and they showed that it has the representation

BWRTf(t)=TTsin(2πW(ts))π(ts)f(s)ds.(2)

We now introduce the concept of time-frequency localization of a function fL1().

Definition 3.1. Let fL1(), T,W be measurable sets, and εT,εW>0. Then, f is called εT-localized if fRTfL1()εT. Let

1(W):={fL1()|supp(f^)W}.

The function f is called εW-band-limited if there is a function g1(W) such that ||fg||L1()εW.

In what follows we are mainly interested in the case where both W and T are symmetric intervals as introduced above. We now aim for a pointwise estimate of the phase difference between the compactly supported signal fL2() and its band-limited version fW. To be more precise, suppose that fL2(),supp(f)[T,T] and let fW:=BWf for some W>0. Then, both the phase function θ(t) of f and the phase function θW(t) of fW are well-defined almost everywhere. We would like to have an estimate for |θ(t)θW(t)|. To this end, we define the following class of functions. For n and K0, let

Gn,K:={fCn():supp(f)[T,T],f(n)BVK},

where BV denotes the total variation norm.

Theorem 3.2. Assume that fGn,K with Knπn+1 and let fW=BWf. Then, for every tD(f) with |f(t)|Wn, we have

|θ(t)θW(t)|2arcsin(12n).

Proof. Since fCn() with supp(f)[T,T], we have f(k)L1(), for all 0kn. Hence, for all ξ, we get

|(2πiξ)n+1f^(ξ)|=|2πiξf(n)^(ξ)|fBV,

where we used the elementary inequality h^(ξ)hBV|2πξ|1 which holds for functions hBV() (see, for instance, [3, pp. 33–34]). If ξ0, this yields the estimate:

|f^(ξ)|f(n)BV(2πξ)n+1.

Since f and fW are pointwise-defined and fGn,K, we obtain

|f(t)fW(t)|=|f^(ξ)e2πiξtdξWWf^(ξ)e2πiξtdξ|[W,W]c|f^(ξ)|dξ2||f(n)||BV(2π)n+1W1ξn+1dξ=||f(n)||BV2nWnπn+1n12nWn,

for all t. We define J:={tsupp(f):|f(t)|Wn}.. Then,

|f(t)fW(t)||f(t)|12n,

for all tJ. Now, Lemma 2.2 yields

|θ(t)θW(t)|2arcsin(12n)

on J.

An immediate consequence of this result shows that, for a certain class of C-functions, we get the equality of the phase functions. More precisely, we have the following.

Corollary 3.3. Let W>1 and let fC() with supp(f)[T,T]. Assume there exists N such that

||f(n)||BVnπn+1(3)

for every nN. Then, θ=θW everywhere on D(f).

Proof. The assumptions on f imply that

fnNGn,Kn,

with Kn=nπn+1. According to Theorem 3.2, we have |θ(t)θW(t)|2arcsin(2n) on Jn:={tsupp(f):|f(t)|Wn}, for every nN. Since W>1, we have Wn0 as n. We observe that arcsin(2n)0 as n yields the following statement.

Corollary 3.3 shows, in particular, that if fC() has compact support and if the sequence (f(n))n satisfies the growth condition

f(n)BV=O(πn+1),

then θ=θW everywhere on D(f). Moreover, one can weaken Corollary 3.3 by requiring estimate 3 not to hold for all nN but just for a subsequence (nk)k with the property nk as k. We further observe that, for a compactly supported function fC(), the BV-norm of f(n) can indeed grow exponentially in n. To see this, we take, for instance, the function f(t)=eet and smooth it down to zero at the boundary of [1,1]. In this case, the sequence f(n)1 grows exponentially and so does f(n)BV.

Let us briefly discuss the case of real-valued functions. First, we observe that, according to identity 2 the operator BW acts as an integral operator with a real-valued kernel on the space of time-limited functions. This implies that fW is real-valued. Consequently, |θ(t)θW(t)|{0,π}, for every tD(f)D(fW), and θ=θW if and only if sgn(f)=sgn(fW). Therefore, to obtain the equality of θ(t) and θW(t) at tD(f)D(fW), it suffices that |θ(t)θW(t)|<π. For a real-valued function f, an ε[W,W]-localization of its Fourier transform automatically implies equality of the phases on a suitable subset of D(f). More precisely we have the following.

Theorem 3.4. Let fL2() be a real-valued function with supp(f)[T,T]. If f^L1() and f^ is ε[W,W]-localized, then

θ(t)=θW(t),

for every t[T,T] with |f(t)|>εW.

Proof. Note that fL1(), and the assumption f^L1() implies that both f and fW are continuous. Hence, we have the pointwise estimate:

|f(t)fW(t)|=|[W,W]cf^(ξ)e2πiξtdξ|||f^RWf^||L1().

Since f^ is ε[W,W]-localized and tD(f) with |f(t)|>εW, we obtain

b:=||f^RWf^||L1()|f(t)|<1.

Now, Lemma 2.1 gives |θ(t)θW(t)|2arcsin(b)<π and since f is real-valued, this implies |θ(t)θW(t)|=0.

In a similar fashion as before, we could add in Theorem 3.4 a certain regularity assumption (C1-regularity suffices) to ensure the integrability of f^. The latter statement reveals the interaction between localization in time and frequency. The smaller the εW is, the better the result is, which means the better f is localized in the frequency domain. However, εW cannot be made arbitrarily small as this would contradict the Heisenberg uncertainty principle. The relation to an uncertainty principle due to Donoho and Stark is, however, more natural in our context. It reads as follows.

Theorem 3.5 (2, Theorem 7). Let fL1() with fL1()=1. If f is ε-localized and εW-bandlimited, then

|W|||1εεW1+εW,(4)

where |W| and || denote the Lebesgue measure of W and , respectively.

In the setting of Theorem 3.4, inequality 4 leads to

W1εW4T.(5)

for every fL2(), supp(f)[T,T] with fL1()=1 such that f^L1() and localization f^RWf^L1()εW. Inequality 5 can be seen as a necessary condition on the band-limiting operator BW, for which a bound on |θθW| is possible.

Remark 3.6. Consider the band-limiting operator BW acting on the space of time-limited functions L2(T), where T=[T,T]. The eigenfunctions {ψj:j} of BW are the so-called prolate spheroidal wave functions (PSWFs) which are highly oscillating for big n. Suppose that the corresponding real eigenvalues {λn:n} are ordered by λ1>λ2>0. One can show that the amplitudes of the PSWFs ψj satisfy a polynomial growth, while the eigenvalues λj decay to zero exponentially [4]. Hence, a perturbation of a signal fL2(T) by a PSWF ψn causes a significant pointwise distortion of the original signal f (and therefore, of the phase θf). If we define hn:=f+ψn, then the band-limited version of hn is given by

BWhn=fW+λnψn.

Observing that λnψn0 exponentially as n, we conclude that the distortion is negliglible after band limiting the perturbed signal hn. In particular, this implies that, in general, one cannot expect a bound on the phase of hn and BWhn. This effect is visualized in Figure 1.

FIGURE 1
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FIGURE 1. Perturbation of a signal f by a prolate spheroidal wave function ψn before and after applying the band-limiting operator. To obtain the above visualization, we made use of approximation techniques of PSWFs as described in [1]

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., θf=θBWf. Moreover, we demonstrated that the equality of the two phases is not always possible, and we gave explicit examples for θfθBWf by using perturbations by PSWFs. Therefore, the following question now comes up naturally.

Which operators T:L2()L2() satisfy θf=θTf, for every fL2()? Can we characterize those operators T?

In this section, we will give a precise characterization of operators on L2() which leave the phase invariant. The results will also be generalized to operators on Lp(),1p<. We start with a definition of a class of operators which we will call phase-preserving operators (PPO for short).

Definition 4.1. We call a bounded linear operator T:L2()L2() to be phase-preserving if

θf=θTf,

for every fL2().

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

Lemma 4.2. If T:L2()L2() is phase-preserving, then, for every fL2() with f0 a.e., there exists an a.e. unique function pf0 such that Tf=pff.

Proof. Let fL2() with f(t)0, for a. e. t. Then, for all t such that f(t)0 and f(t), we have

Tf(t)=eiθTf(t)|Tf(t)|=eiθTf(t)|Tf(t)|eiθf(t)|f(t)|f(t)=|Tf(t)||f(t)|f(t),

which implies

pf(t)=|Tf(t)||f(t)|.

Since {t:f(t)=0}{t:f(t)=} is a zero set, the function pf is unique almost everywhere.

For ϕL(), let the multiplication operator Mϕ:L2()L2() be defined by

Mϕf=ϕf.

The function ϕ is called the symbol of the multiplication operator Mϕ. Clearly, Mϕ is bounded with Mϕ=ϕ. It is obvious that if ϕ is real-valued with ϕ0 a.e., then Mϕ is phase-preserving. In the following, we will prove that the converse of this statement is true as well, i.e., for every PPO T, there exists a ϕL() with ϕ0 a.e. such that

T=Mϕ.

In other words, the function pf from Lemma 4.2 is independent of f. To show this statement, we start by investigating the map fpf, where pf is defined as in Lemma 4.2.

Definition 4.3. Let f: be measurable functions. We call f and g pointwise linear-independent if (Ref(t),Imf(t)) and (Reg(t),Img(t)) are linear-independent vectors in 2, for a.e. t.

Lemma 4.4. Let T:L2()L2() be phase-preserving, fL2() with f0 a.e., and λ×. Then,

pf=pλf.

Moreover, if gL2() is a second function which is pointwise linear independent of f, then

pf+g=pf=pg.

Proof. If f0 a.e., then λf0 a.e. Due to linearity, we have

λpff=T(λf)=pλfλf.

Therefore, pλf=pf. Now assume that gL2() is pointwise linear independent of f. Then, f+g0 a.e. since otherwise f(t) and g(t) would be pointwise linear-dependent on a set of measure greater than zero. Let t s.t. f(t) and g(t) are linear-independent. Due to the linearity of T, we obtain

T(f+g)(t)=pf+g(t)(f+g)(t)=pf(t)f(t)+pg(t)g(t).

Linear independence implies pf+g(t)=pf(t)=pg(t). This equation holds for a.e. t.

Suppose that V={f1,f2,}L2() is a system of functions with the property that, for every n, there exists a k with kn such that fn is pointwise linear independent of fk. In this case, Lemma 4.4 implies that a PPO T:L2()L2() is a multiplication operator on spanV. As a consequence, if V would be complete in L2(), then 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 defined by the functions χc(t)=e2πict,c. The set has the following property.

Lemma 4.5. Let χc1,,χcn,χcn+1 be n+1 distinct exponentials and let λ1,,λn. We define : to be a nonzero linear combination of the first n exponentials:

:=j=1nλjχcj.

Then, is pointwise linear independent of χcn+1

Proof. Without loss of generality, we may assume that λj0 for j=1,,n. Let λj=aj+ibj with aj,bj. Then,

Re(t)=j=1najcos(2πcjt)bjsin(2πcjt)(6)

and

Im(t)=j=1nbjcos(2πcjt)+ajsin(2πcjt).(7)

We define the matrix

M(t)=(Re(t)cos(2πcn+1t)Im(t)sin(2πcn+1t)).

Then, and χcn+1 are pointwise linear-independent if detM(t)0, for almost every t. Plugging Eqs 6and7 into M(t) and using the definition of the determinant as well as elementary trigonometric identities, we arrive at

detM(t)=j=1najsin(rjt)bjcos(rjt)

with rj:=2π(cn+1cj). In particular, rj0, for every j, since cj was assumed to be distinct. We observe that tdetM(t) can be extended to an entire function. Since zeros of entire functions form a discrete set with no accumulation point, it suffices to show that tdetM(t) is not the zero function. To do so, we fix k{1,,n} and let R>0. By using trigonometric identities, we observe that, for jk, the term

(ajsin(rjt)bjcos(rjt))(aksin(rkt)bkcos(rkt))(8)

is a sum of terms of the form

αsin((rj±rk)t+β)

with some α,β. Due to periodicity, this implies that there exists a C>0 such that

|RR(ajsin(rjt)bjcos(rjt))(aksin(rkt)bkcos(rkt))dt|C,(9)

for every R>0 and a constant C>0 independent of R. In case j=k, Eq. 8 becomes

(aksin(rkt)bkcos(rkt))2,

which is a nonzero periodic function. Hence,

RR(aksin(rkt)bkcos(rkt))2dt(10)

as R. Combining Eq. 9 with Eq. 10, it yields

RRdetM(t)(aksin(rkt)bkcos(rkt))dt

as R In particular, detM(t) is not the zero function.

We now give a characterization of PPO.

Theorem 4.6. Let T:L2()L2() be a bounded linear operator. Then, T is phase-preserving if and only if there exists ϕL() with ϕ0 such that

Tf=ϕf,

for every fL2(). In other words, T=Mϕ.

Proof. The fact that every multiplication operator Mϕ with ϕ0 is phase-preserving was discussed above. It remains to show the converse. We fix f(t)=eπt2; then, fL2() and f(ξ)0, for every ξ. Wiener’s Tauberian theorem [8] implies that span{f(c)|c} is dense in L2(). Since the Fourier transform is an isometry and f=f, the set

Q:=span{Mcf|c}

is dense in L2(). Let g,hQ with gh. Then,

g=(jAλjχcj)f

and

h=(iBμiχci)f

for some cj,ci, λj,μi, and finite subsets A,B. We choose an exponential χck with kA and kB. Since f is positive everywhere, Lemma 4.5 implies that χckf is pointwise linear independent of both g and h. Lemma 4.4 implies that ph=pg. Since g and h are arbitrary, we obtain a function ϕL() with

Tg=ϕg

for every gQ. We show that ϕL(). Assume the contrary, i.e., ϕ is not essentially bounded. Then, for every n the set

An:={tR:|ϕ(t)|n}

has a positive Lebesgue measure. We now choose fnL2() with fn=0 on ,An and ||fn||2=1. Furthermore, we choose qnQ such that

fnqn21n.

which is possible due to density of Q in L2(). We estimate the L2-norm of qn as follows:

|qn|L2(An)2=qnL2()2|qn|L2(,An)2
(11n)2(fnqnL2()2fnqnL2(An)2)12n

With this choice of qn, we obtain the following estimate:

T2|qn|22|ϕqn|22n2An|qn|2n2(12n)

Because qn21+1n, we see that

Tndn

where (dn) is a sequence with dn1. This is a contradiction to the boundedness of T. Consequently, ϕL() and

T=Mϕ

on Q. Since Q is a dense subset of L2(), we have T=Mϕ on the closure of Q, i.e., on L2().

The characterization of PPOs in Theorem 4.6 was stated in the space L2(). The first property we used in the proof was the pointwise linear independence of exponentials. This condition is purely algebraic and does not depend on the structure of Lp(). The second crucial property we needed was the density of modulations of a positive function fL2() (in our case, f was a Gaussian) which was based on Wiener’s Tauberian theorem. Equivalently, this means that the set of exponentials

:={e2πic|c}(11)

is complete in L2(,μ), where μ=fdx. In view of this reformulation, the characterization of PPOs can be generalized to operators on Lp() which provided that the set of exponentials is complete in Lp(,μ). It is well known that this is true for every Borel measure µ on which is positive and finite.

Theorem 4.7. Let µ be a positive, finite Borel measure on . We denote by the set of exponentials as defined in Eq. 11. Then, is complete in Lp(,μ) for every p[1,).

Proof. Assume by contradiction that is not complete in Lp(,μ). Then, there exists an 0fLp(,μ),span¯. Moreover, the Hahn–Banach theorem implies that there is a continuous linear functional Lp(,μ)* such that

|span¯=0,(f)=dist(f,span¯)0.(12)

By the Riesz representation theorem, the functional has the form

(g)=g(t)m(t)dμ(t)

for some mLq(,μ) and every gLp(,μ), where q is the Hölder conjugate exponent of p. The identities in Eq. 12 imply that

m(t)e2πictdμ(t)=0

for every c. Let τ:=mμ. Then, τ is a finite signed measure on with the property that its Fourier–Stieltjes transform is zero:

τ^=0.

By the uniqueness theorem of the Fourier–Stieltjes transform of measures, we have τ=mμ=0 which implies that =0, contradicting Eq. 12.

Corollary 4.8. Let p[1,) and T:Lp()Lp() a bounded linear operator. Then, T is phase-preserving if and only if T is a multiplication operator with nonnegative symbol ϕ0.

Proof. It is clear that a multiplication operator on Lp() with a nonnegative symbol is a PPO. For the other directions, we fix f(t)=eπt2. Since f is a Schwartz function, it lies in Lp(), for every 1p. Consider the measure

μ=fdx

where dx stands for the Lebesgue measure. Clearly, µ is a finite, positive Borel measure on . Theorem 4.7 implies that the set of exponentials is complete in Lp(,μ), which is equivalent to say that

Q=span{Mcf|c}

is dense in Lp(). The proof now follows the lines of the proof of the characterization theorem in L2() by replacing the L2-norm by the Lp-norm.

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 fW.

Assume that ϕL1() is nonnegative with ϕL1(). Let Mϕ be the corresponding multiplication operator and for W>0, let BW be the band-limiting operator. By Young’s inequality, the operator Sϕ:L2()L2(),

Sϕh=ϕ*h

is well-defined and bounded. It follows that

Mϕf=1(ϕ*f)=1Sϕf.

Similar to Definition 3.1, we say that a function fL2() is (strictly) εW-localized with respect to the L2-norm if ||fRWf||2<εW. As defined before, W denotes the interval [W,W]. Now, suppose that the Fourier transform of a given compactly supported function fL2(T) is (strictly) εW-localized with respect to the L2-norm, i.e.,

||fRWf||2<εW.

We define δ:=εWfRWf2>0 and choose ϕ in such a way that

||ϕ*f||2δ.

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

||MϕfBWf||2=SϕfRWf2εW.

This proves the following statement.

Corollary 4.9. Let fL2(T) be such that its Fourier transform is (strictly) εW-localized in the L2-sense. Then, there exists a nonnegative function ϕL1()L(), for which

||MϕfBWf||2εW.(13)

Note that an L2-localization of f can be achieved in an analogous manner to Section 3. Suppose that fCn+1() and that we have control over the total variation norm of f(n). We then obtain an εW-localized function (in the L1-sense), and from the control over the BV-norm, we can assume that |f^(ξ)|1, for every |ξ|W. This yields

||f^RWf^||2f^RWf^1εW.

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 |θθW|, then the band-limiting operator BW is close to a multiplication operator (in the L2-sense).

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, Germany

Reviewed by:

Robert Beinert, Technical University of Berlin, Germany
Elijah 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