Reconstruction of complex-valued propagating wave fields using the Hilbert-Hankel transform

Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 247

Reconstruction of complex-valued propagating wave fieldsusing the Hilbert-Hankel transform

Michael S. Wengrovitz and Alan V. Oppenheim

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

George V. Frisk

Department of Ocean Engineering, Woods Hole Oceanographie Institution, Woods Hole, Massachusetts 02543

Received May 2, 1986; accepted August 27, 1986

In this paper the theory of the unilateral inverse Fourier transform and the unilateral Hankel transform isdeveloped. The consistency between each transform and its bilateral version leads to an approximate real-partsufficiency condition for complex-valued one-dimensional even signals and two-dimensional circularly symmetricsignals. The two-dimensional result is used in a reconstruction algorithm that is applied to synthetic -andexperimental underwater acoustic fields.

1. INTRODUCTION

A well-known property in Fourier-transform theory is thatcausality in one domain implies real-part sufficiency in thealternate domain., This property is the basis for the factthat the real and imaginary components of a signal are relat-ed by the Hilbert transform, if the spectrum of the signal iscausal. In wave propagation problems, it is the circularlysymmetric two-dimensional Fourier transform or, equiva-lently, the Hankel transform that is of central importance.Because of the symmetry in such problems, the condition ofcausality is not applicable. In one dimension, the counter-part of the circularly symmetric signal is the even signal.The one-dimensional Fourier transform of an even signal isalso even, and thus, again, the condition of causality is notapplicable.

In our work we have shown that under some conditions itis possible to relate approximately the real and imaginarycomponents of a one-dimensional even signal or a two-di-mensional circularly symmetric signal. The approximationis based on the validity of the unilateral inverse Fouriertransform in one dimension and that of the unilateral Han-kel transform, referred to as the Hilbert-Hankel transform,in two dimensions. In this paper we develop the approxi-mate real-part sufficiency condition in one and two dimen-sions by using these transforms. The two-dimensional re-sult forms the basis for a reconstruction algorithm in whichthe real (or imaginary) component of a complex-valuedpropagating wave field is obtained from the imaginary (orreal) component. The algorithm is illustrated with synthet-ic and experimental underwater acoustic fields.

In Section 2 first a review of one-dimensional exact ana-lytic signals is provided, and then the theory of one-dimen-sional approximate analytic signals is presented. A numberof statements involving the unilateral Fourier transform, theunilateral inverse Fourier transform, causality, and approxi-mate real-part-imaginary-part sufficiency are made. InSection 3 the theory is extended to two-dimensional circu-

larly symmetric signals. These signals, which can be equiv-alently described in terms of the Hankel transform, are di-rectly related to two-dimensional fields propagating in hori-zontally stratified media. We will show that, under someconditions, it is possible to relate approximately the real andimaginary components of these fields. To do this, a unilat-eral version of the Hankel transform, referred to as theHilbert-Hankel transform, is described. The transform canbe used to synthesize approximately an outgoing wave field,and its consistency with the Hankel transform will be shownto imply an approximate relationship between the real andimaginary components of the field. The properties of theHilbert-Hankel transform and the relationship of thistransform to several other transforms will be discussed. InSection 4 an asymptotic version of the Hilbert-Hankeltransform is developed. This transform leads to an efficientalgorithm for reconstructing a complex-valued acoustic fieldfrom its real or imaginary part. The algorithm is applied toboth synthetic and experimental underwater acoustic fields.

2. ONE-DIMENSIONAL THEORY: THEUNILATERAL INVERSE FOURIER TRANSFORM

From the theory of analytic functions, a complex-valuedfunction of a complex-valued variable is analytic at a point ifit is both single valued and uniquely differentiable at thatpoint. A necessary condition for a unique derivative is thatthe real and imaginary components of the function satisfythe Cauchy-Riemann conditions,' which involve the partialderivatives of the function. If these partial derivatives arealso continuous, the Cauchy-Riemann conditions form anecessary and sufficient condition for analyticity at a givenpoint.

The Cauchy-Riemann conditions imply that the real andimaginary components of a function cannot be chosen inde-pendently, if the function is to be analytic. These condi-tions imply that if the real (or imaginary) component isspecified within the region of analyticity, the imaginary (or

0740-3232/87/010247-20$02.00 © 1987 Optical Society of America

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248 J. Opt. Soc. Am. A/Vol. 4, No. 1/January 1987

real) component can be determined. In some cases, knowl-edge of one of the components along only the boundary ofthe analytic region is sufficient to determine the alternatecomponent. 2'3 For example, for the region of analyticitydefined by a circle centered at the origin of the complexplane, integral relationships between the real and imaginarycomponents of the function referred to as Poisson integralshave been developed. Similarly, for the region of analyticitydefined by a half-plane that includes the real or imaginaryaxis, integral relationships between the real and imaginarycomponents of the function along the axis, referred to asHilbert transform integrals, have been developed.

In a signal processing context, there is a related conceptthat states that a one-sided, or causal, condition in onedomain implies a real-part-imaginary-part sufficiency con-dition in the alternate domain; that is, a complex-valuedsignal has a real-part-imaginary-part sufficiency conditionif its Fourier transform is causal and vice versa. A signalthat can be exactly synthesized in terms of a one-sided Fou-rier transform is often referred to as an analytic signal.

To summarize the relationships, let us consider a com-plex-valued signal f(t) with a Fourier transform consisting ofonly positive components, i.e.,

transform. Our primary interest has been in two-dimen-sional circularly symmetric signals, which are related to thetwo-dimensional circularly symmetric Fourier transform or,equivalently, to the Hankel transform. However, the exten-sion of the theory of analytic functions can best be presentedby first considering the one-dimensional case. In the re-mainder of this section the theory of one-dimensional signalsthat are approximately analytic is developed.

We now consider a complex-valued signal f(t) for whichthe Fourier transform exists and which has the Laplacetransform Fo(s) given by

Fo(s) = j.-f(t)e-stdt. (6)

In general, the Fourier transform F(co) is a two-sided func-tion of c, and thus the signal f(t) is represented in terms ofthe inverse Fourier transform as

f(t) = f F(.)e- tdc. (7)

We also define the related signal fu(t) in terms of the unilat-eral inverse Fourier transform as

fAt) =1 f F(o)eJwdw.27r

(8)(1)

It can be shown that the one-sided integral in Eq. (1) impliesthat f(t) is an analytic function in the upper half of thecomplex t plane.4' 5 This one-sided condition connects thetheory of the analytic signal with the theory of the analyticfunction.

Let us express f(t) in terms of its real and imaginarycomponents, for t real, as

f(t) = g(t) + jg^(t), (2)

with G(o) and O(w) denoting the Fourier transforms of g(t)and j(t), respectively. The relationships between the realand imaginary components of f(t) and their Fourier trans-forms can then be summarized as follows:

F(M) = 2G(uM,(w) = -j sgn(w)G(co), (3)

where u(-) is the unit step function. The signals g(t) andi(t) form a Hilbert transform pair.

Although the Hilbert transform relationship between g(t)and j(t) is conveniently summarized in the frequency do-main, it is also possible to use the convolution property ofFourier transforms to develop the relationship in the timedomain. Specifically, using the inverse Fourier transform of-j sgn(w), it can be shown that

gi(t) = * g(t) (4)7rt

and similarly that

g(t) = - it *g(t), (5)Art

where the integrals are interpreted as Cauchy principal val-ued.

It is possible to extend some of the properties of analyticsignals to signals that do not possess a one-sided Fourier

Note that f>(t) is an analytic signal, since its Fourier trans-form is causal.

One way of extending the theory of analytic signals to thesignal f(t), which has a two-sided Fourier transform, is torequire that

(9)

Thus a signal f(t) that can be approximated by a unilateralversion of its inverse Fourier transform can be consideredapproximately analytic.

The condition that a signal can be accurately synthesizedin terms of its unilateral inverse Fourier transform is ratherrestrictive, and it certainly does not apply to any arbitrarysignal. For example, consider a signal, composed of a sum ofcomplex exponentials, that has a rational Laplace trans-form. In Fig. 1 we have indicated the positions of severalpoles in the s plane, corresponding to the arbitrarily chosensignal, as well as the region of convergence for the Laplacetransform. The condition that f(t) f>(t) is equivalent tothe statement that the inverse Laplace transform contour C1+ C2 can be approximately replaced by the contour C1. Theapproximation will be poor if a pole, such as P3 , is located inquadrant III or IV of the s plane. Essentially, the effects ofthis pole, quite important in determining the character ofthe corresponding signal f(t), are only negligibly included byintegrating along the positive imaginary axis only. That is,if f(t) is exactly synthesized as

f(t) = 2rj J Fo(s)estds, (10)

s3 that

f(t) =2rj Jci F(s)estds + 2 j | Fo(s)estds, (11)

the pole at position P3 contributes primarily to the second ofthese integrals. Thus the approximation

Wengrovitz et al.

f"M � ' f F(.)ej-td..27r 0

A~t) - A, M .


P3X

V

'/'

-7,

V//

s plane

Fig. 1. Complex s plane, indicating positions of poles, the inverseLaplace transform integration contour, and the Laplace transformregion of convergence.

f(t) 2 c1A Fo(s)estds = f F(c.)ejItdsc (12)

is not accurate because of the position of pole P3 in the splane. However, if there are no poles in quadrant III or IVof the s plane, the unilateral inverse transform can yield anaccurate version of f(t). This can be argued on the basis ofthe fact that poles such as P1 and P2 contribute primarily tothe contour integral C1 and only negligibly to the contourintegral C2.

This concept of approximate analyticity is intuitive-ifthe signal f(t) can be approximated by the analytic signalf.(t), f(t) is approximately analytic. Although it is possibleto develop an approximate relationship between the real andimaginary components of f(t), there are no further conse-quences of the relationship f(t) - f(t). However, as will bediscussed in the remainder of this section, some interestingrelationships result if additional restrictions are placed onthe signal f(t) and the definition of approximate analyticityis slightly modified. For example, there are interesting con-sequences that arise if f(t) is restricted to be a causal signalor an even signal. Essentially, by considering f(t) to be aneven signal, the case of the causal signal can be treated aswell, since an even (or causal) signal can always be invertiblyconstructed from a causal (or even) signal.

To develop the theory, let us now consider the complex-valued signal f(t) to be even. We denote the Laplace trans-form by Fo(s) and again assume that the Fourier transformF(co) = Fo(s)ls=j1 exists. Since the signal is even, it followsthat its Fourier transform and Laplace transform must alsobe even. We will find it convenient to define not only theFourier transform and the inverse Fourier transform buttheir unilateral counterparts as well. Thus

f(t) -- 11F(co))- J F(.)ej* tdw, (13)

fu(t) - SI-11F(t )2 F(J)ejetd, , (14)

F(co) -- j;f(t)}--f f(t)e-j'tdt, (15)

F,,(w) -gulf(t))- =_fo f(t)e-jwtdt, (16)

where fu(t) represents the unilateral inverse Fourier trans-form of F(.), and Fu(w) represents the unilateral Fouriertransform of f(t). Symbolically, 9 and i-1 represent theFourier transform and the inverse Fourier transform opera-tions and Du and 9>-l represent the unilateral Fourier trans-form and the unilateral inverse Fourier transform opera-tions.

Note that, although 5T and 5T-1 are inverse operations, luand 5,-l are not necessarily inverse operations. Addition-ally, it is recognized that both fu(t) and Fu(w) are analyticsignals, since their Fourier transforms are causal. There-fore the real and imaginary components of fu(t) are exactlyrelated by the Hilbert transform, and the real and imaginarycomponents of Fj(w) are exactly related by the Hilberttransform. Also note that, since f(t) and thus F(w) are evensignals, they can be synthesized in terms of the cosine trans-form as

(17)

(18)

ft) = 2 | F()cos wtdt,7r o

FMX = 2 fo f(t)cos wtdt.

To extend the theory of analytic signals to the signal f(t),we will require that f(t) satisfy the condition

(19)

that is, only those functions f(t) that can be approximated bythe unilateral inverse Fourier transform for positive valuesof t will be considered. Note that this condition differs fromthe condition in relation (9). Specifically, the unilateralinverse Fourier transform is required to synthesize the evensignal f(t) only for positive values of t. The even functionf(t) will be defined as approximately analytic if the conditionin relation (19) is satisfied. To the extent that the approxi-mation in relation (19) is valid, there will also exist an ap-proximate relationship between the real and imaginary com-ponents of f(t), for t > 0. This result is the basis for anumber of statements that will now be made.

If f(t) - fu(t) for t > 0, then:

Statement 2.1 The real and imaginary components off(t) must be approximately related by the Hilbert trans-form for t > 0.

Statement 2.2 The unilateral inverse Fourier transformf>(t) is approximately causal.

Statement 2.3 The unilateral Fourier transform andunilateral inverse Fourier transform are related by the ap-proximation

(20)

Statement 2.4 The real and imaginary components ofF(w) must be approximately related by the Hilbert trans-form for w > 0.

Statement 2.1 follows as a consequence of the fact that thereal and imaginary components of fu(t) are related exactly bythe Hilbert transform.

To justify statement 2.2, we note that f(t) fu(t), t > 0implies that

+ J F(w)ej'tdb - 0, t > 0, (21)

l- - - w * - l --7 - -

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f(t) - f,,(t), t > 0;

gjgU_'jF(w)jj - F(w)u(w).


so that

I f F(.)e-J-do 0, t > 0.2ir J

The latter step follows from the fact that F(w) is even in w.From relation (22) and the definition of the unilateral in-verse Fourier transform, it can be seen that

f,(-t) - 0, t > 0, (23)

and thus

f"(t) - 0, t < 0. (24)

Therefore, under the condition that f(t) fu(t) for t > 0, theunilateral inverse Fourier transform must be approximatelycausal.

Let us next consider statement 2.3. As pointed out earli-er, the unilateral Fourier transform and the unilateral in-verse Fourier transform are generally not inverse operations.However, from the definition of fu(t),

5= Iff,(t)1. (25)Since fu(t) is approximately causal, it follows that

511.0 1f.(t)1.

Additionally, from the definition of fu(t), it is also apparentthat

51fu(t) = F(w)u(w). (27)

When the three relations are combined, statement 2.3 fol-lows.

To justify statement 2.4, we use statement 2.3 to establishthat

WyuJF. 'F(w)J} - F(w)u((o), (28)

so that

Wulfu(t) I F(o) u (w). (29)

The signal F(w)u(w) is approximately analytic, since it isrelated to the one-sided Fourier transform on the left-handside of relation (29). Thus, since F(co)u(w) is approximatelyanalytic, its real and imaginary parts must be related by theHilbert transform.

Although the specific statements made involve the rela-tionships among the Fourier transform, the inverse Fouriertransform, and their unilateral counterparts, it is also possi-ble to derive a number of interesting relationships betweenthe cosine and sine transforms that compose these. Severalof these relationships are derived in Appendix A.

To summarize briefly, in this section we have reviewed theone-dimensional theory of exact and approximate analyticsignals. The theory of an exact analytic signal was present-ed in terms of the properties of the Fourier transform. Thistheory was then extended to develop the notion of a signalthat is approximately analytic. Although such a signal doesnot have a causal Fourier transform, its real and imaginaryparts can be approximately related by the Hilbert trans-form. The necessary condition for an even signal to possessthis property is that its causal portion must be accuratelysynthesized by the unilateral inverse Fourier transform.This condition implies a number of other interesting conse-quences, including an approximate real-part-imaginary-

part sufficiency for both the causal part of the signal and thecausal part of its Fourier transform and an inverse relation-

(22) ship between the unilateral Fourier transform and the uni-lateral inverse Fourier transform.

3. TWO-DIMENSIONAL THEORY: THEHILBERT-HANKEL TRANSFORM

In the previous section, we considered conditions underwhich a one-dimensional complex-valued signal is eitherexactly or approximately analytic. In this section, the the-ory of approximate analyticity will be extended to two-di-mensional circularly symmetric signals. Although the the-ory can be developed for the more general multidimensionalcase by considering the multidimensional version of the uni-lateral inverse Fourier transform, we will focus primarily onthe special case of a two-dimensional circularly symmetricsignal because of its practical relevance to a wave field prop-agating in a horizontally stratified medium.

Consider a two-dimensional signal p(x, y) and its two-dimensional Fourier transform g(kx, ks), which are relatedby

p(X,y) = 1 2 J g(k, ky)expU(kx + kyy)]dkdky.

(30)

If p(x, y) is circularly symmetric, then g(kx, ky) is also circu-larly symmetric, and the relationship in Eq. (30) can bewritten as

(26)

p(r) = j g(kr)JO(krr)krdkr, (31)

where r = (X2

+ y2

)1/2, kr = (kx + ky)'/ 2, g(kr) = g(kr)/(2r), and

J0 (-) is the zero-order Bessel function.The relationship described in Eq. (31) is a Hankel trans-

form. Note that p(r) may be a complex-valued signal, but,because of the circular symmetry involved, the condition ofcausality is not applicable, and the real and imaginary com-ponents of p(r) are generally unrelated. The Hankel trans-form is self-inverse, so that

g(kr) = J p(r)JO(krr)rdr. (32)

The two-dimensional circularly symmetric signal p(r) isanalogous to the one-dimensional even signal f(t) consideredin the previous section.

To develop the property of approximate analyticity fortwo-dimensional circularly symmetric signals, we must de-velop a unilateral version of the Hankel transform. That is,with analogy to the one-dimensional bilateral inverse Fouri-er transform and one-dimensional unilateral inverse Fouriertransform, we wish to develop the Hankel transform and itsunilateral version. In examining Eq. (31) we see that in onesense the Hankel transform is already unilateral, since thelimits of integration are from zero to infinity. However, thisversion of the Hankel transform is actually analogous to theone-sided cosine transform for one-dimensional signals.Here, we want to develop the transform analogous to theFourier transform and then consider its unilateral counter-part.

To this end, the zero-order Bessel function of the firstkind is written in terms of Hankel functions 6 as

Wengrovitz et al.


Jo(krr) = 1/ [Ho l)(krr) + HO(2)(krr)], (33)

so that Eq. (31) becomes

p(r) = 1/2 J g(kr)HO(l)(krr)krdkr + '/2 J g(kr)HO(2

)(krr)krdkr,

(34)

which is valid for both positive and negative values of r. Theexpression can be simplified by using the property 7 that

(35)

to yield

p(r) = '/2 J g(kr)HO(l)(krr)krdkr r > 0. (36)

The signal p(r) can be determined for negative values of r byutilizing this equation and the fact that p(r) = p(-r). How-ever, it is important to recognize that the bilateral transformin Eq. (36) correctly synthesizes the signal p(r) for positivevalues of r only. Specifically, although p(r) is an even func-tion of r, the Hankel function Ho(')(krr) is not an even func-tion of r, nor is g(kr) an odd function of kr, and thus theexpression in Eq. (36) is not correct for r < 0. The correctexpression for r < 0 can be obtained, by using properties ofHO(M)(krr) and Ho(2)(krr), as

p(r) = 1/2 f g(kr)HO( 2 )(krr)krdkr, r < 0. (37)

Alternatively, a bilateral expression that describes p(r) cor-rectly for both positive and negative values of r can be writ-ten as

p(r) =- Yi-g(kr)- 1/2 J g(kr)H0(l)(krIr1)krdkr. (38)

The transform in Eq. (38) will be referred to as the bilateralinverse Hankel transform.

The unilateral version of the transform in Eq. (38) isdefined as

p,(r) =_ -tj7'Ig(kr)I '/2 J g(kr)H 0 (l)(krr)krdkr. (39)

The operator ]f,-' is appropriately thought of as the unilat-eral inverse Hankel transform. However, because the Han-kel transform in Eq. (31) is already unilateral, the nameunilateral Hankel transform is ambiguous. Instead, thetransform defined in Eq. (39) will be referred to as theHilbert-Hankel transform because of its close relationshipto the Hankel transform and because, as will be discussedshortly, the transform implies an approximate relationshipbetween the real and imaginary components of p(r).

The bilateral inverse Hankel transform and the Hilbert-Hankel transform can be written in alternate forms by utiliz-ing the relationship

Ho(')(krr) = JO(krr) + jYO(krr), (40)

where YO(krr) is the zero-order Bessel function of the secondkind, also referred to as the Neumann function.6 We notethat both Jo(krr) and YO(krr) are real-valued functions forreal-valued arguments. By using this relationship, the bi-lateral inverse Hankel transform can be written as

5f-1Jg(kr)} -'/2 J g(kr)[JO(krr) + jYo(krlrl)]krdkr, (41)

and the Hilbert-Hankel transform can be written as

ku '(kr) 'l/2 f g(kr)[Jo(krr) + jY 0 (krr)]krdkr. (42)

It is also possible to develop a bilateral transform for theinverse relationship between p(r) and g(kr). From Eq. (32)and the relationship in Eq. (35), it follows that

g(kr) =- Np(r)l ' 1/2 J p(r)Ho(')(Ikrlr)r dr. (43)

This expression will be referred to as the bilateral Hankeltransform. Note that the bilateral Hankel transform andthe bilateral inverse Hankel transform are identical opera-tors, although they apply to different domains.

The most obvious definition for the unilateral version ofthis tranform is obtained by replacing the lower limit in Eq.(43) by zero. However, we will find it convenient to definethe transform differently. In particular, the unilateral ver-sion of the transform will be defined as

g.(kr) =-IL>p(r)j '-/2 Jp(r)[JO(krr)-jHo(krr)]r dr. (44)

The function Ho(z) is the zero-order Struve function.7' 8

The zero-order Struve function and the zero-order Besselfunction of the first kind, Jo(z), form a Hilbert transformpair.5 To demonstrate this, Jo(z) is expressed in terms of aFourier synthesis integral as

°( ) =27r f1(1 -. 2) 1/2 eZdw (45)

To compute the Hilbert transform J 0(z), the integrand inthis expression is multiplied by -j sgn(w) to yield

°0(z) = J 1 - sgn(w) (1 2)1I2 e@Zdc

I2 1 sin wz dw.

r o (1 -w2)1/ 2 (46)

The last integral is also the integral representation for thezero-order Struve function.7

The transform defined in Eq. (44) has also been consid-ered by Papoulis59 and has been referred to as the complexHankel transform. It is noted thatg"(kr) must be an analyt-ic signal, since its real and imaginary components are relatedby the Hilbert transform. From preceding discussions, thisimplies that the one-dimensional Fourier transform of gu(kr)must be causal. This fact and the use of the projection slicetheorem for two-dimensional Fourier transforms1 0 "' pro-vide the basis for expressing the complex Hankel transformin terms of the Abel transform, which is the projection of thetwo-dimensional circularly symmetric function and the one-dimensional Fourier transform. Specifically, the complexHankel transform of p(r) can be determined by computingthe Abel transform of p(r), retaining the causal portion, andcomputing the one-dimensional Fourier transform. The re-lationships among the Abel, Fourier, Hankel, and complexHankel transforms are summarized in Fig. 2. In this figure,the operator A refers to the Abel transform, defined as

PA(X) = J4{P(r)J I J p(r)dy,

Wengrovitz et al.

HO(2) (ze-j'r) = -H0(1)(Z)

(47)


p(r)

A

p(r)u(x)

A

g(kr) 9u(kr)

Fig. 2. Relationships among the Abel, Fourier, Hankel, and com-plex Hankel transforms.

where r = (x2 + y 2 )1/2 . We note that

.Ajp(r)u(x)l = J I p(r)u(x)dy = pA(X)u(x). (48)

The definitions for the bilateral and unilateral transformsare now summarized:

p(r) _ '-'g(kr)1 '/2 J g(kr)[JO(kr) + jYo(krlrl)]krdkr,

(49)

pu(r) f- 'Ig(kr)l -/2 |'J g(kr) [JO(krr) + jYo(krr)]krdkr,

(50)

g(k,) Y-fp(r)J '/2 7 p(r)[J0 (k.r) + jYo(Ikrlr)]r dr, (51)

g.(k,) I Sjp(r) - /2f p(r) [JO(krr) - jHo(krr)]r dr, (52)

where pu(r) represents the Hilbert-Hankel transform ofg(kr) and gu(kr) represents the complex Hankel transform ofp(r). These equations are analogous to Eqs. (13)-(16),which were developed in the one-dimensional context in thepreceding section. It is also convenient to define the follow-ing transforms and symbolic notation:

closely parallel the one-dimensional versions in the preced-ing section.

If p(r) -pu(r) for r > 0, then:

Statement 3.1 The real and imaginary components ofp(r) must be approximately related for r > 0 by

Re[p(r)] -- Y{Jo{Im[p(r)]}}

(57)

Statement 3.2 The Hilbert-Hankel transform p"(r) isapproximately causal.

Statement 3.3 The Hilbert-Hankel transform and thecomplex Hankel transform are related by the approxima-tion

(58)

Statement 3.4 The real and imaginary components ofg(kr) must be approximately related by the Hilbert trans-form for kr > 0-

Statement 3.1 can be justified as follows. The conditionthat p(r) pu(r), r > 0 can be equivalently written as

(59)

so that

Jofg(kr)) '/2JOtg(kr)l + j'/ 2Yofg(kr)), r > 0. (60)

By using the facts that Jo and Yo are real operators and thatRe[p(r)] = JotRe[g(kr)]}, Im[p(r)] = JotIm[g(kr)]j, the state-ment is established by equating real and imaginary parts onboth sides of relation (60).

Statement 3.2 will now be justified. Note that, from Eq.(50), the Hilbert-Hankel transform is defined for all valuesof r. Thus the causality condition in statement 3.2 is not aconsequence of the definition of the Hilbert-Hankel trans-form but rather is a consequence of the condition that p(r)pu(r), r> 0. We note that p(r) pu(r), r> 0 implies that

Jofg(kr)l - | g(kr)Jo(krr)krdkr, (53)

-YoJg(kd1 - |o g(kr) YO(krr)krdkr, (54)

Jo~ -g(kr)H0 (krr)k~dk,. (55)

To develop the theory of two-dimensional circularly sym-metric signals that are approximately analytic, we will re-quire that

(56)

that is, only circularly symmetric signals p(r) that can beapproximated by the Hilbert-Hankel transform for r > 0will be considered approximately analytic. This conditionis analogous to the condition stated in the previous section,thatf(t) '-f(t), t >0. To the extent that the approximationin relation (56) is valid, there will also exist an approximaterelationship between the real and imaginary components ofp(r), for r > 0. This result is the basis for the first of severalstatements that will now be discussed. The statements will

(61)'/2 J g(k,) Ho(')(krr)krdkr 0, r > 0,

so that

- '/2 J g(kr)HO()(-krr)krdkr 0, r > 0. (62)

The latter step follows from the fact that g(kr) is even in hr.From this equation and the definition of the Hilbert-Hankeltransform, it can be seen that

(63)

and thus

(64)

Therefore, under the condition that p(r) - pu(r), r > 0, theHilbert-Hankel transform must be approximately causal.

Let us next consider statement 3.3. The complex Hankeltransform can be written in terms of the operators Jo, Yo,and Ho as

KIpu,(r)l = 'I4g(k,) + '/4H0JY0Jg(k,)J}

+ '1/4 j[Jo{Yog(kr)11 - H0 JJ0 g(k,)Jf], (65)

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Im[p(r)] - Y0JJ0JRe[p(r)JJJ.

K.JKu-'Jg(kr))1 - g(kr)u(kr)-

J019(hr)) - Kulg(k)), r > 0,

p,,(-r) - 0, r > 0,

p (r) - p,,(r), r > 0;p,,(r) - 0, r <0.


C2 P3X

X X

kr plane

P1 P2x x

X

P3

CI

Fig. 3. Complex kr plane, indicating positions of poles and thebilateral inverse Hankel-transform contour.

which is valid for all kr. From the approximation p(r) -pu(r), r > 0, it then follows that

JOIg(r) '1/2Jotg(kr)) + 1/2jY0og(kr)J, r > 0, (66)

so that

Jotg(kr)) - jY0 1g(kr)}, r > 0.

Substituting this expression into Eq. (65), we find that

Kf/{P,(r)I -14g(kr) + '/4H101YOtg(r)AI

(67)

transforms that compose these. Several of the relationshipsare derived in Appendix B.

The condition that p(r) - pu(r), r > 0 is restrictive in thecontext of the general class of circularly symmetric signalsp(r). For example, in Fig. 3, the positions of several poles inthe kr plane, corresponding to a rational function g(kr), areindicated. The poles labeled P,', P2', and P3' are in symmet-rically located positions with respect to poles P,, P2, and P3because g(kr) is even. The condition that p(r) - pu(r), r> 0is equivalent to the statement that the bilateral inverse Han-kel transform integration contour C, + C2 can be approxi-mately replaced by the contour C1. Clearly, the approxima-tion will be poor if a pole, such as P3 , is located in quadrant IIof the kr plane. Essentially, the effects of this pole, quiteimportant in determining the character of the correspondingsignal p(r) for r > 0, are only negligibly included by integrat-ing along the positive real axis only. Note that the pole atP3' determines the behavior of the signal p(r) primarily forvalues of r < 0. Thus, if p(r) for r > 0 is exactly synthesizedas

p(r) = '/2 J (74)

so that

ality

+ '4j[-jJ0 JJ0lg(kr)} - jHo{Y0{g(kr)}], (68)

which is valid for all kr. Substituting the orthogonrelationships'2,13

JOIJOIg&kM)} = g(kr)

and

Ho{Y0Jg(kd)} = sgn(kr)g(kr) (70)

into relation (68) justifies the statement that

51,,p,,(r)} - g(k,)u(kr). (71)

Statement 3.4 summarizes the real-part-imaginary-partsufficiency condition that occurs in the kr domain. The factthat g(kr) has an approximate real-part-imaginary-part suf-ficiency condition is not completely unexpected, since, aspreviously discussed, there exists an approximate causalitycondition in the alternate r domain. To justify statement3.4, statement 3.3 is used to obtain the expression

KSu{P(r)} - g(kr)u(kr) . (72)

Writing the operator K/f in terms of the operators Jo and Y0yields

Jo1Re[pu(r)]i + Ho{Im[pu(r)]l + j(-HoJRe[pu(r)]J

+ J 01Im[p>(r)]1) - Re[g(kr)] + j Im[g(kr)],

kr > 0. (73)

It is noted that the real and imaginary components on theleft-hand side of this expression form a Hilbert transformpair, so that the real and imaginary components on the right-hand side are also related approximately by the Hilberttransform.

Although the specific statements made involve the rela-tionships between the Hilbert-Hankel transform and thecomplex Hankel transform, it is also possible to derive anumber of interesting relationships among the Jo, Yo, and Ho

p(r) = 1/2 f g(kr)HO(l)(krr)krdkr + 1/2 f g(kd)H0 (l)(krr)krdkrh

(75)

the pole at position P3 contributes primarily to the second of(69) these two integrals for values of r > 0. The approximation

p(r) - 1/2 J g(kr)HO(l)(krr)krdkr = Jf/ -1Jg(kd)1 (76)

is not accurate for r > 0 because of the position of pole P3 inthe kr plane.

Although the condition that p(r) -'pu(r), r > 0 is a restric-tive condition for the general class of two-dimensional circu-larly symmetric functions p(r), the condition is much lessrestrictive in the context of wave propagation. Essentially,the condition p(r) - pu(r), r > 0, when written in the form

p(r) - '/2 J g(kr)HO(l)(hrr)krdkr, r > 0, (77)

can be interpreted as the statement that p(r) is accuratelyapproximated by a superposition of positive, or outgoing,wave-number components only. This is consistent with thefact that in wave propagation problems physically meaning-ful poles do not arise in quadrant II or IV of the kr plane.The term outgoing has been associated with the functionHO(')(krr), as can be justified by asymptotically expandingHO(')(krr), for r >> 0. For example, a propagating wave fieldthat is due to a harmonic point source can be written forr >> 0 as

1 g(kr)exp(-j7r/4)p(r, t) - (27)/2 Ic g(kr)e /) expj(krr - t)]krdkr.

(78)

The field is seen to be composed of the superposition ofoutgoing cylindrical waves of the form expU(krr - t)JI(krr)"2 . It is also possible to write the error in the Hilbert-Hankel transform approximation as

--Ot 0---

Wengrovitz et al.

g(kdH0(1)(krr)krdkr,


e(r) p(r) - p,(r) = 1/2 J g(kr)Ho(2)(krr)krdkr, r > 0.

(79)

The error can be interpreted as a synthesis over the incom-ing wave-number components of p(r). Here, the term in-coming has been associated with the function HO(2)(krr).This association can be justified by asymptotically expand-ing HO(2)(krr), for r >> 0. The error, with temporal variationincluded, can thus be written as

1 rOg(kr)exp0X~/4)E(r, t) (2x)l/2 7o (j r ) exp[-j(khr + wt)]krdkr-

(80)

The error is seen to be composed of the superposition ofincoming cylindrical waves of the form exp[-j(krr + wt)J/(hrr)"2.

The unilateral synthesis implied by the condition p(r) -pu(r), r > 0 is widely used in the area of underwater acous-tics. For example, the unilateral synthesis implied by theHilbert-Hankel transform is an important component in anumber of synthetic data generation methods for acousticfields, such as the fast-field program (FFP).14,15 The impli-cation is that the two-dimensional theory of approximatelyanalytic signals, based on the condition p(r) - pu(r), r > 0, isapplicable to the wide class of outwardly propagating wavefields.

To summarize briefly, in this section the property of ap-proximate analyticity was extended to two-dimensional cir-cularly symmetric signals. To do this, we developed a bilat-eral version of the inverse Hankel transform and its unilater-al counterpart, referred to as the Hilbert-Hankel transform.Under the condition that the two-dimensional circularlysymmetric signal is approximated by the Hilbert-Hankeltransform for r > 0, it was shown that the real and imaginaryparts of such a signal are approximately related. The Hil-bert-Hankel transform was also related to another unilater-al transform, referred to as the complex Hankel transform.A number of consequences based on the validity of the Hil-bert-Hankel transform were developed. The theory is ofparticular importance because of its application to outgoingwave fields.

4. APPLICATION OF THE HILBERT-HANKELTRANSFORM TO THE RECONSTRUCTION OFUNDERWATER ACOUSTIC FIELDS

In the previous section, the Hilbert-Hankel transform wasdefined and a number of its properties were developed. Itwas shown that if the causal portion of a circularly symmet-ric signal, described by the bilateral inverse Hankel trans-form, can be approximated by the Hilbert-Hankel trans-form, there are some important consequences that includean approximate real-part-imaginary-part sufficiency condi-tion. As one application of the Hilbert-Hankel transform,we develop in this section an efficient reconstruction algo-rithm for obtaining a complex-valued underwater acousticfield from its real or imaginary part. Several examples ofthe reconstruction algorithm are provided in this section,and additional examples can be found in Ref. 16.

In underwater acoustics, the spatial part of the acoustic

pressure field due to a point time-harmonic source located atro satisfies the Helmholtz equation

[v2 + h2(r)Jp(r, ro) = -47r(r -ro), (81)

where k(r) = w/c(r), co is the frequency of the source and c(r)is the speed of sound. If horizontal stratification is as-sumed, with the source located at (0, zo), the solution to Eq.(81) can be written as

p(r, z; zo) = J g(kr, z; zo)JO(krr)krdkr. (82)

The function g(kr, z; zo), referred to as the depth-dependentGreen's function, satisfies the equation

+ h2(Z) h2(dz2 + k -r g(kr, z; zO) = -25(z - zo). (83)

In a number of applications, measurements of the com-plex-valued field p(r, z; zo) are required. For example, inthe inverse problem of determining the ocean bottom reflec-tion coefficient, the Hankel transform of the complex-val-ued field, for fixed values of z and z0, is determined to yieldthe depth-dependent Green's function, and the reflectioncoefficient is then extracted.' 7"18 We will show that both themagnitude and the phase of the acoustic field p(r) can beapproximated from measurements of a single quadraturecomponent only.

In Section 3, a real-part-imaginary-part sufficiency con-dition was obtained by requiring that p(r) - pu(r) for r > 0,where

p(r) = J g(kr)JO(krr)krdkr (84)

and

pP(r) = 1/2 J g(kr)HO(')(krr)krdkr- (85)

It was also shown that under the condition that p(r) - pu(r)for r > 0, the function pu(r) is approximately a causal func-tion of r, so that

p(r)u(r) - '/2 f g(kr)H 0(1)(krr)krdkr.Wewil in t onenen osutiteheaypocex(86)

We will find it convenient to substitute the asymptotic ex-pansion for the Hankel function,

z

zoO

J

Z {X

77777W�Fig. 4. Geoacoustic model for the Lloyd mirror field correspondingto the sum of a direct field plus a field that has interacted with apressure-release bottom. Parameters: RB = -1, ZO = 100 m, z = 50m, f = 220 Hz, c = 1500 m/sec, ko = 0.9215 rad/m.

--

Wengrovitz; et al.

I


0 500 1000

0 500 1000

r (meters)Fig. 5. (a) Magnitude and (b) residual phase of the Lloyd mirror field.

HO(')(khr) - 2 1/2 exp(krr -7r/4)]

into relation (86) to yield

p(r)u(r) - 1 X (k ) expU(kr - 7r/4)] k dkp(ru0r) I g(hr h1d h

(87)

(88)

Multiplying both sides of this equation by r1/2 and definingg(kr) as

g(k) = (27rkr)g/2g(kr)exp(-j7r/4),

we find that

1/2 1 ( f ep(Orr u(r) --- g(kr)exp~jhrr)dr,,2ir Jo

(89)

r >> 0. (90)

This expression is the basis of the FFP commonly used inunderwater acoustics for synthetic data generation.'4 "5 "9

Relation (90) also implies that p(r)rl/2u(r) is approximatelyanalytic, since it is approximately synthesized in terms of

0

0

)o

(a)

I-

(b)

ads-I

1-1

1500

1500

Wengrovitz et al.


the unilateral inverse Fourier transform. Thus, by incorpo-rating the additional asymptotic expansion for HO(')(krr),the signals Re[p(r)r'/ 2u(r)] and Im[p(r)r1/ 2u(r)J are seen tobe approximately related by the Hilbert transform for r >> 0.

When sampled versions of these signals are involved, thereexist several methods for determining the Hilbert trans-form.20 ,21 In the method that we chose, a sampled version ofRe[p(r)u(r)r'/2 ] + j Im[p(r)u(r)r'/2] is obtained by comput-ing the fast Fourier transform (FFT) of Re[p(r)u(r)r'12 ] (or jIm[p(r)u(r)r'/2]), multiplying by 2u(kr), and computing theinverse FFT. A discrete Hilbert transform based on an

(a)

Q..

1

0

0.

1

optimal finite impulse response filter design may also haveapplication to this problem.22'23

We will now present three examples of this reconstructionmethod. The first example consists of a simple acousticfield that was synthetically generated by using a closed-formexpression. The second example consists of a realistic deep-water acoustic that was synthetically generated by comput-ing the Hankel transform of the associated Green's function.The third example consists of an acoustic field collected inan actual ocean experiment.

As the first example, consider the Lloyd mirror acoustic

0 500 1000

0 500 1000

r (meters)Fig. 6. (a) Magnitude and (b) residual phase of the reconstructed Lloyd mirror field.

(b)

I-"0

1,..

1500

1500

Wengrovitz et al.


W.0'.4$.4

1e-080 500 1000 1500

r (meters)Fig. 7. Error in the reconstruction of the real component (bottom curve) and true field magnitude (top curve) of the Lloyd mirror field.

field,2 4 which can be written as the sum of a direct field plus areflected field that has interacted with a pressure releaseboundary. The total field can be written in closed form as

p(r) = exp(jk 0RO)/R0- exp(jk 0Rj)/Rj, (91)

where B0 = (r2 + (z - zo)2 )1/2, R, = (r2 + (z + Zo)1/2, and ko =w/co is the water wave number. The geoacoustic model forthis example is summarized in Fig. 4.

In Fig. 5(a) is shown the magnitude of the Lloyd mirrorfield. The peaks and hulls in the magnitude are characteris-tic of the interference between the direct field and the re-flected field. Rather than displaying the rapidly varyingphase of the field, we have chosen to display in Fig. 5(b) therelated quantity 0(r), referred to as the residual phase,16,25

which is defined as

0(r) = Plarg{p(r) -k0r), (92)

where Pi I denotes principal value. The residual phase ismore slowly varying than the total phase because the linearcontribution to the total phase is removed.

To perform the reconstruction, the acoustic field was sam-pled every 1.7 m, corresponding to a rate of approximatelyfour samples per water wavelength. The real part of thisfield was set to zero, and 1024 samples of the imaginary partwere retained. The real part was then reconstructed, andthe magnitude and the residual phase of the reconstructedfield are shown in Fig. 6. The reconstruction is excellent, ascan be seen by comparing Figs. 5 and 6. As further evidenceof the high quality of the reconstruction, the magnitude ofthe difference between the original real component and thereconstructed real component is plotted in Fig. 7 along withthe original field magnitude. From this figure it can be seenthat the error in the reconstruction occurs primarily in thenear field.

The degradation in the reconstructed near field can beattributed to two effects. First, the assumption that theacoustic field can be synthesized by using the Hilbert-Han-kel transform, i.e., using positive wave numbers only, is notstrictly valid at very short ranges. Second, the asymptoticexpressions, obtained from the asymptotic Hilbert-Hankeltransform, that are used in the reconstruction method arenot valid at short ranges. In the following two examples, wewill see that the reconstruction method also yields somedegradation in the near field. In many coherent processingapplications this error may not be significant, but we pointout that it is a limitation of the theory and method forreconstruction that we have proposed.

14-20 1

0

E

ICL

201

40 I

60

80

VELOCITY (m/sec)50 1500 1550 2200

I I Z I/cm

CO p = I g/cm3"A a= OdB/m at 220 Hz

c/\ WATER-BOTTOM INTERFACE

- \ p= 1.6 g/cm3

a= .0039 dB/m

_ _ __ __ C2 0_ 3SUe BOTTOM

p = 1.6 g/cm 3

a = .0039 dB/mC3= 2200 m/sec

1-

Fig. 8. Deep-water geoacoustic model. Additional parameters:zo = 124.9 m, z = 1.2 m, f = 220 Hz.

Wengrovitz et al.

i


(a) 1

-I.

Q.

0.

0

1

1

(b) 4

2

"0-'.

CI%-.

I--

-2 .

-4L0o

Wengrovitz et al.

500 1000

r (meters)Fig. 9. (a) Magnitude and (b) residual phase of realistic deep-water field.

As the second example, we consider a synthetic deep-water acoustic field, which consists of a direct field plus areflected field that has interacted with a horizontally strati-fied ocean bottom with a realistic sound velocity profile.The geoacoustic model for this example is summarized inFig. 8. The synthetic field was generated by first computingthe depth-dependent Green's function'6 and then comput-ing its numerical Hankel transform. The resulting reflectedfield was then added to the closed-form expression for thedirect field to obtain the total field. The magnitude and theresidual phase of this field are shown in Fig. 9.

To perform the reconstruction, the real part of the fieldwas set to zero, and 1024 samples of the imaginary part,which was sampled every 3.14 m, were retained. The realpart was reconstructed, and the corresponding magnitudeand residual phase are shown in Fig. 10. Comparing Figs. 9and 10, we see that the reconstructed field is very similar tothe original field. As further evidence of this, the error inthe reconstructed real component is plotted in Fig, II, alongwith the magnitude of the original field. The error signal issubstantially smaller than the original field magnitude forranges greater than about 25 m. The error in the recon-

0 500 1000 1500

0

1500


struction in the near field can again be attributed to the factthat the Hilbert-Hankel transform does not correctly syn-thesize the field at short ranges and also to the fact that theasymptotic expansion of the Hankel function is not valid atshort ranges.

As the final example, we will consider the reconstructionof an acoustic field collected in a shallow-water ocean experi-ment, conducted in Nantucket Sound in May 1984 by theWoods Hole Oceanographic Institution. 2 6 The experimen-

tal configuration is shown in Fig. 12. The procedure for

acquiring the acoustic data consisted of towing a harmonic

(a)

0 1

0.01

Q.. 0.001

0.0001

le-05

0

(b)

11__

I-,

_1..

500

source at a fixed depth away from the moored hydrophonereceivers over an aperture of approximately 1320 m. Thehydrophone receivers demodulated the harmonic pressuresignal and digitally recorded both the real and the imaginarycomponents of the spatial part of the field.

The magnitude and the unwrapped residual phase of theacoustic field recorded at the upper hydrophone are shownin Fig. 13. The interference pattern in the magnitude is dueto the constructive and destructive interference of severaltrapped and partially trapped modes in the shallow-waterwaveguide.26 The original acoustic field was sampled non-

1000 1500

0 500 1000

r (meters)Fig. 10. (a) Magnitude and (b) residual phase of reconstructed realistic deep-water field.

,...........................................................................

f\

1500

-11K.| -^ -

Wengrovitz et al.

........................................................................... ........................................................................... ................................................. ........................

........................................................................... .......................................................................... ..........................................................................

.................................................

................................................................................................................................................................................................................................

........................................................................... .......................................................................... ............................... ..........................................

I.....................................


1

S.401'.45.4

Wengrovitz et al.

0 500 1000 1500

r (meters)Fig. 11. Error in the reconstruction of the real component (bottom curve) and true field magnitude (top curve) for the realistic deep-waterfield.

Radar Ranging System

1))PRESSURERELEASESURFACEr

-- Synthetic Aperture--.

C z 1500 m/sec

p a 1.0 g/cm 5A

Anr~hnr -

/ / / / /_ / / / / / / / / l / 7 / / / -7/HORIZONTALLY STRATIFIED BOTTOM

Fig. 12. Configuration of the ocean experiment.

SURFACE BUOY

RECEIVER 2

Z2 a 7.1 m

RECEIVER 1

Zi u 12.5mI ---

/ / / / / / /7'/7

uniformly in range because of the difficulty in maintaining afixed spatial sampling interval in the ocean experiment.First the field was interpolated to a uniformly sampled gridbefore the reconstruction experiment.

To demonstrate the reconstruction algorithm, the realpart of this experimental field was set to zero, and 1024samples of the imaginary part were retained. The real partwas then reconstructed from the imaginary part, and thecorresponding magnitude and the unwrapped residual phase

of the reconstructed field are shown in Fig. 14. The similar-ity of the reconstructed field with the original and resampledfields is evident. In particular, the behavior of the un-wrapped residual phase is preserved, as are the peaks andnulls in the magnitude of the field. If the curves are exam-ined in further detail, it can be seen that some of the finerdetails differ. These slight differences occur not only atnear ranges, as was the case for the previous two examples,but at far ranges as well. These differences are also evident

m

SOURCE

Z a 6.1 m

f, * 140 Hzf 2 -220 Hz

. ._. . A,

Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 261Wengrovitz et al.

from examination of the error signal for this example, shown

in Fig. 15.There are several factors that may explain the reconstruc-

tion error, which is present not only at small values of range

but at large values of range as well. Theoretically, the re-

construction algorithm is based on the real-part-imaginary-part sufficiency condition that applies to outgoing fields. Inan ocean environment, the effects of velocity changes within

the water column, as well as violations of both the horizontal

stratification and radial symmetry, may yield acoustic fields

1

0.1

_ UI . 0.01Q.

0.001

0.0001

(b) 5

0

0_

- -

-10

-4=

0

that are not completely outgoing. Additionally, in shallowwater, the effects of a nonperfect surface may yield scatteredacoustic components that are not outgoing.

Nevertheless, the reconstructed experimental field isquite similar to the original field. In fact, the agreementbetween the curves suggests that this method might also beused as a measure of the quality of the experimental datacollected. In this approach, both components are recordedand resampled to a uniform grid. One component is then setto zero and reconstructed from the alternate component. A

500 1000 1500

15000 500 1000

r (meters)Fig. 13. (a) Magnitude and (b) unwrapped residual phase of the interpolated 140-Hz experimental shallow-water field.

(a)

_ I,;


(a) I

0.1

1 -0 . 01

0.001

0.0001

(b)

1-%

"0Ce1.4

-6

0

Wengrovitz et al.

500 1000

U 500

1500

1000 1500

r (meters)Fig. 14. (a) Magnitude and (b) unwrapped residual phase of the reconstructed 140-Hz experimental shallow-water field.

similar procedure can be performed with the alternate com-ponent. Presumably, if the reconstructed and originalfields differ substantially, there is an implication that effectssuch as errors in range registration, surface scattering, andmedium inhomogeneity cannot be treated as negligible.Such a procedure may also provide important informationabout the scattering properties of the medium as a functionof position. In numerical experiments, we have seen a corre-lation in the quality of the reconstruction with the accuracyof the ranging method established independently. Thus wesee that, in addition to providing a means for eliminating

hardware in the data acquisition system, the reconstructionmethod can provide an important consistency check on thequality of the sampled field.

5. SUMMARY

In this paper we developed the theory for one- and two-dimensional unilateral transforms. In one dimension, it wasshown that if the causal portion of an even signal can beapproximately synthesized by the unilateral inverse Fouriertransform, there are important consequences. Specifically,

3

0 . .

5O X \ .............................................

Z) '

4

-

_1


such a signal possesses an approximate real-part-imaginary-part sufficiency condition, as does its Fourier transform.Additionally, the approximation was shown to imply an in-

verse relationship between the unilateral inverse Fouriertransform and the unilateral Fourier transform as well asother relationships between the cosine and sine transformsof the real and imaginary components of the signal and itsFourier transform.

In extending the theory to two dimensions, we developed a

bilateral version of the Hankel transform and a unilateralversion, referred to as the Hilbert-Hankel transform. Itwas shown that if the causal portion of a circularly symmet-

ric signal can be approximately synthesized by the Hilbert-

Hankel transform, there are important consequences. Spe-cifically, such a signal possesses an approximate real-part-

imaginary-part sufficiency condition, as does its Hankel

transform. Additionally, the approximation implies an in-verse relationship between the Hilbert-Hankel transformand the complex Hankel transform as well as other relation-ships among the Jo, Yo, and Ho transforms of the real and

imaginary components of the signal and its Hankel trans-form. As was pointed out, the unilateral synthesis in twodimensions is particularly applicable to circularly symmet-ric fields that are outwardly propagating. The two-dimen-sional result led to the development of a reconstruction

algorithm for obtaining the real (or imaginary) componentfrom the imaginary (or real) component of a circularly sym-

metric field. This algorithm was applied to three examplesof underwater acoustic pressure fields.

Although our interest has been primarily in the applica-tion of the two-dimensional theory to the underwater acous-tics problem, the general nature of the theory suggests that itmay be applicable to other problems as well. For example,

0

0

0.001

0.000

1

one application of the one-dimensional theory might be tothe propagation of a wide-band acoustic pulse in a seismic

borehole in which the upgoing field is approximately synthe-

sized by the unilateral inverse Fourier transform. 27

Also, the development of the general multidimensionaltheory incorporating different symmetry conditions is sug-gested. This includes, for example, two-dimensional evensignals that can be approximated by the unilateral inverse

two-dimensional Fourier transform. Although we have de-veloped the theory for an approximate real-part-imaginary-part sufficiency condition, it is also possible to develop theanalogous theory for an approximate magnitude-phase suf-ficiency condition by applying the complex logarithm to theFourier transform.21 Essentially, in one dimension thereexists duality between the even signal and the even cepstrumand between the approximate real-part-imaginary-part suf-ficiency condition and the approximate magnitude-phasesufficiency condition. The duality suggests that there existsan approximate Hilbert transform relationship between themagnitude and the phase for a special class of mixed-phasesignals. The relationship between the unilateral transformand cepstrum may have some important consequences in

applications that include reconstruction of a signal from itsmagnitude only, phase unwrapping, and general homomor-phic signal processing.

APPENDIX A

To develop additional relationships between the cosine andsine transforms, we introduce the following notation:

CIF(co)) = I F(w)cos cotdco,7r 0

(Al)

15000 500 1000

r (meters)Fig. 15. Error in the reconstruction of the real component (bottom curve) and true field magnitude (top curve) for the experimental 140-Hz

shallow-water field.

Wengrovitz et al.

264 J. Opt. Soc. Am. A/Vol. 4, No. 1/January 1987 Wengrovitz et al.

StF(w)j-f F(c)sin tdw. (A2)

Note that since the signal f(t) is even, and thus its Fouriertransform F(w) is also even, Eqs. (13) and (15) in the text canbe written in terms of cosine transforms as

f(t) = 5f-1'F(w)j = CIF(w)J, (A3)

F(w) = WV(t)l = 27rCVf(t)j. (A4)

Additionally, the unilateral inverse Fourier transform andthe unilateral Fourier transform, in Eqs. (14) and (16) of thetext, can be written in terms of cosine and sine transforms as

fu(t) = S>-{F(c8 o)) = '/2C{F(w)j + j'/2 StF(w)j (AS)

and

F (w) = 5uff(t)1 = 7rCVf(t)I - j.rStf(t)j. (A6)

Two statements involving the relationships between the var-ious cosine and sine transforms are now made.

StatementAl If f(t) '-fu(t) fort > O then the cosine andsine transforms of the real and imaginary components ofF() are related, for t > 0, by

ClRe[F(w)] -- S{Im[F(o)Jf, (A7)

CfIm[F(w)]j - S{Re[F(w)]}. (A8)

0, in terms of either the real or the imaginary components off(t), as

(A15)

(A16

These relationships may be of importance if only onecomponent of the signal (or Fourier transform) is availableand it is desirable to determine the Fourier transform (orsignal). To obtain expressions (A13) and (A14), the factthat f(t) - fu(t), t > 0 implies that

f(t) - 9 '-1PF(&)J, t > 0is used. The right-hand side of this relation is expressed interms of cosine and sine transforms, as in Eq. (AS). Therelationships stated in the first part of statement Al are thensubstituted to derive expressions (A13) and (A14). To ob-tain the second pair of equations, we utilize statement 2.3,which relates the unilateral inverse Fourier transform andthe unilateral Fourier transform, to derive that

The right-hand side of this relation is then expressed interms of cosine and sine transforms, using Eq. (A6). Therelationships stated in the second part of statement Al arethen substituted to derive expressions (A15) and (A16).

Additionally, the cosine and sine transforms of the real andimaginary components of f(t) are related, for co > 0, by

CJRe[f(t)]J - SfImVf(t)]J, (A9)

C{ImVf(t)]J - -S{ReV[(t)] }- (A10)

To derive the first pair of equations, the fact that f(t) -fu(t), t > 0 implies that

r-'{F(w)}) - r-1{F(w)J, t > 0 (All)is used. Substituting Eq. (A3) into the left-hand portion ofthe expression and Eq. (A5) into the right-hand portion ofthe expression and equating real and imaginary parts onboth sides yields the first pair of equations. To derive thesecond pair of equations, we utilize statement 2.3, whichrelates the unilateral inverse Fourier transform and the uni-lateral Fourier transform, to derive that

(A12)

Substituting Eq. (A4) into the left-hand portion of the ex-pression and Eq. (6) into the right-hand portion of the ex-pression and equating real and imaginary parts on both sidesyields the second pair of equations in the statement.

Another interesting consequence of the validity of theunilateral synthesis of f(t) for t > 0 is presented in thefollowing statement.

Statement A2 If f(t) - fu(t) for t > 0, then f(t) can beapproximately synthesized, for t > 0, in terms of either thereal or the imaginary components of F(w), as

f(t) - 251,j'JRe[F(co)]J,

f(t) - 2jW,,_1JIm[F(w)]}.

(A13)

(A14)

APPENDIX B

To develop additional relationships among the Jo, Yo, andHo transforms, we write the Hilbert-Hankel transform andthe complex Hankel transform as

p>(r) = WHu71Jg(kd)1 = '/2J0Jg(k,)J + j'/2Y0 Jg(kd)1 (Bi)and

g (kr) = W1fup(r)j = 1/2 JO{p(r)} - j'/2 HOtp(r)j. (B2)

Two statements involving the relationships among the Jo,Y0, and Ho transforms are now made.

Statement Bi If p(r) -'pu(r) for r> O then the JO and Yotransforms of the real and imaginary components of g(kr)are related for r > 0 by

J0Re [g(k,)]} -- Y 0{Im[g(k,)JJ,

JofIm[g(kd) '- YofRe[g(kd)]j.

(B3)

(B4)

Additionally, the JO and Ho transforms of the real andimaginary components of p(r) are related, for kr > 0, by

J0IRe[p(r)]f - H0fIm[p(r)]J,

JofIm[p(r)]f -- HotRe[p(r)]I.

(B5)

(B36)

To justify the first part of this statement, the conditionp(r) - pu(r), r > 0 is written as

Jog(k,)} - '12Jolg(kr)} I+ j'/ 2Y 0tg(kd)j, r > 0, (B7)so that

J01g(kr)) '-iYofg(k,)J, r > 0.If the real and imaginary parts on both sides of this expres-

Additionally, F() can be approximately analyzed, for w >

F(w) - 2,7,JReV(t)JJ,

F(w) - 2j51.JImV(t)]J.

(A17)

F(w) - 51,,V(t)), w > o. (A18)

511A01 - 9"W01, W > 0.

(B8)


sion are equated, the first pair of expressions in the state-ment are obtained. To derive the second pair of expres-sions, we use statement 3.3, which relates the Hilbert-Han-kel transform and the complex Hankel transform, to derivethat

Michael S. Wengrovitz was with the Massachusetts Insti-tute of Technology/Woods Hole Oceanographic InstitutionJoint Program, Woods Hole, Massachusetts 02543. He isnow with Atlantic Aerospace Electronics Corporation, Lex-ington, Massachusetts 02173.

(B9)J,)Ip(r)} - Wj{p(r)), kr > O.

Using Eq. (B2), this expression becomes

Jst g(kh) I -t/2JOtp(r)I- iHotp(r) ,so that

kr > 0, (B10)

JQfp(r)f -- jHl0 p(r)), kr > 0. (B11)

Equating the real and imaginary parts on both sides of thisexpression yields the second pair of expressions.

An additional consequence of the validity of the unilateralsynthesis of p(r) for r > 0 is summarized in the followingstatement.

Statement B2 If p(r) - p(r) for r > 0, then p(r) can beapproximately synthesized, for r > 0, in terms of either thereal or the imaginary components of g(kr), as

p(r) - 2WjU'IRe[g(kr)]I, (B12)

p(r) - 2jW,,-11Im[g(kr)]}. (B13)

Additionally, g(kr) can be approximately analyzed, for kr >0, in terms of either the real or the imaginary components ofp(r), as

g(k,) - 2JVJRe[p(r)]} (B14)

g(kh) - 2jY6uIm[p(r)]}. (B15)

To develop the first pair of expressions, we use the factthat

p(r) - /2JO1g(kr)) + j1/2Y01g(kr)1, r > 0. (B16)

If expressions (B3) and (B4) are substituted into the right-hand side of this expression, the first pair of expressions isobtained. To derive the second pair, we use statement 3.3,which relates the Hilbert-Hankel transform and the com-plex Hankel transform, to derive that

g(kr) - Wjfp(r)f, kh > 0. (17)

If expressions (B5) and (B6) are substituted into the right-hand side of this expression, the second pair of expressions isobtained.

ACKNOWLEDGMENTS

This work has been supported in part by the AdvancedResearch Projects Agency monitored by Office of Naval Re-search under contract N00014-81-K-0742 NR-049-506 at theMassachusetts Institute of Technology and contractN00014-82-C-0152 at Woods Hole Oceanographic Institu-tion and in part by the National Science Foundation undergrant ECS-8407285. M. Wengrovitz gratefully acknowl-edges the support of the Fannie and John Hertz Foundation.This paper is Woods Hole Oceanographic Institution Con-tribution number 6175.

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Documents

Reconstruction of complex-valued propagating wave fields using the Hilbert-Hankel transform