Physical aspects of the hartley transform

Physical aspects of the Hartley transform R. N. BRACEWELL Space, Telecommunications

and Radioscience

Laboratory,

Stanford.

California

94305, U.S.A.

Abstract-The close links between Fourier analysis and physics are shared by the Hartley transform which, as its implications are worked out, will offer alternative approaches. A distinguishing feature is that the Hartley transform represents real data by real transform values, rather than complex, a feature that carries over into optical interferometry. Some objects however are characterized by phase as well as brightness. In such cases the Hartley transform acquires an imaginary part while the Fourier transform loses its Hermitian property. The interrelation between the two complex planes suggests an instrumental means for object phase determination starting from only amplitude information in the transform domain.

1. INTRODUCTION

connections between the Hartley transform and physics and promise that other connections will follow. This contribution interprets the optical and microwave experiments in terms of the complex plane of the Fourier transform. The Hartley transform, when real, is the projection on an axis at -45 degrees. There is a second projection at +45 degrees, and this construction enables us to see why it is that, very often but not always, the single real projection on its own suffices for our needs. This, of course, is in contrast with Fourier analysis, where complex coefficients are the rule rather than the exception. Nevertheless, the one-to-one correspondence makes the two transforms fully equivalent. It is only an accident of history, by now quite irreversible, that Fourier’s original choice of cosx and sinx as basis functions, rather than cosxfsinx and cosx-sin x, led later to complex transform values for the representation of real data. Following Hartley, we use the notation

HARTLEY (1942) originally introduced his transform as a mathematical entity, distinguished by the two properties of being real and of having an inverse transformation that is the same as the direct transformation. These two properties account for the computational success achieved by the discrete Hartley Transform when it later became the basis of a fast algorithm (BRACEWELL, 1986) that challenged the speed of the Fast Fourier Transform algorithm. But, computing aside, Fourier analysis has had a long and honorable association with physics, memorialized by Thompson and Tait’s view that “Fourier’s theorem is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics”. Examples from atmospheric physics would include the elegant representation of diffracted waves by BOOKERand CLEMMOW (1950) as an angular spectrum, the application cas x = cos x+ sin x. of the spatial generalization of the notion of spectrum to the scattering of waves by the aurora and the The complementary function cos x - sin x is cas’ x, the ionosphere (BOOKER, RATCLIFFE and SHINN, 1950; derivative of cas x, just as cos x is the derivative of BOOKER, 1956; RATCLIFFE, 1956) and the elegant sinx. But, in addition, the complementary function theory of incoherent back-scatter from the ionosphere cosx-sin x is expressible as cas( -x). This symmetry, (FEJER, 1960). Because of the intimate connection not shared by cosx and sin x, turns out to be between Fourier analysis and the world of physics significant. there is a presumption (DUHAMELand VETTERLI,1987) that the Hartley transform might represent merely a computational trick without the deeper physical 2. NOTATION associations of the Fourier transform. This has turned For continuity with the dominant trend in the literaout not to be the case. The Hartley transformation has now been implemented with light (VILLASENOR ture of Fourier analysis the following notation will be H(s) of a given and BRACEWELL, 1987) and microwaves (VILLASENOR adopted for the Hartley transform functionf(x) : and BRACEWELL,1988). developments which establish 791

R. N. BRACEWELL

792

f(x)

cas 2asx dx.

Then the inverse transformation H(s) is given is

that yields f(x) when

0, f(x)=

x < 0

l-(x-I)“,

o 2

i 0, whose Fourier transform is

.f(x) = = H(s) cas 2nsx ds. s -% As may be seen, the absence of a complex exponential kernel means that H(s) will be real if ,f’(x) is real. In addition the transformation is seen to be perfectly reversible, not being concerned with reversal of the sign of i that is obligatory with inversion of the Fourier transform. 3.COMPLEX

RELATIONSHIP

TO THE FOURIER

TRANSFORM

One may express the Fourier transform F(s) of a function f(x) by graphing the real and imaginary parts separately as functions of the transform variable s (which often represents frequency, or spatial frequency) but it is equally possible to graph F(S) on the complex plane. This sort of representation is familiar from the Cornu spiral and the related diagrams commonly used for discussing diffraction. Let F(S) = R(s) +2(s). The example of Fig. 1 uses the particular function

F(s) = R(.s)+iZ(s) =

cos 2ns sin 271s - cos 27l.r L c 271.s > sin 3~s sin 2ns _+_i_~ . ?I:s2 cos 2ns - ---~27ls > C nlrZ

The real and imaginary parts R(s) and I(s) are shown at the top right and the corresponding complex locus is shown in the upper left corner. The locus shows the Hermitian property, characteristic of functions f(x) that are real, that reversing the sign of s converts F(S) to its complex conjugate ; thus the locus is symmetrical about the real axis. Now introduce a coordinate system (R’, Z’) which is rotated throughout -45 degrees with respect to (R,I). We can show that the projection of the complex-plane locus onto the P-axis yields l/J2 times the Hartley transform H(S). This is because the Hartley transform of a real function is given by R(s) -Z(s), which is proportional to the projection of F(z) onto the K-axis. The factor fi arises because by definition 2

R(s)

f- R(s)

Fig. 1. The complex Fourier locus R(s) + if(s) (top left), the real and imaginary parts versus s (top right) and the Hartley transfo~ H(s) constructed by projection onto the R’-axis (bottom) for a selected real function .f(~).

Physical

aspects of the Hartley

transform

L-P

T

Fig. 2. Buneman’s construction for the Hartley transform, which associates the real line segment OH with the complex Fourier transform value F(s) = R(s) + if(s).

the cas function does not have unit ~mplitnde, as do cos and sin, but has an amplitude J2. The conclusion from this construction is that there is a complex (R’,i’)-plane, rotated by -45 degrees with respect to the complex Fourier plane, and defined by the transformation R’(s) + Z(s) = [R(s) + U(s)] x ern’4 with the property that the Hartley transform of any real function ,f(x) is distributed along the K-axis. For the example in hand, this Hartley transform is shown in Fig. 1 extended along a diagonal s-axis which is a prolongation of the negative I’-axis. The even and odd symmetries possessed by R(s) and i(s) are not present ; indeed it is the absence of redundancy associated with symmetry that permits a full representation of f(x) to be condensed onto one real line. In this example, the projection of the locus onto the r-axis gives no additional information about the chosen Y(X); because of symmetry about the R(s)axis the second projection is simply the reverse of the first as can be seen from Fig. 1. By contrast, both projections of the complex Fourier locus are essential.

In the event that the original function S(.X) were complex, then the information content would be doubled and both projections would be required on the compiex Hartley plane also. The complex-plane relationship described above can be expressed in a fully equivalent way by a construction due to Professor Oscar Buneman. In Fig. 2 the real distance OH directly represents the Hartley transform, free from any factor $, while OH’ represents the complementary Hartley transform (the transform with the cas( -x) kernel, or the projection onto the P-axis).

Fig. 3. This optical Hartley transformer, having a cube corner C in one arm of a Michelson interferometer, operates on an input transparency T that imposes phase and amplitude variations on an incident wavefront. Path inequality of the arms is introduced by axial adjustment of mirror h4, and in the switchable case the mirror is given a reciprocating location. Amplitude information alone, dist~buted over the output P, suffices to determine the input.

lens is also one focal length then the phase distribution over the Fraunhofer plane [the (u,v)-plane] will be such that the field distribution there is just the twodimensional Fourier transform T(u, v) of t(x.~). While the electric field in the transform plane is strictly speaking real nevertheless the conventional phasor representation is necessarily complex, even if the transmission factor f(x,y) is real, because fields at different points (u,v) do not have to be in the same phase. One consequence of the complex character of the Fourier transform is that a record made with a photographic plate in the transform plane is inadequate for recovering t(x,_~) because the photographic emulsion responds to the square of electric field and ignores that part of the information carried by phase. This circumstance makes the idea of a two-dimensional optical Hartley transform plane interesting because the whole of the information about a real transmission factor t(x,y) is encoded in real form, not requiring phase at all. The key to constructing a Hartley plane is to evolve a spatial analogue to the 45 degree projection discussed earlier. The procedure is to split the beam emerging from the transparency into two beams, rotate these two beams by 180 degrees relative to each other about their optical axes and to recombine them with a relative phase shift of 90 degrees. This prescription is summarized in the easily verified relation (BRACEWELLet al., 1985) ein,4 H(u,c)

=

If a plane transparency 7’(Fig. 3) with transmission factor t(x, y) is illuminated at normal incidence by a

uniform plane wave then the Fraunhofer diffraction pattern can be obtained in the front focal plane of a lens L. If the distance from the transparency to the

[F(~,ti)+e~‘“‘*F(

-u,

-r)].

J2

The unimportant 4. OPTICAL FOURIER AND HARTLEY TRANSFORMS

-;-

phase factor exp (i7c/4) results from of time should have

the choice that a cosine function

zero phase. Many optical embodiments can be imagined. The first one to be implements DILUTOR and BRACEWELL. 1987) utilized the beamsplitter technique of the Michelson interferometer and the image inversion property of the Galilean telescope for the 180 degree

194

R. N.

BRACEWELL

rotation. Figure 3 illustrates a modification in which the image rotation is performed with a cube corner C functioning as one mirror and M as the other. The beamsplitter is B and P is the output plane. In this configuration the 90 degree phase shift could be obtained from axial position adjustment of one of the interferometer mirrors. An interesting question of sign determination arises which can be handled in more than one way, for example by adding a perturbation. When the perturbation is a reference beam of finite strength the optical Hartley transformer produces a new sort of hologram. Although the optical configurations suggested by the search for a Hartley transform plane are simple they are novel and may have a variety of applications. The apparatus was found in practice to be tolerant to adjustment errors (VILLASENOR,1989). One category of applications would be in wavelength ranges other than optical, for example microwaves. There is a close resemblance between optical and microwave elements but because of the great wavelength ratio the details of a Michelson interferometer are necessarily different. The first version of a microwave Hartley transformer substituted a corner reflector for the Galilean telescope and used a suitably proportioned sheet of plastic as a beamsplitter (VILLASENORand BRACEWELL, 1988). Both the optical and microwave demonstrations represent instruments that are capable of measuring electromagnetic phase by means of square-law detectors. In the microwave range, where it is possible to bring a phase reference signal to the measuring point by coaxial cable, phase measurement does not present great difficulty, but provision of an alternative phasemeasuring procedure in optics could be more interesting. Finally, in the X-ray domain, where squarelaw detection is virtually obligatory, any new way of phase determination would be of great interest. However, the transmission factor t(-u,y) referred to the plane of emergence of X-rays scattered from a crystalline structure under study cannot be thought of as real. To discuss the situation one must take into account the r-axis of the (R’, I’)-plane.

5. THE COMPLEX HARTLEY PLANE All the machinery for considering a complex input function f(x) or t(x,y) has already been introduced.

The only new element is that the projection on the raxis is now no longer to be neglected. To acquire the full information implied by complex input we might begin by recording in the Hartley plane with the apparatus previously described ; but information theory tells us that in addition a second recording would be needed. This could take the form of a second record in the same plane but with the 90 degree phase shift reversed. In other words projections on both the R’- and r-axes would be taken. When the optical Hartley apparatus was experimented with it could be brought into absolute adjustment by noting the position of interference fringes due to a test object in the form of an off-axis point source. In the scheme just introduced absolute adjustment would be circumvented ; it would be necessary only to make two records corresponding to a switchable phase shift within the instrument. In the optical arrangement this would simply mean advancing a mirror through an exact fraction of a wavelength using a screw or a piezoelectric support. At X-ray wavelengths the small distances involved would require a different approach. Split-beam interferometers have been demonstrated with X-rays using monolithic construction in silicon and so in principle a Hartley plane could be constructed. To readjust the instrument that did this by a fractional wavelength change in path difference between the arms might be possible by straining the silicon substrate, mechanically, thermally, or by use of piezoelectric or magnetostrictive deformation.

6. CONCLUSION Clearly the Hartley transform has turned out to be more than a mathematical curiosity. The idea has led to novel laboratory apparatus, one of whose applications is the determination of phase from intensity measurements with square-law detectors. Extensions to other wavelength ranges may be expected. The full equivalence with the Fourier transform that has been established in this paper indicates that every process describable by the Fourier transform possesses an alternative decription in terms of the Hartley transform. Thus the physical significance of the Fourier transform is shared equally with the Hartley transform, contrary to intuitive opinion. This work was supported by the Office of Naval Research under contract N00014-85-K-0544.

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Physical

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transform

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1986 1985

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