Aspects of the Hartely Transform

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Aspects of the Hartley Transform

RONALD N. BRACEWELL, LIFE FELLOW, IEEE

Invited Paper

Known

in

the signal processing literature mainly as a compu-

tational technique, the Hartley transform has proven to possess

attributes in the physical wor ld of experiment, especially in optics

and microwaves. The Fourier transform, which has constituted

the standard approach to spectral analysis, is understood both

in the domain of numerical analysis and in the world of natural

phen omen a, gaining its cen tral significance from the ubiquity

of sinusoidal natural behavior; it also gains appeal from the

convenience offered by complex variable analysis, which is a

basic part of technical edu cation. Computers, however, prefer real

numbers, an attribute that first ope ned a niche fo r the Hartley

transform.

A s

applications have multiplied, and others open up,

it

will be helpful to understand the Hartley idea from more than

one point of yiew. Several topics, including multidimensional

transforms, the complex generalization, and microwave phase

measurement by amplitude measurement only, are discussed with

the intention of throwing light on the numerical and physical

relationships between the Hartley and Fourier transforms.

I. INTRODUCTION

When Hartley’s integrals [11were first adapted for use on

a computer [2],

[3]

the Hartley transform was thought of, by

analogy with the Fourier transform, as providing a basis for

a discrete transform with application purely in the numerical

world. But the Fourier transform, apart from its numerical

side, was already known for its wide applicability in the

world of physics. Once you understand the mathematical

aspect of the transform and, in addition, one application to

some branch of physics, then you can translate problems

from a field such as mechanical vibration or acoustics with

which you may be not too familiar into the field with which

you are familiar, say waveforms and their spectra. The

problem then becomes a different one but one that perhaps

you can solve because of your familiarity with the field;

you then translate the solution back into the field of origin.

One is able to apply this technique when the underlying

mathematical theory is the same; and it so happens in the

physical world that there is a good deal of overlap of theory

between various fields when linearity and time invariance,

or space invariance, coexist. This combination of properties

is met, to greater or lesser accuracy, in

so

many everyday

The author is with the Space, Telecommunicationsand Radioscience

Manuscript received October

8, 1993.

Laboratory, Stanford University, Stanford CA94305.

IEEE

Log

Number 9215541.

problems, that Thompson and Tait in their

Treatise

on

Natural

Philosophy

state that “Fourier’s theorem is not

only one of the most beautiful results of modem analysis,

but may be said to furnish an indispensable instrument in

the treatment of nearly every recondite question in modem

physics.” William Thompson, or Lord Kelvin as he later

became, was one of the first people in Britain to read

Fourier’s work and was no doubt influential in bringing

attention in Britain to what was ostensibly a treatise on the

theory of heat. I mention this background to recall the close

identification of Fourier’s theory with heat conduction first

of all, and then with nearly every other branch of physics.

Now in the case of the Hartley transform there was a

presumption that the physical connection, if any, would

exist via a conversion to the Fourier transform. There were

reasons for thinking that the Hartley transform was less

fundamental in some way than the Fourier transform; for

example, the ear hears acoustic power spectra, expressible,

to the extent that physiological sensations are expressible

mathematically, by the squared modulus of the Fourier

transform of the acoustic pressure waveform. One does

not hear Hartley transforms. Likewise, a lens spaced at

one focal length between a plane source and a screen

generates an optical intensity distribution on the screen

that is the squared modulus of the Fourier transform of

the light amplitude distribution over the source. The Fraun-

hofer diffraction pattern was well known from the time

of Rayleigh and Michelson to have this Fourier transform

relationship. Explicit opinions that “the Hartley transform

has no physical significance” were expressed in print [4]

Nature certainly prefers sine waves. If you shake a

mechanical structure sinusoidally the response in some

other part of the structure will also be sinusoidal, unless you

shake so hard as to exceed the range of linear deflection.

A

second condition is that the constants of the structure must

not alter as time elapses.) If you shake the structure with a

force varying as a Walsh function

of

time the response at

some other location will certainly not be a Walsh function;

the same is the case for Bessel function inputs, or any other

input function except sinusoidal. It is because sine waves

have this special standing in nature that Fourier analysis

turned out to be so widely useful.

0018-9219/94$04.00 1994 IEEE

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82, NO.

. MARCH 1994

38 1



As the Hartley kemel, or cas function, is itself sinusoidal

it is to be expected that the physical applicability of the

Hartley analysis will be correspondingly wide. R. V. L.

Hartley (1890-1970) introduced the cas function [11, saying

that the name is an abbreviation of cosine and sine. After

fifty years of obscurity the cas function

cas t = cos t + sin t

has become widely known through more than one hundred

papers stimulated by one paper in 1983. One might have

thought that this publication in the Journal of the Optical

Society

of

America

[2] would prove just as obscure as

Hartley’s paper in the

Proceedings

of

the Institute ofRadio

Engineers [11; the change in the computational environment

in the interim made the difference.

Programming details are not a main concem of this

paper but some incidental impact on computational spectral

analysis occurred as a consequence of development of the

fast Hartley program. Permutation is a step which replaces

the index of an array by its bit-reversed value. Timing

analysis [3], [5], which is fundamental but less popular than

complexity analysis, revealed that T programs in com-

mon use were penalized by slow permutation [3]. Attention

to this topic resulted in substantial improvement [6]-[8].

Another incidental segment of code pretabulates needed

trigonometric functions; a fast way of avoiding repetitive

fetches was evolved by Buneman [9] in connection with

the installation of a fast Hartley routine on the CRAY-2 for

use by plasma scientists dependent on supercomputers.

In the present paper the relationship between the Fourier

and Hartley transforms is presented in Section I in terms

of a resolution of the redundancy possessed by the Fourier

transform, the property characterized by the term Her-

mitian. In Section 111 the complementary cas function is

introduced and in Section IV a mathematical interpretation

on the complex plane is described. Section V discusses

the microwave embodiment of the Hartley transform and

the interesting possibility opened up whereby amplitude

measurements alone permit the measurement of phase,

a desirable consideration in laboratory study of antenna

models. Computing multidimensional Hartley transforms,

which is explained in Section VI, offers another interesting

window on the relationships with Fourier practice.

11.

REDUNDANCY

IN THE

FOURIER RANSFORM

Since the Fourier transform is by definition complex,

while data can usually be counted on to come in real

form, there is a fundamental asymmetry between the data

domain and the Fourier transform domain. This feature is

evidenced by the Hermitian property according to which,

if you reverse the function, you conjugate the Fourier

transform of that function. This is only true if the function

is real, which is so often the case that we just accept

the Hermitian property as one of the consequences of

mathematical theory. For example, a waveform

f(t )

=

te-tH(t)

where H(t) s Heaviside’s unit step function, has a Fourier

transform

If we reverse the waveform to get -t exp( t)H( -t ), then

the real part of the Fourier transform remains the same

but the imaginary part changes sign. Thus F ( s ) becomes

What this implies is that the whole of the Fourier

transform is not needed in order to fully specify the original

real function. But that is not the message conveyed by the

inversion integral, which says integrate from minus infinity

to infinity. The fact is that if we are given F(f ) rom zero

frequency to infinity then we do not need to be given the

values for the other semi-infinite range from minus infinity

to zero frequency, because they are deducible from the

Hermitian property that F(-f) s the complex conjugate

The corresponding thing applies in two dimensions (or

more): the two-dimensional Fourier transform F ( u , I) f

given in two adjacent quadrants suffices to determine the

values in the remaining half plane.

In a way this is an advantage flowing from the complex

nature of the Fourier transform; the two values, one real

and one imaginary, associated with each value of the inde-

pendent variable, allow an infinite amount of information

to be packaged into a semi-infinite range. A more cautious

wording of this thought is facilitated by the finite situation

that we deal with in computing with

N

data values.

By the same token, the fact that the Hartley transfor-

mation of real data results in real values requires that one

knows the Hartley transform over the full infinite range in

order to have the function fully specified. The advantages

of this representation are that the Hartley transform is not

redundant and that it is correspondingly real, just what we

want for numerical computing.

Looking back on the choice which was made in favor

of the familiar Fourier approach long ago we see an

interesting thing. Fourier himself introduced his theory in

terms of analysis into sines and cosines. He also developed

his theory in a physical way by thinking in terms of a

temperature distribution on a conducting bar which could

be bent into a circular arc of indefinitely great radius and

closed on itself, thus rendering any initial heat distribution,

no matter how extensive, periodic. This permitted analysis

into a trigonometric series of harmonically related terms,

leading to a set of coefficients

a,

for cosines and a second

set b for sines. Both of these coefficients of course are

real, and may be considered as separate lists, as indeed

they are for the purposes of numerical harmonic analysis.

By packing them into one complex coefficient a, - b,,

we gained an important advantage, namely, the ability to

bring to bear our theoretical ability to think in terms of

complex analysis, which has been a staple of mathematical

education throughout the century. However, the theory of

the Fourier integral, as distinct from the Fourier series for

a periodic function, has been developed entirely in terms of

(1

-

27Tf)-2.

of F ( f ) .

382

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MARCH 1994



the complex variable and not in terms of the real cosine and

sine transforms into which the lists

a,

and

b,

generalize as

the radius of the closed arc approaches infinity.

Ironically, the conversion to complex thinking disen-

gaged the theory from the physical world of oscillations. An

oscillation a, cos 27rnt

+ b,

sin 27rnt is perfectly real and

physically measurable; the “oscillation”

F ( )

xp ( j 2 7 r f t )

is fully equivalent but is a mental mathematical construct

that takes advantage of the beauty of complex algebra. N o

one suggests that the square root of minus one is a physical

entity or that complex numbers exist in-the physical world.

Even in quantum mechanics the wavefunctions

$

that are

complex are carefully distinguished from observables such

as $$*.

What the Hartley transform does is to package the sine

and cosine transforms in a new way, that does not appeal to

complex algebra, and indeed does not simplify theoretical

thinking, but does simplify instruction to the computer,

machine or human. This package, call it

H ( f ) ,

s the linear

superposition of the real part of the Fourier transform and

the negative imaginary part. Thus

or

explicitly

~ ( f )

/

f ( t ) as

27rf t

d t

--M

where cas t = cost

+

sint.

It seems puzzling to some that both the cosine transform

and the sine transform are needed to analyze a given

function whereas one cas function transform suffices; after

all, the cas function is only a cosine function shifted

45’ in phase. An explanation that may appeal to such

a person is provided by pointing out that, as frequency

changes, a cosine function remains an even function, while

a cas function continues to sense both an odd and an

even component, continually adjusting itself as frequency

changes to ensure that the two components are kept in

quadrature, with equal magnitudes.

111.

THE

COMPLEMENTARYas FUNCTION

It has been mentioned that two simulated functions

are required in Fourier analysis, both the cosine and the

sine, while in Hartley analysis the cas function alone

seems to suffice. But there is indeed another function

complementary to the cas function; the fact that we can

get along without it comes from the frequency with which

we work with real data. Fourier analysis, by contrast, even

when applied to real data, does not escape the need for

a pair of basis functions, essentially because the Fourier

kemel exp

( - j 2 7 r f t )

is by definition complex. Ordinarily

we do not notice that the one complex kemel stands for

two things; only when we switch from algebra to numerical

implementation does this two-fold structure obtrude.

One way of arriving at the complement of the cas

function is to note that cosine is the derivative of sine.

Let us therefore define the complementary cas function as

the derivative of cast, namely, cas’t. We see that this is

given by

cas’ t = cos t

-

sin

t .

But this is just another cas function, namely, cas

(-t).

Just

as cosine and sine are related by a shift,

so

the cas function

and its complement are related by mirror reflection about

the origin of the independent variable. A special name for

the complementary cas function is therefore not usually

needed.

One does, however, need the complementary cas function

in order to analyze given functions that are complex. Data

of this type can occur occasionally, even though measured

data will normally be regarded as real, and in fact the

owner can always elect to treat data as real. Conversely,

the owner can elect to treat data as complex, superposing

a mathematical or mental operation upon the data for a

desired purpose.

IV. THE COMPLEX ARTLEY RANSFORM

At first sight there would appear to be no need for a

complex Hartley transform because the favorable properties

of Hartley analysis relative to Fourier analysis arise from

the simplified way in which Hartley analysis treats real

data. This advantage will be lost when complex data are

encountered and the two sorts of analysis should become

equivalent. However, an interesting way of looking at Hart-

ley analysis emerges, which has utility in two-dimensional

applications.

The real part of the complex Hartley transform will be

by definition the ordinary Hartley transform of the real part

of the given complex function. The imaginary part of the

complex Hartley transform will be the complementary Hart-

ley transform of the imaginary part of the given complex

function. By complementary Hartley transform of g t ) I

mean

g(t)(cos27rft

-

sin27rft) d t .

1,

Thus the complex Hartley transform H,( t ) of a complex

function

g t ) = ~ ( t ) i ( t )

is

3

H, ( t )

=

~ ( t )

as 27rft d t

1,

+

/ i t) (cos 2.rrft - sin 27rft) d t .

--M

If the given function has no imaginary part [ i t) = 01, then

H , ( t ) reduces to the ordinary Hartley transform.

Of course a complex function g t ) has an ordinary Hart-

ley transform, which is itself complex, but the definition

of

H,( t )

given above differs by the presence of the minus

sign in the second integral.

To discuss the complex transforms it will be convenient

to deal in terms of parametric representation on the complex

plane, a technique familiar from the complex-plane loci of

linear system theory and from the Comu spiral and similar

BRACEWELL: ASPECTS

OF

THE HARTLEY TRANSFORM

383



,I(, 4;

2

2

Fig.

1.

Parametric representation of F ( f )=

R(f)

Z f )

on

the complex plane (upper left). The real and maginary parts of the

Fourier transform for any valu e off are the components measured

on

the

R-

and I-axes. The Hartley transform is the component

measured

on

the R'-axis (bottom right).

diagrams used to discuss optical diffraction gratings and

apertures.

To

illustrate the basic idea geometrically, consider the

special waveform

t O

f ( t ) =

l - t - l ) 2 ,

O < t < 2 ,

I t > 2

whose Fourier transform is

F (

f)

=

R(

f)

j1

cos 27rf

sin 27r f sin 27rf

+j-

(cos

27rf-

~

7 r 2 f 2

27rf

This example has been used previously [lo] for a similar

discussion.

The real and imaginary parts

R ( f )

nd

1 f)

re shown

at the top right of Fig. 1. The corresponding complex locus

is shown in the upper left comer. The locus shows the

Hermitian property, characteristic of functions f z)hat

are real, that reversing the sign of f converts F ( f ) o its

complex conjugate; thus the locus is symmetrical about the

vertical axis. Note that the real axis is plotted vertically.

A new coordinate system (R ,1 ) has been introduced

which is rotated throughout

-45

with respect to (R , ). It

can be shown that the component of the complex-plane

locus along the R'-axis yields 1/f i times the Hartley

transform

H ( f ) .

This is because the Hartley transform of a

real function is given by

R ( f ) - I ( f ) ,

which is proportional

to the component of

F ( f )

long the R'-axis. The factor fi

arises because by definition the cas function has amplitude

fi

ather than unit amplitude as for cos and sin.

The $onclusion from this construction is that there is a

complex (R , ')-plane, rotated by -45 with respect to the

complex Fourier plane, and defined by the transformation

~ ' ( f ) j ~ ( f )[ ~ ( f ) 1 ( f ) ] ejnI4 .

384

H

H

Fig. 2.

Given a point F(f)

R(f)

+ I ( f )

on

the com-

plex Fourier plane, the real line segment OH gives

the

Hartley

transform and OH gives the complementary Hartley transform.

This construction is due to 0: uneman. If I ( f ) were negative, H

would fall where

H

is sholm.

The Hartley transform of any real function

f x)

will be

found distributed along the R'-axis. For the example in

hand, this Hartley transform is shown in Fig. 1 extended

along a diagonal f-axis which is a prolongation of the

negative 1'-axis. The even and odd symmetries possessed

by R ( f ) and 1( f ) are not present in H ( f ) ; ndeed, it is

the absence of redundancy associated with symmetry that

permits a full representation of f(z)o be condensed onto

one real line.

In this example, the component of a locus point along

the I'-axis gives no additional information about the chosen

f(z);ecause of symmetry about the R(f )-axis the second

projection is simply the reverse of the first as can be seen

from Fig.

1.

By contrast, both components of the complex

Fourier locus are essential. In the more general case where

the original function

f(z)

s complex, then the information

content is doubled and components are required along both

axes of the complex Hartley plane.

A construction due to Professor Oscar Buneman ex-

presses the complex-plane relationship described above

in a fully equivalent way. In Fig.

2,

the real distance

O H directly represents the Hartley transform, free from

any factor fi hile OH represents the complementary

Hartley transform as the component along the I'-axis of

Fig. 1.

In a way it seems an accident that the cas function kernel

proved to be

so

special in trapping the essentials of a real

function, not requiring

a

complementary basis function

to

do so.

On

second thoughts, the view from 45 is not critical;

40

would

be

just as effective

[2].

If instead of using

fi in

( t

+

7r/4)

= cas t as a kemel we use fi in

( t+T)

no information will

be

lost unless

T = 0 , 7 r / 2 , . .

The

inversion kemel will be

i

in

t

+&cos t . What is

an accident is that it is precisely the Fourier axis orientation

on the complex plane that fails to avoid the necessity for

a pair of basis functions. More general kernels have been

studied by Evans [111.

v.

A MICROWAVEEALIZATION

As

mentioned earlier, diffraction pattems, including the

far-field radiation pattems of antennas, are representable

by the Fourier transform, which is a simple expression of

Huygens' principle. Each element

of

a radiating surface

contributes by linear superposition to the distant radiation

field at some point with a phase factor appropriate to the

path length from the element to that point. The phase

PROCEEDINGS OF THE IEEE, VOL. 82, NO.

3,

MARCH 1994



factor, of the form exp [-j27r(uz

+

wy ], where U and

'U

are direction cosines, is precisely the Fourier kemel.

When an antenna radiation pattem is measured, or an

optical diffraction pattem, special arrangements are neces-

sary if phase is to be determined. The ordinary square-law

detector wastes the phase information that is present prior to

detection. It would be useful in situations where an aperture

distribution is to be determined from measurement of its

radiation field, if phase were more readily accessible.

Now the absence of redundancy in the Hartley transform

means that transform amplitude values alone fully specify

the original function. There is no need to measure transform

phase because the transform is real; this is unlike the Fourier

transform which, because of redundancy by a factor of

two, requires both amplitude and phase (or equivalently

real and imaginary parts) to be ascertained, though only

one-half of the transform domain need

be

explored. One

expression of this fundamental distinction is seen in X-ray

diffraction pictures, or photographs of optical diffraction

pattems, which exhibit rotational symmetry. The Hartley

transform is not symmetrical and the additional information

encodes the phase in the form of Hartley amplitude.

How to produce the Hartley transform plane optically has

been discussed in the literature

[6], [12]-[15]

and an up-

to-date synthesis is given in this issue by Villasenor [16].

Here we introduce the subject in the simpler form of a

one-dimensional microwave experiment for the purpose of

commenting on the significance for physical applications.

A way of constructing the one-dimensional Hartley trans-

form, starting from the Fourier transform is to make use

of a property that may be seen from the complex-plane

diagram, namely, that applying

f 4 5

phase shifts applied

to the Fourier transform yields two terms which, if com-

bined, cancels the imaginary parts and leaves the Hartley

transform. The algebraic expression of this property is

[121

The second form can be implemented with microwaves

by separating a microwave field into equal parts with

a beamsplitter, subjecting one of the parts to a quarter-

wavelength delay and left-to-right reversal, and recombin-

ing the parts to reconstitute a new wavefront which will

then express the Hartley transform.

Various ways are available for performing these oper-

ations, that depend on the wavelength in use and other

considerations. The apparatus used in the original demon-

stration [17] is shown in Fig. 3.

Radiation from a transmitting hom T was emitted from

a reference plane in which the field was a function of x

and allowed to impinge on a transparent dielectric sheet

B proportioned so as to let about half the incident energy

pass through toward a plane mirror

M I

while half was

reflected at right angles toward a comer reflector Mz. After

reflection, the two beams were recombined at the beam

splitter and transmitted to a receiving plane

x

which could

be explored with a receiving horn R. The effect of the

+

s

I

Fig. 3. Method for demonstrating the Hartley transform of a

microwave aperture distribution in a plane

s.

Dimensions

are

in

millimeters.

comer reflector was to reverse the field left-to-right as

compared with the field retuming from the plane mirror,

while a quarter-wave delay was injected into the comer

reflector arm by physical displacement through one-eighth

of a wavelength. The wavelength used was

2.8

cm.

More

details and altemative experimental arrangements are found

in

[14], [17].

Suffice it to say that the setup functions as

planned and shows clearly that a source located on-axis at

0 produces in the output plane x a casoidal interference

pattem whose peak is offset by 45' (or X / 8 from the axis.

In practice, it is convenient to put a compact source in

the source plane and to utilize the 45 displacement as a

sensitive means of getting the apparatus into adjustment.

Instead of a point source it is perfectly effective to use a

small hom, which does not destroy the

45

spatial phase in

the receiver plane but simply causes the amplitude to fall

off to left and right.

Experiments with asymmetrical aperture distributions

confirmed that the unsymmetrical Hartley transform was

being observed, rather than the symmetrical pattern that

would be seen in the far field or Fraunhofer plane. The

Fraunhofer diffraction pattem can be compared simply

by removing one of the mirrors; in that condition the

symmetrical far-field pattem is received on the output

plane and the phase information is lost.

Among the possible applications for a laboratory arrange-

ment of this kind is use in an anechoic chamber for antenna

radiation pattem studies where phase is important and

where piping a phase reference signal into the radiation field

is undesirable. Field applications at different wavelengths

can also be thought of.

VI. COMPUTING ULTIDIMENSIONAL

HARTLEY

RANSFORMS

The kemel for the two-dimensional Fourier transform

exp [--j27~(ux wy ] is separable in the sense that it can

be factored into one-dimensional functions of ux and

wy

alone; this property is the basis for applying the Fast

Fourier Transform algorithm to two-dimensional data by

BRACEWELL

ASPECTS

OF THE HARTLEY TRANSFORM

385



carrying out a series of one-dimensional transforms. The

kemel cas[2r(uz + wy ] does not permit separation into

factors in this way and it did not seem that an economical

analogy with the two-dimensional Twould exist. How-

ever, methods were developed [181, including progressive

quartering; the preferred method takes the row transforms

one by one, replacing the rows of the data matrix as each

one-dimensional transform is completed. Then the column

transforms are taken one by one with replacement as before.

A simple linear recombination of some of the elements, a

procedure known as “oscarization,” then leaves the two-

dimensional Hartley matrix. The operation is much tidier

than for the FFT because it is carried out on a single

matrix rather than on the two needed for managing real

and imaginary parts. The program for the inverse two-

dimensional Hartley transform is identical.

With the

FFT

one thinks in terms of two matrices, one

for reals and one for imaginaries. Some versions of the two-

dimensional T exploit the two-fold redundancy to pack

the reals and imaginaries into one matrix only, and cope

with the change of the sign of j in various ways. These

complications arise from the fact that exp [ - j 2 7 r (u z + wy ]

is separable in algebra only; when expressed in terms of the

sines and cosines that will be needed in a program the kemel

expands into four terms, exactly as with cas [2 7 r (u x +w y) ] .

The elegant, reversible way in which the four terms are

handled by the step of oscarization, is characteristic of the

comparisons [5] that arise between the Hartley and Fourier

approaches to real data.

Some thought has been given to an alternative kemel cas

2 r u x

cas

27rwy,

which generates what is called the cas-cas

transform [2], [19], [20], but it has not been used much.

Extension to three and n dimensions [21], [22] was soon

made; an interesting difference was found depending on

whether the number of dimensions was odd or even [22].

VII.

FINAL HOUGHTS

There is no question that the Hartley transform received

its impetus from the changed state of present-day computing

relative to the year 1942 in which Hartley’s publication

appeared. That the transform has now turned out to be

multifaceted, with various applications including physical

applications, is a bonus.

It quickly became apparent that the computing advantage

was valuable, as evidenced by the flow of publications

and by incorporation of the code into high-performance

instrumentation such as the Hewlett-Packard digitizing os-

cilloscope which depends on computing speed to sample

and operate on signals ranging from dc to 20 GHz. It was

soon appreciated that the various nonreciprocal or unilateral

transforms, those taking complex input and giving real

output and those for doing the opposite, were obsoleted by

the elegant Hartley technique which, with real arithmetic,

takes you to the other domain, from whatever domain you

happen to be in, by one and the same subprogram call.

On the analog side, future applications to optical signal

processing, holography, interferometry, and antenna prac-

tice have been opened up. In principle, X-ray phase can

be

measured by Hartley interferometry. In general, future

developments seem to lie in the direction of the many fields

in which wave phenomena are important.

REFERENCES

R.

V. L. Hartley, “A more symmetrical Fourier analysis applied

to transmission problems,” Proc. IRE, vol. 30, pp. 14&150,

Mar. 1942.

R. N. Bracewell, “The discrete Hartley transform,” J . Opt. Soc.

Amer., vol. 73, pp. 1832-1835, Dec. 1983.

-

“The fast Hartley transform,”

Proc. IEEE,

vol. 72, pp.

1010-1018, Aug. 1984.

P. Duhamel and M. Vetterli, “Improved Fourier and Hartley

transform algorithms: Application to cyclic convolution of

real data,”

IEEE Trans. Acou st. Speech Signal Pro cess.,

vol.

ASSP-35, pp. 818-824, June 1987.

R. N. Bracewell, “Assessing the Hartley transform,” IEEE

Trans. Acoust. Speech Signal Proc ess.

vol. 38, pp. 2174-2176,

Dec. 1990.

-,The Hartley Transform.

New York Oxford Univ. Press,

Mar. 1986.

D. M. W. Evans, “An improved digit-reversal permutation algo-

rithm for the fast Fourier and Hartley transforms,” IEEE Trans.

Acou st. Speech Signal Proce ss., vol. ASSP-35, pp. 1120-1 125,

Aug. 1987.

J.

S. Walker, “ A new bit reversal algorithm,”

Trans. Acoust.

Speech Signal Process. vol. 38, pp. 1472-1473, 1990.

0. Buneman, “Hartley transforms for the CRAY-2,”

Buffer,

vol.

13, July 1989.

R.

N . Bracewell, “Physical aspects of the Hartley transform,”

J. Atmosph. Terrestrial Ph ys., vol. 51, pp. 791-795, 1989.

D . M. W . Evans, “ n improved approach to harmonic spec-

tral analysis, and the canonical transform,” Ph.D. disse rtation,

Stanford Univ., 419 pp., Dec. 1989.

R.

N. Bracewell, H. Bartelt, A. W. Lohmann and N. Streibl,

“Optical synthesis of Hartley transform,” Appl. Opr. vol. 24,

J. D. Villasenor and

R.

N. Bracewell, “Optical phase obtained

by analogue Hartley transformation,”

Nature,

vol. 330, pp.

735-737, Dec. 24, 1987.

J Villasenor, “Two-dimensional digital and analogue Hartley

transforms,” Ph.D. dissertation, Stanford Univ., 1988.

-,“Two-dimensional analog Hartley transforms in the pres-

ence of errors,”

Appl. Opt.,

vol. 28, pp. 2671-2676, July 1989.

-, “Optical Hartley transforms,” this issue, pp. OOO-000.

J Villasenor and

R. N .

Bracewell, “Lensless microwave imag-

ing using the Hartley transform,”

Nature,

vol. 335, pp. 617-619,

Oct. 13, 1988.

R.N. Bracewell, 0. Buneman, H. Hao, and J Villasenor, “Fast

two-dimensional Hartley transforms,”

Proc. IEEE,

vol. 74, pp.

1283-1284, Sept. 1986.

M. G. Perkins, “A separable Hartley-like transform in two

or

more dimensions,”

Proc. IEEE,

vol. 75, pp. 1127-1 129, Aug.

1987.

I.-L. Wu and S. C.-Pei, “The vector split-radix algorithm for

2-D DHT,” IEEE Trans. Sig. Process., vol. 41, pp. 960-965,

1993.

H. Hao and

R. N.

Bracewell, “A three-dimensional DFT algo-

rithm using the fast Hartley transform,” Proc. IEEE, vol. 75,

pp. 264-266, Feb. 1987.

0. Buneman, “Multidimensional Hartley transforms,” Proc

IEEE.

vol. 75, p. 267, Feb. 1987.

pp. 1401-1402, 1985.

Ronald

N.

Bracewell (Life Fellow, IEEE) re-

ceived the B.Sc. degree in mathematics and

physics from the University

of

Sydney, Sydney,

Australia, and the Ph.D. degree in physics from

Cambridge University, Cambridge, England.

He learned about spectral analysis in 1940

by studying H. S. Carslaw’s

introduction

t

the

Theory of Fourier’s S eries and integrals work-

ing towards the B.Sc. degree, and later while

working on wartime radar development at the

CSIR Radiophysics Laboratory, was influenced

386

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OF THE

IEEE,

VOL.

82.

NO.

3,

MARCH

1994



by association with J.

C.

Jaeger. Moving to the Cavendish Laboratory,

Cambridge, in 1946, he benefited from the lectures of J.

A.

Ratcliffe,

whose way of interpreting mathematics in physical terms, and especially

Fourier analysis in terms of diffraction, has been widely promulgated

by his students. On gaining the Ph.D. degree, he retumed to Australia

(1950-1954), published several papers on instrumental aspects of radio

astronomy and co-authored (with J. L. Pawsey) the first textbook on

radio-astronomy (1955) After a year as a visiting assistant proessor in

the Astronomy Department at Berkeley, CA, he moved to the Electrical

Engineering Department at Stanford, where he has been ever since,

producing a text on the Fourier transform (1965) among other activi-

ties. Becoming interested in medical imaging after his inversion of the

tomographic transform was incorporated into CAT scanners, and as a

founding member of

the

editorial board of the Journal of Compurer

Assisted

Tomography

he published papers on tomography and developed a

lecture course on imaging. This material. which draws on two-dimensional

spectral analysis, and reflects much experience with digital aspects, will

appear as a textbook. Meanwhile, The Harrley Transform, which appeared

in 1986, has re -i nd uc ed Hartley’s name to new generations of students

who no longer hear about the Hartley oscillator or Hartley’s law.

In 1992,Dr. Bracewell was elected to the Institute of Medicine of the

National Academy of Sciences and received an inaugural alumni award

from the University of Sydney. He is also the recipient of

the

IEEE

1994

Heinrich Hertz Medal “for pioneering work in antenna aperture

synthesis and image reconstruction as applied to radio-astronomy and to

computer-assisted tomography.”

BRACEWELL: ASPECTS OF THE HARTLEY TRANSFORM

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Documents

Aspects of the Hartely Transform