Electronically controllable primary feed for profile-errorcompensation of large parabolic reflectorsA. W. Rudge, Ph.D., and Prof. D. E. N. Davies, D.Sc, C.Eng., M.I.E.E.

Abstract

The paper shows that it is possible to compensate for the effects of mechanical profile errors in large parabolicreflectors, by means of a specially designed multielement feed array situated in the focal plane of the reflector.The factors relating the magnitude and period of these profile errors to the size and complexity of thecompensating feed are discussed, and a novel method is described for achieving this compensation bycontrolling the phasing of the array. The system employs a spatial Fourier-transforming device, and its use ismainly restricted to long-period profile errors. Experimental results of this process, derived from a small-scale laboratory system employing a 1-dimensional 5 ft parabolic reflector incorporating an 8-element feed,are presented. These results demonstrate compensation for profile errors up to TT/2 radians. The possibleapplication of this technique to existing and new reflectors is discussed.

List of principal symbols/ = focal length of parabolic reflectord = diameter of reflector6 = halfangle subtended at the focus by point on reflectorB = maximum value of 6u = sin d

p,<j> — aperture plane polar co-ordinatest, <f>' = focal-plane polar co-ordinatesp, q — sines of rectangular-aperture angular co-ordinates

(p = u cos (f), q = u sin <f>)x, y = focal-plane rectangular co-ordinates

E = focal-plane field-distribution functionF = aperture-plane field-distribution functionG = modified aperture-plane field-distribution function

pe = peak magnitude of aperture periodic phase errorTV — spatial-recurrence frequency across reflector of a

periodic phase error8 = array-element spacingn = number of array elementsA = wavelengthk = 2TT/A

1 IntroductionA difficult problem associated with the construction of

large parabolic-reflector antennas is that of maintaining thesurface profile of the reflector to a high accuracy under alloperating conditions. Profile inaccuracies introduce phaseerrors across the antenna aperture,1-2 and such errors reducethe gain, spoil the side-lobe structure of the antenna direc-tional pattern and determine the upper frequency limit ofoperation.

Although high accuracies have been achieved with somelarge steerable parabolic reflector antennas, the economicproblems involved in the reflector construction and themaintenance of the reflector profile under operational condi-tions, which may include variations in wind, temperature andgravitational loads, still present a major problem.

Considerable improvement in the performance of largesteerable paraboloids has been achieved by improved struc-tural design and alignment techniques.3 The British PostOffice modified Goonhilly antenna and the AustralianCSIRO Parkes antenna are examples of such improvements.Both antennas operate without radomes, relying on theirrigid structure to reduce the effects of meteorological varia-tions. In the class of radome-protected antennas, the MITHaystack facility4 represents an outstanding example of the

Paper 6078 E, first received 29th July and in revised form 19thNovember 1969Dr. Rudge and Prof. Davies were formerly with the Department ofElectrical Engineering, University of Birmingham. Birmingham,England. Dr. Rudge is now with I1T Research Institute, 10 West 35Street, Chicago, 111. 60616, USA. Prof. Davies is now with BritishRailways Board, Research Department, Derby, England, and theDepartment of Electronic & Electrical Engineering, University of Tech-nology, Loughborough, Leics., England

PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

application of modern design techniques to achieve a highprofile accuracy for all elevation angles. In general, however,antennas of both types involve very high initial costs; and,while the latter system gives the best approach to profilemaintenance, it is still prone to the effects of temperaturevariations, and considerable controversy still exists over theamount of performance deterioration introduced by theradome cover.3

Some recent activity has also been stimulated in the field ofadaptive profile control for large reflectors. With thisapproach, it is accepted that varying profile errors will beintroduced into the reflector during operation. Compensationfor such errors can, in principle, be achieved by mechanicalcontrol of the reflector profile,5 or by electronic processing ofthe signals associated with a multielement feed array.6'7 Theformer approach has received some attention but littlereported success. This paper described work based on thesecond approach.

It is shown here that it is possible, within certain limits, tocompensate for such profile errors in the design of a multi-element primary feed for the reflector. It is convenient tothink of the process of profile compensation in terms of itsoperation on a reflector used as a receiving antenna. Thesignals collected by the array situated near the focus of thedish are combined coherently in a way which tends to restorethe full gain of the undistorted reflector. Since the profileerrors of the reflector may vary with the temperature and thewind loading on the antenna structure, it is worth consideringthe further possibility of making the process automaticallyadaptive, to compensate for the errors presented at anyrequired pointing direction due to variable wind loading.

2 Physical description of profilecompensation

2.1 Parabolic reflectors with distorted profilesThe idealised operation of a parabolic reflector may

be thought of in terms of ray optics; i.e. as the focusing of anormally indicident plane wave down to a single point. Themore accurate application of diffraction theory to the situa-tion shows that the distribution of the focused energy acrossthe focal plane of the reflector when it is receiving a normallyincident plane wave is a function quite similar to the far-fielddirectional pattern of the reflector when it is fed from a pointsource at the focus. For reflectors with large f/d ratios (focal-length/diameter ratio greater than about 0-5), the approxima-tion is close.

Fig. 1 shows such a field distribution, plotted across thefocal plane of a reflector which is parabolic in one plane only.To demonstrate the principles of electronic profile compensa-tion, initial consideration will be limited to this 1-dimensionalparabolic reflector, though the basic principle will be ex-tended to the doubly curved surface.

Since we are going to analyse the antenna behavtbur for

351

reception, we may note a useful analogy between trans-mission and reception. During transmission, the far-fieldpolar diagram (strictly in sin 6 measure) is the Fourier trans-form of the field distribution across the aperture plane. When

reflectorfocalplane

axis

fielddistribution

Fig. 1Field distribution in focal plane of parabolic reflector illustrated by aplane wave arriving from a direction normal to aperture of reflector

the above approximation is used, i.e. f\D > 0 • 5, we mayinfer that the focal-plane distribution when a plane wave isincident on the boresight is the Fourier transform of the fielddistribution across the aperture.

Consequently, the field distribution in the focal plane for aplane wave incident along the boresight will change in thepresence of profile errors, in the same way as does the farfield directional pattern set up by a point-source feed in thepresence of the same profile errors. We shall consider later ananalysis which permits the calculation of the form of suchpatterns, but content ourselves for the time being with somewell known effects of profile errors on antenna radiationpatterns.2 Errors with long spatial periods, i.e. long correla-tion intervals, tend to increase the near-in sidelobe level,errors with short spatial periods tend to produce an increasein the far-out sidelobe levels, and in the limit, when theperiod is smaller than a wavelength, these errors will causescattering over the entire range of angles.

2.2 Focal-plane distributions

If the results of Section 2.1 are applied to the fielddistribution set up in the focal-plane region on reception of anormally incident plane wave, it is seen that the misfocusedenergy due to the profile errors will be concentrated near thetrue focus for long-period errors, but it will be scattered somedistance from the focus for short-period errors. The problem(on reception) of compensating for the effect of these profileerrors may be regarded as that of collecting this misfocusedenergy and coherently combining it. An array of feed hornslocated in the focal plane could, in principle, be used tointercept that portion of the misfocused energy due to thelong-period errors.

The amplitude and phase distributions across the array inthe focal plane will clearly be complex functions, and it will benecessary to correct the phases of the signals received at eachelement and then to combine the signals coherently withoutloss, a process which involves suitable amplitude weightingand therefore requires impedance-changing circuits. Thus,although compensation for the long-period errors is, inprinciple, feasible by the use of a correctly designed multi-element feed, it appears to be extremely difficult to make thecoherent phasing and lossfree combining processes variablefor differing profile errors.7

The solution proposed in this paper for overcoming theworst of these difficulties lies in performing a second spatialFourier transform on the signals received by the multielementarray in the focal plane. This produces a new set of signalswhich correspond directly to the field distribution across theaperture plane. This distribution approximates to a uniform-amplitude constant-phase function, but with the phase depart-ing from constancy to an extent governed by the errors in thereflector profile.352

One method of obtaining a spatial Fourier transform thathas been successfully adopted in the field of linear arrays isthe multiple-beam-forming matrix network, e.g. the Butlermatrix.8 Such networks have n input ports and n output ports.They contain only phase shifters and hybrid circuits and willbe discussed in more detail later. Other methods of obtainingsuch transforms include the use of lenses or further reflections,but, for brevity, the matrix approach alone will be consideredin this paper. Fig. 2 shows the output of a focal-plane array

reflector

focal plane

Fouriei—transformmatrixnetwork

Onsignal -combiningnetwork

combinedoutput

n-elementarray

phaseshifters

Fig. 2Schematic of system for electronic profile compensation for parabolicreflector used for reception

feeding such a matrix network. Owing to the Fourier trans-form performed by the matrix, the outputs from the portsOj-O,, will be constant in amplitude but will vary in phase indirect sympathy with the phase errors across the reflectorcontour. It is convenient to bear in mind a simple (thoughapproximate) physical explanation of this process, in whicheach output of the matrix network may be regarded as havingformed a receiving beam via the array which samples theillumination of a particular section of the reflector.

Profile compensation has now been reduced to the coherentcombination of these variable phase signals. When the profileerrors are primarily due to gravitational stresses, it might beadequate to insert a set of variable phase changers whosevalues are controlled in a predetermined way in sympathywith the inclination of the reflector to the horizontal. It isfurther possible to use adaptive combining techniques similarto those employed in either adaptive arrays9 or diversity-combining communication systems. Alternatively, an auto-matic system could be installed for continuous recording ofthe reflector deformations and the servomechanism control-ling the compensating phase shifters.

However, the purpose of this paper is to analyse the processof profile compensation and determine any basic limitationson its performance. This will be undertaken in the followingSections and confirmed by some experimental results.

3 Focal plane field distributions forparabolic reflectors

3.1 Undistorted parabolic reflector

Recent work,10 using an extension of the conventionalscalar diffraction theory to remove the normal restrictionsimposed by the fid ratio, has shown that the principal

Fig. 3Co-ordinates and notation for analysis of field distributionsx and y axes lie in focal plane

PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

component E of the electric-field distribution in the focalplane of a large parabolic reflector is related to the corres-ponding component F of the electric-field distribution in theaperture plane by a scalar equation of the form (see Fig. 3).

ZTT Jn J2H -wo-"2 exp {jktu cos (</> — (f>')}udud<f>

. . • • ( 1 )

where k = 2rr/A, u = sin 6, u = sin §, and 6 is the maximumangle subtended by the reflector rim from the focus. Thederivation of eqn. 1 is given in Appendix 10.

In this paper, we will consider a parabolic reflector having arectangular aperture. The co-ordinate system is shown inFig. 3: x, y are rectangular co-ordinates with the origin at thefocus. For a point on the reflector, p and q are defined by

• (2)

(3)

p = 6 cos sin (f> = u cos <f>

q — s in 6 sin <f> = u s in <f>

(p and q denote the maximum values of/? and q at the reflectorrim). Then, provided that the aperture distribution of acomponent F of the field is separable into factors F(p) andF(q), which are, respectively, functions of p and q only, it ispossible to express eqn. 1 as the product of functions of(x, p) and (y, q) only. The process is valid if we can approxi-mate the denominator y/{\ — u2) = V O — P2 — q2) of

2 2eqn. 1 in the form y/{\ — p2) x \/(l — q2), which introducesnegligible error in all practical configurations. We then obtain

F{p)exp (jkxp)dp

r. F{q)exp (jkyq)dq (4)

The resultant similarity of the x and yplane distributionspermits the consideration of either one of the integralsindependently without loss of generality. With a distant trans-mitter located along the antenna boresight, and with an un-distorted reflector, the aperture-plane field distribution of thelatter will be uniform in phase and amplitude. Under suchconditions, we may put F(p) = F(q) = 1-0. Hence, if thefocal field distribution is expressed as

E(x,y)=j—E'(x)E'(y)LIT

(5)

the solutions to eqn. 4 take the following form:

and E'(x) = 7rJ0(kx) for p = 1 0

(6a)

(6b)

and similarly for E'(y). The characteristics of these resultingdistributions are such that interpolation for intermediatevalues of p or q is easily carried out. Owing to the comparativeinsensitivity of the focal-plane distribution to small-amplitudetapers in the aperture plane, eqn. 6a has been found10 to givereasonable approximations to the measured focal-plane fielddistributions, down to fid ratios of 0-5 (p = 0-8). The resultsof eqn. 6a with fid ratios of 1 0 and 0-5, and of eqn. 6b withan f]d ratio of 0 • 25, are shown in Fig. 4.

3.2 Distorted reflectorSmall profile deformations in the reflector surface may

be conveniently considered as effective phase errors.1 Suchphase errors may be theoretically treated by projecting theminto the aperture plane of the reflector, thereby modifying theeffective field distribution in that plane. The magnitude of thephase errors, being proportional to the operating frequency,impose an upper frequency limitation above which theantenna is inoperable.

Profile deformations may be either regularly (i.e. period-ically) or randomly distributed over the reflector surface, theformer distribution being attributable to the regular mode ofconstruction employed in the reflector surface and supportstructure. It is relevant to examine the differing effects on thefocal-plane field of both forms of error distribution.

PROC. IEE, Vol. 117, No. 2, FEBRUARY 197015 E9

3.2.1 Periodic errorsConsider a periodic profile deformation across the

'/?' plane of a rectangular parabolic reflector. When referredto the aperture plane, the phase error can be expressed as

mwhere /3e is the maximum departure from the reference (error-free) phase, and Af corresponds to the spatial recurrence of the

2-Or

1 0 -

0-1

20-

10

0-1

0-1-10

Fig. 4Functions representing eqns. 6a and b(a) fid = 1 (b) fid = 0-5 (c)fld = 0-25

error across the reflector aperture. With a uniform incidentwavefront, the effective aperture-plane field distribution nowbecomes

^p)} (8)On inserting this aperture distribution into the first integralin eqn. 4 and carrying out the integration, we obtain

/wS

sin (s+nNn) sin (s — nNir)\ |s + UNTT s - nNir ) \ * '

for h = 0 - 5 , and

E\x) = 7r[I- 7J = 0

{JoCr + nN-ri) + Jo(* - «Mr)}] (10)

for p — 1-0, where s = kpx.

Comparison of this result with the undistorted case given

353

by eqns. da and b shows that tne general effect of a periodic-error distribution is to reduce the undistorted focal-planedistribution by the Bessel coefficient J0(j8e). The displacedenergy is redistributed in a series of grating lobes, each lobebeing the shifted image of the undistorted focal-plane pattern

2N

Fig. 5A

Focal-plane field distribution with simple periodic aperture phaseerrorMagnitude of errors = peSpatial frequency of errors = N

weighted by a Bessel coefficient of increasing order, andspaced at intervals governed by the spatial recurrence of thephase error across the reflector aperture. The effect is illu-strated in Fig. 5a.

3.2.2 Random errors

While constructional techniques have a tendency toproduce regular or periodic profile-error distributions, ingeneral, the actual profile of a given reflector is an unknown

Fig. 5BProbable focal-plane field distribution in presence of random profile

Effective r.m.s. value of error = eCorrelation interval = 2«/25

factor which can only be treated satisfactorily on a statisticalbasis.

Let us define a function G so that

G{p,q) =

G(p,q) = 0

F(p,q)~P2-q2 q2)

for \p\ <p and \q\ < q

for \p\ >p and \q\ >q

(11)

G will be termed the 'modified' aperture distribution, since itcan be considered as an amplitude-tapered version of the trueaperture distribution. In the rectangular co-ordinates pre-viously defined, eqn. 1 may now be rewritten as

.00 «,

E(x, y) = \ G(p, q) exp { jk(xp + yq)} dpdq (12)J — oo J —oo

In this form, it can be seen that a spatial 2-dimensionalFourier-transform relationship exists between the modifiedaperture-plane field distribution G and that of the focal plane(£•). For reflectors having f/d ratio greater than about 0-5,the transform relationship can be considered to exist directly354

between the reflector aperture-plane field and that of thefocal plane. This spatial relationship is identical to that relat-ing the reflector aperture-plane field with the antenna far-fieldradiation pattern.1 The focal-plane field distribution and thatof the antenna radiation pattern are therefore identical, beingrelated by a double Fourier transformation. Departure fromthis relationship at very small f/d ratios can be observed bycomparison of eqns. 6a and 6b and allowed for by applicationof eqn. 11.

The existence of this transform relationship permits thedirect application of the extensive work carried out on theaperture/far-field relationship to the modified aperture/focal-plane case. Hence, in order to study the effects of randomaperture phase errors, we may conveniently adapt the classicalwork of Ruze2 to determine the probable form of the focal-plane field. Ruze has employed a mathematical modelcomprising error regions statistically uniformly distributedover the aperture of a paraboloidal reflector with correlationradius c'. Each region has a Gaussian distribution and is in-dependent of any other region. The work is well covered in theliterature, and it will be sufficient to adapt the relevant resultsto indicate the probable focal-plane field distribution of aparaboloidal reflector in the presence of small random profileerrors.

With the statistical error distribution given above, theprobable power distribution in the focal plane can be ex-pressed, with the notation of Fig. 1, as

+ k2c2 exp - (2ke)2 £ (2ke)2» exp ( - ( - ^n=\ { 4n

. . . . (13)

where PQ = error-free focal-plane power distribution

e = effective surface r.m.s. error

2uc'and c = ——

d

Some probable focal-plane power distributions are illustratedin Fig. 5B.

3.3 Conclusions drawn from error analysis

Examination of the effects of periodic and random-profile errors has indicated that the presence of either distortsthe field distribution in the focal plane by a factor proportionalto the magnitude of the error. The way in which the dis-placed energy is redistributed in the focal plane is dependenton the magnitude of the error and the correlation radius orspatial period of its distribution across the reflector surface.It can be seen that, for errors of the same magnitude, shorter-period errors redistribute the energy farther from the geo-metric focus. In addition, regularly distributed profile errorsmay be more detrimental than random errors of similarmagnitude—ambiguous lobes are introduced rather than amore gradual smearing of the field distribution.

4 Second spatial transformationFrom an examination of eqns. 6a and b, the focal region

of the parabolic reflector can be seen to contain an electric-field distribution which has drastic fluctuations of both phaseand amplitude, even in the absence of profile errors. Thepresence of such errors, which may vary with the operatingconditions and the positioning of the reflector, further compli-cates the situation. Direct replacement of a conventionalprimary feed with a more extensive multielement primary feedwill thus involve a time-varying system with the capability ofnon-uniform signal combination. To avoid a worsening ofthe signal/noise ratio as a result of the combining process, itbecomes necessary to weight each array-element gain with itsown signal/noise ratio.11

However, eqn. 12 also suggests the alternative approachdescribed in Section 2. The equation shows that a spatial2-dimensional Fourier transform relationship exists betweenthe modified aperture distribution and the focal-plane field. Ifa second spatial Fourier transformation is carried out on thefocal-plane field, the resulting distribution would be identical

PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

to the orginal modified aperture distribution (i.e. a constant-amplitude distribution as given by eqn. 11, with a phasedistribution corresponding to the reflector profile-errordistribution). This amplitude distribution, with a normallyincident plane wave, is shown in Fig. 6.

Profile-error compensation now becomes a problem of

20

10

-10 1-0

Fig. 6

Amplitude distribution after second spatical transformG(u) = 1/VO - "2)

phase correction and subsequent summation. The field-amplitude distribution remains constant and independent ofsmall varying profile errors. From Fig. 6, it can be seen that,over a considerable range of f/d ratios, the amplitude distri-bution is uniform. For very small fid ratios, a power loss willbe incurred owing to the tapering characteristic. However, thisloss is less than that encountered in a conventional single-element primary feed at small fid ratios, and this fact can beused to advantage by tapering the reflector aperture illumina-tion to obtain a reduction in the sidelobe levels of the antennaradiation pattern.

To achieve a spatial transform on the focal-plane fielddemands, ideally, an infinite transforming device. However,from examination of eqns. 6a and b, it can be seen that theoff-axis decay of the focal-plane field is such that a closeapproximation to the infinite transform may be obtained witha transforming device having a collecting aperture withdimensions of the order of a few wavelengths.

Eqns. 9 and 10 indicate that, for small phase errors, thespread of the error lobes is directly proportional to the periodof their spatial distribution across the aperture plane. Hence,if the aperture phase-error distribution is represented by aFourier series, the dimensions of the collecting aperture of thetransforming device will govern the number of terms in theseries which appear at the output of the device and which arethus available for compensation. Higher-order terms willcreate lobes outside the collecting aperture which will not beavailable for compensation. For larger errors, a peN productcan be defined as a figure of merit for a given collectingaperture. Aperture phase-error components with pe N pro-ducts exceeding the given value will not be available forcompensation.

Similarly, a figure of merit for the dimensions of theprimary-feed collecting aperture can be expressed, in thepresence of random profile errors, in terms of the effectiver.m.s. profile error c and the correlation radius c'.

5 Experimental systemSeveral practicable methods of achieving the second

transformation have been studied, including the use of asecond smaller parabolic reflector and the application of amicrowave lens. For experimental purposes, a 1-dimensional10GHz system has been constructed;* it comprises a lineararray of eight waveguide feeds followed by an 8-port Butlermatrix to provide the Fourier transformation. The Butlermatrix consists of a theoretically lossless matrix of fixed phaseshifters and directional couplers, interconnected so that thefield distribution across the output ports effectively comprisesa sampled spatial Fourier transformation of that field distri-* at the University of Birmingham

PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

bution applied across the input ports.8> 10> 12 Fig. 7 illustratesthe operation of the Butler matrix.

Each of the matrix output ports is taken via adjustable

\yf\

I 1 I IButler

I rmatrix

i

TTl U ITT ss?I f

Fig. 7

Sampled Fourier-transform properties of Butler-matrix networkE represents sampled values of field at input ports of matrix networkF represents corresponding outputf sampled values

continuous function

phase shifters, providing a range of greater than 360° to asumming network. The adaptive primary-feed system fromthe array input to the final output from the summing networkis constructed completely in Xband waveguide components.The Butler matrix and summing network are constructedseparately as one-piece units by a dip-brazing technique.13

While the system is theoretically lossless, the experimentaladaptive primary feed introduces an insertion loss of the orderof ldB, this being largely attributable to the imperfectconstruction of the r.f. components forming the matrix andsumming networks.

The optimum spacing So of the feed-array elements, interms of the error compensation, is10

(14)

This spacing represents the maximum separation of the arrayelements which excludes the formation of grating lobes overthe angular spectrum subtended by the reflector from thefocus. Such a spacing thus provides the primary feed with the

primary-feed array

Butler matrix

•\r-i adjustable|\J phase shifters

2nd Butler matrix

matched loads

outputFig. 8Block schematic of experimental 8-element feed system for electronicprofile compensation

3ft

largest permissible intercepting area for a given number ofelements.

This experimental primary-feed system with adjustableprofile compensation was employed in conjunction with avariable-profile parabolic reflector, curved in one dimensionwith a nominal fid ratio of 0-5. The reflector is adjustable at17 points across its diameter, to permit the superposition ofprofile errors on the parabolic contour. The system is shownin block schematic form in Fig. 8.

6 Measurements and discussionThe performance of the experimental adaptive primary

feed was evaluated by comparing its performance with theantenna gain and directivity achieved with a single-elementwaveguide primary feed. Three results are illustrated inFig. 9 for the undistorted parabolic-reflector profile and for

10

0-5

VOr

0-5

1-0

0-5

-2O 0 2-0

Ix , cm

a

with the single feed. Of this, 1-OdB is due to the systeminsertion loss and 0-5dB to the additional aperture blocking.

With the larger-amplitude periodic error, a 6dB reductionin gain was measured for the single-element primary feed.This loss was completely compensated by the adaptive feed.The antenna radiation pattern can be seen to be restored toits error-free condition. The smaller-amplitude higher-perioderror reduced the antenna gain by — 3-5dB with the single-element primary feed, and, while the adaptive feed restoredthe gain, the sidelobes in the antenna radiation pattern be-came assymetrically distributed, resulting in a 3 dB increase inthe peak sidelobe level on one side of the main beam.

The determination, from eqns. 8-10 of the focal-plane fielddistribution resulting from the two periodic-error distri-butions reveals that, with the smaller-amplitude higher-perioderror, a significant amount of the displaced energy will bedistributed outside the adaptive-feed collecting aperture and

( i )

( i i )

-2° 6 2°

(iii)

6 ,degb

-2° 0 2«

9,degc

Fig. 9

Experimental results for focal-pane distributions and directional patterns for 180 cm 1-dimensionalparabolic reflector with differing profile errors, showing effect of profile compensationa Field distribution measured in focal plane of reflectorb Directional pattern of reflector with single-element feedc Directional pattern of reflector with multiple-element feed, adjusted for profile compensation

(i) No profile errors(ii) Sinusoidal profile distortion with P« = 0-5^ radians and N = 2

(iii) Sinusoidal profile distortion with (?„ = 0-4TT radians and N — 3

two cases of regular profile deformations having peN pro-ducts of 0-5TT x 2 and 0-4TT X 3, respectively. In each case,the antenna far-field radiation pattern is shown, first with thesingle-element primary feed and then with the adaptiveprimary feed. For the multielement feed, the phase shifterswere adjusted in turn to maximise the output power with thetarget transmitter located along the antenna boresight, priorto the radiation-pattern measurement.

With the error-free reflector, the multielement feed can beseen to introduce slightly higher sidelobe levels in the antennaradiation pattern. This is a consequence of the aperture block-ing introduced by the array. For a large practical reflector,this effect would be negligible. The gain of the error-freesystem with the adaptive feed is 1 • 5 dB below that obtained356

will thus not be available for compensation. From eqns. 8-10,a guide to the compensating range of this type of adaptiveprimary feed may be expressed as follows. For regular com-ponents of aperture phase error having magnitudes in therange O-TT/2 radians, and with spatial-recurrence frequenciesacross the aperture not exceeding (n — l)/2, the adaptivefeed is capable of achieving complete compensation providedthat

where n is the number of phase-compensated inputs to thesumming network and S is the array-element spacing.

For planar-array primary fees, n will correspond to thePROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

number of inputs to the summing network relating to thesame plane as the phase-error distribution. In a practicalsystem, the maximum value of n will be restricted by theincreasing complexity of the system, and thus the compensat-ing capacity of the adaptive primary feed will be restricted toths lower spatial-frequency components of the aperture-planephase-error distribution. However, statistical surveys of somelarge reflector profiles2'14 have indicated that the probabledistribution of power in the far field on transmission (andhence the probable focal-plane distribution on reception) is ata maximum in the region close to the main lobe and can thusbe considered to be the result of an aperture phase-errordistribution with significant lower-spatial-frequency com-ponents. In particular, time-varying profile deformations ofthe type incurred by wind and gravitational effects tend toproduce phase-error distributions with a large lower-spatial-frequency content.

7 ConclusionsThe experimental model has demonstrated the validity

of the second spatial Fourier-transform technique, developedin the text as a means of compensating for profile errors withlow spatial recurrence frequencies across the reflector witheffective magnitudes of up to approximately TT/2 radians. Theprimary-feed system developed readily lends itself to adaptivetechniques, in that the problem has been reduced to onerequiring the adaptive control of a number of phase shifters.A considerable amount of work has been devoted to theproblems associated with adaptive phase control, andmany of the techniques developed will be directly applic-able.9-12

Certain practical problems exist in the application of Butlermatrixes in planar primary-feed systems with regard toinsertion loss and spillover. Neither problem incurs a funda-mental limitation on the transform technique, though forlow-noise applications, improvement in the manufacturingtolerances on matrixes presently commercially available arerequired to reduce the insertion losses introduced by theimperfect matrix components. Microwave-lens or reflectortechniques may also be used to provide the second trans-form, and they may prove to have advantages in this respect,particularly with regard to planar systems. Butler-matrixsystems are also expensive, and this is another reason forstudying alternative Fourier transformers.

The results shown in Fig. 9 indicate that profile compensa-tion can be successfully applied to profile errors corres-ponding to a peak phase error of 77/2 radians. This is at leastfour times the normal limits specified for a good reflectorsurface, and this suggests that the application of such atechnique could, in principle, extend the upper frequencylimit of a reflector by two octaves of more.15 However, it isalso necessary to bear in mind the effect of different spatialperiods of the error function, since the compensation islimited to a given jSe N product, as discussed in Section 4.Consequently, there would be little value in applying thetechnique to a very rigid reflector with short-perioderrors, since a very large number of elements would berequired.

The above situation frequently occurs in modern reflectors,but a recent paper16 has shown that the effects of longer-period phase errors on directional patterns is not nearly assevere as previous work has suggested. This is an interestingdevelopment, since it suggests that some cheapening of thestructure might be possible as a result of this relaxation on thedesign. Furthermore, this type of reflector profile would beparticularly suitable for the application of electronic profilecompensation, resulting in the possibility of increasing theupper frequency limit on the reflector or further relaxing thephase-error requirements for long-period errors. Thus,although the application of the above profile-compensationsystem is, in practice, limited to reflectors with predominantlylong-period errors, it nevertheless appears that there is astrong case for further study of the technique in conjunctionwith the structural design of the reflector system. This couldresult in useful savings of cost and an increase in the upperfrequency limit of the reflector.PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970

8 AcknowledgmentsThis work was undertaken while the authors were in

the Department of Electronic & Electrical Engineering,University of Birmingham, and the authors wish to acknow-ledge the valuable assistance and advice of their formercolleagues, in particular Prof. E. D. R. Shearman andProf. M. J. Withers. Thanks are also due to the ScienceResearch Council who provided financial support for thework.

9 References1 SILVER, s.: 'Microwave antenna theory and design' (McGraw-Hill,

1949)2 RUZE, J.: 'The effect of aperture errors on the antenna radiation

pattern', Nuovo Cimento SuppL, 1952, 9, pp. 364-3803 'Design and construction of large steerable aerials', IEE Conf.

Publ. 21, 19664 WEISS, H. c : 'A description of a high performance microwave

experimental facility.' MIT Lincoln Laboratory technical report,1964

5 BURR, D. w.: 'Improvements in, or relating to, aerial structures.'British Patent 1030872, March 1962

6 DAVIES, D. E. N. : 'Proposals for electronic compensation of surfaceprofile errors in large reflectors' in 'Design and construction oflarge steerable aerials', IEE Conf. Publ. 21, 1966

7 LOUX, p. c , and MARTIN, R. w.: 'Efficient aberration connectionwith a transverse focal plane array technique.' IEEE Internal.Convention Record, 1964, 12, pp. 125-127

8 BUTLER, J., and LOWE, R.: 'Beam forming matrix simplifies designof electronically scanned antennas', Electron. Des., 9, pp. 170-173

9 WITHERS, M. J., DAVIES, D. E. N., and APPERLEY, R. H. : 'Self focussingaerial arrays for airborne communications.' Presented at the12th symposium of the avionic panel of the Advisory Group forAeronautical Research & Development, Dusseldorf, 1966

10 RUDGE, A. w.: 'Adaptive primary feeds for phase error compensa-tion in parabolic reflector antennae.' Ph.D. thesis, University ofBirmingham, 1968

11 BRENNAN, D. G. : 'On the maximum signal/noise ratio realizablefor several noisy channels', Proc. Inst. Radio Engrs., 1955, 43,p. 1530

12 ALLEN, J., et al.\ 'Phase array radar studies.' MIT Lincoln Lab-oratory technical report 236, 1961

13 THRAVES, J.: 'An X-band Butler matrix array.' Presented at the12th symposium of the avionics panel of the Advisory Group forAeronautical Research & Development, Diisseldorf, 1966

14 RUZE, J.: 'Antenna tolerance theory—a review', Proc. Inst. Elect.Electron. Engrs., 1966, 54, pp. 633-640

15 DAVIES, D. E. N., and RUDGE, A. w.: 'Some results of electroniccompensation for surface-profile errors in parabolic reflectors',Electron. Lett., 1968, 4, pp. 433-434

16 vu THE BAO: 'Influence of correlation interval and illuminationtaper in antenna tolerance theory', Proc. IEE, 1969, 116 (2),pp. 195-202

17 MATTHEWS, P. A., and CULLEN, A. L. : 'Field distribution at the focusof a microwave lens', ibid., 1956, 103C, pp. 449-456

10 AppendixTo an observer situated in the focal region of a large

paraboloidal reflector and facing the vertex, the apertureplane of the reflector appears as a spherical surface, angularlylimited by the reflector rim. A mathematical model of thissituation may be established by considering a spherical waveconverging on a point O and incident on an aperture in aninfinite opaque screen (Fig. 10). If an equivalent reflectoraperture is defined over the spherical surface of the incident

infinite screen

incidentspherical

wave

Fig.10Notation for analysis in Appendix 10

357

wave, angularly limited by the edge of the opaque screen,can be shown17 that, provided that kR > 1 and kr > 1, theprincipal compnent of the electric-field vector at a point Pon the blind side of the screen can be expressed, with thenotation of Fig. 10, by the scalar equation

EP=J F-^- (cos a + cos j8) expjk(R - r)dS (16)

Rr

where F(s) is a complex term which may be employed toaccount for nonuniform phase and amplitude field distri-butions over the aperture plane, and k = 2TT/2.

Provided that the distance from O to P is small comparedwith r, the approximation cos a = cos (—j8) is not too inac-curate. Restricting the analysis to the focal plane of a circularparaboloid, and employing the notation of Fig. 1, we have

and, provided that t/R < 1, as is the case here,

R — r = t sin 6 cos ((f> — <f>') . . . . (18)

- 2Rt sin 9 cos (<f> - (17)

From Fig. 1, p = f tan 0, so that dp = f sec20 dd. With theapproximation that Rr ~ R2 in the amplitude term is reason-able since the difference in path length will be negligible, andincorporating these modifications in eqn. 16, the focal-planefield distribution for a paraboloidal reflector of any fjd ratiomay be expressed as

u

. ' 2 exp {jktu cos (<f> — </>')) udud(f>

. . . . (19)

where u — sin $, u = sin 6, and 6 is the maximum halfanglesubtended by the reflector rim at the focus. This is the ex-pression for E(t, 6') used in Section 3.1.

358 PROC. IEE, Vol. 117, No. 2, FEBRUARY 1970