the peak sidelobe of the phased array having randomly located elements

7/29/2019 The Peak Sidelobe of the Phased Array Having Randomly Located Elements

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IEEE TRANSACTIONS O N AN TE NK ~S m- PROPAGATION, VOL. a-20, O. 2, MARCH 1972 12

The Peak Sidelobeof the Phased Array HavingRandomly Located ElementsBERNARD D. STEINBERG, FELLOW, mm

Abstract-A formula is derived for the peak sidelobe level of aphased array in which theelements are randomly located. Theparameters of the formula are the number and size of the arrayelements, size of the array, wavelength, beamsteering angle, andsignal bandwidth. The theory is tested by measurement of thepeak sidelobe of several undred computer-simulated randomarrays. Unlike thecase for the conventional array the effect ofspatial taper (nonuniform density of element location) upon thepeak sidelobe level is minor. The peak sidelobe of the two-dimensional planar array is approximately 3 dB higher than thetof the inear ar ray of the same lengthand amenumber ofelements.

T I. INTRODUCTIONHE COST of a large phased array which is designedprimarily for high angular resolution rather than forweak signal detection may be reduced manyfold throughthinning, i.e., reducing the number of elements in heaperture below that of the filled array in which the inter-element spacing is nominally onehalf-wavelength. In-creasing the interelement spacing hasanothersalutoryeffect; a separation of a few wavelengths reduces mutualcoupling to negligible proportions. Thinning, therefore, isattractive from both points of view. But these benefitsare not free of penalty. Unless t.he element locations arerandomized or made otherwise nonperiodic, grating lobesappear.' Also, the reduction in the number of elementsreduces the designer's control of the radiation pattern in

' the sidelobe region, which in turn influences the level ofthe largest, or peak, sidelobe.

In thispaper the peak sidelobe of the random array isstudied. The paper does' not argue the merits of randomarray design; its purpose is to offer a tool to the arraydesigner who, whether by necessity or choice, is designinga random array.

Since an a priori description of a random array canonlybe given statistically, it is logical to seek a solution in aprobabilistic sense. Such a solution yields an estimator of

Manuscript received March 17, 1971;evised October 19, 1971.This paper is part of a dissertation submitted t.o the Universityof Pennsylvania, Philadelphia, Pa., inpartial fulfillment of th erequirements for the Ph.D. degree.University of Pennsylvania, Philadelphia, Pa. 19104.The author i.; with the Moore School of Electrical Engineering,array is found in [1]-[7] deterministic aperiodic array), 181-1151Extensive theoretical work of t he last decade on the aperiodicstatistical array). The random m a y is characterized by element(nondeterministic randomarray),and [lS]-[19] nondeterministiclocat,ions chosen by some random process; in the stati stica l a m yis removed at random. In all cases a set of phase shiftersis assumeda conventional array is designed, and a given fraction of the elementsso as to eophase the elements at the beamsteering angle.

the peak sidelobe in terms of a probability or confidencelevel tha t thepredicted value will not be exceeded. Specifcally, a probabilistic estimator of the peak sidelobe of thelinear uniform random array with equally-weighted ele-ments is derived and ested by Monte Carlo computesimulations. The result is a practical design theory whichplaces the pertinent design parameters in evidence andmakes their relative weights apparent.Theparametersare the number and size of the array elements, size of thearray, wavelength, beamsteering angle, and signal band-width.

11. LINEAR ANDOM m yConsider an a rray of N unit, isotropic and monochro-

matic radiators a t locations xn on a line of length L. Thex,, are chosen from a set of independent random variablesdescribed by some first probability density distribution,initially assumed to be uniform overL. It is assumed thaeach element, irrespective of its location, is properlyphased so that a main lobe of maximum strength is formeda t e which s measured from the normal to the array.The reduced angular variable u = sin e - in 0, containsthe beamsteering information. The complex far-field radia-tion pattern f(u) s the Fourier transform of the currendensity. Since the latter is a set of delta functions, f(u)is proportional to the sum of unit vectors having phaseangles kxnu, k = 2u/X. The array factor is

Nf(u>= C exp (5 k u ) . (1)

-1

The main lobe amplitude is N , occurring at u =0. Outsidthe neighborhood of the main lobe, the unit vectors combine withrandom phases.Hence thermsamplitude isN112 nd its quare, which is the mean of the power patternis N . Thus the power ratio of the average sidelobe to themain lobe is N / N 2 = 1/N.

Averages alone, however,may be somewhat misleadingFig. 1 illustrates the situation. It is the calculated powerpattern for a random array of 30 elements in which theprobability density function (pdf) of element location isuniform over L = 70X, the element positions having beenchosen bya andomnumber generation program. Thebeamsteering angle O0 =0. The ordinate scale is in dB ,and the abscissa is linear in degrees. The sidelobe evelaverages about -1 5 dB, which is consistent with thepreceding theorem (- 10 log 30N - 5 dB). The theory


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- .30

1. Pattern of 70-wavelength random array of 30 isotropicelements.

inadequate, however, in not alerting he designer to the-7 dB, or 8 dB higher than

111. PEAKSIDELOBESTIMATOREstimation of the peak sidelobe requires knowledge ofe pdf of pattern magnitude. By the centralimit theorem

N ) the quadrature components of the array factorthe sidelobe region are asymptotically normal [ S I and,

distribution is Ilayleigh. Henceis

=Lo ( A ) d A = expmthe probability that he magnitude of an arbitraryof the radiation pattern, away from the region of

lobe, exceeds some hreshold A,. Its complement,- , s the probability that such a sample is less thano .If n independent samples are taken

the probability that none exceeds A o.From ( 3 ) , Ao2=In ( 1 - l n ) . It is convenient to normalize this ex-N , the average sidelobe level, and to give the

ionless power ratio Ao2/N a new symbol, B . ThusB = -In (1 - n ) v nn - n n ,9-l. (4 )

B may be interpreted as a statistical estimator of theo of the peak-to-average sidelobe of a set of nB is a confidence level; it is the prob-

that none of n independent samples of the sidelobeer pattern exceeds the mean value by the factor B .

is an array parameter, which is a function of all theN . It is proportionalthe number of sidelobes in the visible region. It mayAn interesting method

t the number of samples required t o specify a wave

IEEE ?RANSAClIONS O N ANTENNAS AND PROPAGATION, MARCH 1972which is limited in bandwidth to W and in time durationto T s 2WT [20]. The complex radiation pattern of arandom array is such a band-limited function, thelimit being due to the finite length of the array. Thecomparison is evident from the Fourier transforms re1a.t-ing signal spectrum F ( f ) and timefunction f ( t ) , andcurrentdensity i (s /X) and far-fieldcomplex ra,diat,ionpattern f(u) :

where t t) , f w x/X, W t) /X, and Tw nonredundantport,ion of the visible domain of u. The latter interval isusually less than two because the power pat.tern is sym-metrical about t,he beamsteering angle u =0.This fol-lows from (1) and the defining expression for the powerpattern of a.n array of unit radiators

h Nf(u>f*(u)=c c exPMz72 - Gn)u

n mwhichalso equals f( -u)F (-u) since the index n isindistinguishable from m. The visible domain is-1 - in eo 5 u 5 1 - in eo . The length of the non-redundant portion is 1 + sin 00 I. Consequently, the num-ber of independent samples needed to specify the complexradiation pattern is 2(L/X) 1+ sin eo ) . Ha.lf this num-ber may be associated with the amplitude of the arrayfa,ctor and half with its phase. Therefore, the power pat-tern is uniquely specified by

independent samples, the average angular nterval be-tween samples being AIL. n is dominated by t.he length ofthe array n units of wavelength and secondarily influencedby the beamsteering angle.

Equations (4 ) an d (5 ) , however, are insufficient to pro-vide an unbiased estimate of the peak sidelobe. The prob-ability is zero that any finite set of samples of a powerpattern falls exactly upon the crest of the largest sidelobe.Hence such estimation is downward biased. A correctiont.0 (4 ) maybe obtainedby calculating thedifferencebetween the largest of a set of samples and the height ofthe lobe from which t,he sample is taken. Consider a. ran-dom sample point a t u1 lose to a lobe crest. I n t.he neigh-borhood of ul, he power pattern can be represented bythe Ta.ylor expansion


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STEMBERG: PEAH SIDELOBE O F PHASED ARRAY 131

where Pt(ul) is the first, derivative with respect to u. ofP (u = ul), tc. For sufficiently small int,ervals the firstt.hree terms of t,he series suffice to describe t.he lobe crest..Its peak (at. u =up ) s located by sett,ing the derivat.iveof t,he t.runca.t,ed eries to zero. Thus dP/du = 0 =P' +P"(uP- l), nd up- u1 = -Pt/P". Insert,ing up nto(6) (truncated)andsubtract,ing P ( u I )gives the differ-ence AP between the peak value P(u,) and t.he samplevalue P(u l ) in t.erms of the first txo derivatives at thesample point,:

To evaluate t,his expression the array fact.or is repre-sented asa sum of quadrature comp0nent.sa ( u ) and b (u )the power pattern P(u) = a2 ( u )+b 2 ( u ) , nd the deriva-tives P' = 2(aa' +bb')and Prt=2(m" +aJ2+ b" + '2),where a = a(u,),etc. The increment

a2a'z -+2aa'bb' +b2bf2aa" +at2+bb" + I 2AP =-the average value of which can be est,imat,ed. n the.side-

lobe region, a and b are asymptot.ically Gaussian randomvariables [8] 6hich ca.n be shown to be independent. andstationary [21]. The pdfof the six variables in (7) areknown (e.g., [22, ch. 31). The quadrat.ure components a-nd b are uncorrelated with their fist derivatives, i.e.,aa' = =0, but are highly correlat.edwith their secondderivatives. In [22, eq. (3.16)] it. is shown that for largethresholds t,he pdfof ar r (or b") is a narrowly peakedGaussian distribution. Consequently, a" (or b") can beregarded as nonrandom and equal to it,s condit.iona1 mea.n-&a (or -a b ) . The coeEcient

where R ( u ) s t.he aut.ocorrelation function of t.he quadra-ture components. Thus aa" = -&a2, and aa" +bb" =- 2(a2+b2) =- 2P. The ot.her t.erms in the denomi-nator areaf 2 nd bt2.Their pdf are exponent-ial (calculatedfrom [22, eq. (3.15)])

and, therefore, are not narrowly peaked. However, for athreshold large compared t.oN , the contributions of theseterms are not large so that each may be represented inthe denominaaor by it.s average value. From (8) 3

= NR2. Hence the denominator of (7) may be approxi-mated by the nonrandom quantit.y -R2P +2NR2 =The average value of (7 ) is approximately tjhe average

of its numerator divided by this quantity. The a.verage ofthe middle numerator t.ermis zero, as observed previously.The remaining numera.tor terms average to N%(u2+ 2 ) =N&P, and t,he average of (7) becomeswell approxi-

-R!P( 1 - N/P).

mated by

where it is understood t.hat P =P(u1).The sum of (9)and P(u1) is the approximate height of the lobe crest,If P(u1) is t


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IEEE TPLNSACTIONS ON ANTEWAS AND PROPAGATION,MARCH 1972

0 = ~ . Q Q Q

4 I I I l l l l ' I I I I I l l 1 I I I I IIII10 100 1000ARRAY PARWETER n = ( L I ~ ) c ~ + I s I N E ~ J )

2. Probabilistic est.imator of peak sidelobe of random array.N is number of array elements, PSL/ML is power ratio of peaksidelobe to main lobe, 6 is probability or confidence level that. nosidelobeexceeds ordinate, L is array length, X is wavelength,en is beamsteering angle.

to eo = 0",15", 30",45", and 60". The median of the peaksof each set is plott,ed against the arra.y pa.rameter andshown in relation to the theoretical curve, for p = 0.5.The data points for this particular se t of random arra.ysfall above the curve; nevertheless, it is evident t.hat t.hegrowth in the median peak is consistent with the theory.

I V. NONISOTROPICLEMENTSThe foregoing theory assumes isotropic array elements.

When the elements a,re not isotropic, the element patternsuppresses sidelobes of the array factor which are outsidethe central portion of the main lobe of the element pat-tern.The effect is to reduce t.he number of sidelobeswhich can compete t o be t.he largest, thereby decreasingthe a.rray pa.rameter n a.nd the peak sidelobe.

Consider a sidelobe of the arra.y factor at some angleel #0. Let. eo = 0. If the elements radiated isotropicallythe probability that hat sidelobewouldexceed omethreshold B is a =exp (- B ) , as before. However, thelobe st.rength at 81 is reduced by Gl iSince B is on he order of 5 or more,exp (- B ) andexp -B/Gi are smallenough .o permit, replacement oft.he logarithm by he first term of t,heseriesexpansionIn ( 1 +x ) N x leading to

G is the normalized square magnit.ude of the element fa.ctorg ( u ) which, near the origin, may be approximated by thequadratic g ( u ) 'v g(0) +(1/2)g"(0)uz. The curvatureis relat.ed to t.he current density i ( x ) across the element.It is calculat.ed by twice differentiating tlhe Fourier t.rans-form relation g(u)=Ji( x) exp ( jksu) dl : and evaluatingthe second derivative g " ( u ) = J- zs2i(x ) exp ( jksu) dxat the origin. Thus g"(0) = -k2Jx2i( x) d x p -kzIZg(O),where P is the second moment of the current excitation


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STEINBERG: PEAK SIDELOBE O F PH&SED A R R A Y 13across the element., and

G ( u ) = I g ( u ) / g ( O ) l2 =1 - 2Pu2, forSimilarly, 1/G N 1 +k212u2 nd 1/Gi - small angles.1 2r: k 2 h i 2 =4d12u:/X2. Since ui =Xi/L, X/L being the sampling inter-val, (13) becomes

Equation (15) may be written

where 2 =L2/8P12B. The first termntegrates toa(?r/2)I2 . The second term is less t.han 5 percent of thefirst term for L/Xa 2 2 (evaluatedrom [23, eq.(26.2.12)]. Since the dependence of B upon n is logarith-mic (see (4)),a 5-percent error is exceedingly small. Thus

As an example, consider an element, of length I =sXuniformly illuminated over it.s length. Thus

and he condition on B becomes B 2 6/.K2.9, which iseasily satisfied for s 2 /2. For smaller e1ement.s the effect,upon t.hesideIobe pattern disappears and he isot.ropiccase pert.ains. The condition is similarly satisfied for mostpracticalcurrent distributions. The restrict.ion on (16),t.herefore, may be dropped. For the uniformly illuminatedelement

Fig. 5 shows two test.s of (17), one for L = lOOX andN = 100, and t.he other for L = 4001 and N = 30 . Ineach test ten uniform random arrays were generat.ed. Theelement,wasassumed t.0 be illuminated uniformly. Theva,lues of s were 1/2, 1, 2, 4, 8, and 16 for t.he 4oox arrayand 1, 2, 4 , 8 for bhe lOOX array. The st.eering angle waseo =0. The peak sidelobe of each r ray wa,s found for eachvalue of s , and t.he median of each group is plotked in thefigure along with the t,heoretical curve for /3 = 0.5. Theisotropic case (s =0) is also included. The variation ofpeak sidelobe with element size appears consistent wit.htheory.

To account, for eo #0 t.he lower limit in ( 15 ) is shift.edby m, he number of independent samples between theorigin and sin eo, i.e., 1 sin 00 I = m ( X / L ) .Equation (15)becomes

n N J exp-dy =LIX -y

--m 2 2 0/-:..+1...

I I I I I I I I I I I I I I I I I I I I I I I I I L j10 100 l0OCAU R AY P A R W , E T E R n = L / ~ > . s B ~

Fig. 5 . Effect, of element ize 1 = S A uponpeak idelobe. Eacsamplepoint, is median of experimentaldktribution of peasidelobe of ten linear random arrays.

The second integral is (15) which eva,luat,es to (16). Thfirst integral represents the increase in n. due to steerinoff t,he array normal. For small steering anglesIoxp(5)y =m --+ - *?n3

-7n 6a2

and

V. WIDE-BAND IGNALSpreading the signal energy over a band of frequencie

further reduces n. Let there be two frequency componentw1 and w2 in the signal passing t.hrough the array. Thecomposit,e complex pat.tern is the weighted sum f(u)=f ( u , w ~ )+Q f ( u , ~ ) ,here q is t.he relat,ive strengths othe two omponent,s.More generally,f(u) f s (w ) f ( u ,w ) dwwhere s ( w ) ,w =2 r v , is hhe Fourier spectrumof th e signalThis process leaves the centra1 port,ion of the patatern ntac t while smoothing the portions away from the mailobe. Consequently, the sidelobecrest,s shrink and t.hnulls fill in, t.hereby diminishing t,he probability tha,t. t,hdistant sidelobes contribute the pea.k of the random sidelobe pattern.

For this calculation it is useful to make (1) an explicifunct,ion of w ; replacing k by w / c , c = speed of light

Consider a crest in the sidelobe pat,tern occurring at UO,WOwhere wo is the nomina,l cent,erof the signal spect.rum. I nthe neighborhood of wo and at u =uo,f(u,w) may b


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4L-

IEEE TRANSACTIONS ON A K ~ N N A S ND PROPAGATIOK, MIARCH 1972

(w - o) 2 is proportional to thew. This quantit.y,

f a random variable, is it.self a randomle, the dist.ribution of which is needed for the calcu-

f bandwidth upon the peak sidelobe.because of the weak (approximately 1ogarit.h-B upon n, an estimate of t.he effect ofh can be obtained by replacing the ra.ndom vari-a typical value, a.s, for example, that which is

on is the Fourier t.ransform of t.he first probabilityy distribution of element location [SI. For the uni-random array of length L it isfo 1 L l 2 exp f?) - sin d u / 2 cOLu/2C * (19)2 -LIZa lobe crest (u = uo, w =wo +So)

+Sw) = os LuoSw/2c 1 - 1/2) Ww/2c)2w&uo/2c wolu0/2c-(20)

e magnitude of the crest,, compared to its magnitudeo bandwidth, is .he integral of (20)weighted by

signal spect,rum and normalized to the same weightedwit.hSw = 0. This quantit,y s the gain g(%) byuo is reduced due to hhe nonzero

,h of the signal spectrum.s an example, for the uniform spectrum of width Aw

(21) has the same form as (14).By equatingcoeEcients of t,he angular variables, solving for I and( lS), the array parameter is found to beoo/AwB112+( L / X ) sin eo 1 subject to the conditionit not exceed the number of independentsample

( 5 ) . Thus n becomes the smaller of the(5). Inclusion of the curvature distri-on n he derivation wouldmodify the bandwidth

a coefficient but would not alter its functionalA similar comment pertains to the shape of the

np: t8-0.5

0

0LoI I I I I I IIII I I I 1 1 1 1 1 l I I I I I I l l

10ARRAY PARAMETER n C L t a ~ c l ~ l s r t a o ~ )100 1000

Fig. 6. Effect of number of elements hT upon peak sidelobe. Eachsamplepoint is median of experimentaldistribution of peaksidelobe of ten linear random arrays.

signal spectrum. Hence

under fairly general conditions. The crossover occurs near( J , , / A W B ~ ~L/X or at abandwidth which is approxi-mately the reciprocal of the travel time f light across thearray.

VI. FINJTENTT~WERF ELE~~ENTSThe maximum possible sidelobe amplitude is N , which

is equa.1 to the main lobe. On the other hand, theasymptotic pdf (2) on which the theory is based is unbounded.Hence discrepancies may be expected for small N . It ispertinent to determine the smallest value of N for whichthe asymptotic theorys an adequate representation.A measure of the difference between the finite sampledistribution and the asymptotic istribution is the integralof (2) from N to a,which is exp (- N ) This quantityshrinks rapidly with N , e.g., xp (-5)


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S T E I N B E R G : P E d g SIDELOBE O F PHASED ARRAY 135should drop below the curve. The data for N = 100 andN =30 do continue to increase with n over the entirerange shown. In both cases the limiting va.lue N is notapproached. The data for N = 10, on the otherhand,drop and remain below the theoretical curve when theyreach within a factor of two of the ma.ximumpossiblevalue, suggesting that the umber of elements rat.her thanthe number of sidelobes has begun to dominate. Theseobservations lead to the conclusion that the minimumnumber of elements for which the theory is satisfactory isth e larger of 15 or 2B(n,B),or

N,, = (15,2Bj. (23)VII. SPATIALAPERR NONUNIFORMLEMENT

DISTRIBUTIONTapering the current density across an array is a com-

monly used procedure for reducing the sidelobe level. Itmight be expected, therefore, also to be useful in randomarrays. This appears not to be the case.

The analog of durrent density for the random array isthe pdf of element location [SI. Its effect upon the peaksidelobe is through the array parameter n,which is pro-portional to the -number of sidelobes. With a spatiallytapered aperture, i.e., higher density of elements near thecent,er than near the edges, the beam cross section is in-creased, and n decreases proportionally. However, theslope of the peak sidelobe level with n is small (see (4 ) ) .tapering can usually be neglected.

For example, changing from a uniform aperture to atriangular excitation in a conventional array increases thelobe cross section by 44 percent and reduces the peak side-lobe by approximately 13 dB. The lobe width of a randomarray with triangula.r pdf is similarly increased by 44 per-cent, and ita array parameter n is reduced by the sameamount. However, this reduction in n has a verymodest effect upon the peak sidelobe. Consider a lOOXarray teered to eo = 2 6 O . Thearrayparameter n =(L/X) (1 + sin eo ) = 144. The array parameter for atriangular array of the same length is 100. At the 90-per-cent level of confidence (from Fig. 2) B , = 8.51 and 8.18,respectively, for the two arrays, which amounts to 0.2 dBdifference, demonstrating tha t the effect of spatial taperupon peak sidelobe level is slight.

VIII. PLANARRRAYExtension of the peak sidelobe theory to two- andthree-dimensional arrays requires only a reevaluation ofthe array arameter n. Consider aa an examplea rectangu-lar planar array having sidesL nd Lz and uniform pdfof element location. The average patterns in lanes throughthe center of the array and parallel to either of the twosides is given by (19). The angular interval for independ-ent sampling of the patternamplitude in these orthogonalplanes is A/L and X/& (Section 111). The area in theUI-wplane associated with each sample point is on theorder of X2/LL. he visible area of the plane, which is acircle of unit radius, is T . Hence the maximum number of

independent samples over the hemisphereis approximatelyrLLz/X2.The same result pertains to a three-dimensionaarray in which LlLz is the projected area upon a planeperpendicular to the axis of th e main lobe of the elementfactor.

Symmetry in the pattern reduces the number of in-dependent samples. With the array steered to the zenith(0, =0) each lobe in every polar cut has an image lobein the same p h e . Thus the range of variation of n witheo is a factor of two. The logarit,hmicrelation (4 ) betweenpeak sidelobe and the array para.meter minimizes the im-portance of the detailed variation. The dominant featureis the approximate squaring of n when a. fixed number o felements N is spread from a linear array toa planar arrayof the same length and width. The result is (approxi-mately) a doubling, or %dB increase, in the peak side-lobe.

IX. CONCLUSIONS1) N (number of elements) is the dominant quantity,

influencing the peak sidelobe linearly.2) X (wavelength), L (array size), I (element size),and Au (signal bandwidth) have logarithmic effect8 uponthe peak and, therefore, are less influential in determiningthe peak value than N .3) 0, (beamteering angle) has a minor influence.4) The effect of spatial taper of the element density

upon the peak sidelobe is slight.5 ) The peak sidelobe of ~ the random planar array is

approximately 3 dB larger than that of the ra.ndom lineararray of the same length and same number of elements.

ACKNOWLEDGMENTTheauthor wishes to a.cknowledge with appreciation

the many valuable suggestions made by Prof. R. SBerkowitz of the the Moore School of Electrical Engineer-ing. In addition, discussions with A. E. Zeger of GeneraAtronics Corporation, and the fruitful ideas which devel-oped from them, are much apprecia,t,ed,as is the machineand program aasist,ance of other former colleagues atGeneral Atronics, particularly J. T. Beardwood, 111M. L. Cohen, and E. D. Banta.

REFERENCES[I ] D. D. King, R . F. Packard,and R. K. Thomas,Unequally-spaced,broad-bandantennaarrays, IRE Trans. AntennasPropagat., vol. AP-8, pp. 380-384, July 1960.[2] S. S. Sandler,Someequivalencebetweenequallyandun-equally pacedarrays, IRE Trans. Antennas Propagat., vol[3] A. L. Maffett, (Array factors with nonuniform spacing param-Ap-8, pp. 496500, ept. 1960.eters, IR E Trans.AntennasPropagat., vol. AP-10, pp. 131-136,Mar. 1962.[4] R E. Willey,Space aDeringof inearandplanararrays,

July 1962.IRE Trans. Antennas Propagat. , vol. AP-IO, pp. 369-357,[5] A. Ishimaru, Theory of unequauy-spaced arrays, IR E Trans161 M. I. Skolnikand J. N. Sherman,Planararrays mth un-Antennas Propagat., vol. AP-10, pp. 691-702, Nov. 1962:equally spaced elements, Rudw Electron. Eng., vol. 2s , no. 3,Sept. 1964.[7] A. Ishimam and Y. S. Chen,Thinningandbroadbandingantenna arrays by unequal spacings, IEE E Trans. Antenrm[SI Y. T. Lo, A mathematical heory of antennaarrayswith

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IEEE TRANSACTIONS ON ANTl3NNASAND PROPAGATION, VOL. AP-20, NO. 2, MARCH 1972randomly spaced elements, IR E Tr am . Antennas Propagat.,vol. AP-12, pp. W7,268, May 1964.-, A probabilatlc approach o the problem of large antennaarrays, Rudw Sci., vol. 68D, pp. 1011-1019, Sept. 1964.Y. T. Lo and S. W. f;ee, Sidelobe level of nonuniformlyspaced antennarrays, IEE E rans . Antennas Propagd.-, A study of space-tapered arrays, IEEE Trans .Antennas( C m m u n . ) ,vol. AP-13, pp. 817-818, Sept. 1965.Y. T.Lo and R. J. Simcoe, An experiment on antenna arraysPropagat., vol. AP-14, pp. 22-30, Jan. 1966.Propagat., vol. AP-15, pp. 231-235, Mar. 1967.withandomlypaced elements, I E E ETrans. AntennasY. T. Lo and V. D. Agrawal, A method for removing blindnessin phased arrays, Proc IEEE (Lett.), vol. 56, pp. 1586-1588,A. R. Panicaliand Y. T. Lo,A erobabilisiticapproach toSept. 1968.largecircularand phencalarrays, I E E ETrans. AntennasPropagat., vol. AP-17, pp. 514-522, July 1969.V. D. Agrawal and Y. T. Lo, Distribution of sidelobe levelin random arrays, Proc I E E E (Lett..), vol. 57, pp. 1764-1765,Oct. 1969.

[16] J. L. Allen,Some extensions of the theory of random erroreffects onarraypatterns, in Phased Array. Radar Studies.Lexington, Mass.:Lincoln Lab.Rep. 236, ch. 3, pt. 3, Nov.1961.[17] T. M. Maher and D. K . Cheng, Random removal of radiatorsfrom large linear arrays, IEEE Trans. Antennas Propagat.,[18] Y. T. Lo, Random periodic arrays, Rudw Sci., vol. 3, pp.vol. AP-11, pp. 106-112, Mar. 1963.425-436. Mav 1968.[19] M.R. I. SkOl&,Collin and Nonu.hiformF. J. Zucker,arrays, in AntennaEds. New York: McGraw-Kill,Theury, pt. 1,1969, ch. 6.[20] P. M. Woodward, Probabiilityand Infmmation Them y, with

Applications to Rcrdap. New York: McGraw-Hill, 1953.[21]R. . Berkowit.z, Moore School Elec. Eng., Univ. Pennsylvania,Philadelphia, 1971, unpublished memo.1221 R. L. Stratonovitch, Topics in the Theory of Random Noise ,vol. 2. New York: Gordon and Breach, 1967, ch. 3.1231 M. Abramowitz and I. A. Stegun, Eds., Handbook of Math-ington, D. C . :U.S. Govt. Print. Off., 1964.matieal Fundions (Applied Mathematics Series 55). Wash-

n Measurements of Standard ElectromagneticHorns in the K and K, Bands

re escribed in which the gains ofin the K andK , bands were determined at aumberwavelengths. The two-antenna method was used. It is believed,

adetailed error analysis, hat the gain is known within aof about 0.06 dB ateachwavelengthmeasured. Ais described for measuring the ohmic losses in the horn,

that the measuredgainsmay be compared with theoreticalfrom the formula of SchelkunoffFriis and corrected for ohmic losses agree within about 0.1 dBthe average with the measured gains.

I. INTRODUCTIONHE ACCURATE calibration of a radio telescope an-tenna by the gain comparison method relies on the

ecise gain det,ermination of the standard antenna usedThis art.icle summarizes the calibra-f two standard horns in K band and K , band whichused in t,he calibrat.ion of a 20-ft diameter millimet,er-

ve ntenna. Previous accurate horn gain determinationslonger Wavelengths have been described by Chu and

[l], Slayton [2], Jull and Deloli [3], and Jakes

18, 1971;revised September 20, 1971.T. Wrixonwas with the Radio Astronomy Laboratory, theSciences Laboratory, nd theDepartment of ElectricalHill Laboratory, Bell Telephone Laboratories, Inc.,University of California, Berkeley. He is now with

J. 07733.J. Welch is with the Radio Astronomy Laboratory, the Spaceof California, Berkeley, Calif. 94720.ences Laboratory, and the Department of Electrical Engineering,

[4]. Beatty [5] has given a general discussion of theerrors that arise in horn gain measurements, and Bowman[6] has critically reviewed work on this problemprior to1968, giving estimates of theaccuracy that one mightexpect to at,t,ain n such determinations.

In order that t,he horns could be calibrated by the two-ant,enna technique, two nearly identical pairs of hornswere constructed for each band. The two K-band horns,which differ by a small amount in length were electro-formed. The K,-band horns, which have identical dimen-sions, consist, of smooth brass sheets soldered together ona mandrel. All t.he horns have their inner surfaces silver-plated. The dimensions of the horns are summarized inFig. 1.

The calibrat.ion is based on accurate measurements ofthe traiumission loss between the pairs of horns with d i f -ferent separations. The gains are then worked out withaid of the proximity corrections of Chu and Sempiak [l].

Finally, the measured gains, a t 4 frequencies in K,, bandand 3 frequencies in K band, are compared with calcu-lated values from the formula of Schelkunoff and Friis[7]. Before the comparison is made, the estimated ohmiclosses in eachhornaresubtractedfrom the calculatedgain. These loss estimates are based on measurements ofthe Q of each horn in a resonant cavity configuration.The generally good agreement between theory and meas-urement suggests that Schelkunoffs formula is accurateto within about 0.1 dB for horns with flare angles andgains comparableto the ones used in thepresent study.

the peak sidelobe of the phased array having randomly located elements

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