IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 2 , MARCH 1 9 8 2
273
Synthesis of Antenna Patterns with Prescribed
Nulls
HANS STEYSKAL, MEMBER,IEEE
Abmucr-The least meansquare pattern synthesis method is extendedtoincludeconstraintssuchas
pattern nulls orpatternderivative nulls at a given set of angles.
The problem is formulated as
a constrained approximation problem which is solved exactly, and a
clear geometrical interpretation of the solution in a multidimensional
vector space is given. The relation of the present method to those of
constrained
gain
maximization
and
signal-to-noise
ratio
(SNR)
maximization is discussed and conditionsfor their equivalence stated.
For a linear uniform ,%‘-elementarray it is shown that, when A4 single
nullsareimposedonagiven“quiescent”pattern,
theoptimum
solution for the Constrained pattern is the initial pattern and a set of
”weighted (sin :\!r)/sin x-beams. Each beam is centered exactly at the
corresponding pattern null, irrespective of its relative location. For
the case of higher order nulls, the nth pattern derivative is similarly
canceled by the nth derivativeofa
(sin ;V.x)/sin x-beam. In addition,
simple quantitative expressions are
derived for the pattern change and
gain cost associated with the forced pattern nulls. Several illustrative
examples are included.
metrical interpretation in a multidimensional vector space. It
will beshownthatundercertainconditionsthepresent
approach yields the same result as the methods of constrained
gain maximization and as SNR maximization. The least mean
squareerrorcriterionwith
single-null constraintshasbeen
lucidlydiscussed [ 4 ] in very general terms and as applied to
satellite multiple-beam antennas. In contrast we
will study the
classic problem of pattern synthesis for a linear array
ofisotropic elements, which leads t o a slightly different formulation
and some complementary viewpoints and results. The use of
the derivative to broaden a pattern null was first suggested in
[ 2 ] and recently applied in the context of pattern synthesis
in [ 51 and adaptive systems in [ 61.
11. FORMULATION OF THE PROBLEM
We considerasituationwhereanarrayantenna
is being
illuminated by desiredsignalsand
also byhighlydominant
1. INTRODUCTION
interference
signals
from
certain
discrete
directions.
The
optimum antenna pattern for this case is reasonably defined as
HE PROBLEM of forming nulls in the radiation pattern of
so called
an antenna, in order to suppress interference from certain t h e desired pattern in the absence of the jammers, the
quiescent
pattern,
suitably
modified
so
as
to
form
pattern
directions, presently receives much attention. Most work is in
the area of adaptivenullingsystems,asdiscussed
by Apple- nullsintheinterferencedirections.Thedegreesoffreedom
baum [ 1 1 whereaperformanceindexsuch
as the signal-to- available in the antenna pattern are thus used in first place t o
of freedom
noise ratio (SNR) is maximized. In the case where jammers are formthepatternnulls,withremainingdegrees
being
used
for
approximation
of
the
quiescent
pattern.
thedominantnoisesourcethisprocessautomaticallyplaces
The corresponding antenna pattern synthesis problem conpatternnulls in thedirectionsofthejammers.Aseemingly
p a to a given
differentapproach
is that of Drane-McIlvenna [ 2 ] where sists of determining the closest approximation
quiescent
pattern
P
O
,
subject
to
a
set
of
null
constraints.
The
another index, antenna gain, is maximized, subject to a set of
solution
of
this
problem
requires
a
definition
of
“distance”
null constraints on the pattern. In both methods the performbetween two patterns and this will be defined in Gauss’s sense
ance index is the quantity of prime interest, whereas the role
as
themeansquaredifferencebetweenthepatterns.This
of the antenna pattern is n o t too clear, which to an antenna
particular
metricprovidesanoverallmeasureofapproximaengineer is unsatisfactory. The purpose of this paper is to show
that the problem can be formulated as a direct pattern synthe- tion, and in contrast to, for instance, the Chebyshev approximation,places no explicit bound on the maximum deviation
sis problem which includes the pattern nulls.
from the desired function at any particular point. However it
Anarrow-bandinterferencesourcecan
be suppressedby
is the only metric that allows the approximation problem to
imposingasinglenullintheantennapatternattheproper
be solved with any senseof generality.
angle.Awide-bandjammer,however,appearstocoveran
Forsimplicity we consideralineararrayof
111 isotropic
angular sector of the pattern because of the frequency dependantennaelementswithuniformhalf-wavelengthspacing.Setence of the antenna, and therefore it may be required to null
ting u = sin 0 where 0 is defined in Fig. 1 , the antenna far-field
anentiresector.Thiscanbedoneeither
by imposingsingle
pattern is described by the array factor
pattern nulls on a set of closely spaced points over the sector
or by imposing nulls in the pattern and its derivatives at the
center point of the sector.
In circuit theory this is analogous
to the Chebyshev and Butterworth filter alternatives.
We will
consider the synthesis problem for both cases and also derive
expressions for the effects of the imposed nulls,
in terms of
where x, denotesthecomplexexcitation
of thentharray
pattern change and gain cost.
Thesynthesismethod
is basedonleastmeansquare
or element.
The synthesis problem can now be stated mathematically:
Gaussianapproximation [ 3 ] , whichallowsanattractivegeofind the pattern pa(u), such that the mean square difference
T
hlanuscripf received September 18, 1980;revised June 19, 1981 and
August 13. 1981.
The author is with the ElectromagneticSciences Division, Rome Air
Development Center. Hanscom AFB, MA 01731.
U.S. Government work not protected by U.S. copyright
274
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 2 , MARCH 1982
FIELD POINT
‘I; ‘I; Y
x,
-
x2
y
‘I;
where X0 and Fa are the excitation vectors corresponding to
the patterns p o and p a , respectively. Likewise the pattern constraints(2b)canbeexpressed
as constraintsonthearray
excitation. The mathematical expressions simplify somewhat
if we first multiply the pattern function
p by a phase factor
e x p (i$u), where
ARRAY
ANTENNA
- - XN
N+ 1
$ =-,
EXCITATION Ixnl
2
which shifts the phase center of the pattern
to the array center.
Substituting this new function in (2b), we find in view of (1)
that a single or zero-order null a t u = up requires
APERTURE
FIELD Ip(u1l
FAR FIELD
and in general, for a vth order null
u =sin 8
-1
0
1
N
Fig. 1. Array antenna: its aperture and far field.
subject to the constraints
dV
- p a ( u p ) = 0,
duV
p = 1, .-, M I , v = 0, .-*,M,,
(2b)
}
~ the
~ angular
1
location of the M I interwhere { ~ ~ denotes
ferencesources.Sincethetotalnumber
of constraintsmust
notexceedthenumberoffreevariablesitisrequiredthat
M,(M,
1) < N .
We assume that the desired quiescent pattern
is given as a
sum of N harmonics, as represented by
(1). For the general
case where p0(u) has any functional form, p o may be simply
approximated by the first N terms of its Fourier-series expansion.Althoughthesynthesisprocedurethen
involves t w o
subsequent approximations, it iseasily shown to lead to the
correct least mean square approximation of the initial pattern
171.
+
111. METHOD O F SOLUTION
Thesynthesisproblemposedabove
is mostconveniently
formulated inamultidimensionalvectorspace,whereeach
pointrepresentsonearrayexcitation,whichallowsaclear
geometricalinterpretation
of theapproximationsinvolved.
From the solution for the array excitation the desired pattern
is then simply given by (1). The underlying principle for this
equivalence between array excitation and radiation pattern is
of course Parseval’s theorem.
We introduceanN-dimensionalexcitationspace
X, in
which the array excitation{ x , } ~is represented
~
by the vector
= (xl,
X N ) . The inner product of two vectors we define
as (2,7)= Z,xny,*, wheretheasteriskdenotescomplex
conjugate, and the norm
= (F,
In order to express the mean square errorE in terms of the
arrayexcitationwesubstitute(1)
in(2a)andobtainafter
integration
x
a**,
llxll
finallylets us write ( 6 ) asorthogonalityconditionsonthe
array excitation
Notethatwenowhavecharacterizedeachcombinationof
jammerdirection u p andpatternderivationorder
v by one
constraint vector. Since we have a total of M = Ml(1 + M 2 )
such combinations, see (2b), we have
M different constraint
vectors. In the following we will suppress the superscriptv and
y m , where m runs from 1
denote the constraint vectors just by
to M.
In view of (3) and (8) the synthesis problem, as expressed
by (2), now becomes
E=IIx~-x,II~=~~~
(Fa,&)
= 0,
(9a)
m = 1, ...)M ,
(9b)
where Eo and Ea denotetheunconstrainedandconstrained
array excitation, respectively.
Equation (9) shows that the desired solution
Fa is orthogonal to the constraint vectors { U m } l M . A geometrical interpretation of this relation is obtained if the excitation space X
is divided into an M-dimensional subspace Y , spanned by the
vectors
and
its
(N-M)-dimensional
orthogonal
complement 2. Any vector now has a unique decomposition
{u,x}lM
[81
x
STEYSKAL:
where 7E
Y ,
275
OF ANTENNA PATTERNS
Z E Z , FLY. and because of this orthogonality
11x1I2 = llB1I2 + 1 1 ~ 1 1 2 .
Usingthisdecompositionfor
(lO),and(ll)
(1 1)
X0
and X a we getfrom
(91,
Equation (1 2b) yields
Y Q
Fig. 2. Geometrical illustration of approximationproblem: desired
point = xo, closest approximation confined to subspace 2 is x, =
= 0,
ZO.
and therefore E in (1 2a) is minimized by setting ?, = TO leading to the sought constrained excitation
becomes
‘and the least mean square error
When N is large, ( 1 8) can be approximated as
Equations ( 1 4) and (1 5) constitute the mathematical solution
totheposedproblem.Itsproperties
will nowbediscussed
from various points of view.
The method of solution is illustrated in Fig. 2. The excitation
which is t o be approximated, has the projections Yo
and To in subspaces Y and Z . Equation(9b)impliesthatthe
to the constraint vector set
approximation ‘ 0 is
{Ym}lX’ which spans
and therefore ‘ 0 is ‘Onfined to the
subspace Z . Under these circumstances the best approximation
t o X0 is obtained b)! setting Xa = 20, since of ail elements 2 E
Z this point is closest to -TO.
Returning to the solution for the constrained excitation as
7 0 is a linear
Of the
given
by
( I 4 ) we
note
that
vectors yi,, and therefore Fa may be written
:‘
Thus the pattern p o is simply given by the quiescent pattern
p o and M superimposed sinc-beams. This result agrees with the
single-jammerconsideredin
[ 11 andthe general conclusion in [ 4 1 ,
The Jf cancellation
beams
represent
,If degrees
offreedom,
andclearlyitshouldbe
possible to realize A4
nulls
with these. However it is noteworthy that each of these beams
is centered exactly on the corresponding null, irrespective of
theirrelativelocationandthatthebeamshape,givenby
sin ((S.rru12)!sin (nu,2). is fixed regardless of how much the individual
beams
over]ap, Similar
hold
for the
can-
cellation beams corresponding to higher order pattern
derivatives. These properties are consequences
of the isotropic array
elementsandtheleastmeansquareapproximation
we have
(I6)
adopted.
Thepresentsynthesismethodwith
single-null constraints
will yield the same pattern as does SNR maximization
[ 11 in
where the coefficients a,, willbe determined later. Presently
the limiting case, where the jammers become infinitely strong.
we infer from (16) that the sought excitation X a is composed
ThislatterconditionforcestheoptimumSNRpatternto
of the-quiescent excitation -TO and a weighted sum of the vecmaintaintruenulls,ratherthanshallowdips,
in the jammer
. thedualrole
of thesevectors:initiallythey
tors J ’ , ~ ~Note
directions and then the two methods are comparable,
as shown
characterized a constraint, now they represent an array excitain theappendix.Further,we.alsofindequivalencewith
tion.
constrained gain maximization [ 21 in the specialcasewhere
As for the resultant antenna pattern. it follows from
(1 6)
p o is a maximum gain pattern, on which a set of single nulls
and the linear relation between the array excitation and the
is imposed.Minimizingthepatternchange
E then simultanepattern. that the constrained pattern p,(u)will be the quiescent
ously minimizes the gain cost. as shown below, in conjunction
pattern Po(2‘) with 11;1 beamssuperimposed.Thebeamcorwith (26), and thus the constrained least mean square pattern
responding to the excitation ?,,, we call a cancellation beam,
coincides with the constrained maximum gain pattern.
denoted by 4,7J(u).and it is easily shown by using (7) in (1)
Compared to these methods, however, the pattern synthesis
that
method is a more direct and therefore conceptually more appealing approach, which provides valuable insight into fundamental pattern properties.
IV. THE SYNTHESIZED PATTERN
where p and v are determined b>/ the index J f 7 . For the case of
N single nulls in the pattern ( V = 0 ) the constrained pattern
The pattern p a . which satisfies t h e desired null constraints
is given b>f (18) where, however, the coefficients a,, so far are
276
IEEE TRANSACTIONS
ANTENNAS
ON PROPAGATION,
AND
VOL. AP-30,
NO.
2, MARCH 1982
unknown.Theymaybedeterminedfrom(16)and(12b)
which leads to the following system of equations:
'
- '30,
0
G = G(71, -., 7 ~is t)h e coefwhere the Gram determinant
ficient matrix in (20): see [ 81, and Dm is the determinant of
the same matrix with the mth column replaced by the column
vector ((ul, TO), -., (UM,XO)).Notethatthereareonlyas
many equations as there are constraints and usually therefore
(20) will represent a small system of equations, which
will be
I\
I
easy t o invert.
To illustratethesynthesismethod
we haveprogrammed
(21) on adigitalcomputerandcalculatedafewactualpatterns. We considered "sinc-patterns" defined by the function
sin (Nm/2)/hr sin (nu/2) and Chebyshev patterns, since they
of
areinasensecomplementary-theformerhavesidelobes
constant width and varying height, the latter have sidelobes
of varying width but constant height. In the first examples we
chose the original pattern P O t o b e a sinc-pattern and imposed,
respectively, a zero-, a first-, and a second-order null in the pattern at the rather arbitrary value u = 0.22. Fig. 3 shows that
thenullwidthindeedincreasesasexpectedandalsothat
the changes in the pattern are rather local.
As a comparison to the latter case, which uses three degrees
offreedomforsidelobecancellation,
we alsocalculatedthe
pattern with three single nulls located atu1 = 0.21, u2 = 0.22,
u 3 = 0.23. Fig. 4 shows that in this case we do get somewhat
less sidelobe cancellation but over a wider sector. In contrast
t o derivative nulling the placement of single nulls thus provides
-IC94
:
:
.
)
I
.
,
efficient control over the trade-off between cancellation ratio
I
0
I
u=sin8
and sector width.
(d)
In the next two examples the unconstrained pattern
is a
Fig. 3. (a) Initial sinc-pattern. (b) Pattern with a null of zero order
40-dB Chebyshev pattern in which we place four
single nulls
imposed at u = 0.22. (c) With null of first order. (d) With null of
over a narrow sector (0.22, 0.28) and eight single nulls over a
second order. Pattern change = 0.01, 0.03, 0.13, andgaincost =
wider sector (0.22, 0.36), respectively. In both cases the nulls
0.06, 0.11, 0.59 dB, respectively. Twenty-one array elements.
areequallydenselyspaced
Au = 0.02apart.Theresultant
patterns, given in Figs. 5 and 6, show a sidelobe cancellationof
30 dB and 51 dB, respectively, over the sectors. This
is a surprising fact. Intuition would lead us to expect less cancellation
for the wider sector, which contains a larger number of nulls,
i.e.,alarger
number ofsuperimposedsinc-beams,whoseuncontrolled sidelobes we might expect to add up to a relatively
higher average sidelobe level between the nulls.A related problem of practical interest concerns the numberof nulls required
to suppress a jammer over
a given bandwidth, which will be
addressed in the future.
u=slna
Finally, we showasinc-patternanda20-dBChebyshev
Fig. 4. Sinc-pattern with three nulls
equispaced
over
the sector
pattern in Figs. 7 and 8, again with four single nulls equispaced
(0.18, 0.26). Sidelobe Cancellation = 36 dB, pattern change =
0.12, gain cost = 0.55 dB. Twenty-one array elements.
over the interval (0.22, 0.28). The sidelobe cancellation in this
,
'
277
STEYSKAL: SYNTHESIS OF ANTENNA PATTERNS
case is 34 dB and 32 dB respectively, which is of roughly the
same magnitudeas t h e 30 dB obtained for the 40-dB Chebyshev
patternabove.Thisindicatesthatthecancellationratio
is
rather independent of the actual sidelobelevel. It thus takesas
many degrees of freedom to suppress the sidelobe level, for
example, from 20 dB to60 dB, as from 40 dB to 80 dB.
V. THE EFFECTSOF NULL CONSTRAINTS ON
THE PATTERN
-100;
1111
:
c
,
,
U'SlP3
I
I
Fig. 5 . Initial40-dBChebyshev
pattern with four nullsequispaced
over thesector(0.22,
0.28).Sidelobe cancellation = 30dB, pattern change = 0.001, gain cost = 0.04 dB. Forty-one array elements.
I
It is clearthatforcingtheantennapatterntozeroat
certaindirectionsdoesaffectthepatternovertheentire
angular region, and the extent of these effects
is a matter of
practicalaswellastheoreticalinterest.Thefollowingtwo
measuresforthedifferencebetweenthequiescentandthe
constrained pattern seem natural.
1 ) Pattern Change E = 1/2 J I PO - p a l2 d u : This overall
measure is relevant for shaped beam patterns or for patterns
described over the entire angular sector.
2) Gain Cost eg = G,(ul) - G,(ul): This is the reduction in
maximum directivity in the look direction ul and is of interest
particularly for pencil beams.
The pattern change which by definitionis minimized by the
pattern p a is given by (1 I), (14): and (1 5) as
I
0
u=%n3
Fig. 6. Initial 40-dBChebyshev patternwith eightnullsequispaced
over the sector (0.22, 0.36). Sidelobe cancellation = 5 1 dB, pattern
change = 0.004, gain cost = 0.15 dB. Forty-one array elements.
I
-!
This form is convenient when the constrained excitation x, is
explicitly known.
A simple estimate for emin is obtained for thecase of single
nulls when the constraint vectors
y m arenearlyorthogonal,
so that
@m,~n)<tlFm
II lI7nl1,
W If n .
(23)
This condition issatisfied as soon as the jammers are more
than a beamwidth (2/N) apart. In this case the vectors{?m}lM
afternormalizationby
IlF, = fi formanapproximately
orthonormal basis for the subspace Y , and thus we find
-I
0
u=sme
I
Fig. 7. Initial sinc-pattern with four nullsequispacedover the sector
(0.22, 0.28). Sidelobe cancellation = 34 dB, pattern change = 0.03,
cost gain
= 0.13 dB. Forty-one
elements.
array
Emin = 1170
1 1
l
=N
~
1
N
lM
2I
(FO,F~)I~
I
2Ml Ip0(um)l2.
(24)
Equation. (24) for emin seems reasonable since 1) the cancelp o to produce the
lation beams, which are superimposed on
nulls, are proportional to
p ~ ( u , ~and
) , 2) the beamwidth of
the cancellation beams, and therefore, their power content
is
inversely proportional t o N .
The quantity emin,normalized to the total radiation power
11 50 ]I2, is given in the text accompanying Figs. 3-8. Clearly
the pattern changeis insignificant for all cases considered.
The gain cost eg evaluated at u = ul is
Fig. 8. Initial 20-dBChebyshev pattern with four nullsequispaced
over the sector (0.22, 0.28). Sidelobe cancellation = 32 dB, pattern
change = 0.04, gain cost = 0.03 dB. Forty-one array elements.
Equation(25)canbeexpressedinterms
of thearray
excitation if we, for the look directionuI:define a vector XI=
27 8
IEEE TRANSACTIONS ON ANTENNAS
AND
For the particular case where the quiescent pattern
p o is
asinc-patterndirectedat
u = ui, itis foundthatthecorrespondingexcitation
= x l and eg(u1) = 2(llX0 112 11’) = 2e. Inthiscasethereforeminimizingthepattern
change E simultaneously minimizes the gain cost. as mentioned
earlier.
Anestimate for Eg can be obtainedwhentheconstraint
vectorsareapproximatelyorthogonalagain.Assuming
single
patternnulls,setting
for simplicity U I = 0 andneglecting
higher order terms of
11 x 0 - x a II/IIxo 11, itiseasily shown
from (26) that
x.
For the examples discussed earlier, the gain cost, measured
indecibels, is givenin the captions of Figs. 3-8. Apparently
the cost, which ranges between 0.03 and 0.6 dB, is n o t severe
althoughinsomecases
we havesuppressedthesidelobes
substantially.
VI. SUMMARY AND CONCLUSION
We have extended the general method of least mean square
pattern synthesis [ 3 ] t o include null constraints on the pattern
anditsderivatives.Theproblemhasbeenposed
as aconan exactsolutionhas
strainedapproximationproblemand
been obtained. The relation to other methods to achieve patternnullsundermathematicallywell-definedconditionshas
been discussed. For a linear uniform array we have shown that,
when M single nulls are imposed on a pattern, the constrained
pattern is the sum of the original pattern and 11-1 weighted sincbeams.Eachbeam
is centeredonthecorrespondingnull,
o r how much the
irrespective of how closely spaced they are
beams overlap. In the general case
of higher order nulls, the
mth pattern derivative iscanceledwithabeam
givenb>/ the
m t h derivativeofasinc-beam.
In addition we havederived
simple quantitative expressions for the pattern change and the
gain cost associated with the forced pattern nulls.
Severalillustrativeexamplesofpatternswithsinglenulls
and higher order nulls are given. These indicate that when
we
impose a set of closely spaced single nulls on a pattern sector
the sidelobes are reduced by a rather constant factor, which is
independent of the actual sidelobe level. The relation between
thenumber ofnullsandthesidelobecancellation
is more
complex.Forinstance,doublingsimultaneouslythesector
widthandthenumberofnullsproduces,contrarytointuition,anincreasedcancellationratioforthewidersector.
The gaincostsseemsratherinsignificant
f o r all examples
considered.
Finally,it is worthnotingthat.although
wehave formulated the constrained synthesis method for a linear array with
MARCH 19S2
isotropic half-wavelength spaced elements, it is not limited to
these cases. It can readily be formulated in more general terms:
in which case any desired linear passive beamforming network
may be included in the antenna. It is hoped that this approach
can contribute to an understandingof the fundamental properties and limitations of an adaptive antenna.
APPENDIX
We consider Applebaum’s approach in which we seek a set
of arrayelementweights,denoted
by therowvector
=
(wl,-., w , ~ )thatmaximizesthegeneralizedsignal-to-noise
( S / N ) ratio
Here t is a generalized signal vector and 1l.I the noise-covariance
matrix. The latter consists of two parts:
(29)M = A,
(27)
PROPAGATION, VOL. AP-30,2,NO.
+ B,
where A represents the quiescent environment (receiver noise
only) and B represents the statistically independent, external
interferencesources(jammers).
For thepresentcomparison
withpatternsynthesis,
we canassume all arrayeleFentsto
contribute uncorrelated noise
of equal power I Vo I L , which
leads to
where I is the identity matrix.
The matrix B is derived as per Applebaum [ 11. AssumingM
jammers located in the directions { u ~ ~ }the~ total
’ ~ interfer!
ence signal at the kth elementis
1
where V , denotesthecomplexamplitudeoftheindividual
jammer. The terms u k l of the matrix B are given by
where E denotes “expected value,“ and in view of the fact that
the jammers are statistically independent,
(33)
I
A pleasant consequence of (33) is that B can be written as the
sum of thecovariancematrices
of theindividualjammers;
therefore,
M
(34)
1
with ( B , ) p y = exp [ i k ( q - p)u,].
(34) in (28) yields
(S/N) =
Substitution of (30) and
I (ir,I * )12
(G*[ Vo2 + V I ’ B , + ... 4- V,142B,14] !w*)
(3 5 )
279
STEYSKAL: SYNTHESIS O F ANTENNA PATTERNS
Inthelimitofinfinitelystrongjammers,anecessarycon-obtain
dition for a nontrivial resultis
-
m = 1,
(Z*B,, w * )= 0 ,
...,M .
(36)
Noting that B , can be rewritten as the outer product of the
vectors TmTand
where T
, is given by (7) and the dagger
denotes the complex conjugate transpose, it
iseasilyshown
that (36) is equivalent to
T,,
F,,,)
fix,
m
= 0,
= 1,
.-.,M .
(37)
Summarizing ( 3 9 , (36), and (37) we thus arrive at the problem to seek
Thus in the case of infiitely strong jammers, the pattern that
is the least mean square apmaximizes the generalized SNR
proximation to the desired quiescent pattern with single-null
constraints.
VII. ACKNOWLEDGMENT
The author wishes to thank Dr. Robert Mailloux for suggesting the problem, Ms. Yelena Gersht for the computer programming, and the reviewers for some valuable comments.
REFERENCES
“Adaptive
arrays,”
IEEE Trans.
Antennas
111 S . Applebaum,
Sept. 1976.
Propagat., vol.AP-24:pp.585-598,
P I C. DraneandJ.McIlvenna,“Gainmaximizationandcontrolled
[31
subject to the constraints
[41
As beforewedecomposethevectors
iT and i intotheir
componentsinthesubspace
Y spannedby
andits
orthogonal complement Z to obtain
=
-I- w z ,i = i,, -II,.In view of (38b) the component = 0 and (38a) become
fi,>,”
I(
IIZ z * II
.>I
w wy
wy
2
= max,
(39)
151
161
[71
L81
which leads t o Gz* = hiz, where h is an arbitrary proportionality constant. To compare this result to that of pattern synthesis (14) we set h = 1, note that i =
and invoke the
phase conjugacy between a weight distribution
in the receive
modeandanaperturedistribution
in thetransmitmodeto
null placement simultaneously achieved in aerial array patterns,”
Radio Electron. Eng., vol. 39, no. 1 , pp. 49-57, Jan. 1970.
A . Ishimaru,
“Antenna
pattern
synthesis,”
in
A. Schell
and
AntennaTheory! CollinsandZucker,Ed.
NewYork:McGrawHill, 1969, part I.
J . IMayhan,“Nulling limitations for amultiple-beam antenna,”
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1976.
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x,,
Hans Steyskal (”76).
for a photograph and biography please see page
831 of the November 1979 issue of this TRANSACTIONS.