243
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 44, NO. 2, FEBRUARY 1996
A Gibbsian Model for Finite Scanned Arrays
R. C. Hansen, Life FeZZow, IEEE, and Daniel Gammon
Abstract- A finite-by-infinite array of thin half-wave dipoles
with H-plane scan is used to show the existence of a Gibbs’
phenomenon-type standing wave in scan impedance (normalized
by the infinite array value) over the elements of the array. The
period of this wave is .SA at broadside for X/2 array spacing
and increases as the scan angle increases by a grating lobe-type
expression. A simple empirical model based on Gibbs oscillations
is fitted to the scan-impedance wave; the model predicts the
1/(1- sin&) period variation, and should be useful for systems
trades and for preliminary design purposes.
I
cu
I-
I. INTRODUCTION
“The purpose of computing is insight, not
numbers.”
Richard W. Hamming
I
T is important to understand how the edges of a finite
array affect its performance. These “edge effects” typically
reduce the performance of arrays, especially at angles near
grazing, for both radiation and scattering. These effects were
recognized and analyzed many years ago through direct solution of the impedance matrix equations. Unfortunately, these
methods are limited to small finite arrays because the number
of equations to be solved is equal to the number of elements
in the array times the number of Moment-Method expansion
functions needed for convergence. These approaches are limited in their ability to accurately model large arrays because
of computer resource limitations. Other approaches solve the
finite equations in the spectral domain or use a Wiener-Hopf
approach. Still another approach embeds the finite array in a
matrix of identical arrays with blank space in between. To
reduce errors introduced by aliasing, it is necessary for the
blank spaces to be greater than the width of the finite array. But
as the ratio of blank to array becomes larger, the convergence
deteriorates. Geometric theory of diffraction (GTD) can be
used to include gross-edge effects in patterns, but offers no
help in impedance.
It is desirable to have a way of estimating finite array edge
effects without involving a large numerical exercise each time.
Such a method would be very useful for systems trades and
for preliminary design. The purpose of this paper is to present
a simple method that does this.
Section I1 discusses the finite-by-infinite array simulator
used in the numerical experiments. In Section 111, the Gibbs
phenomena are discussed. Section IV proposes a simple model
Manuscript received January 25, 1995; revised September 18, 1995.
R. C. Hansen is a consulting engineer, Tarzana, CA 91357 USA.
D. Gammon is with the United States Air Force, assigned to ARPA,
Washington, DC USA.
Publisher Item Identifier S 0018-926X(96)01209-4.
/ / / / /
/ / / / I
CO
Fig. 1. Finite-by-infinite dipole array with H-plane scan.
that utilizes the Gibbs parameters. Conclusions are given and
areas of investigation in work are discussed in Section V.
11. FINITE-BY-INFINITE
ARRAYSIMULATOR
Scan impedance’ is a key design parameter for
electronically-scanned phased arrays-it
provides array
gain versus beam direction and includes all mutual coupling
and impedance mismatch effects. Extensive material on scan
impedance is given by [4] and [6]. Finite array effects are
determined by comparing the scan impedances with that of
an infinite array for the same beam direction. Thus, the finite
array scan impedances are divided by that of the infinite
array. One objective of this study has been to extract the edge
effects with a minimum of computational baggage. To this
end, complex array elements and Moment-Method solutions
are not needed at this stage. As has been shown many times,
half-wave thin dipole arrays exhibit all the essential features
of mutual coupling.
To investigate the effect of the edge a simple strategem is
adopted-analysis of a finite-by-infinite array with scan only
across the finite dimension. By considering parallel or collinear
Formerly called active impedance; this term is now deprecated.
0018-926X3/96$05.00 0 1996 IEEE
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 44, NO. 2, FEBRUARY 1996
244
1.2
.
semi-infinite array
... .... ..........
1.1
Gibbsian model
c
.-C
I
N
\
N
O
0)
r
1
0.9
Fig. 2. 201 linear infinite arrays of collinear H W dipoles, 8 = Oo
dipoles in the scan plane, both H-plane and E-plane scans
can be accommodated. However, in this paper only parallel
dipoles (H-plane scan) are utilized, as this represents the case
where mutual impedance effects are most pronounced. For
the same reason, no back screen is included (see Fig. 1).
The computational simulator then consists of N infinite linear
arrays. Each infinite array (“stick”) is composed of collinear
half-wave thin dipoles. The mutual impedance [3]between a
dipole and the dipoles in one infinite “stick” was computed
in the spatial domain using the Levin acceleration method.
Less than 21 Levin cycles were required for four-figure
accuracy. In addition, the Levin method avoids a concatenation
of roundoff errors [7]. For larger array spacings the spatial
domain convergence improves.
Thus, the spatial-domain Levin formulation provided the
mutual impedance between a dipole and all of the dipoles in
an infinite stick. The set of these for all the sticks is used to
form the usual impedance matrix, with the solution providing
the scan impedance of a dipole in each stick. An array with
a square lattice with A/2 spacing, and with a wire radius of
0.001X was simulated. To insure that the simulation results
were not a computing artifact, a spectral formulation was also
used [9]. This code, although slightly slower than the spatial
code, produced scan impedance plots within a line width of
the spatial domain plots. Thus, the computer simulator was
validated.
The reference scan impedance for the infinite array is
provided by the spectral domain formulation pioneered by
Oliner [6].This gives the scan resistance as a single term for
each main lobe or grating lobe and a spectral summation of
trigonometric type functions for the scan reactance. For array
spacings close to A/2 the formulation converges extremely
rapidly.
Calculations have been made for 201 linear arrays [5],
and the results are repeated in Figs. 2-6, for subsequent
comparisons, Shown in the solid line is normalized scan
impedance for the 201 dipoles, except that in Figs. 2 and
3 only half of the dipoles are shown for an expanded plot.
The dipoles are numbered 1-201 starting at the edge, with
beam scan toward element 1. These figures show apparently
continuous curves, but note that they have been plotted from
201 points. The similarity to Gibbs phenomenon is readily
apparent; a model based upon this will next be discussed after
a look at the Gibbs phenomena.
111. GIBBS’PHENOMENA
It is well known that a band-limited approximation to
square-wave exhibits oscillations that are entitled Gibbs’ phe-
nomenon. The peaks and dips of the oscillations are regularly
spaced, and the amplitudes of the first, second, third, etc.,
peaks are the same regardless of the upper-band limit. Here,
the first peak occurs at each edge of the square wave, with the
second peak adjacent, etc. As the upper-band limit increases,
the total number of peaks increases, being roughly the upper
limit divided by T . Of course the spacing changes, but the
amplitudes remain the same. Little mention is in the literature
regarding how to calculate these peaks for a square wave, but
[2] quotes results derived by [l].The extrema locations from
,
the edge of a pulse of width r / 2 are given by m ~ / 2 Nwhere
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245
semi-infinite array
................
0.9
Gibbsian model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3 5 7 9 11 13 15 17 19 21 2 3 2 5 2 7 29 3 1 3 3 3 5 3 7 3 9 4 1 4 3 4 5 4 7 4 9 51 5 3 5 5 5 7 5 9 61 6 3 6 5 6 7 69 71 7 3 7 5 7 7 7 9 8 1 8 3 8 5 8 7 8 9 9 1 9 3 9 5 9 7 9 9 1 0 1
Element Number
Fig. 3. 201 linear infinite arrays of collinear HW dipoles, 0 = 30'.
1.2
semi-infinite array
~
.................
1.1
Gibbsian model
r
8
._
I
N
\
N
cn
U
5
1
0.9
1
11
21
31
A1
51
61
71
91
81
101
111
121
131
141
151
161
171
181
191
201
Element Number
Fig. 4.
201 linear infinite arrays of collinear HW dipoles, 6' = 45'.
m is the extremum index, and N is the upper limit on the
summation. The amplitudes at these points are immediately
expressed as an integral which, for large N , can be expressed
as a Sine Integral; the result is 2/7r Si(m7r). Thus, for large
band limit values, the peak amplitudes are readily determined
to be 1.1790, 0.9028, 1.0062, 0.9499, etc.
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 44, NO. 2, FEBRUARY 1996
246
1.2
1.1
L
C
._
I
1
N
\
N
CT)
0
=
0.5
semi-infinite array
....... .. .. . . .. ..
Gibbsian model
0.t
1
11
21
31
41
51
61
71
81
91
101
121
111
131
141
151
161
171
181
191
201
Element Number
Fig. 5. 201 linear infinite arrays of collinear HW dipoles, 6 = 60°.
One might expect that the Gibbs oscillations for a bandlimited approximation to a single pulse would be different than
those for a square wave. These are determined by taking the
Fourier transform of the sinc spectrum with finite limits. Using
upper and lower limits of U r / 2 and 0, and using variable
T = 2t/r, where r is the pulse width, the expression for the
band-limited pulse is
+
f ( t ) = si U ( 1 + T) S i U ( 1 - T ) .
(1)
The Gibbs oscillations again are produced by Sine Integrals,
but with different arguments. Following [l], the extrema
positions are determined by first differentiating the transform
integrals, which fortunately can be integrated directly. The
result is a set of sinc functions which is then set to zero
sinc U ( 1
+ T ) - sinc U ( l - T ) = 0.
(2)
The actual values for extrema locations are found from a root
solver which employs analytical derivatives. This yields results
for any value of upper band limit. As the upper limit becomes
large, the extrema locations approach T = 1- n n / U , where
T is measured from the pulse center. To allow an extremum at
T = 0, the upper limit should be U = Nn.Then T = I - n / N ,
and there are 2N 1 equally spaced oscillations across the
pulse. Amplitudes are given by
The amplitudes from these equations are about half those for
the square wave; the latter are given by 2/7r Si(nn),but the
physical reasoning why this is so is not known to this author.
It might be expected that a single pulse with an exponential
linear phase across it would represent the scanned finite array.
The aperture distribution should be exp-jtU sin BO for ltl 5 1,
and zero for It( > 1. The transform of this aperture distribution
is the spectrum, which is [SI
F ( t ) = sinc (w
= [Si(2iV - n)n
+ Sznn]/n.
(3)
When (2N - n ) is large, an approximation can be used
fmax 2:
.5
1 .
+ -Sznn.
7r
(4)
a)r/2.
(5)
Here, a = Usindo. This spectrum is then transformed with
finite limits to produce the scanned pulse with Gibbs type
oscillations. The transform is
U
F(t)=
.1,
~ s i n c ( w- a ) r
I
2expjwtdw.
(6)
After various changes of variable the result is, for convenience,
written in two parts: f l and f 2 . These are
fi
=expjat[SiU(I+T)(l+sinBo)
+ S i U ( I + T ) ( I -sin&)
+ S i U ( I -T)(I+sinBo)
+ S i U ( 1 - T ) ( I- sinBo)]
+
fmax
-
fi
+
+
exp jat [CiU(1 T )(I sin 6'0)
-CiU(I+T)(I-sin&)
- C z U ( 1 - T ) ( I+ s i n & )
Ci U ( I - T)(I - sinBo)].
=j
(7)
+
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(8)
HANSEN AND GAMMON: GIBBSIAN MODEL FOR FINITE SCANNED ARRAYS
241
12
._
= 0.9
I
L
N
\
N
0
0)
0.8
semi-infinite array
I
..... . . . .. .....
Gibbsian model
0.7
0.6
0.5
1
1
11
21
31
41
51
61
71
Fig. 6. 201 linear infinite arrays of collinear HW dipoles, 0
81
91
101
111
121
131
141
151
161
171
181
191
201
= 75'.
period is not a perfect match; from center to edge there is
a shift of a fraction of a cycle which is fitted by AT =
.5 tan2 00. The amplitudes are related to the single Gibbs
amplitude and an excellent fit is obtained by using half of the
Gibbs values, except for the very edge elements which appear
T t a n U ( 1 + s i n & ) + T t a n U ( 1 -sin&)
to behave differently. For these, an empirical trig function fit
- U (I
sin 00) sec' U T ( 1 sin 0,)
has been used.
Figs. 2-62 show the normalized scan impedance calculated
- ~ ( - 1sin 0 0 ) sec' UT(^ - sin 0 0 ) = 0.
(9)
from the simple model described above (dotted line), for scan
Using the previous value for U , and dropping the phase term, angles of 0", 30°, 45", 60", and 75". It can be seen that the
f l becomes
agreement is excellent for all of the scan angles except 75",
where it is good. Note that the match between the discrete
f l = Si(2N - n)7r(1+ s i n & )
Si(2N - n).rr(l- sindo)
array points and the dotted curve (as in Fig. 3) is important,
Si2727r(1+ sindo) S i 2 n ~ (-l sin6'o). (10)
rather than the comparison between the dotted peak and the
For large (2N - n ) ~the
, first three terms are approximately array solid line. This simple model will be adequate for system
6/7r; in comparison with ( 3 ) , it can be seen that the period trades and for preliminary design.
A question that frequently arises is, "How many 'edge
of the oscillations has been increased by 1 / ( 1 - sin&), a
elements' exist?' Using a 1% variation in scan impedance,
pseudograting lobe type factor.
the number of edge elements at each edge is given in Table I.
At broadside, roughly six edge elements exist, while at 75"
I v . A GIBBSIAN
MODELFOR FINITE ARRAYS
scan, more than 100 exist (entire array is "edge elements"). An
The computer experiment showed that the oscillations in approximate edge element number formula is N = 6/c0s2 6'0.
scanned impedance for a finite array are regularly spaced;
their amplitudes correspond roughly to the Gibbs amplitudes
V. CONCLUSION
for a single pulse. The period of this Gibbs standing wave is
a function of the scan angle as one might expect. It has been
The existence of a Gibbs type standing wave in scan
determined from the array simulation to be given closely by
impedance across the elements of a finite array has been
demonstrated. Smaller scan angles from broadside produce
1/(1 - sin eo), in wavelengths
more cycles; these oscillations exist even at broadside. A
where 00 is the scan angle from broadside. This result matches
*Prepared by Computer Associates DISSPLA.
that obtained from the Gibbs analysis. However, the Gibbs
f1 represents a pulse with oscillations, while f 2 contains higher
order delta functions at the pulse edges, along with oscillations.
Thus, f1 is the part of interest. The extrema locations are found
by setting the derivative of f l wrt T equal to zero
+
+
+
+
+
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248
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,VOL. 44, NO. 2, FEBRUARY 1996
TABLE I
NUMBEROF EDGEELEMENTS
VERSUSSCAN
scan angle
number of
edge elements each side
6
0
30
45
8
12
26
60
75
all
simple approximate formulation for the standing wave period
has been derived. An excellent fit to the computer experimental
results is obtained from a model based upon Gibbs oscillations
for a single pulse; the model is extremely simple, and requires
only a sine integral subroutine.
Work underway includes arrays with E-plane scan, both
planes with back screen, and array spacings that allow grating
lobes. A rationale for the atypical behavior of elements at the
edge is needed.
REFERENCES
H. S. Carslaw, An Introduction to the Theory of Fourier’s Series and
Integrals. New York: Macmillan, 1930; New York: Dover, 1950.
R. W. Hamming, Numerical Methods for Scientists and Engineers.
New York McGraw-Hill, 1973.
R. C. Hansen, “Formulation of echelon dipole mutual impedance for
computer,” IEEE Trans. Antennas Propagat., vol. AP-20, pp. 78Cb781,
Nov. 1972.
R. C. Hansen, “Planar arrays,” in Handbook of Antenna Design, IEE
Peregrinus, vol. 2, ch. 10, A. W. Rudge, et al., Eds., 1983.
R. C. Hansen and D. Gammon, “Standing waves in scan impedance
of finite scanned arrays,” Microwave Opt. Technol. Lett., vol. 8, pp.
175-179, Mar. 1995.
A. A. Oliner and R. G. Malech, “Mutual coupling in infinite scanning arrays,” Microwave Scanning Antennas-Volume II. New York:
Academic, R. C. Hansen, Ed., ch. 3, 1966; Peninsula Pub., 1985.
S. Singh and R. Singh, “On the use of Levin’s t-transform in accelerating
the summation of series representing the free-space periodic Green’s
functions,” IEEE Trans. Microwave Theory Tech., vol. 41, pp. 884-886,
May 1993.
I. N. Sneddon, Fourier Transforms. New York McGraw-Hill, 1951.
J. M. Usoff and B. A. Mu&, “Edge effects of truncated periodic surfaces
of thin wire elements,” IEEE Trans. Antennas Propagat., vol. 42, pp.
946-953, July 1994.
R. C. Hansen (S’47-A’49-M’55-SM’56-F’62LF’92) received the B.S.E.E. degree from the
Missouri School of Mines, Rolla, MO, 1949, and
the Ph.D. degree from the University of Illinois,
Urbana, in 1955. He was awarded an honorary
Doctor of Engineering degree by the University of
Missouri-Rolla in 1975.
From 1949-1955, he worked in the Antenna
Laboratory of the University of Illinois Urbana on
ferrite loops, streamlined airborne antennas, and
DF and homing systems. In 1960, he became a
Senior Staff Member at the Telecommunications Laboratory of STL, Inc.
(now TRW), engaged in communication satellite telemetry, tracking, and
command. Earlier, he was Section Head of the Microwave Laboratory of
Hughes Aircraft Co., working on surface wave antennas, slot arrays, nearfields, electronic scanning and steerable arrays, and dynamic antennas. From
19641966, he formed and was Director of the Test Mission Analysis Office,
responsible fro computer programs for the planning and control of classified
Air Force Satellites. Prior to that he was an Associate Director of Satellite
Control where he was responsible for converting the Air Force satellite control
network into a realtime computer-to-computer network. From 1966-1967, he
was Operations Group Director of the Manned Orbiting Laboratory Systems
Engineering Office of the Aerospace Corp. and was responsible for mission
profiles and mission control center equipment and displays. From 1967-1970,
he was with KMS Industries.
Dr. Hansen’s professional activities include being Chair of the U S .
Commission B of URSI (1967-1969), Chair of the 1958 WESCON Technical
Program Committee, President of IEEE Antennas and Propagation Society
(1963-1964 and 1980), Chair of Standards Subcommittee 2.5, Editor of
the AF’S Newsletter (1961-1963), Member of APS AdCom (1959-1974),
Member of E E E mtblishing Board (1972 and 1974), and Director of E E E
(1975). He is a registered Professional Engineer in California and England and
is a member of the American Physical Society, Tau Bela Pi, Sigma Xi, Eta
Kappa Nu, and Phi Kappa Phi. The University of Illinois Urbana Electrical
Engineering Department gave him a Distinguished Alumnus Award in 1981,
and the College of Engineering awarded him a Distinguished Alumnus Service
Medal in 1986. The IEEE AESS Barry Carlton Best Paper Prize was awarded
him in 1991. He received the E E E A P S Distinguished Achievement Award,
in 1994, and was elected to the National Academy of Engineering, in 1992.
He has written over 100 papers on electromagnetics, has been an Associate
Editor of Microwave Joumal since 1960, was Associate Editor of Radio
Science from 1967-1969, and of Microwave Engineer’s Handbook (1971),
and was Editor of Microwave Scanning Antennas-Vols. I, II, and III (1964
and 1966), Signifcant Phased Array Papers (1973), Geometric Theory of
Diffraction (1981), and Moment Methods in Antennas and Scattering (1990).
Daniel Gammon received the B.S. degree in chemistry, in 1977, from
Middle Tennessee State University, Murfreesboro, TN, where he also attended
graduate school. He received the B.S.E.E. degree from the University of South
Florida, Tampa, E,in 1984, and the M.S.E.E. degree from the U S . Air Force
Institute of Technology, Wright Patterson Air Force Base, OH, in 1989.
He is cunecntly assigned to the Advanced Research Projects Agency,
Arlington, VA.
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