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7/27/2019 Tolerance Analysis of Antenna Arrays
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2013 1
Tolerance Analysis of Antenna Arrays through
Interval ArithmeticN. Anselmi, L. Manica, Member, IEEE , P. Rocca, Senior Member, IEEE , and A. Massa, Member, IEEE
Abstract— An analytical method based on Interval Analysis(IA) is proposed to predict the impact of the manufacturingtolerances of the excitation amplitudes on the radiated arraypattern. By expressing the array factor according to the rulesof the Interval Arithmetic, the radiation features of the lineararray are described in terms of intervals whose bounds areanalytically determined as functions of the nominal value andthe tolerances of the array amplitudes. A set of representativenumerical experiments dealing with different radiated beams andlinear array sizes is reported and discussed to point out thefeatures and potentials of the proposed approach.
Index Terms— Antenna Arrays, Linear Arrays, Tolerance
Analysis, Interval Analysis, Interval Arithmetic.
I. INTRODUCTION
MODERN wireless communication systems require antennas
able to guarantee high-quality and reliable data links. In many
practical applications, arrays generating patterns with high
peak directivity and low secondary lobes are necessary [1].
Towards this end, several strategies have been proposed to syn-
thesize suitable values for the control points (i.e., amplifiers,
phase shifters, time delays) of the corresponding beamforming
network (BF N ) to guarantee the desired radiation features
[2][3][4][5]. However, each element of the antenna system,both the radiating elements and the BF N control points,
can differ from the ideal one because of the manufacturing
tolerances. Such errors unavoidably cause some modifications
of the radiated beam and performance degradations usually
arise. For example, tolerances in the implementation of the
levels of amplification and/or in the delays introduced by the
phase shifters could result in a pattern with relatively high
sidelobes [4]. To prevent such a circumstance, the antenna
system must be suitably calibrated before being installed with
a consequent waste of time and human resources.
In the past, methods to estimate the impact of manufacturing
tolerances on the pattern features, namely the sidelobe level(SLL), the peak directivity, and the mainlobe direction, have
been proposed. In this framework, errors on the amplitude
and phase values of the excitations as well as on the array
element positions have been taken into account [6][7][8][9] by
means of statistical techniques where random and statistically-
independent errors among the array elements have been as-
sumed. Another probabilistic method based on white noise
gain and expected beam-power pattern has been proposed
Manuscript received January 1, 2013Dr. Anselmi, Dr. Manica, Dr. Rocca, and Prof. Massa are with
the ELEDIA Research Center@DISI, University of Trento, Via Som-marive 5, Povo 38123 Trento - Italy (e-mail: nicola.anselmi, luca.manica,
[email protected]; [email protected])
N−1N−2N−3nn−1 n+1
E l e m e n
t A m p
l i t u d e
Element Index, n
0 1 2
supAn
inf An
αn
ωAn−1
µA1
ǫ(sup)n+1
ǫ(inf )n+1
Fig. 1. IA-based Approach - Reference amplitudes (αn, n = 0,...,N −1),upper (sup An, n = 0,...,N −1) and lower (inf An, n = 0,...,N −1) bounds of the corresponding tolerance interval (An, n = 0,...,N − 1).
in [2] for assessing the robustness of the array performance
against the statistics of the errors on the array sensors.
Moreover, in the field of superdirective arrays, great attention
has been paid to the synthesis of solutions that are robust
with respect to the errors in the amplitude gains and phases
[12][13][14]. The problem of quantization errors due to the
use of digitally-controlled phase shifters or amplifiers with alimited number of possible values has been addressed [4][5],
as well.
As for the robust array design, probabilistic methods [10] have
been investigated since no antenna can be realized in practice
by continuously and arbitrarily reconfiguring the values of
the control points. Moreover, statistical approaches have been
recently applied to estimate the maximum tolerance of each
array element to generate a beam pattern having prescribed
radiation performance [11]. However, such approaches cannot
be considered as completely reliable and guaranteeing a robust
array realization because of the impossibility to test the infinite
number of combinations of the control point values within a
given error threshold.
In this work, an innovative method based on the Interval Anal-
ysis is presented to evaluate in a deterministic, exhaustive, and
analytic fashion, unlike the probabilistic approaches already
proposed [2][6][7][8][9], the effects of the manufacturing
tolerances in the control points of the BF N on the radiation
pattern of linear antenna arrays. Firstly proposed to determine
the error bounds on the rounding operations in numerical
computation [15][16], IA has then been applied to solve linear
and non-linear equations [17] as well as optimization problems
[18][19]. In electromagnetics, the use of IA has been limited
to some pioneering works dealing with the robust design of
magnetic devices [20][21] and reliable systems for target track-
7/27/2019 Tolerance Analysis of Antenna Arrays
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2013 2
ing based on range-only multistatic radar [22]. Recently, IA
has been also applied to inverse scattering problems [23][24]
where both the problem and the cost function, quantifying the
mismatch between measured and reconstructed data, have been
formulated according to the arithmetic of intervals.
Thanks to the intrinsic capability of the IA to deal with uncer-
tainties, the rules of interval arithmetic are here considered for
determining the tolerance, namely the upper and lower bounds,of the array factor and the corresponding power pattern when
manufacturing tolerances arise on the excitation amplitudes.
Towards this aim, the array analysis is addressed firstly by
expressing the quantities affected by errors or tolerances in
terms of interval numbers and then the arithmetic of intervals
is used to analytically determine the bounds of the radiation
functions.
The outline of the paper is as follows. The problem is
mathematically formulated in Sect. II where, after introducing
the interval representation for the values of the antenna control
points and of the array factor, the upper and lower bounds
of the array factor (Sect. II.A) and the power pattern (Sect.
II.B) are analytically determined by means of IA. A set of
representative numerical examples is reported in Sect. III to
assess the reliability of the proposed method (Sect. III.A) as
well as its effectiveness when dealing with arrays different in
the aperture size or the radiated SLL (Sect. III.B). Eventually,
some conclusions are drawn in Sect. IV where the advantages
of the proposed IA-based method for array tolerance analysis
are highlighted, as well.
II . MATHEMATICAL FORMULATION
Let us consider a linear array of N isotropic radia-
tors uniformly-spaced along the x-axis and the correspond-
ing amplitude weights characterized by known or measur-
able tolerances, ε(sup)n and ε
(inf )n , around the nominal (ex-
pected/reference) value αn (n = 0,...,N − 1). Accordingly,
the levels of amplification or attenuation actually realized
through the BF N can be represented by the intervals An
defined by their bounds (Fig. 1), inf An αn−ε(inf )n and
sup An αn + ε(sup)n ,
An = [inf An ; sup An] , n = 0,...,N − 1 (1)
or, in a dual fashion, in terms of the interval midpoint,
µ An inf An+supAn2
= αn +ε(sup)n −ε(inf )n
2, and width,
ωA
n sup
A
n −inf
A
n= ε
(sup)n + ε
(inf )n ,
An =
µ An − ωAn2
; µ An + ωAn2
,
n = 0,...,N − 1.(2)
By considering the amplitude tolerances (1), the array factor
can be mathematically expressed as follows
AF (θ) =N −1n=0
AnejΘn(θ) (3)
where j =√ −1 is the complex variable and Θn (θ) =
(nkdsinθ + ϕn), k = 2πλ being the free-space wavenumber, λ
the wavelength, (ϕn, n = 0,...,N
−1) the phase weights of the
array elements, d the inter-element spacing, and θ ∈ −π2 ; π2
the angular direction measured from boresight (i.e., θ = π2
).
As it can be noticed (3), the array factor is a (complex)
interval number (see Appendix I ) for each angular direction,
θ ∈ −π2
; π2
whose bounds can be analytically computed by
using the arithmetic of complex intervals [25] according to the
mathematical rules summarized in Appendix II and exploited
in Sect. II.A.
A. Array Factor Interval Definition
In order to determine an explicit expression for AF (θ),
θ ∈ −π2
; π2
, as a function of the amplitude tolerances,
ε(sup)n and ε
(inf )n , and nominal values αn (n = 0,...,N − 1),
let us rewrite the complex interval (3) in terms of its real,
AFR (θ)=N −1
n=0 An cosΘn (θ), and imaginary, AFI (θ)=N −1n=0 An sinΘn (θ), components
AF (θ) = AFR (θ) + jAFI (θ) . (4)
With reference to AFR (θ), let us substitute (1) within the
expression of the n-th term of AFR (θ), (AFR (θ))n. It
follows that
AFR (θ) =N −1
n=0 (AFR (θ))n=N −1
n=0 [inf An cosΘn (θ) ; sup An cosΘn (θ)] .(5)
Since the following relationships hold true
sup An cosΘn (θ) ≥ inf An cosΘn (θ)if cosΘn (θ) ≥ 0
sup An cosΘn (θ) < inf An cosΘn (θ)if cosΘn (θ) < 0,
(6)
the width of the n-th term of AFR (θ), ω (AFR (θ))n=sup
An cosΘn (θ)
−inf
An cosΘn (θ)
, turns out being
equal to
ω (AFR (θ))n =
ε(sup)n + ε(inf )n
|cosΘn (θ)| , (7)
while the corresponding interval mid-point, µ (AFR (θ))n=inf An cosΘn(θ)+supAn cosΘn(θ)
2, is equal to
µ (AFR (θ))n = αn cosΘn (θ) +
+ε(sup)n −ε(inf )n
2
cosΘn (θ)
(8)
Since the sum of the widths/mid-points of real intervals
is equal to the width/mid-point of the interval sum (see
Appendix III ), ω AFR (θ) =
N −1n=0 ω (AFR (θ))n and
µ AFR (θ) = N −1
n=0 µ (AFR (θ))n, hence
ω AFR (θ) =
N −1n=0
ε(sup)n + ε(inf )n
|cosΘn (θ)| (9)
andµ AFR (θ) =
N −1n=0 αn cosΘn (θ) +
+ε(sup)n −ε(inf )n
2
cosΘn (θ)
(10)
Analogously, the width and midpoint of AFI (θ) can be
proved being equal to
ω
AFI (θ)
=
N −1
n=0 ε(sup)n + ε(inf )n |sinΘn (θ)
|(11)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0, 2013 3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1 2 3 4 5 6 7 8 9 10
E x c
i t a t i o n
A m p
l i t u d e
Element Index, n
Taylor - δαn = 1%
Interval Amplitude, An
Reference Amplitude, αn
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1 2 3 4 5 6 7 8 9 10
E x c
i t a t i o n
A m p
l i t u d e
Element Index, n
Taylor - δαn = 5%
Interval Amplitude, An
Reference Amplitude, αn
(a) (b)
-35
-30
-25
-20
-15
-10
-5
0
5
-1 -0.5 0 0.5 1
R e
l a t i v e
P o w e r
P a
t t e r n
[ d B ]
u
supP(u)
infP(u)
P(u)(q)
-35
-30
-25
-20
-15
-10
-5
0
5
-1 -0.5 0 0.5 1
R e
l a t i v e
P o w e r
P a
t t e r n
[ d B ]
u
supP(u)
infP(u)
P(u)(q)
(c) (d)
Fig. 2. Example 1 (N = 10, d = λ2
, δαn = 1, 5%; Taylor pattern: SLLref = −20 dB, n = 2) - Reference amplitudes (αn, n = 0,...,N − 1) and
amplitude tolerance intervals (An, n = 0,...,N −1) (a)(b) and bounds (supP (u) and (inf P (u)) of the corresponding power pattern P (u) intervals(c)(d ) when (a)(c) δαn = 1% and (b)(d ) δαn = 5%. Although the interval amplitudes are not appreciable for small values of δαn [Fig. 2(a)], the toleranceeffects on the power pattern are evident in dB scale [Fig. 2(c)].
andµ AFI (θ) =
N −1n=0 αn sinΘn (θ) +
+ε(sup)n −ε(inf )n
2
sinΘn (θ)
(12)
respectively.
By substituting (9)(10) and (11)(12) in (2)
inf
AFR (θ)AFI (θ)
= µ
AFR (θ)AFI (θ)
−
ω
AFR (θ)AFI (θ)
2
(13)
sup
AFR (θ)AFI (θ)
= µ
AFR (θ)AFI (θ)
+
ω
AFR (θ)AFI (θ)
2
(14)
the bounds of the two components of AF (θ) assume the
following mathematical expressions
sup
AFR (θ)AFI (θ)
=N −1
n=0
αn +
ε(sup)n −ε(inf )n
2
× cosΘn (θ)
sinΘn (θ)+
ε(sup)n −ε(inf )n
2
|cosΘn (θ)|
|sinΘn (θ)
|
(15)
inf
AFR (θ)AFI (θ)
=N −1
n=0
αn +
ε(sup)n −ε(inf )n
2
× cosΘn (θ)
sinΘn (θ)−ε(sup)n −ε(inf )n
2
|cosΘn (θ)||sinΘn (θ)|
.
(16)
Finally, the interval of the array factor (4) in terms of ε(sup)n ,
ε(inf )n , and αn (n = 0,...,N − 1) is then obtained by
substituting (15) and (16) in the following
AF (θ) = [inf AFR (θ) ; sup AFR (θ)]+ j [inf AFI (θ) ; sup AFI (θ)]
(17)
being θ ∈ −π2 ; π
2
.
B. Power Pattern Interval Definition
In many practical applications (e.g., communications and
radars), the dependence on the manufacturing tolerances of
the bounds of the power pattern, P (θ) = |AF (θ)|2 =AF (θ) AF ∗ (θ), instead of the array factor, AF (θ), are pre-
ferred. To determine upper and lower values of the (real) power
pattern interval, let us consider that the complex interval,
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−30
−25
−20
−15
−10
−5
0
5
−1 −0.5 0 0.5 1
N o r m a l i z e d P o w e r P a t t e r n [ d B ]
u
inf SLL supSLL
supP(u)inf P(u)
(a)
3 [ d B ]
3 [ d B ]
−5
−4
−3
−2
−1
0
1
2
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
N o r m a l i z e d P o w e r P a t t e r n [ d B ]
u
inf BW
supBW
supP(u)inf P(u)
(b)
Fig. 3. IA-based Approach - Upper and lower bounds of the intervals ( a)SLL and (b) BW.
TABLE I
Example 1 (N = 10, d = λ2
; TAYLOR PATTERN: SLLref = −20 dB,
n = 2 ). AMPLITUDEDISTRIBUTION.
n αn n αn
0 0.542 5 1.0001 0.629 6 0.9132 0.771 7 0.7713 0.913 8 0.6294 1.000 9 0.542
P (θ) AF (θ)AF∗ (θ), according to (43) and (4), can be
rewritten as
P (θ) = AF2R (θ) +AF2I (θ) (18)
where AFR (θ) and AFI (θ) are real number intervals. The
analysis of the two terms in (18) is dual, then only the deriva-
tion of AF2R (θ) will be detailed, while the expressions for
AF2I (θ) will be obtained by trivial extension. With reference
to (44) and concerning the real term, two different conditions
have to be taken into account:
• Case (inf AFR (θ) > 0 or
sup AFR (θ) < 0) - From (44), AF2R (θ) =min
(inf AFR (θ))2 , (sup AFR (θ))2
;
max(inf AFR (θ))2
, (sup AFR (θ))2, then
let us analyze the square value of (13) and (14) given bysup
inf AFR (θ)
2
= µ2 AFR (θ) +
+ω2AFR(θ)4
± µ AFR (θ) ω AFR (θ) .
(19)
Since µ2 AFR (θ),ω2AFR(θ)
4, and ω AFR (θ) are
all positive quantities, it follows that
(sup AFR (θ))2 ≥ (inf AFR (θ))2
if µ AFR (θ) ≥ 0
(inf AFR (θ))2 ≥ (sup AFR (θ))2
if µ AFR (θ) < 0 .
(20)
Consequently, it turns out that
AF2R (θ) =
|µ AFR (θ)| − ωAFR(θ)
2
2;
|µ AFR (θ)| + ωAFR(θ)2
2;
(21)
• Case (inf AFR (θ) ≤ 0 ≤sup AFR (θ)) - From (44), AF
2R (θ) =
0; max
(inf AFR (θ))2 , (sup AFR (θ))2
,
then it results that (13)(14)
AF2R (θ) =
0;|µ AFR (θ)| + ωAFR(θ)
2
2(22)
As for the term AF2I (θ),
• Case (inf
AFI (θ)
> 0 or sup
AFI (θ)
< 0) -
Analogously to (21)
AF2I (θ) =
|µ AFI (θ)| − ωAFI(θ)
2
2;
|µ AFI (θ)| + ωAFI (θ)2
2;
(23)
• Case (inf AFI (θ) ≤ 0 ≤ sup AFI (θ)) - Analo-
gously to (22)
AF2I (θ) =
0;|µ AFI (θ)| +
ωAFI(θ)2
2(24)
By combining the four previous cases and applying (38), it
results that
P (θ) = [inf P (θ) ; sup P (θ)] (25)
where
inf P (θ) =|µ AFR (θ)| − ωAFR(θ)
2
2+
+|µ AFI (θ)| − ωAFI (θ)
2
2sup P (θ) =
|µ AFR (θ)| + ωAFR(θ)
2
2+
+|µ AFI (θ)| +ωAFI (θ)
2 2(26)
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0.16
0.18
0.2
0.22
0.24
0.26
0.28
0 1000 2000 3000 4000 5000
B W
[ u ]
Solution Index, q
infBW
supBW
BW(q)
BWref
-25
-20
-15
-10
0 1000 2000 3000 4000 5000
S L L [ d B ]
Solution Index, q
infSLL
supSLL
SLL(q)
SLLref
8.5
9
9.5
10
10.5
11
0 1000 2000 3000 4000 5000
D
[ d B ]
Solution Index, q
infD
supD
D(q)
Dref
(a) (c) (e)
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0 1000 2000 3000 4000 5000
B W
[ u ]
Solution Index, q
infBW
supBW
BW(q)
BWref
-25
-20
-15
-10
0 1000 2000 3000 4000 5000
S L L [ d B ]
Solution Index, q
infSLL
supSLL
SLL(q)
SLLref
8.5
9
9.5
10
10.5
11
0 1000 2000 3000 4000 5000
D
[ d B ]
Solution Index, q
infD
supD
D(q)
Dref
(b) (d) (f )
Fig. 4. Example 1 (N = 10, d = λ2
, δαn = 1, 5%; Taylor pattern: SLLref = −20 dB, n = 2) - Bounds and nominal value of the intervals (a)(b)BW, (c)(d ) SLL, and (e)( f ) D when (a)(c)(e) δαn = 1% and (b)(d )( f ) δαn = 5%.
if (inf AFR (θ) > 0 or sup AFR (θ) < 0) and
(inf AFI (θ) > 0 or sup AFI (θ) < 0),
inf P (θ) =|µ AFR (θ)| − ωAFR(θ)
2
2sup P (θ) =
|µ AFR (θ)| + ωAFR(θ)
2
2+
+|µ AFI (θ)| + ωAFI (θ)
2
2 (27)
if (inf AFR (θ) > 0 or sup AFR (θ) < 0) and(inf AFI (θ) ≤ 0 ≤ sup AFI (θ)),
inf P (θ) =|µ AFI (θ)| − ωAFI(θ)
2
2sup P (θ) =
|µ AFR (θ)| + ωAFR(θ)
2
2+|µ AFI (θ)| + ωAFI (θ)
2
2 + (28)
if (inf AFR (θ) ≤ 0 ≤ sup AFR (θ)) and
(inf AFI (θ) > 0 or sup AFI (θ) < 0), and
inf P (θ) = 0
supP
(θ
) = |µ AF
R (θ
)| +
ωAFR(θ)
2 2
++|µ AFI (θ)| + ωAFI (θ)
2
2 (29)
if (inf AFR (θ) ≤ 0 ≤ sup AFR (θ)) and
(inf AFI (θ) ≤ 0 ≤ sup AFI (θ)).
The final expression of P (θ) in terms of ε(sup)n , ε
(inf )n ,
and αn (n = 0,...,N − 1) is then obtained by substituting
(9)(10)(11)(12) in (25) throughout (26)(27)(28)(29).
III. NUMERICAL RESULTS
In the following, the proposed IA-based analysis method is
assessed by reporting and discussing the most representative
results of a wide set of numerical simulations. Besides the
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
E x c i t a t i o n A m p l i t u d e
Excitation Index, n
Dolph - SLL=-30 [dB]
αn
δαn=1%
δαn=5%
δαn=10%
(a)
-40
-35
-30
-25
-20
-15
-10
-5
0
5
-1 -0.5 0 0.5 1
R e
l a t i v e
P o w
e r
P a
t t e r n
[ d B ]
u
Dolph - SLL=-30 [dB]
P(u)
P(u) - δαn=1%
P(u) - δαn=5%
P(u) - δαn=10%
(b)
Fig. 5. Example 2 (N = 20, d = λ2
, δαn = 1, 5, 10%; Dolph-Chebyshev pattern: SLLref = −30 dB) - Reference amplitudes (αn,n = 0,...,N − 1) and amplitude tolerance intervals (An, n = 0,...,N − 1)(a) and bounds (sup P (u) and (inf P (u)) of the corresponding power
pattern intervals P (u) (b) when δαn = 1, 5, 10%.
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-60
-50
-40
-30
-20
-10
0
10 15 20 25 30 35 40 45 50 55 60
S L L [ d B ]
|SLLref| [dB]
SLLref
SLL - δαn=1%
SLL - δαn=5%
SLL - δαn=10%
(a)
0.05
0.1
0.15
0.2
10 15 20 25 30 35 40 45 50 55 60
B W
[ u ]
|SLLref| [dB]
BWref
BW - δαn=1%
BW - δαn=5%
BW - δαn=10%
(b)
9
10
11
12
13
14
15
16
10 15 20 25 30 35 40 45 50 55 60
D [ d B ]
|SLLref| [dB]
Dref
D - δαn=1%
D - δαn=5%
D - δαn=10%
(c)
Fig. 6. Example 3 (N = 20, d = λ2
, δαn = 1, 5, 10%; Dolph-Chebyshev pattern: SLLref ∈ [−60;−10] dB) - Bounds and nominal
value of the intervals (a) SLL, (b) BW, and (c) D versus SLLref whenδαn = 1, 5, 10%.
method validation, a numerical study on the bounds of some
key pattern features [i.e., SLL, half-power beam width (BW ),
and peak directivity (D)] is presented dealing with arrays
of different sizes or radiating different SLLs. Without loss
of generality, the condition ε(inf )n = ε
(sup)n = δαn, n =
0,...,N −1 has been assumed throughout the whole numerical
assessment.
A. Method Validation
Let us consider a linear array of N = 10 elements uniformly-
spaced by d = λ2 . The reference/nominal amplitudes, αn,
n = 0,...,N − 1 given in Tab. I, generate a Taylor pattern
with SLLref = −20dB and n = 2, n − 1 being the number
of sidelobes on each side of the mainlobe with peaks at
SLLref [3][5]. To investigate on the effects of amplitude
tolerances on the radiation pattern, the following two cases
have been considered: δαn = 1100
αn and δαn = 5100
αn.
The corresponding intervals An = [inf An ; sup An],n = 0,...,N −1 are indicated with the bars in Figs. 2(a)-2(b),
where the nominal amplitudes are reported, as well. Moreover,the power pattern intervals P (u) [u sin(θ)], u ∈ [−1, 1]computed by means of interval arithmetic are shown in Fig.
2(c) and Fig. 2(d ) in terms of their upper, sup P (u), and
lower, inf P (u), bounds. For a preliminary indication that
the patterns potentially radiated by the array with manu-
facturing tolerances lay within the IA bounds, Q = 5000different beams have been generated by randomly selecting the
amplitude values as α(q)n = αn± r
(q)n δαn, q = 1,...,Q, r
(q)n ∈
[0, 1] being a random variable with uniform distribution.
Although a complete analysis is not possible since it would
require the generation of all (infinite) patterns generated by
the arrays whose amplitudes can vary with continuity within
the intervals of tolerance inf An ≤ An ≤ sup An,
n = 0,...,N − 1, the fact that all Q beams are within the
IA analytically-defined bounds (inf P (u) ≤ P(q) (u) ≤sup P (u), q = 0,...,Q − 1) [Figs. 2(c)-2(d )] fully confirm
the theoretical proof given by the Inclusion Function Theorem
[19]. Customized to array analysis, the theorem states that,
when the condition An ⊂ An, n = 0,...,N − 1 is verified,
namely that interval An is included within interval An for all
n = 0,...,N −1, the interval power pattern P (θ) yielded from
the interval amplitudesA0, A1, ..., AN −1
and computed
according to the rules of Interval Arithmetic is included within
the bounds of P (θ) generated with
A0, A1, ..., AN −1
,
namely P (θ) ⊂ P (θ). The same conclusion holds true whenAn is a degenerate interval, namely an interval containing
a single amplitude value αn where ε(inf )n = ε
(sup)n = 0,
n = 0,...,N − 1. As a consequence, it follows that P (θ)contains all power patterns generated by amplitude valuesαn ∈ An, n = 0, . . . , N − 1.
To better quantify the effects of the amplitude tolerances on
the array radiation properties, the intervals related to SLL,
BW , and D have been defined and computed. Towards this
end, let the end-points of SLL defined as follows [Fig. 3(a)]
inf SLL = maxθ /∈Ω inf P (θ)
−max
θ∈Ω sup
P
(θ
) [dB
]
(30)
sup SLL = maxθ /∈Ω sup P (θ)−maxθ∈Ω inf P (θ) [dB]
(31)
where Ω denotes the mainlobe region, while those of BW
[Fig. 3(b)]
inf BW = θ(inf )3dB,r − θ
(inf )3dB,l (32)
sup BW = θ(sup)3dB,r − θ
(sup)3dB,l (33)
θ(inf )3dB,l = min θ : inf P (θ) = sup P (θ) − 0.5 and
θ(inf )3dB,r = max
θ : inf
P (θ)
= sup
P (θ)
−0.5
be-
ing the angular directions corresponding to the two points
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0 Re
Im
inf X supX
inf Y
supY
ReC
C
I m C
Fig. 10. IA-based Approach - Complex interval number.
• naturally cope with the uncertainties and the tolerance
errors in the values of the array amplitudes;
• define exact and analytical bounds of the array factor and
the corresponding power pattern by exploiting the rulesof interval arithmetic;
• be robust and reliable thanks to the inclusion function
property of IA.
As for the numerical assessment, the effectiveness of the
proposed approach in evaluating the impact on the pattern
features of the manufacturing tolerances has been studied by
addressing several tolerance-analysis problems concerned with
different beams and array dimensions. More in detail, the
numerical results have shown that
• the higher the amplitude errors, the larger are the admis-
sible deviations from the reference/nominal values of the
pattern features;• the errors on the amplitude excitations limit the possi-
bility to arbitrary reduce the SLL also increasing the
tapering of the amplitude coefficients;
• the method efficiently deals with the analysis of large an-
tenna arrays, as well, thanks to the analytical formulation.
Further advances will consider the extension of the proposed
IA-based method to the analysis of the tolerances when errors
are present on the excitation phases and/or on the positions of
the array elements. Towards this aim, both the Cartesian (as
done in this paper) and polar representation [25] of complex
intervals will be taken into account and compared. Moreover,
the formulation will be extended to deal also with planar (2D)or conformal (3D) array configurations.
APPENDIX I - COMPLEX INTERVAL DEFINITION
A complex intervals C (Fig. 10) is defined by a
pair of ordered intervals C = X + jY, where X
Re C = [inf X ; sup X] and Y Im C =[inf Y ; sup Y] are real-valued intervals1. Accordingly,
C contains the complex values x + jy where
1The two real intervals, namely X and Y, defining C are ordered because
the complex intervals C′ = Y + jX obtained by inverting the order of thetwo real intervals identifies a different region of the complex plane. Hence,
C′ = C.
C =
x + jy
inf X ≤ x ≤ sup Xinf Y ≤ y ≤ sup Y
. (36)
APPENDIX II - COMPLEX INTERVAL ARITHMETIC
The operations defined for the arithmetic of complex inter-vals (e.g., addition, subtraction, multiplication, conjugation)
are extensions of those used for the arithmetical operations
between real intervals [18][19]. Since the arithmetic of real
intervals just requires the knowledge of the end points of the
two intervals involved in the operations, the same holds for the
arithmetical operations of complex intervals. In the following,
the key operations of the complex interval arithmetic are
summarized:
• Addition of Complex Intervals
The sum of the complex intervals C = X + jY and
C′ = X′ + jY′ is
C+C′ = (X+X′) + j (Y +Y′) (37)
where X,Y and X′,Y′ are real intervals. Moreover, the
sum X +Y, with X = [inf X ; sup X] and Y =[inf Y ; sup Y], turns out being [19]
X+Y = [inf X + inf Y ; sup X + sup Y] .(38)
The subtraction of complex intervals is a particular case
of addition where the sign of an interval is inverted. The
negative of C, namely −C, is defined as
−C =
−X
− jY (39)
where −X = [−sup X ; −inf X] and −Y =[−sup Y ; −inf Y], respectively.
• Multiplication of Complex Intervals
The product of the complex intervals C = X + jY and
C′ = X′ + jY′ is
CC′ =
XX
′ −YY′
+ jXY
′ +YX′
(40)
where the product between two real interval
numbers, X = [inf X ; sup X] and Y =[inf Y ; sup Y], is
XY = [min
inf
X
inf
Y
, inf
X
sup
Y
,
sup X inf Y , sup X sup Y ;max inf X inf Y , inf X sup Y ,
sup X inf Y , sup X sup Y] .(41)
• Complex Conjugation of a Complex Interval
The complex conjugate of the complex interval C = X+ jY, namely C∗, is defined as
C∗ = X− jY. (42)
• Multiplication of a Complex Interval and its Complex
Conjugate
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The product between the complex interval C = X+ jY
and its complex conjugate C∗ = X − jY is the real
interval
CC∗ = X2 +Y2 (43)
where
X2 =
min(inf
X
)2 , (sup
X
)2 ,
max(inf X)2 , (sup X)2if inf X > 0 or sup X < 00; max
(inf X)
2, (sup X)
2
if inf X ≤ 0 ≤ sup X .(44)
• Division of Complex Intervals
The division of the complex intervals C = X+ jY and
C′ = X′ + jY′ is
C
C
′=
XX′ +YY′
X
′2
−Y′2
;YX
′ +XY′
X
′2
−Y′2 . (45)
APPENDIX III - SUM OF INTERVAL WIDTHS /M ID-POINTS
It is proved in the following that the sum of the
widths/mid-points of real intervals is equivalent to the
width/mid-point of the interval sum. Towards this aim, let
us consider two real intervals X = [inf X ; sup X]and Y = [inf Y ; sup Y], having widths ω X =sup X − inf X, ω Y = sup Y − inf Yand mid-points µ X = inf X+supX
2, µ Y =
inf Y+supY
2
, respectively. According to (38), the sum
of two real intervals turns out equal to X + Y =[inf X + inf Y ; sup X + sup Y]. The width of
the interval sum, ω X+Y, is defined as the distance
between the right and left end-point of X+Y as
ω X+Y = (sup X + sup Y)−(inf X + inf Y) .
(46)
Since all terms in (46) are real numbers, it can be rewritten as
ω X+Y = (sup X − inf X) + (sup Y − inf Y)
= ω X + ω Y (47)
thus assessing that the width of an interval sum is equivalent
to the sum of the interval widths.
As for the mid-point µ X+Y, it is given in terms of the
end-points of X+Y as
µ X+Y =(inf X + inf Y) + (sup X + sup Y)
2(48)
or
µ X+Y =(inf X + sup X)
2+
(inf Y + sup Y)
2= µ X + µ Y (49)
thus proving that the mid-point of an interval sum is equivalent
to the sum of the interval mid-points.
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[2] H. L. Van Trees, Optimum Array Processing (Part IV). New York, NY:Wiley & Sons., 2002.
[3] R. S. Elliott, Antenna Theory and Design, 2nd ed. Hoboken, NJ: Wiley& Sons., IEEE Press., 2003.
[4] R. J. Mailloux, Phased Array Antenna Handbook , 2nd ed. Norwood,MA: Artech House, 2005.
[5] R. L. Haupt, Antenna Arrays - A Computation Approach. Hoboken, NJ:
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[6] J. Ruze, “The effect of aperture errors on the antenna radiation pattern,” Nuovo Cimento (Suppl.), vol. 9, no. 3, pp. 364-380, 1952.
[7] R. E. Elliott, “Mechanical and electrical tolerances for two-dimensional
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[8] M. I. Skolnik, “Nonuniform arrays,” Ch. 6 in Antenna Theory, R. E.Collin and F. J. Zucker, New York, NY: McGraw-Hill, 1969, pp. 227-
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[10] W. F. Richards and Y. T. Lo, “Antenna pattern synthesis based on
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[11] J. Lee, Y. Lee, and H. Kim, “Decision of error tolerance in array elementby the Monte Carlo method,” IEEE Trans. Antennas Propag., vol. 53,no. 4, pp. 1325-1331, Apr. 2005.
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[17] A. Neumaier, “Interval iteration for zeros of systems of equations,” BIT ,vol. 25, no. 1, pp. 256-273, 1985.
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Nicola Anselmi received the Bachelor De-
gree and the Master Degree in Telecomunication Engineering
from the University of Padova, in 2009 and from the University
of Trento, Italy, in 2012, respectively. From November 2012 he
is a member of the ELEDIA Research Center and his research
interests are mainly focused on optimization techniques and
antenna array design and synthesis.
Luca Manica was born in Rovereto, Italy.
He received his B.S. and M.S. degrees in Telecommunication
Engineering both from University of Trento, Italy, in 2004 and
2006, respectively. He received the PhD degree from the Inter-
national Graduate School in Information and Communication
Technologies, University of Trento, Italy. He is a member of
the ELEDIA Research Center and his main interests are the
synthesis of the antenna array patterns and fractal antennas.
Paolo Rocca received the MS degree
in Telecommunications Engineering from the University of
Trento in 2005 (summa cum laude) and the PhD Degree in
Information and Communication Technologies from the same
University in 2008. He is currently an Assistant Professor
at the Department of Information Engineering and Computer
Science (University of Trento) and a member of the ELEDIA
Research Center. He is the author/co-author of over 180 peer
reviewed papers on international journals and conferences. He
has been a visiting student at the Pennsylvania State University
and at the University Mediterranea of Reggio Calabria. Dr.
Rocca has been awarded from the IEEE Geoscience and
Remote Sensing Society and the Italy Section with the best
PhD thesis award IEEE-GRS Central Italy Chapter. His main
interests are in the framework of antenna array synthesis and
design, electromagnetic inverse scattering, and optimization
techniques for electromagnetics. He serves as an Associate
Editor of the IEEE Antennas and Wireless Propagation Letters.
Andrea Massa received the "laurea"
degree in Electronic Engineering from the Uni versity of
Genoa, Genoa, Italy, in 1992 and Ph.D. degree in electronics
and comp uter science from the same university in 1996. From
1997 to 1999 he was an Assis tant Professor of Electromag-
netic Fields at the Department of Biophysical and Electronic
Engineering (University of Genoa) teaching the university
course of Electromagnetic Fields 1. From 2001 to 2004,
he was an Associate Professor at the University of Trento.
Since 2005, he has been a Full Professor of Electromagnetic
Fields at the University of Trento, where he currently teaches
electromagnetic fields, inverse scattering techniques, antennas
and wireless communications, and optimization techniques.
At present, Prof. Massa is the director of the ELEDIA Re-
search Center at the University of Trento and Deputy Dean of the Faculty of Engineering. Moreover, he is Adjunct Professor
at Penn Stat University (USA), and Visiting Professor at the
Missouri University of Science and Technology (USA), at
the Nagasaki University (Japan), at the University of Paris
Sud (France), and at the Kumamoto University (Kumamoto
- Japan). He is a member of the IEEE Society, of the
PIERS Technical Committee, of the Inter-University Research
Center for Interactions Between Electromagnetic Fields and
Biological Systems (ICEmB), and he has served as Italian
representative in the general assembly of the European Mi-
crowave Association (EuMA).
His research work since 1992 has been principally on electro-
magnetic direct and inverse scattering, microwave imaging,
optimization techniques, wave propagation in presence of
nonlinear media, wireless communications and applications
of electromagnetic fields to telecommunications, medicine and
biology. Prof. Massa serves as an Associate Editor of the IEEE
Transactions on Antennas and Propagation.