tolerance analysis of antenna arrays

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-


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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) +


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)





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) +


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)+


2

|cosΘn (θ)|

|sinΘn (θ)

|

(15)

inf

AFR (θ)AFI (θ)

=N −1

n=0

αn +


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,





−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)





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) =


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) =


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%.





-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


http://slidepdf.com/reader/full/tolerance-analysis-of-antenna-arrays 7/12





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





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.

REFERENCES

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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:

Wiley & Sons., 2010.

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[7] R. E. Elliott, “Mechanical and electrical tolerances for two-dimensional

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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.

tolerance analysis of antenna arrays

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