Real array pattern tolerances from amplitude excitation errors · Marco Salucci2, and Paolo...

Real array pattern tolerancesfrom amplitude excitationerrors

Toshifumi Moriyama1a), Lorenzo Poli2, Nicola Anselmi2,Marco Salucci2, and Paolo Rocca2b)1 Graduate School of Engineering, Nagasaki University,

1–14 Bunkyo-machi, Nagasaki 852–8521, Japan2 ELEDIA Research Center @ DISI, University of Trento,

Via Sommarive, 38123 Trento, Italy

a) [email protected]

b) [email protected]

Abstract: The impact on the nominal power pattern of random and not

a-priori known errors affecting the excitation amplitudes of real linear arrays

is analyzed by means of an analytic approach based on the interval analysis

math. Starting from the expressions of the pattern bounds as a function of the

excitation tolerances modeling the saturation in the amplifiers of the array

feeding network, the effects on the radiation characteristics and the pattern

descriptors are evaluated.

Keywords: pattern tolerance, antenna arrays, linear arrays, excitation am-

plitude errors, interval analysis

Classification: Electromagnetic theory

References

[1] E. Brookner: Practical Phased Array Antenna Systems (Artech House, Norwood,MA, 1991).

[2] R. J. Mailloux: Phased Array Antenna Handbook (Artech House, Norwood, MA,2005).

[3] J. Lee, Y. Lee and H. Kim: IEEE Trans. Antenn. Propag. 53 (2005) 1325. DOI:10.1109/TAP.2005.844444

[4] J. Ruze: Nuovo Cim. 9 [3] (1952) 364. DOI:10.1007/BF02903409[5] R. L. Haupt: Antenna Arrays—A Computation Approach (Wiley & Sons,

Hoboken, NJ, 2010).[6] N. Anselmi, L. Manica, P. Rocca and A. Massa: IEEE Trans. Antenn. Propag. 61

(2013) 5496. DOI:10.1109/TAP.2013.2276927[7] R. Moore: Interval Analysis (Prentice-Hall, Englewood-Cliffs, NJ, 1966).[8] G. Alefeld and J. Herzberger: Introduction to Interval Computations (Academic

Press, New York, NY, 1983).[9] P. Rocca, L. Manica, N. Anselmi and A. Massa: IEEE Antennas Wirel. Propag.

Lett. 12 (2013) 639. DOI:10.1109/LAWP.2013.2261912

© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014

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LETTER IEICE Electronics Express, Vol.11, No.17, 1–8

http://dx.doi.org/10.1109/TAP.2005.844444




http://dx.doi.org/10.1007/BF02903409

http://dx.doi.org/10.1007/BF02903409





http://dx.doi.org/10.1109/LAWP.2013.2261912




1 Introduction

The analysis of the tolerances on the power pattern generated by an array when

manufacturing errors affect the excitation coefficients and/or the antenna control

points is a problem of great interest in the antenna community. As a matter of fact,

such a problem has been studied for decades [1, 2] and several different methods

have been proposed still recently [3]. Moreover, the knowledge of the average

radiation performances and the potential deviations from the corresponding mean

values is of interest to predict the antenna behavior when used in real operating

conditions. On the other hand, the availability of fast analytic techniques for pattern

tolerance analysis enables the synthesis of robust and reliable antenna arrays

avoiding and/or simplifying complex and time consuming trial-and-test calibration

processes that, nowadays, are usually mandatory.

In the past, statistical approaches have been developed [4, 5] for dealing with

the analysis of the array pattern tolerances. By considering random errors around

the nominal excitations, the impact on the radiation characteristics has been studied

as the superposition of an additional power pattern to the nominal/expected one.

By virtue of their intrinsic statistical nature and the fact that a-priori assumptions

are required on the error distributions, the arising tolerance estimations turn out

being only probabilistically verified.

In [6], an innovative method for the analytic and exact computation of the

power pattern tolerance for errors on the array amplitude coefficients has been

proposed. Based on the use of Interval Analysis (IA) [7, 8] and without a-priori

hypotheses on the amplitude error distributions, the effects on the arising power

pattern of the deviations of the array coefficients from the nominal ones are

expressed through the rules of interval arithmetic [7, 8].

In this paper, starting from the knowledge of the mid-point and width of each

error interval on the excitation amplitudes, the behavior of the average power

pattern radiated by a realistic array is evaluated by extending the preliminary

analysis for ideal cases proposed in [9] and further assessing its effectiveness and

reliability.

2 IA-based approach for pattern tolerance evaluation

Let us consider a N-element linear array radiating a reference/nominal power

pattern PðuÞ, u 2 ½�1;1� being the directional cosine, by feeding the antenna

elements with amplitude weights an, n ¼ 0; . . . ; N � 1. Since the amplifiers are

affected by unknown error tolerances, each n-th excitation can assume a value

within the range

mn � 12wn; mn þ 1

2wn

� �n ¼ 0; . . . ; N � 1 ð1Þ

where

mn ¼ asupn þ ainfn

2; wn ¼ asupn � ainfn ; n ¼ 0; . . . ; N � 1 ð2Þ

are the mid-point and the width of the n-th real amplitude interval ½an� ¼ ½ainfn ;asupn �(n ¼ 0; . . . ; N � 1), ainfn and asupn being its lower and upper bounds supposed given

or estimated.


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IEICE Electronics Express, Vol.11, No.17, 1–8

According to the IA-based approach in [6], the average power pattern,

mfPðuÞg, and the corresponding pattern tolerance width, wfPðuÞg, assume the

following closed-form expressions

mfPðuÞg ¼ M2RðuÞ þW2

RðuÞ þM2I ðuÞ þW2

I ðuÞ ð3aÞwfPðuÞg ¼ 2 MRðuÞWRðuÞf g þ 2 MIðuÞWIðuÞf g ð3bÞ

when [MRðuÞ þWRðuÞ < 0 or MRðuÞ �WRðuÞ > 0] and [MIðuÞ þWIðuÞ < 0 or

MIðuÞ �WIðuÞ > 0], otherwise

mfPðuÞg ¼ M2RðuÞ þW2

RðuÞ þ 12MIðuÞ þWIðuÞf g2 ð4aÞ

wfPðuÞg ¼ 2 MRðuÞWRðuÞf g þ 12MIðuÞ þWIðuÞf g2 ð4bÞ

when [MRðuÞ þWRðuÞ < 0 or MRðuÞ �WRðuÞ > 0] and [MIðuÞ �WIðuÞ � 0 �MIðuÞ þWIðuÞ], otherwise

mfPðuÞg ¼ 12MRðuÞ þWRðuÞf g2 þM2

I ðuÞ þW2I ðuÞ ð5aÞ

wfPðuÞg ¼ 12MRðuÞ þWRðuÞf g2 þ 2 MIðuÞWIðuÞf g ð5bÞ

when [MRðuÞ �WRðuÞ � 0 � MRðuÞ þWRðuÞ] and [MIðuÞ þWIðuÞ < 0 or MIðuÞ �WIðuÞ > 0], and

mfPðuÞg ¼ 12MRðuÞ þWRðuÞf g2 þ 1

2MIðuÞ þWIðuÞf g2 ð6aÞ

wfPðuÞg ¼ 12MRðuÞ þWRðuÞf g2 þ 1

2MIðuÞ þWIðuÞf g2 ð6bÞ

when [MRðuÞ �WRðuÞ � 0 � MRðuÞ þWRðuÞ] and [MIðuÞ �WIðuÞ � 0 � MIðuÞ þWIðuÞ], being

MRðuÞ ¼XN�1n¼0

mn cos�n

��; WRðuÞ ¼

XN�1n¼0

wn cos�nj j ð7Þ

and

MIðuÞ ¼XN�1n¼0

mn sin �n

��; WIðuÞ ¼

XN�1n¼0

wn sin �nj j ð8Þ

where �n ¼ kndu, k is the wave-number and d is the inter-element distance.

Starting from the expressions of mfPðuÞg and wfPðuÞg, the bounds of the power

pattern interval ½PðuÞ� ¼ ½Pinf ðuÞ;PsupðuÞ� turn out to be Pinf ðuÞ ¼ mfPðuÞg �12wfPðuÞg and PsupðuÞ ¼ mfPðuÞg þ 1

2wfPðuÞg, respectively.

In order to quantify the effects of the amplitude tolerances on the power pattern,

the functional extensions of the main beam descriptors to the pattern interval ½PðuÞ�(namely, the sidelobe level interval ½SLL� ¼ ½SLLinf ; SLLsup�, the mainlobe half-

power beamwidth interval, ½BW� ¼ ½BW inf ;BW sup�, and the directivity interval,

½D� ¼ ½Dinf ;Dsup�) are evaluated as follows

SLLinf ¼ maxu=2� Pinf ðuÞ� � �maxu2� PsupðuÞf g ð9ÞSLLsup ¼ maxu=2� PsupðuÞf g �maxu2� Pinf ðuÞ� � ð10Þ

BW inf ¼ usup!inf�3dB;r � usup!inf

�3dB;l ð11ÞBW sup ¼ uinf!sup

�3dB;r � uinf!sup�3dB;l ð12Þ


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Dinf ¼ infXN�1n¼0

an½ ��

��2� XN�1

n¼0an½ �j j2

( )ð13Þ

Dsup ¼ supXN�1n¼0

an½ ��

��2� XN�1

n¼0an½ �j j2

( )ð14Þ

where Ω identifies the main-lobe region, usup!inf�3dB;r=l (u

inf!sup�3dB;r=l) is the angular direction

on the left (l) or the right (r) side of the main-lobe peak where Pinf ðuÞ [PsupðuÞ]turns out to be 3 dB below the peak of PsupðuÞ [Pinf ðuÞ].

3 Real-array pattern tolerance prediction

To investigate the behavior of the pattern tolerance in real arrays by extending the

analysis carried out in [6] and limited to ideal cases in [9], let us consider the

following benchmark examples. With reference to a linear array of N ¼ 21

elements uniformly-spaced by d ¼ �=2, λ being the wavelength at the working

frequency, let us assume control points generating amplitude excitations an,

n ¼ 0; . . . ; N � 1 in the range ainfn � an � asupn (Fig. 1) depending on the nominal

value. As it can be observed, an upper threshold for the excitations, athn ¼ 1:0, has

been set to model the saturation effect of real amplifiers.

Fig. 1. Amplitude excitation tolerance versus the value of the nominalamplitude.

Fig. 2. Example 1 - Mean value, mn, and width, wn, of the toleranceinterval of the amplitude excitations and nominal values, an,n ¼ 0; . . . ; N � 1.


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In the first example (Example 1), the nominal excitations have been chosen to

generate a Dolph-Chebyshev sum pattern [2, 5] with sidelobe level 20 dB below the

peak of the main beam (SLLref ¼ �20 dB).

By considering the tolerance profile in Fig. 1 and the nominal excitations, an,

n ¼ 0; . . . ; N � 1, in Fig. 2, the mid-points and widths of the excitation error

intervals, ½an�, n ¼ 0; . . . ; N � 1, turn out as in Fig. 2. It is interesting to notice

the saturation of the upper bound asupn of the elements n ¼ 3; . . . ; 17. The average

function mfPðuÞg of the power pattern interval ½PðuÞ� and its bounds, Pinf ðuÞ andPsupðuÞ, are shown in Fig. 3.

As expected, the nominal beam, PðuÞ, as well as mfPðuÞg lay within the boundsof the power pattern interval [Pinf ðuÞ � mfPðuÞg � PsupðuÞ and Pinf ðuÞ � PðuÞ �PsupðuÞ]. The beam descriptors of the average power pattern, mfPðuÞg, the nominal

one, PðuÞ, and the power pattern interval ½PðuÞ� (i.e., Eqs. (9)–(14)) are given in

Table I. As it can be noticed, the values of the descriptors of mfPðuÞg and PðuÞ arealso contained within the interval descriptors of ½PðuÞ� (Table I). As for the average

deviation from the nominal beam (i.e., mfPðuÞg vs. PðuÞ), it turns out that there isan increment of the SLL of about 3.97 dB, while the directivity has a variation of

0.15 dB (Table I).

Fig. 3. Example 1 - Plot of the mean value, mfPðuÞg, and bounds,Pinf ðuÞ and PsupðuÞ, of the tolerance interval of the powerpattern, ½PðuÞ�, together with the nominal beam pattern, PðuÞ.

Fig. 4. Example 2 - Mean value, mn, and width, wn, of the toleranceinterval of the amplitude excitations and nominal values, an,n ¼ 0; . . . ; N � 1.


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In the second example (Example 2), the same antenna geometry is dealt with,

but the values of the nominal excitations have been set to afford a Taylor sum

pattern [2, 5] still with SLLref ¼ �20 dB and n ¼ 3 (Fig. 4). The values of the

excitation amplitude mid-points and widths when considering the tolerance profile

of Fig. 1 result as in Fig. 4. Analogously to the first example, the upper bounds of

the excitation amplitude intervals asupn , n ¼ 0; . . . ; N � 1 saturate at the elements

n ¼ 3; . . . ; 17. The plots of mfPðuÞg, PðuÞ, and the bounds of ½PðuÞ� are shown in

Fig. 5, while the corresponding beam descriptors are reported in Table II.

By comparing the values in Table I and Table II, the pattern indexes turn out

quite similar. However, the average power pattern of the Dolph-Chebyshev array

has a sidelobe level (SLLðmfPðuÞgÞjDC ¼ �16:03 dB) lower than that of the Taylor

array (SLLðmfPðuÞgÞjTaylor ¼ �15:74 dB). On the other hand, the directivity of the

average beam of the Taylor pattern (DðmfPðuÞgÞjTaylor ¼ 13:17 dB) is slightly

above that of the Dolph-Chebyshev one (DðmfPðuÞgÞjDC ¼ 13:14 dB). This is

caused by the decreasing behavior of the secondary lobes of the nominal pattern

(Fig. 5). It is also worthwhile to point out that the beam pattern descriptors of

mfPðuÞg and PðuÞ belong to the intervals of the descriptors of the pattern interval

further confirming the validity of (9)–(14).

The last example (Example 3) is concerned with the analysis of the pattern

tolerances when varying the dimension of the array, while generating the same

nominal patterns of the previous examples. Towards this end, the number of array

elements has been varied between N ¼ 10 up to N ¼ 100 elements. For each array

configuration, starting from the distribution of the nominal amplitude coefficients,

the corresponding amplitude errors have been determined according to the rules

pictorially summarized in Fig. 1.

Table I. Example 1 - Power pattern indexes.

SLL [dB] BW (�10�1) [u] D [dB]

mfPðuÞg −16.03 0.87 13.14½PðuÞ� [−38.51;−7.39] [0.00;1.37] [9.83;16.49]PðuÞ −20.00 0.89 12.99

Fig. 5. Example 2 - Plot of the mean value, mfPðuÞg, and bounds,Pinf ðuÞ and PsupðuÞ, of the tolerance interval of the powerpattern together, ½PðuÞ�, with the nominal beam pattern, PðuÞ.


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Successively, the interval power patterns and the intervals of the beam descrip-

tors have been computed. Figs. 6, 7, and 8 show the behavior of the lower and

upper bounds of the descriptor intervals [SLL], [BW], and [D] together with the

value of the sidelobe level, half-power beamwidth, and directivity of the nominal

power pattern as a function of the number of elements in correspondence with each

pattern type (i.e., the Dolph-Chebyshev pattern with SLLref ¼ �20 dB and the

Taylor pattern with SLL ¼ �20 dB, n ¼ 3).

As it can be observed (Fig. 6), the values of upper bound of the sidelobe level,

SLLsup, turns out being almost equal between the Dolph-Chebyshev beam and

the Taylor one. As for SLLinf , the Dolph-Chebyshev index rapidly decreases when

enlarging the array, while the variations for the Taylor one are limited to few

Table II. Example 1 - Power pattern indexes.

SLL [dB] BW (�10�1) [u] D [dB]

mfPðuÞg −15.74 0.89 13.17½PðuÞ� [−33.34;−7.05] [0.00;1.42] [9.70;16.67]PðuÞ −20.00 0.95 13.01

Fig. 6. Example 3 – Behavior of the bounds of [SLL] together with thenominal value.

Fig. 7. Example 3 – Behavior of the bounds of [BW] together with thenominal value.


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decibels whatever N. Unlike Fig. 6, Fig. 7 shows that both half-power beamwidth

bounds mainly depend on the array size and slightly from the pattern type.

Finally, Fig. 8 gives an indication on the dependence of the directivity bounds

on the number of array elements. As expected [2], the directivity of the nominal

Taylor patterns is higher than that of the Dolph-Chebyshev ones, especially when

dealing with large arrays. Moreover, the interval widths of [D] result narrower

when using Taylor arrays.

4 Conclusion

Estimates of the pattern tolerances in realistic linear arrays presenting bounded

errors in the amplitude excitations have been yielded by exploiting IA-derived

analytic expressions. Starting from a realistic model of the amplitude tolerances,

numerical bounds for the power patterns have been analyzed to give the array

designer suitable indications on the arising performances thus avoiding time-

expensive trial-and-test numerical predictions.

Acknowledgments

This work was supported in part by a Grant-in-Aid for Scientific Research (C)

Number 25420411.

Fig. 8. Example 3 – Behavior of the bounds of [D] together with thenominal value.


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Real array pattern tolerances from amplitude excitation errors · Marco Salucci2, and Paolo...

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