real array pattern tolerances from amplitude excitation errors · marco salucci2, and paolo...
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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
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
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.
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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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
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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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.
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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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.
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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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.
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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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.
© IEICE 2014DOI: 10.1587/elex.11.20140571Received June 13, 2014Accepted July 14, 2014Publicized August 1, 2014Copyedited September 10, 2014
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