# Tolerance Analysis of Antenna Array Pattern and Array Synthesis in the Presence of Excitation Errors

Authors: Ying Zhang; DanNi Zhao; Qiong Wang; ZhengBin Long; Xiaofeng Shen
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

This paper analyzes array pattern tolerance against excitation errors. The nonprobabilistic interval analysis algorithm is used for tolerance analysis of the nonideal uniform linear array in this work. Toward this purpose, corresponding interval models of the power pattern functions are established, respectively, with the consideration of the amplitude errors, phase errors, or both simultaneously, in antenna arrays. The tolerance for the amplitude-phase error of the main function parameters including the beamwidth, sidelobe level, and the directivity is simulated by computer according to the indicators and the actual requirements. Accordingly, the worst admissible performance of an array can be evaluated, which may provide theoretical reference for optimal antenna array design. As for the problem of array synthesis in the presence of various array errors, interval analysis-convex programming (IA-CP) is presented. Simulation results show that the proposed IA-CP based synthesis technique is robust for the amplitude and phase errors.


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## Body

## 1. Introduction

Antenna arrays are widely used for their flexibility and reliability in wireless data transmission. For array design, an important performance indicator is the radiation pattern. In real applications, excitation errors including amplitude error and phase error widely existed due to manufacturing errors. Excitation errors affect radiation pattern of arrays, which usually deteriorates performance of the designed arrays. Therefore, the effect of excitation errors on array radiation pattern should be analyzed for robust array design [1].Traditional error analysis strategies are generally based on the statistical theory. For instance, under the assumption that the excitation errors are normally distributed, Ruze proposed analyzing the association between the radiation pattern and the random excitation errors [2]. In [3], Hsiao obtained the statistical regularities of sidelobes distributed in different regions by studying the impacts of stochastic independence and corresponding amplitude-phase errors on the sidelobe level. However, these statistical methods cannot be applied into engineering design directly as the distribution of amplitude and phase errors may be unknown.To resolve this issue, a novel approach based on interval arithmetic (IA) was proposed in [4]. IA was originally introduced for the computation of the rounding errors when using numerical resolution strategies [5, 6]. It was then extended to deal with nonlinear equations [7] as well as optimization problems [8, 9]. Recently, this method has been successfully applied to compute the bounds of the radiation pattern when array excitation errors exist [4, 10–13]. Existing IA-based analysis methods showed satisfactory results. Nevertheless, there has been no literation considering the case where excitation amplitude and phase errors simultaneously exist.In this paper, the excitation amplitude and phase errors are considered simultaneously to achieve the upper and lower bounds of array radiation patterns. Tolerance of array radiation pattern is analyzed by interval arithmetic (IA). Since excitation phase error has different influence on the real and imaginary parts of the radiation pattern, the optimal bounds are hard to achieve. Instead, a relaxed version bound is presented. Accordingly, the worst admissible performance of an array can be evaluated, which may provide theoretical reference for optimal antenna array design. As for the problem of array synthesis in the presence of various array errors, a method based on interval analysis-convex programming (IA-CP) is given. Simulation results show that the IA-CP-based synthesis technique is not only robust for the amplitude and phase errors, but also suitable for large arrays.


## 2. Radiation Pattern Tolerance Analysis in the Presence of Excitation Amplitude and Phase Errors

### 2.1. Analysis Based on Interval Arithmetic

Consider a uniform linear array withN elements. Assume the real excitation amplitude and phase on the nth element are denoted by αn and φn, respectively. The array factor interval is given by(1)AFIθ=∑n=0N-1AFnIθ=∑n=0N-1αnIejΘnIθ,where αnI=αnL,αnU denotes the amplitude interval. ΘnIθ=nkdsin⁡θ+φnI, where φnI=φnL,φnU denotes the phase interval. θ∈[-π/2,π/2] is the angular direction measured from boresight, j=-1 is the unit complex number, d denotes element space, and k=2π/λ is the wavenumber.The interval of power pattern is given by(2)PIθ=AFRIθ2+AFIIθ2,where AFRIθ and AFIIθ denote the real and imaginary part of AFIθ, respectively.In the presence of amplitude and phase error, it is difficult to determine the upper and lower bound of  (2), since the real and imaginary parts of power pattern are coupled with phase. Use the fact that(3)maxAFR2θ+AFI2θ≤maxAFR2θ+maxAFI2θ,minAFR2θ+AFI2θ≥minAFR2θ+minAFI2θ.A relaxed version of the upper and lower bound of power pattern is proposed as(4a)PUθ=AFRUθ2+AFIUθ2,(4b)PLθ=AFRLθ2+AFILθ2.Then, similar to [4], the upper and lower bounds can be expressed by the midpoint and the interval width of AFRIθ and AFIIθ:(5a)PUθ=μAFRIθ+ωAFRIθ22+μAFIIθ+ωAFIIθ22,(5b)PLθ=μAFRIθ-ωAFRIθ22,0∉AFRI,0∈AFIIμAFIIθ-ωAFIIθ22,0∈AFRI,0∉AFIIμAFRIθ-ωAFRIθ22+μAFIIθ-ωAFIIθ22,0∉AFRI,0∉AFII0,otherwise,where μAFRIθ=∑n=0N-1μAFn,RIθ, ωAFRIθ=∑n=0N-1ωAFn,RIθ, μAFIIθ=∑n=0N-1μAFn,IIθ, and ωAFIIθ=∑n=0N-1ωAFn,IIθ.The midpoint and the interval width ofAFn,RIθ and AFn,IIθ are given by(6a)μAFn,RIθ=12AFn,RLθ+AFn,RUθ,μAFn,IIθ=12AFn,ILθ+AFn,IUθ,(6b)ωAFn,RIθ=AFn,RUθ-AFn,RLθ,ωAFn,IIθ=AFn,IUθ-AFn,ILθ.Based on the rules of interval algorithm,AFn,RIθ and AFn,IIθ can be computed as (7a)AFn,RUθ=AnLcosU⁡ΘnI,cosU⁡ΘnI<0,AnUcosU⁡ΘnI,otherwiseAFn,RLθ=AnLcosL⁡ΘnI,cosL⁡ΘnI>0,AnUcosL⁡ΘnI,otherwise.(7b)AFn,IUθ=AnLsinU⁡ΘnI,sinU⁡ΘnI<0,AnUsinU⁡ΘnI,otherwiseAFn,ILθ=AnLsinL⁡ΘnI,sinL⁡ΘnI>0,AnUsinL⁡ΘnI,otherwise.The lower and upper bounds ofsinΘnI and cosΘnI are given by(8a)cos⁡ΘnIθ=cos⁡ΘnLθ,cosΘnUθ,∂cos⁡ΘnI∂ΘnI>0,cos⁡ΘnUθ,cos⁡ΘnLθ,∂cos⁡ΘnI∂ΘnI<0,mincos⁡ΘnLθ,cos⁡ΘnUθ,1,2iπ∈ΘnI,-1,maxcos⁡ΘnLθ,cos⁡ΘnUθ,π+2iπ∈ΘnI,(8b)sin⁡ΘnIθ=sinΘnLθ,sinΘnUθ,∂sin⁡ΘnI∂ΘnI>0,sinΘnUθ,sinΘnLθ,∂sin⁡ΘnI∂ΘnI<0,minsinΘnLθ,sinΘnUθ,1,π2+2iπ∈ΘnI,-1,maxsinΘnLθ,sinΘnUθ,-π2+2iπ∈ΘnI.


## 2.1. Analysis Based on Interval Arithmetic

Consider a uniform linear array withN elements. Assume the real excitation amplitude and phase on the nth element are denoted by αn and φn, respectively. The array factor interval is given by(1)AFIθ=∑n=0N-1AFnIθ=∑n=0N-1αnIejΘnIθ,where αnI=αnL,αnU denotes the amplitude interval. ΘnIθ=nkdsin⁡θ+φnI, where φnI=φnL,φnU denotes the phase interval. θ∈[-π/2,π/2] is the angular direction measured from boresight, j=-1 is the unit complex number, d denotes element space, and k=2π/λ is the wavenumber.The interval of power pattern is given by(2)PIθ=AFRIθ2+AFIIθ2,where AFRIθ and AFIIθ denote the real and imaginary part of AFIθ, respectively.In the presence of amplitude and phase error, it is difficult to determine the upper and lower bound of  (2), since the real and imaginary parts of power pattern are coupled with phase. Use the fact that(3)maxAFR2θ+AFI2θ≤maxAFR2θ+maxAFI2θ,minAFR2θ+AFI2θ≥minAFR2θ+minAFI2θ.A relaxed version of the upper and lower bound of power pattern is proposed as(4a)PUθ=AFRUθ2+AFIUθ2,(4b)PLθ=AFRLθ2+AFILθ2.Then, similar to [4], the upper and lower bounds can be expressed by the midpoint and the interval width of AFRIθ and AFIIθ:(5a)PUθ=μAFRIθ+ωAFRIθ22+μAFIIθ+ωAFIIθ22,(5b)PLθ=μAFRIθ-ωAFRIθ22,0∉AFRI,0∈AFIIμAFIIθ-ωAFIIθ22,0∈AFRI,0∉AFIIμAFRIθ-ωAFRIθ22+μAFIIθ-ωAFIIθ22,0∉AFRI,0∉AFII0,otherwise,where μAFRIθ=∑n=0N-1μAFn,RIθ, ωAFRIθ=∑n=0N-1ωAFn,RIθ, μAFIIθ=∑n=0N-1μAFn,IIθ, and ωAFIIθ=∑n=0N-1ωAFn,IIθ.The midpoint and the interval width ofAFn,RIθ and AFn,IIθ are given by(6a)μAFn,RIθ=12AFn,RLθ+AFn,RUθ,μAFn,IIθ=12AFn,ILθ+AFn,IUθ,(6b)ωAFn,RIθ=AFn,RUθ-AFn,RLθ,ωAFn,IIθ=AFn,IUθ-AFn,ILθ.Based on the rules of interval algorithm,AFn,RIθ and AFn,IIθ can be computed as (7a)AFn,RUθ=AnLcosU⁡ΘnI,cosU⁡ΘnI<0,AnUcosU⁡ΘnI,otherwiseAFn,RLθ=AnLcosL⁡ΘnI,cosL⁡ΘnI>0,AnUcosL⁡ΘnI,otherwise.(7b)AFn,IUθ=AnLsinU⁡ΘnI,sinU⁡ΘnI<0,AnUsinU⁡ΘnI,otherwiseAFn,ILθ=AnLsinL⁡ΘnI,sinL⁡ΘnI>0,AnUsinL⁡ΘnI,otherwise.The lower and upper bounds ofsinΘnI and cosΘnI are given by(8a)cos⁡ΘnIθ=cos⁡ΘnLθ,cosΘnUθ,∂cos⁡ΘnI∂ΘnI>0,cos⁡ΘnUθ,cos⁡ΘnLθ,∂cos⁡ΘnI∂ΘnI<0,mincos⁡ΘnLθ,cos⁡ΘnUθ,1,2iπ∈ΘnI,-1,maxcos⁡ΘnLθ,cos⁡ΘnUθ,π+2iπ∈ΘnI,(8b)sin⁡ΘnIθ=sinΘnLθ,sinΘnUθ,∂sin⁡ΘnI∂ΘnI>0,sinΘnUθ,sinΘnLθ,∂sin⁡ΘnI∂ΘnI<0,minsinΘnLθ,sinΘnUθ,1,π2+2iπ∈ΘnI,-1,maxsinΘnLθ,sinΘnUθ,-π2+2iπ∈ΘnI.


## 3. Array Synthesis in the Presence of Array Excitation Errors

For ideal array, array synthesis has been studied in many literatures. However, array errors bring bad influence on the performance of the designed array pattern, for example, decrease of array gain and increase of sidelobe level. To avoid these problems, a solution is to take array error into account when designing the array. In [14], particle swarm optimization (PSO) technique is combined with interval arithmetic to realize array synthesis in the presence of array excitation amplitude error. However, PSO suffers from high computational loads. In this paper, array excitation in the presence of array excitation amplitude and phase errors is optimized by convex programming which can decrease computational loads.The interval of array power pattern considering excitation amplitude and excitation phase error can be expressed as(9)Pα,φIθ=∑n=0N-1αnIejΘnIθ2,where Pα,φI(θ)=Pα,φL(θ),Pα,φU(θ).To optimize the array excitation, the following cost function is constructed:(10a)αopt=argmaxα⁡Pα,φLθ0,(10b)subjecttoPα,φUθsα=αopt≤Mθs,θs∉Ω,where θ0 denotes the look direction, Ω denotes the main-lobe region, and θs denotes the sidelobe region. The proposed cost function aims to maximize the lower bound of the array radiation pattern at the look direction and simultaneously guarantees that the upper bound of the array pattern in sidelobe region does not exceed a preset value.Based on (4b), the objective function can be equivalently expressed as(11)Pα,φLθ0=∑n=0N-1αnLcosLΘnθ02+∑n=0N-1αnLsinLΘnθ02=∑n=0N-1αnLcosΘnUθ02+∑n=0N-1αnLsinΘnLθ02=∑n=0N-1αn-εncosΘnUθ02+∑n=0N-1αn-εnsinΘnLθ02,where εn is the excitation amplitude error of the nth element.Following (5a), (10b) can be expressed as(12)Pα,φUθs=∑n=0N-1μαnIcosΘnIθs+12∑n=0N-1ωαnIcosΘnIθs2+∑n=0N-1μαnIsinΘnIθs+12∑n=0N-1ωαnIsinΘnIθs2≤Mθs.It is observed from (11) and (12) that Pα,φL(θ0) and Pα,φU(θs) are quadratic functions with respect to αn. Therefore, convex programming can be used to solve (10a) and (10b).


## 4. Computer Simulations

### 4.1. IA-Based Method

A uniform linear array (ULA) withN=10 half-wavelength spaced sensors is considered. The nominal amplitudes αn (n=0,1,…,N-1) based on Chebyshev synthesis method are given in Table 1, which generates a power pattern with the sidelobe level −20 dB. The interval of the excitation amplitude and phase are assumed to be αnI=αn-δαnαn,αn+δαnαn and φnI=φn-δφn,φn+δφn.Table 1
Amplitude distribution (N=10, d=λ/2; Chebyshev: SLLref=-20dB).
α
0
=
α
9
α
1
=
α
8
α
2
=
α
7
α
3
=
α
6
α
4
=
α
5
1.000
0.926
1.213
1.436
1.559Q
=
5000 array patterns Pu, where u=sin⁡θ, have been generated by randomly selecting the amplitude and phase from these intervals with uniform distribution. Figures 1–3 show all Q beams as well as the IA-computed power pattern bounds with different amplitude and phase intervals.Figure 1
Bounds of the power patternPu based on IA with δαn=1%, δφn=1°.Figure 2
Bounds of the power patternPu based on IA with δαn=1%, δφn=3°.Figure 3
Bounds of the power patternPu based on IA with δαn=3%, δφn=1°.It can be observed from the figures that the IA-based methods which consider the amplitude or phase error only are unable to consistently give correct bounds of the power pattern. For the proposed method, the bounds are correct for all situations.For different expected sidelobe level (ISLLref), Figures 4–6 show the derived SLL interval, main-lobe bandwidth (BW) interval, and the directivity (D) interval with different amplitude and phase intervals, respectively.Figure 4
SLL interval based on IA againstISLLref.Figure 5
BW interval based on IA againstISLLref.Figure 6
Directivity interval based on IA againstISLLref.It can be concluded from these figures that synthesis of the desired array pattern becomes difficult when the excitation errors are large, since higher excitation errors lead to larger interval width, for example, forSLLref=-25dB, ωSLLδαn=1%,δφn=1°=6.54dB, ωSLLδαn=3%,δφn=3°=29.75dB, and ωSLLδαn=5%,δφn=5°=∞. From Figure 4, as ISLLref increases, the upper bound of the SLL approaches a constant higher than the expected SLL, which means it is unguaranteed to achieve an array pattern with low sidelobe level in the presence of excitation error. Also, from the figure, the precision of the excitation to achieve relative satisfactory performance can be derived; for example, for ISLLref = 20 dB, δαn=1%, δφn=1° guarantees that the derived SLL is within [-21.71,-18.03] dB.


### 4.2. IA-CP Based Array Synthesis

In this simulation, array excitation amplitude and phase errors are taken into account. The interval of the excitation amplitude and phase are assumed to beαnI=αn-δαnαn,αn+δαnαn and φnI=φn-η,φn+η. A ULA with 20 half-wavelength spaced elements is considered. The constraint of the sidelobe level M(θs) is set to 0 dB. Figure 7 shows the derived array excitation amplitudes with δαn=1%, η=1° and δαn=3%, η=3°. Figures 8 and 9 show the generated array radiation patterns using the derived excitation amplitudes.Figure 7
Optimized array excitation amplitudes using IA-CP method.Figure 8
Array radiation patterns using the optimized array excitation (δαn=1%, η=1°).Figure 9
Array radiation patterns using the optimized array excitation (δαn=3%, η=3°).It can be observed from Figure7 that when excitation errors become large, the derived array excitation amplitudes decrease. It is natural to see this since larger errors yield higher upper bound of array pattern. Therefore, in order to satisfy sidelobe constraint, excitation amplitudes become smaller. Figures 8 and 9 validate this conclusion. It can be observed from Figure 9 that the main-lobe gain with δαn=3%, η=3° is smaller than that with δαn=1%, η=1°. For the two cases, the upper bounds of the radiation pattern in sidelobe region meet the constraint, that is, smaller than 0 dB.


## 4.1. IA-Based Method

A uniform linear array (ULA) withN=10 half-wavelength spaced sensors is considered. The nominal amplitudes αn (n=0,1,…,N-1) based on Chebyshev synthesis method are given in Table 1, which generates a power pattern with the sidelobe level −20 dB. The interval of the excitation amplitude and phase are assumed to be αnI=αn-δαnαn,αn+δαnαn and φnI=φn-δφn,φn+δφn.Table 1
Amplitude distribution (N=10, d=λ/2; Chebyshev: SLLref=-20dB).
α
0
=
α
9
α
1
=
α
8
α
2
=
α
7
α
3
=
α
6
α
4
=
α
5
1.000
0.926
1.213
1.436
1.559Q
=
5000 array patterns Pu, where u=sin⁡θ, have been generated by randomly selecting the amplitude and phase from these intervals with uniform distribution. Figures 1–3 show all Q beams as well as the IA-computed power pattern bounds with different amplitude and phase intervals.Figure 1
Bounds of the power patternPu based on IA with δαn=1%, δφn=1°.Figure 2
Bounds of the power patternPu based on IA with δαn=1%, δφn=3°.Figure 3
Bounds of the power patternPu based on IA with δαn=3%, δφn=1°.It can be observed from the figures that the IA-based methods which consider the amplitude or phase error only are unable to consistently give correct bounds of the power pattern. For the proposed method, the bounds are correct for all situations.For different expected sidelobe level (ISLLref), Figures 4–6 show the derived SLL interval, main-lobe bandwidth (BW) interval, and the directivity (D) interval with different amplitude and phase intervals, respectively.Figure 4
SLL interval based on IA againstISLLref.Figure 5
BW interval based on IA againstISLLref.Figure 6
Directivity interval based on IA againstISLLref.It can be concluded from these figures that synthesis of the desired array pattern becomes difficult when the excitation errors are large, since higher excitation errors lead to larger interval width, for example, forSLLref=-25dB, ωSLLδαn=1%,δφn=1°=6.54dB, ωSLLδαn=3%,δφn=3°=29.75dB, and ωSLLδαn=5%,δφn=5°=∞. From Figure 4, as ISLLref increases, the upper bound of the SLL approaches a constant higher than the expected SLL, which means it is unguaranteed to achieve an array pattern with low sidelobe level in the presence of excitation error. Also, from the figure, the precision of the excitation to achieve relative satisfactory performance can be derived; for example, for ISLLref = 20 dB, δαn=1%, δφn=1° guarantees that the derived SLL is within [-21.71,-18.03] dB.


## 4.2. IA-CP Based Array Synthesis

In this simulation, array excitation amplitude and phase errors are taken into account. The interval of the excitation amplitude and phase are assumed to beαnI=αn-δαnαn,αn+δαnαn and φnI=φn-η,φn+η. A ULA with 20 half-wavelength spaced elements is considered. The constraint of the sidelobe level M(θs) is set to 0 dB. Figure 7 shows the derived array excitation amplitudes with δαn=1%, η=1° and δαn=3%, η=3°. Figures 8 and 9 show the generated array radiation patterns using the derived excitation amplitudes.Figure 7
Optimized array excitation amplitudes using IA-CP method.Figure 8
Array radiation patterns using the optimized array excitation (δαn=1%, η=1°).Figure 9
Array radiation patterns using the optimized array excitation (δαn=3%, η=3°).It can be observed from Figure7 that when excitation errors become large, the derived array excitation amplitudes decrease. It is natural to see this since larger errors yield higher upper bound of array pattern. Therefore, in order to satisfy sidelobe constraint, excitation amplitudes become smaller. Figures 8 and 9 validate this conclusion. It can be observed from Figure 9 that the main-lobe gain with δαn=3%, η=3° is smaller than that with δαn=1%, η=1°. For the two cases, the upper bounds of the radiation pattern in sidelobe region meet the constraint, that is, smaller than 0 dB.


## 5. Conclusion

In this paper, a novel approach based on interval analysis has been proposed to efficiently access the impact on power pattern when random manufacturing errors are present on both the array weights and the element position. The upper and lower bounds of the corresponding radiated array pattern are deduced according to the rules of the interval arithmetic. The obtained bounds can be used to facilitate array synthesis when array weights error and elements position error are considered in real applications. Simulation results demonstrate the robustness and reliability of the deduced new bounds. Furthermore, with the derived bounds, array synthesis is realized in the presence of excitation errors.



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