<trans-title xml:lang="en">A class of approximate computation method for antenna directivity</trans-title>

Jun-Qun Liu

doi:10.7498/aps.69.20191268

Acta Physica Sinica, Volume. 69, Issue 2, 028401-1(2020)

A class of approximate computation method for antenna directivity

Jun-Qun Liu^*

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Fig. 1. The solution coordinate for the pattern of planar array antennas.矩形阵列天线方向图函数求解坐标系

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Table 1. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation.

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Table 1. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(20)式理论真值D_pa	40.9512	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为40.9512, INVuv误差界1.247607 × 10^–8; 积分真值区间: [40.95119, 40.95129], 方法4)计算结果与被测天线理论真值近乎完全吻合, 就积分本身结果而言, 方法4)可作为积分真值参考基准	21.51	解析解, 公式应用范围受限
1) PNF	41.13	比真值偏大约0.18 dB	1.11	速度最快
2) 本文算法基础积分求和估算	40.9522	比真值偏大约0.001 dB	20.56	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	40.9522	比真值偏大约0.001 dB, INVuv误差界1.247085 × 10^–8, 积分真值区间: [40.9521, 40.9523]	211.78	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	40.95218	比真值偏大约0.001 dB, quad2d()算法本身误差, 不存在被积函数值误差, INVuv误差界9.120334 × 10^–9, 积分真值区间: [40.952139, 40.952217]	1895.52	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 2. Comparison and evaluation of four computation methods of directivity in uniform amplitude & linear scanning phase distribution of electromagnetic excitation.

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Table 2. Comparison and evaluation of four computation methods of directivity in uniform amplitude & linear scanning phase distribution of electromagnetic excitation.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(20)式理论真值D_pa	38.9473	0; 应用方法(4)对被测天线阵列本身进行数值积分, 结果为38.9474, INVuv误差界2.257101 × 10^–8; 积分真值区间: [38.94733, 38.94746], 方法4)计算结果与被测天线理论真值近乎完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准	20.92	解析解, 公式应用范围受限
1) PNF	39.33	比真值偏大约0.38 dB	1.12	速度最快
2) 本文算法基础积分求和估算	39.2175	比真值偏大约0.27 dB	20.92	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	39.2142	比真值偏大约0.27 dB, INVuv误差界1.248032 × 10^–7, 积分真值区间: [39.21385, 39.21457]	201.61	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	39.2023	比真值偏大约0.25 dB, quad2d()算法本身误差, INVuv误差界2. 873319 × 10^–8, 积分真值区间: [39.20225, 39.20241]	1276.35	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 3. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation.

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Table 3. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(20)式理论真值D_pa	37.8093	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为37.809329, INVuv误差界6.831371 × 10^–10; 积分真值区间: [37.809327, 37.809330], 方法4)计算结果与被测天线理论真值近乎完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准	21.47	解析解, 公式应用范围受限
1) PNF	38.40	比真值偏大约0.59 dB	1.04	速度最快
2) 本文算法基础积分求和估算	38.1953	比真值偏大约0.39 dB	20.65	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	38.2152	比真值偏大约0.4 dB, INVuv误差界1.249154 × 10^–8, 真值区间: [38.21517, 38.21523]	189.59	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	38.190513	比真值偏大约0.38 dB, quad2d()算法本身误差, INVuv误差界1.248422 × 10^–9, 真值区间: [38.19051, 38.190516]	1465.8	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 4. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with raised precision of FFT algorithm.

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Table 4. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with raised precision of FFT algorithm.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(20)式理论真值D_pa	37.8093	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为37.809329, INVuv误差界6.831371 × 10^–10; 积分真值区间: [37.809327, 37.809330], 方法4)计算结果与被测天线理论真值近乎完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准.	21.18	解析解, 公式应用范围受限
1) PNF	38.36	比真值偏大0.56 dB, 结果因FFT点数变化与表3相比略有变化	1.09	速度最快, 本例FFT点数为2¹¹ × 2¹¹, 其他算例FFT点数都为2¹⁰ × 2¹⁰
2) 本文算法基础积分求和估算	38.1533	比真值偏大0.34 dB	82.18	计算时间加长, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	38.1535	比真值偏大0.34 dB, INVuv误差界1.805049 × 10^–8, 真值区间: [38.153468, 38.153549]	900.46	计算时间长, 计算时间依赖于FFT点数和测试数据矩阵大小, 实用性减弱
4) 累加求和被积函数解析值	38.13822	比真值偏大0.33 dB, quad2d()算法本身误差, INVuv误差界1.247355 × 10^–9, 真值区间: [38.138217, 38.138222], 结果因FFT点数变化造成最大值略有变化, 最终结果与表3相比略有变化	1468.83	速度较慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 5. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with uniform random errors for amplitude & phase.

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Table 5. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with uniform random errors for amplitude & phase.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(21)式理论真值D_pa	13.3604	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为13.3604, INVuv误差界3.473624 × 10^–5; 积分真值区间: [13.3602, 13.3607], 方法4)计算结果与被测天线理论真值近乎完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准.	25.09	解析解, 公式应用范围受限
1) PNF	13.63	比真值大约0.27 dB	1.03	速度最快
2) 本文算法基础积分求和估算	13.6866	比真值偏大约0.33 dB	20.25	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	13.6806	比真值偏大约0.32 dB, INVuv误差界7.892290 × 10^–4, 真值区间: [13.6742, 13.6870]	279.98	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	13.6580	比真值偏大约0.30 dB, quad2d()算法本身误差, INVuv误差界1. 382844 × 10^–4, 真值区间: [13.6569, 13.6592]	1259.79	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 6. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with normal random errors for amplitude & phase.

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Table 6. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with normal random errors for amplitude & phase.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(21)式理论真值D_pa	12.6206	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为12.6213, INVuv误差界1.249209 × 10^–4; 积分真值区间: [12.6205, 12.6220], 方法4)计算结果与被测天线理论真值近乎完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准	24.84	解析解, 公式应用范围受限
1) PNF	12.90	比真值偏大约0.28 dB	1	速度最快
2) 本文算法基础积分求和估算	13.0038	比真值偏大约0.38 dB	20.40	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	13.0030	比真值偏大约0.38 dB, INVuv误差界1.249323 × 10^–3, 真值区间: [12.9943, 13.0116]	297.41	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	12.9737	比真值偏大约0.35 dB, quad2d()算法本身误差, INVuv误差界2.116856 × 10^–4, 真值区间: [12.9722, 12.9751]	1561.10	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

Table 7. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation with normal random errors for amplitude & phase.

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Table 7. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation with normal random errors for amplitude & phase.

方法	方向系数计算值/dB	误差分析、估计	计算时间/s	方法特点和局限
(21)式理论真值D_pa	13.5284	0; 应用方法4)对被测天线阵列本身进行数值积分, 结果为13.5299, INVuv误差界1.248662 × 10^–4; 积分真值区间: [13.52897, 13.53091], 方法4)计算结果与被测天线理论真值接近完全符合, 就积分本身结果而言, 方法4)可作为积分真值参考基准.	24.84	解析解, 公式应用范围受限
1) PNF	13.73	比真值大约0.2 dB	1.03	速度最快
2) 本文算法基础积分求和估算	13.8459	比真值偏大约0.33 dB	20.34	速度较快, 计算时间依赖于FFT点数和测试数据矩阵大小
3) 二维插值估计被积函数	13.8598	比真值偏大约0.33 dB, INVuv误差界1.249746 × 10^–3, 真值区间: [13.8493, 13.8703]	310.81	速度较慢, 计算时间依赖于FFT点数和测试数据矩阵大小
4) 累加求和被积函数解析值	13.8184	比真值偏大约0.29 dB, quad2d()算法本身误差, INVuv误差界3.546120 × 10^–4, 真值区间: [13.8154, 13.8213]	1611.23	速度最慢, 计算时间依赖于近场测试数据矩阵大小, 本例为104 × 156, 最佳逼近积分真值

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Jun-Qun Liu. A class of approximate computation method for antenna directivity[J]. Acta Physica Sinica, 2020, 69(2): 028401-1

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Paper Information

Category:

Received: Aug. 21, 2019

Accepted: --

Published Online: Nov. 9, 2020

The Author Email:

DOI:10.7498/aps.69.20191268

Topics

Nuclear physics

Particles and fields

Particle and nuclear astrophysics and cosmology

Table 1. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation.

Table 1. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation.

Table 2. Comparison and evaluation of four computation methods of directivity in uniform amplitude & linear scanning phase distribution of electromagnetic excitation.

Table 2. Comparison and evaluation of four computation methods of directivity in uniform amplitude & linear scanning phase distribution of electromagnetic excitation.

Table 3. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation.

Table 3. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation.

Table 4. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with raised precision of FFT algorithm.

Table 4. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with raised precision of FFT algorithm.

Table 5. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with uniform random errors for amplitude & phase.

Table 5. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with uniform random errors for amplitude & phase.

Table 6. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with normal random errors for amplitude & phase.

Table 6. Comparison and evaluation of four computation methods of directivity in Taylor distribution of electromagnetic excitation with normal random errors for amplitude & phase.

Table 7. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation with normal random errors for amplitude & phase.

Table 7. Comparison and evaluation of four computation methods of directivity in uniform distribution of electromagnetic excitation with normal random errors for amplitude & phase.

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