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Article(id=1239167204185395875, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239167201161302658, articleNumber=null, orderNo=null, doi=10.12347/j.ycyk.20241216001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1734278400000, receivedDateStr=2024-12-16, revisedDate=1736524800000, revisedDateStr=2025-01-11, acceptedDate=null, acceptedDateStr=null, onlineDate=1773370084091, onlineDateStr=2026-03-13, pubDate=1741968000000, pubDateStr=2025-03-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773370084091, onlineIssueDateStr=2026-03-13, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773370084091, creator=13701087609, updateTime=1773370084091, updator=13701087609, issue=Issue{id=1239167201161302658, tenantId=1146029695717560320, journalId=1238841944844054536, year='2025', volume='46', issue='2', pageStart='1', pageEnd='142', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773370083370, creator=13701087609, updateTime=1773370146323, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1239167465285014189, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239167201161302658, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1239167465285014190, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1239167201161302658, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=124, endPage=128, ext={EN=ArticleExt(id=1239167205934420657, articleId=1239167204185395875, tenantId=1146029695717560320, journalId=1238841944844054536, language=EN, title=Near-Field Calibration Algorithm for Millimeter-Wave Array Antennas Based on Plane Wave Expansion, columnId=1239167205829563052, journalTitle=Journal of Telemetry, Tracking and Command, columnName=Microwave, Antenna, Servo and Feed, runingTitle=null, highlight=null, articleAbstract=

The development of wireless communication technology has imposed strict requirements on antennas, including high-frequency operation, miniaturization, and strong directivity, to support higher data transmission rates in the future. Due to inherent amplitude and phase differences among channels in millimeter-wave chips and multi-antenna arrays, as well as additional amplitude and phase differences caused by factors such as temperature characteristics and aging during operation, beamforming accuracy can be significantly degraded. To mitigate the impact of inter-channel amplitude and phase differences on beam control, this paper proposes a near-field calibration algorithm based on plane wave spectrum expansion. The algorithm constructs a signal model between the antenna under test (AUT) and the probe using plane wave spectrum theory. Based on this model, a virtual signal for calibration measurement is generated and compared with the measured signal. A genetic algorithm is used to optimize the difference between the virtual and measured signals, and the initial excitations of the millimeter-wave array antenna are determined when the difference reaches its minimum. To validate the effectiveness of the proposed calibration algorithm, a four-element millimeter-wave linear array was calibrated. The probe scanning range was approximately one-third of the array aperture. The obtained amplitude calibration error range was ±0.43 dB, and the phase calibration error range was ±4.6°, achieving high-precision calibration.

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随着无线通信技术的发展，人们对天线提出了高频、小型化、强定向性的要求，以支持未来更高的数据传输速率。由于毫米波芯片、多天线阵列中固有的通道幅度和相位差，使用过程中受温度特性、老化等因素影响，会产生附加幅度相位差。为消除通道间幅度相位差对波束控制的影响，本文提出一种基于平面波谱展开的近场校准算法。基于平面波谱理论构建待测阵列天线与探头之间的信号模型，通过该信号模型建立校准测量的虚拟信号，并将虚拟信号与实测信号作比较。通过遗传算法优化虚拟信号和实测信号的差别，当其达到最小值时，求解出毫米波阵列天线的初始激励。为验证该校准算法的有效性，文中对一个四单元的毫米波线阵进行了校准测量，其中探头扫描范围约为1/3阵列天线口径。所得幅度校准误差范围为±0.43 dB，相位校准误差范围为±4.6°，实现了高精度的校准。

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唐元华　1999年生，博士研究生。

王正鹏　1981年生，副教授，博士生导师。。

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唐元华　1999年生，博士研究生。

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唐元华　1999年生，博士研究生。

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王正鹏　1981年生，副教授，博士生导师。。

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王正鹏　1981年生，副教授，博士生导师。。

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The comparison of near-field calibration methods

, figureFileSmall=null, figureFileBig=null, tableContent=
校准方法探头测量方式探头扫描范围校准精度
平面扫描法探头扫描远大于阵列天线口径高
内插法探头扫描不小于阵列天线口径较高
外插法探头扫描约1/2阵列天线口径较高
文中的方法探头扫描约1/3阵列天线口径高
), ArticleFig(id=1239285168163311893, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1239167204185395875, language=CN, label=表1, caption=

近场校准方法比较

, figureFileSmall=null, figureFileBig=null, tableContent=
校准方法探头测量方式探头扫描范围校准精度
平面扫描法探头扫描远大于阵列天线口径高
内插法探头扫描不小于阵列天线口径较高
外插法探头扫描约1/2阵列天线口径较高
文中的方法探头扫描约1/3阵列天线口径高
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基于平面波谱展开的毫米波阵列天线近场校准算法
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唐元华 , 王正鹏
遥测遥控 | 微波天伺馈 2025,46(2): 124-128
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遥测遥控 | 微波天伺馈 2025, 46(2): 124-128
基于平面波谱展开的毫米波阵列天线近场校准算法
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唐元华, 王正鹏
作者信息
  • 北京航空航天大学 北京 100191

    • 唐元华　1999年生，博士研究生。

    王正鹏　1981年生，副教授，博士生导师。。

Near-Field Calibration Algorithm for Millimeter-Wave Array Antennas Based on Plane Wave Expansion
Yuanhua TANG, Zhengpeng WANG
Affiliations
  • School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
出版时间: 2025-03-15 doi: 10.12347/j.ycyk.20241216001
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摘要
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随着无线通信技术的发展，人们对天线提出了高频、小型化、强定向性的要求，以支持未来更高的数据传输速率。由于毫米波芯片、多天线阵列中固有的通道幅度和相位差，使用过程中受温度特性、老化等因素影响，会产生附加幅度相位差。为消除通道间幅度相位差对波束控制的影响，本文提出一种基于平面波谱展开的近场校准算法。基于平面波谱理论构建待测阵列天线与探头之间的信号模型，通过该信号模型建立校准测量的虚拟信号，并将虚拟信号与实测信号作比较。通过遗传算法优化虚拟信号和实测信号的差别，当其达到最小值时，求解出毫米波阵列天线的初始激励。为验证该校准算法的有效性，文中对一个四单元的毫米波线阵进行了校准测量，其中探头扫描范围约为1/3阵列天线口径。所得幅度校准误差范围为±0.43 dB，相位校准误差范围为±4.6°，实现了高精度的校准。

毫米波阵列天线  /  近场校准算法  /  平面波谱展开

The development of wireless communication technology has imposed strict requirements on antennas, including high-frequency operation, miniaturization, and strong directivity, to support higher data transmission rates in the future. Due to inherent amplitude and phase differences among channels in millimeter-wave chips and multi-antenna arrays, as well as additional amplitude and phase differences caused by factors such as temperature characteristics and aging during operation, beamforming accuracy can be significantly degraded. To mitigate the impact of inter-channel amplitude and phase differences on beam control, this paper proposes a near-field calibration algorithm based on plane wave spectrum expansion. The algorithm constructs a signal model between the antenna under test (AUT) and the probe using plane wave spectrum theory. Based on this model, a virtual signal for calibration measurement is generated and compared with the measured signal. A genetic algorithm is used to optimize the difference between the virtual and measured signals, and the initial excitations of the millimeter-wave array antenna are determined when the difference reaches its minimum. To validate the effectiveness of the proposed calibration algorithm, a four-element millimeter-wave linear array was calibrated. The probe scanning range was approximately one-third of the array aperture. The obtained amplitude calibration error range was ±0.43 dB, and the phase calibration error range was ±4.6°, achieving high-precision calibration.

Millimeter-wave array antennas  /  Near-field calibration algorithm  /  Plane wave expansion
唐元华, 王正鹏. 基于平面波谱展开的毫米波阵列天线近场校准算法. 遥测遥控, 2025 , 46 (2) : 124 -128 . DOI: 10.12347/j.ycyk.20241216001
Yuanhua TANG, Zhengpeng WANG. Near-Field Calibration Algorithm for Millimeter-Wave Array Antennas Based on Plane Wave Expansion[J]. Journal of Telemetry, Tracking and Command, 2025 , 46 (2) : 124 -128 . DOI: 10.12347/j.ycyk.20241216001
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0　引言
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以往文献提出了多种阵列天线校准算法，根据测量距离可以分为远场校准和近场校准两类。开关法[1]在远场范围内使用探头依次正对测量各个阵列天线单元。开关法由于其测试操作简单、成本低而被广泛应用于工业界，然而其要求探头扫描系统具有很高的定位精度。旋转矢量法[2-6]通过给每个阵列单元配置不同的相位，并记录阵列天线的辐射功率。阵列单元的初始激励由测量的功率变化曲线确定，该曲线可以用余弦曲线拟合。换相法[7-12]通过为阵列天线配置小条件数的相位矩阵，通过测得的信号求解线性方程组得到阵列天线的初始激励。然而，这两种远场校准方法的信号模型都是基于格林函数建立的，该信号模型的精度随着测试距离的减小而降低。
1988年，LEE提出对待测天线平面近场扫描，通过快速傅里叶变换反演待测天线远场方向图并重建口面场，完成对待测天线的校准[13]。该校准算法为了减小平面扫描引起的截断误差，需要远大于阵列天线口径的扫描范围，因此具有测试成本高的特点[14-16]。2022年，文献[17]在远场换相法的基础上引入了内插算法，通过探头近场扫描测量阵列天线单元，并对测得的信号使用内插算法计算阵列天线单元正前方的信号。该算法要求探头扫描范围不小于阵列天线口径。2023年，文献[18]提出了外插法，使用二次曲线拟合探头近场扫描测量信号的幅度和相位曲线，并通过外插算法计算阵列单元正前方的信号，该算法需要的探头扫描范围约等于一半阵列天线口径。本文提出一种基于探头近场扫描测量的校准算法，基于平面波谱展开理论建立了新的校准信号模型，用于高精度校准毫米波阵列天线，并降低了对探头扫描范围的要求。
1　校准信号模型
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校准算法的核心是校准信号模型，信号模型的精度也决定了校准算法的精度。以往文献中校准算法广泛应用的信号模型是基于格林函数构建的，具体如下：
其中，S表示探头在待测天线近场范围的测量信号，F表示待测天线到探头方向的远场方向图，G为探头到待测天线方向的远场方向图。为待测天线和探头之间的自由空间格林函数，其中k为自由空间波常数，r为待测天线和探头之间的距离。该信号模型主要应用于远场测量，在近场情况下信号模型精度会显著下降，这主要是因为它是基于平面波谱展开的探头补偿理论公式的远场形式。接下来将说明这一点，探头补偿理论公式于1978年由D T Paris在其论文[19]中提出，具体如下：
其中，F(kx,kz)为待测天线的平面波谱，G(kx,-kz)为探头的平面波谱，G(kx,-kz)代表的平面波传播方向与F(kx,kz)相反。k为自由空间波矢量，r代表待测天线和探头之间的位移矢量。该信号模型在近场条件和远场条件下都适用，对式(2)在r→∞的远场条件下应用驻定相位法[20]，可以得到：
式(1)与式(3)等效，证明了基于格林函数的信号模型为探头补偿理论公式在远场条件的表现形式。在近场条件下，不满足r→∞，式(1)的精度将会下降。
校准系统如图1所示，假设待测阵列天线是一个由N个天线单元组成的均匀线阵，各个阵列天线单元的远场方向图具有一致性。探头位于阵列天线的近场范围，同时位于阵列天线单元的远场范围。探头逐一对阵列天线各个单元进行扫描测量，其扫描范围为L，共M个采样点。其中，探头测量第n个阵列天线单元接收到的信号可以表示为：
h n表示第n个阵列天线单元的初始激励，xm为探头在第m个位置时跟第n个阵列天线单元之间沿x轴的距离，其中kx被平均分为Q份，y为阵列天线单元和探头的平面波谱积分组成的向量，代表了这两个天线的辐射特性。考虑待测天线所有的N个阵列天线单元，式(4)可以组成一个线性方程组，如下式所示：
其中，sn是通过探头近场扫描测量第n个阵列天线单元得到的信号；hn为待求解的初始激励，由第n个阵列天线单元和探头之间的几何关系决定，可以提前测量计算得到；y由阵列天线单元和探头的平面波谱构成，该向量也是未知的。
2　校准算法
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校准算法的目标是求解线性方程组(7)，并计算出各个阵列天线单元的初始激励hn,n∈[1,N]。在校准系统几何关系已知的基础上，可由计算得到An，探头测量信号sn由初始激励hn和待测阵列天线和探头的辐射特性y决定。当y跟待测阵列天线和探头的辐射特性一样，hn也跟待测阵列天线的初始激励一样时，由hn和y可根据式(4)计算出一个虚拟的信号s'n，其与探头对待测阵列天线的实测信号sn相同。因此，可以定义一个评价函数f来描述虚拟信号与实测信号sn之间的差异，如式(8)所示：
其中，||x||2代表求向量x的长度，因此函数f≥0始终成立。当f=0时，由其确定的初始激励hn和辐射特性y并不是唯一解，而是一个集合。在f=0的解集中，主要是由于天线辐射特性y有多种可能性，而初始激励hn的变化都在一个小范围内变动。在允许的±0.5 dB校准幅度误差范围和±5°校准相位误差范围内，f→0时初始激励的解可作为待测阵列天线的初始激励。此外，f=0是评价函数f的极小值点，根据费马引理，其极值点的必要条件为。根据该必要条件，求得评价函数f关于y*的导数，如式(9)所示：
根据费马引理，可以求得式(10)：
其中，为hn的共轭，为An的共轭转置矩阵。为寻求合适的hn和y，可使用遗传算法求解，使得评价函数f接近极小值点。首先初始化hn=1，根据初始条件通过式(10)求解y，根据这一组参数hn和y,通过式(4)求sn，并且根据式(8)求得评价函数f。通过遗传算法，迭代初始激励hn，继而求解y和sn，再进一步得到评价函数f。当评价函数f逐渐减小并收敛时，得到最终的初始激励hn。
3　校准算法在毫米波阵列天线中的应用
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为验证该校准算法，设计的实验系统如图2所示。待测的毫米波阵列天线为一个由4单元的开口波导天线组成的均匀线阵，毫米波线阵口径为42 mm，其工作频率范围为22 GHz~33 GHz。探头也为一个相同的开口波导天线，且距离毫米波阵列天线6.5 cm，位于毫米波线阵的近场范围。
首先，使用开关法校准该毫米波线阵，使用扫描架控制探头逐一正对毫米波阵列天线各个单元，同时通过控制电路关闭其余单元，进行校准测量。其中，开启的天线单元通过幅相调控网络设置该通道的衰减为0 dB，相移为0°。关闭单元通过设置该单元通道衰减为110 dB，相移为0°。其次，使用基于平面波谱展开方法校准该毫米波线阵，逐一开启毫米波线阵各天线单元，并保持其余单元关闭。探头以毫米波线阵正中心为原点，从-6 mm运动到6 mm，其中运动间隔为1 mm，并在每个位置处接收开启的天线单元发出的信号。将开关法校准实验结果作为对照，与基于平面波谱展开的校准算法结果对比，对比结果如图3所示。其中，幅度校准误差为±0.43 dB，相位校准误差为±4.6°。该实验结果说明：基于平面波谱展开的校准方法实现了对毫米波阵列天线的高精度校准。
4　结束语
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本文基于平面波谱展开理论，率先提出一种全新的校准信号模型。依托该校准信号模型，构建出虚拟接收信号。通过细致对比虚拟信号与实测信号，进一步提出基于遗传算法求解的近场校准算法。为验证该校准算法的有效性，针对由四个开口波导天线组成的毫米波线阵，开展近场探头扫描测量。以开关法校准结果作为基准，基于平面波谱展开的近场校准算法，校准幅度误差范围为 ±0.43 dB，校准相位误差范围为 ±4.6°。这一校准结果充分表明，基于平面波谱展开的校准算法实现了对毫米波阵列天线的高精度校准，有力地证明了该校准算法的有效性。此外，将本文校准算法与文献中近场校准方法的对比结果列于表1。结果显示，本文校准算法的校准模型更为精准，所需的探头扫描范围更小，在毫米波阵列天线高精度校准领域，有望发挥重要作用。
基金
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  • 国家重点研发计划(2022YFB2902403)
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2025年第46卷第2期
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doi: 10.12347/j.ycyk.20241216001
  • 接收时间：2024-12-16
  • 首发时间：2026-03-13
  • 出版时间：2025-03-15
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  • 收稿日期：2024-12-16
  • 修回日期：2025-01-11
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    北京航空航天大学 北京 100191
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