Amirreza Khajehnasiri
IMAGRE LICENSED BY INGRAM PUBLISHING
This article provides the theory behind the two-step over-the-air (OTA) test method widely used in 3rd Generation Partnership Project (3GPP) test specifications and, in particular, the testing and characterization of 5G millimeter-wave user equipment (UE) in compact antenna test ranges (CATRs). Special attention is paid to receiver (Rx) characteristics and the polarization mismatch problem, which is encountered in spherical coverage tests. The received power of a UE with a dual-polarized antenna system is derived, and the output signal-to-noise ratio (SNR) is calculated using the maximal ratio combining (MRC) principle. And finally, flaws in the dominant interpretation of this test method in the development of 5G millimeter-wave standards and their consequences are addressed.
As the operating frequency of mobile phones and other UE extends into millimeter-wave frequencies, hardware (HW) developers and network operators face new challenges for the testing, characterization, and certification of such devices. Whereas wireless standards up to and including 4G supported only frequency bands up to, at most, 6 GHz, 5G New Radio defines the following two distinct frequency ranges (FRs) for supported devices:
Millimeter-wave frequencies and FR2 are often used interchangeably. Due to higher path loss in FR2, phased arrays integrated with low-noise amplifiers (LNAs) and power amplifiers (PAs) are a necessity, which means antenna connectors are also integrated and not accessible for connecting external probes and cables. Besides, microwave switches used for connecting probes to antenna connectors are not available at millimeter-wave frequencies. For these reasons, the 3GPP has designated OTA testing as the baseline method for FR2 UE.
In conducted testing, splitter/combiners are used to split/combine signals between system simulator (SS) and UE transmitter (Tx)/Rx ports. Millimeter-wave UE replaces multiple Tx/Rx ports with two orthogonal polarizations of its antenna system. In OTA testing, there is no easy way of splitting and combining signals to two orthogonal polarizations. It will be shown that this problem is overcome by conducting all tests in two steps with two known orthogonal polarizations and using an equation to combine the results. This equation should mimic the actual modem processing of two orthogonal polarizations as if they were present at the same time. As an example, consider the measurement steps for effective isotropic sensitivity (EIS). Let ${EIS}_{\theta}$ and ${EIS}_{\varphi}$ represent the results with ${\theta}$ and ${\varphi}$ polarizations, respectively. It is shown here that the total EIS can then be calculated as \[\frac{1}{EIS} = \frac{1}{{EIS}_{\theta}} + \frac{1}{{EIS}_{\varphi}}{.} \tag{1} \]
The same formula is used in earlier versions of 3GPP specification 38.810 [1], in the Cellular Telecommunications Industry Association (CTIA) certification test plan for wireless device OTA performance [4], and in equations for calculating the total isotropic sensitivity(TIS)/total radiated sensitivity (TRS) in 3GPP specifications 34.114 [2] and 37.544 [3]. As shown in the following, Equation (1) represents the MRC of the ${\theta}$- and ${\varphi}$-polarized receive chains as if both polarizations were applied at the same time. In RAN4#89 (meeting #89 of Radio Access Network Workgroup 4— WG4) [5], this equation was modified to the following form: \[\frac{1}{EIS} = \frac{\frac{1}{{EIS}_{\theta}} + \frac{1}{{EIS}_{\varphi}}}{2}{.} \tag{2} \]
Compared to the legacy equation, the new one results in a 3-dB performance loss and is equivalent to tightening the EIS specification by 3 dB. This change, which was unfortunately based on erroneous arguments, is not entirely inconsequential.
The development of a new wireless standard, HW design, and network planning/deployment are all years-long endeavors that happen almost simultaneously despite being intricately dependent on one another. Mobile UE is designed to be compliant with a wireless standard which, in turn, cannot be developed without knowing the capabilities and limitations of the HW. Engineers are often forced to make judgment calls and use small margins for predictions that may not come to fruition. The situation is even more complicated with OTA tests, where measurement errors can be as high as several decibels. There is certainly no room for errors due to negligence.
While two-step test methods have been in use for well over a decade in 3GPP radiated test specifications, no detailed explanation appears in the literature. For this reason, this article is written in a tutorial format and intended to help wireless test engineers navigate OTA test methods. It starts by defining the Rx sensitivity and describing the OTA test setup commonly used in wireless test labs and laying the groundwork for the derivation of the EIS equation. This is followed by a brief treatment of EIS measurement steps in the lab and a short discussion of polarization diversity and the reason for the definition of the EIS in terms of the power per receive polarization. The final section is dedicated to refuting the arguments for the introduction of a factor of two in the EIS equation is intended to draw attention to the broader issues The arguments for the introduction of the factor of 2 has created in FR2 standards. Mathematical details appear in the “Calculation of EIS from v and h component polarizations†and “User Equipment Received Power†sections.
Rx sensitivity is defined as the downlink (DL) signal level at UE for which the throughput is 95% of the maximum throughput of the reference measurement channels [6]. As evident in Figure 1 looking from the direction of the DL, the DL signal level at the UE can be measured either before the UE antenna (the power of the incident plane wave) or after it (the power delivered to the antenna feed/terminal/connector). The latter is known as conducted sensitivity, whereas the former is sometimes referred to as radiated sensitivity. It is important to emphasize that a plane wave carries infinite power. So, when we talk about the power of a plane wave, it is only in the context of far-field approximation, where a spherical wavefront is considered flat. Conducted sensitivity can be measured only when antenna connectors are physically accessible. In millimeter-wave UE, antenna connectors are integrated and not accessible for connecting cables.
Figure 1. The Rx sensitivity.
For any given ${\left({\theta},{\varphi}\right)}$, the angle of arrival (AoA) of the incident plane wave, the radiated sensitivity is referred to as the EIS and represented by ${EIS}{\left({\theta},{\varphi}\right)}$. In the absence of beamforming, it is more likely that UE receives the DL signal from many directions. In such scenarios, ${EIS}{\left({\theta},{\varphi}\right)}$ is integrated over the entire sphere to calculate the TIS or TRS as the Rx sensitivity performance indicator. Note that EIS and TIS/TRS can be measured for both 4G/FR1 5G as well as millimeter-wave UE. Since phased arrays and beamforming are rarely used in FR1, the DL signal arrives at the UE from many directions scattered across the entire sphere, rendering the EIS corresponding to a single AoA meaningless. Similarly, at millimeter-wave frequencies, where beamforming is commonplace, resulting in a single, dominant AoA, integrating the EIS across the entire sphere, and hence the TIS/TRS, does not seem like a useful performance indicator. The distinction between conducted versus radiated sensitivity is where the DL signal level is measured. Unless otherwise stated, the material presented in the following applies equally to FR1 and FR2 UE OTA tests.
Calculation of EIS From v AND h COMPONENT POLARIZATIONS
Let us assume the following:
The power of the incident ${v}$- and ${h}$-polarized plane waves can be written as \begin{align*}{P}_{v}^{\text{inc}} & = {\cos}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\theta}^{\text{inc}} + {\sin}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\varphi}^{\text{inc}}, \tag{S1} \\ {P}_{h}^{\text{inc}} & = {\sin}^{2}\,{\varphi}\,{\cdot}\,{P}_{\theta}^{\text{inc}} + {\cos}^{2}\,{\varphi}\,{\cdot}\,{P}_{\varphi}^{\text{inc}}{.} \tag{S2} \end{align*}
From the definition of ${\text{EIS}}_{v}$ and ${\text{EIS}}_{h}$ in terms of 95% of the maximum throughput of the reference measurement channels and corresponding ${\text{SNR}}_{\text{th}}$, \[{P}_{v}^{n} = {G}_{v}{\text{EIS}}_{v} / {\text{SNR}}_{\text{th}},{P}_{h}^{n} = {G}_{h}{\text{EIS}}_{h} / {\text{SNR}}_{\text{th}}{.} \tag{S3} \]
The output signal-to-noise ratio of each Rx chain and maximal ratio combiner can then be calculated as \begin{align*}{\text{SNR}}_{v} & = {\text{SNR}}_{\text{th}} \frac{{\cos}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\theta}^{\text{inc}} + {\sin}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\varphi}^{\text{inc}}}{{\text{EIS}}_{v}}, \tag{S4} \\ {\text{SNR}}_{h} & = {\text{SNR}}_{\text{th}} \frac{{\sin}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\theta}^{\text{inc}} + {\cos}^{2}{\varphi}_{o}\,{\cdot}\,{P}_{\varphi}^{\text{inc}}}{{\text{EIS}}_{h}}, \tag{S5} \\ {\text{SNR}}_{\text{MRC}} & = {\text{SNR}}_{v} + {\text{SNR}}_{h}{.} \tag{S6} \end{align*}
If the incident plane wave is only ${\theta}$ polarized, it can be easily shown that when ${\text{SNR}}_{\text{MRC}} = {\text{SNR}}_{\text{th}}$, we have \[{\text{EIS}}_{\theta} \,{≜}\,{P}_{\theta}^{\text{inc}} = \frac{{\text{EIS}}_{v}\,{\cdot}\,{\text{EIS}}_{h}}{{\sin}^{2}{\varphi}_{o}\,{\cdot}\,{\text{EIS}}_{v} + {\cos}^{2}{\varphi}_{o}\,{\cdot}\,{\text{EIS}}_{h}} \tag{S7} \] since ${P}_{\theta}^{\text{inc}}$ is the power of the ${\theta}$-polarized plane wave where threshold condition is met. Similarly, if the incident plane wave is only ${\varphi}$ polarized, \[{EIS}_{\varphi} \,{≜}\,{P}_{\varphi}^{\text{inc}} = \frac{{\text{EIS}}_{v}\,{\cdot}\,{\text{EIS}}_{h}}{{\cos}^{2}{\varphi}_{o}\,{\cdot}\,{\text{EIS}}_{v} + {\sin}^{2}{\varphi}_{o}\,{\cdot}\,{\text{EIS}}_{h}}{.} \tag{S8} \]
The overall effective isotropic sensitivity is then calculated using (1): \[\frac{1}{\text{EIS}} = \frac{1}{{\text{EIS}}_{\theta}} + \frac{1}{{\text{EIS}}_{\varphi}} = \frac{1}{{\text{EIS}}_{v}} + \frac{1}{{\text{EIS}}_{h}}{.} \tag{S9} \]
User Equipment Received Power
Using the reference coordinate system in Figure S1(a), let ${\vec{E}}_{\text{UE}}$ and ${\vec{E}}_{\text{inc}}$ represent far-field electric field patterns of the user equipment (UE) antenna system and incident plane waves: \begin{align*}{\vec{E}}_{\text{UE}}{\left({\theta},{\varphi}\right)} & = {E}_{h}^{\text{UE}}{\left({\theta},{\varphi}\right)}{\hat{h}} + {E}_{\nu}^{\text{UE}}{\left({\theta},{\varphi}\right)}{\hat{\nu}}, \tag{S10} \\ {\vec{E}}_{\text{inc}}{\left({\theta},{\varphi}\right)} & = {E}_{\theta}^{\text{inc}}{\left({\theta},{\varphi}\right)}{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\left({\theta},{\varphi}\right)}{\hat{\varphi}}{.} \tag{S11} \end{align*}
Figure S1. The (a) reference coordinate system and (b) polarization misalignment between the UE and feed antenna.
It is assumed that the UE antenna system is designed with two feeds, designated by ${v}$ and ${\bar{h}}$. Depending on the excitation method, a voltage or current applied to the ${v}$ - feed generates a ${v}$ -polarized electric field in the ${\theta}{\varphi}$-plane in the far field. Similarly, a voltage or current applied to the ${h}$ feed generates an ${h}$ -polarized electric field in the ${\theta}{\varphi}$ -plane in the far field.
Without a loss of generality, hereafter, we assume that the UE array elements are excited with voltage; ${\hat{v}}$ and ${\hat{h}}$ represent these two directions and, by design, are assumed to be orthogonal to each other. Note that whereas ${\left({\hat{v}},{\hat{h}}\right)}$ are in the transverse ${\theta}{\varphi}$ -plane, they are not necessarily aligned with ${\left({\hat{\theta}},{\hat{\varphi}}\right)}$ [see Figure S1(b)]. By reciprocity, the ${v}$ - and ${h}$ -polarized incident plane waves will induce a voltage across terminals of the ${v}$ and ${h}$ feeds, respectively. Any cross polarization between the ${v}$ and ${h}$ feeds is assumed to be negligible. In other words, voltage induced across terminals of the ${v} / {h}$ feed due to the incident ${h} / {v}$ - polarized plane wave is considered negligible. A ${\theta} / {\varphi}$ -polarized incident plane wave will induce a voltage across terminals of the ${v}$ and ${h}$ feeds.
Using the equivalent circuit representation of a matched receiving antenna (Figure 4), the voltage across a unity load resistance can be written as [9] \begin{align*} & {v}_{v}{\left({t}\right)} = {\text{Re}}{\left\{{V}_{v}{\left({t}\right)}{e}^{{j}{\omega}{t}}\right\}}, \quad {v}_{h}{\left({t}\right)} = {\text{Re}}{\left\{{V}_{h}{\left({t}\right)}{e}^{{j}{\omega}{t}}\right\}}, \tag{S12} \\ & {V}_{v}{\left({t}\right)} = {S}_{v}{\left({t}\right)} + {N}_{v}{\left({t}\right)}, \quad {V}_{h}{\left({t}\right)} = {S}_{h}{\left({t}\right)} + {N}_{h}{\left({t}\right)}, \tag{S13} \\ & \quad \quad {S}_{v}{\left({t}\right)}\,{\propto}\,{E}_{v}^{\text{UE}}{\left({\Omega}\right)}{\hat{v}}\,{\cdot}\,{\vec{E}}_{\text{inc}}{\left({\Omega}\right)}, \tag{S14} \\ & \quad \quad {S}_{h}{\left({t}\right)}\,{\propto}\,{E}_{h}^{\text{UE}}{\left({\Omega}\right)}{\hat{h}}\,{\cdot}\,{\vec{E}}_{\text{inc}}{\left({\Omega}\right)}, \tag{S15} \end{align*} where ${S}_{v}{\left({t}\right)} / {S}_{h}{\left({t}\right)}$ and ${N}_{v}{\left({t}\right)} / {N}_{h}{\left({t}\right)}$ are signal and noise components of ${V}_{v}{\left({t}\right)} / {V}_{h}{\left({t}\right)}$ and ${\Omega}$ denotes the angle of arrival (AoA) of the incident plane wave. Expressions for ${S}_{v}{\left({t}\right)}$ and ${S}_{h}{\left({t}\right)}$ may include summation or integration to account for all AoAs. Here, we consider only a single AoA and, for brevity, drop ${\Omega}$ going forward.
Using (S11) to expand the expressions for ${S}_{v}{\left({t}\right)}$ and ${S}_{h}{\left({t}\right)}$, we have \begin{align*}{S}_{v}{\left({t}\right)} & = {c}_{v}{E}_{v}^{\text{UE}}{\left\{{E}_{\theta}^{\text{inc}}{\hat{v}}\,{\cdot}\,{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\hat{v}}\,{\cdot}\,{\hat{\varphi}}\right\}}, \tag{S16} \\ {S}_{h}{\left({t}\right)} & = {c}_{h}{E}_{h}^{\text{UE}}{\left\{{E}_{\theta}^{\text{inc}}{\hat{h}}\,{\cdot}\,{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\hat{h}}\,{\cdot}\,{\hat{\varphi}}\right\}}, \tag{S17} \end{align*} where ${c}_{v} / {c}_{h}$ are proportionality constants. At the output of the maximal ratio combiner (MRC), we can write \[{V}{\left({t}\right)} = {S}{\left({t}\right)} + {N}{\left({t}\right)}, \tag{S18} \] \begin{align*}{N}{\left({t}\right)} = & {\alpha}_{v}{N}_{v}{\left({t}\right)} + {\alpha}_{h}{N}_{h}{\left({t}\right)}, \tag{S19} \\ {S}{\left({t}\right)} = & {\alpha}_{v}{S}_{v}{\left({t}\right)} + {\alpha}_{h}{S}_{h}{\left({t}\right)} \\ = & {\alpha}_{v}{c}_{v}{E}_{v}^{\text{UE}}{\left\{{E}_{\theta}^{\text{inc}}{\hat{v}}\,{\cdot}\,{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\hat{v}}\,{\cdot}\,{\hat{\varphi}}\right\}} \\ & + {\alpha}_{h}{c}_{h}{E}_{h}^{\text{UE}}{\left\{{E}_{\theta}^{\text{inc}}{\hat{h}}\,{\cdot}\,{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\hat{h}}\,{\cdot}\,{\hat{\varphi}}\right\}}{.} \tag{S20} \end{align*}
As discussed in the article, the effective isotropic sensitivity is tested separately, with two orthogonal polarizations in two steps. Whereas it is possible to derive a general expression for the MRC-output signal-to-noise ratio (SNR) with both polarizations present, the derivation of equations becomes unnecessarily cumbersome. For this reason, we consider the cases ${\vec{E}}_{\text{inc}} = {E}_{\theta}^{\text{inc}}{\hat{\theta}}$ and ${\vec{E}}_{\text{inc}} = {E}_{\varphi}^{\text{inc}}{\hat{\varphi}}$ separately. When ${\vec{E}}_{\text{inc}} = {E}_{\theta}^{\text{inc}}{\hat{\theta}}$, the MRC coefficients become [25] \begin{align*}{\alpha}_{v} & = {K}{c}_{v}^{*}{E}_{v}^{{\text{UE}}{*}}{\tilde{E}}_{\theta}^{{\text{inc}}{*}}{\hat{v}}\,{\cdot}\,{\hat{\theta}} / {\left\langle{\left\vert{N}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}, \tag{S21} \\ {\alpha}_{h} & = {Kc}_{h}^{*}{E}_{h}^{{\text{UE}}{*}}{\tilde{E}}_{\theta}^{{\text{inc}}{*}}{\hat{h}}\,{\cdot}\,{\hat{\theta}} / {\left\langle{\left\vert{N}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle}, \tag{S22} \end{align*} where ${K}$ is an arbitrary complex constant and ${\tilde{E}}_{\theta}^{\text{inc}}$ is modem estimates of the channel gain (e.g., through the transmission of pilot channels).
Replacing for ${\alpha}_{v}$ and ${\alpha}_{h}$ in (S20) and noting that \[{\cal{R}}{e}{\left\{{\alpha}_{v}{c}_{v}{\alpha}_{h}^{*}{c}_{h}^{*}{E}_{v}^{\text{UE}}{E}_{h}^{{\text{UE}}{*}}\right\}} = {\left\vert{\alpha}_{v}{c}_{v}\right\vert}{\left\vert{E}_{v}^{\text{UE}}\right\vert}{\left\vert{\alpha}_{h}{c}_{h}\right\vert}{\left\vert{E}_{h}^{\text{UE}}\right\vert}, \tag{S23} \] the expression for ${\left\langle{\left\vert{S}{\left({t}\right)}\right\vert}^{2}\right\rangle}$ can be simplified as \[{\left\langle{\left\vert{S}{\left({t}\right)}\right\vert}^{2}\right\rangle} = {\left\langle{\left\vert{E}_{\theta}^{\text{inc}}\right\vert}^{2}\right\rangle}{\left\{{\left\vert{\alpha}_{v}{c}_{v}\right\vert}{\left\vert{E}_{v}^{\text{UE}}\right\vert}{\left\vert{\cos}\,{\varphi}_{o}\right\vert} + {\left\vert{\alpha}_{h}{c}_{h}\right\vert}{\left\vert{E}_{h}^{\text{UE}}\right\vert}{\left\vert{\sin}\,{\varphi}_{o}\right\vert}\right\}}^{2}{.} \tag{S24} \]
Similarly, the MRC output noise can be derived as \begin{align*}{\left\langle{\left\vert{N}{\left({t}\right)}\right\vert}^{2}\right\rangle} & = {\left\vert{\alpha}_{v}\right\vert}^{2}{\left\langle{\left\vert{N}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle} + {\left\vert{\alpha}_{h}\right\vert}^{2}{\left\langle{\left\vert{N}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle} \\ & = {\left\vert{K}\right\vert}{\left\vert{\tilde{E}}_{\theta}^{\text{inc}}\right\vert}{\left\{{\left\vert{\alpha}_{v}{c}_{v}\right\vert}{\left\vert{E}_{v}^{\text{UE}}\right\vert}{\left\vert{\cos}\,{\varphi}_{o}\right\vert} + {\left\vert{\alpha}_{h}{c}_{h}\right\vert}{\left\vert{E}_{h}^{\text{UE}}\right\vert}{\left\vert{\sin}\,{\varphi}_{o}\right\vert}\right\}} \tag{S25} \end{align*}
Using (S24) and (S25), the MRC-output SNR becomes \begin{align*}{\text{SNR}}_{\text{MRC}} & = {\left\langle{\left\vert{S}{\left({t}\right)}\right\vert}^{2}\right\rangle} / {\left\langle{\left\vert{N}{\left({t}\right)}\right\vert}^{2}\right\rangle} \\ & = \frac{\left\langle{\left\vert{E}_{\theta}^{\text{inc}}\right\vert}^{2}\right\rangle}{{\left\vert{K}\right\vert}{\left\vert{\tilde{E}}_{\theta}^{\text{inc}}\right\vert}} {\left\{{\left\vert{\alpha}_{v}{c}_{v}\right\vert}{\left\vert{E}_{v}^{\text{UE}}\right\vert}{\left\vert{\cos}\,{\varphi}_{o}\right\vert} + {\left\vert{\alpha}_{h}{c}_{h}\right\vert}{\left\vert{E}_{h}^{\text{UE}}\right\vert}{\left\vert{\sin}\,{\varphi}_{o}\right\vert}\right\}}{.} \tag{S26} \end{align*}
Equation (S26) can be further processed into a more familiar form: \[{\text{SNR}}_{\text{MRC}} = \frac{\left\langle{{\left\vert{{S}_{v}{\left({t}\right)}}\right\vert}^{2}}\right\rangle}{{\left\langle{\left\vert{N}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}} + \frac{\left\langle{{\left\vert{{S}_{h}{\left({t}\right)}}\right\vert}^{2}}\right\rangle}{\left\langle{{\left\vert{{N}_{h}{\left({t}\right)}}\right\vert}^{2}}\right\rangle} = {\text{SNR}}_{v} + {\text{SNR}}_{h}, \tag{S27} \] where \begin{align*}{\text{SNR}}_{v} & = {\left\langle{\left\vert{E}_{\theta}^{\text{inc}}\right\vert}^{2}\right\rangle}{\cos}^{2}{\varphi}_{o}{\left\vert{c}_{v}\right\vert}^{2}{\left\vert{E}_{v}^{\text{UE}}\right\vert}^{2} / {\left\langle{\left\vert{N}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}, \tag{S28} \\ {\text{SNR}}_{h} & = {\left\langle{\left\vert{E}_{\theta}^{\text{inc}}\right\vert}^{2}\right\rangle}\,{\sin}^{2}{\varphi}_{o}{\left\vert{c}_{h}\right\vert}^{2}{\left\vert{E}_{h}^{\text{UE}}\right\vert}^{2} / {\left\langle{\left\vert{N}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle}{.} \tag{S29} \end{align*}
Expressions for ${\left\langle{\left\vert{S}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}$ and ${\left\langle{\left\vert{S}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle}$ can be put into a more familiar form by noticing that \[{G}_{v}\,{\propto}\,{\left\langle{\left\vert{E}_{v}^{\text{UE}}\right\vert}^{2}\right\rangle}, \quad {G}_{h}\,{\propto}\,{\left\vert{E}_{h}^{\text{UE}}\right\vert}^{2}, \tag{S30} \] \[{P}_{\theta}^{\text{inc}}\,{\propto}\,{\left\langle{\left\vert{E}_{\theta}^{\text{inc}}\right\vert}^{2}\right\rangle}, \quad {P}_{\varphi}^{\text{inc}}\,{\propto}\,{\left\langle{\left\vert{E}_{\varphi}^{\text{inc}}\right\vert}^{2}\right\rangle}, \tag{S31} \] where ${G}_{v} / {G}_{h}$ and ${P}_{\theta}^{\text{inc}}$ are the UE antenna system gain for the ${v} / {h}$ polarization and the power of the ${\theta}$-polarized incident plane wave, respectively. From the equivalent circuit representation of a matched receiving antenna (Figure 4), it can be seen that ${\left\langle{\left\vert{S}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}$ is, in fact, the power delivered to the receive chain for ${v}$ polarization. Therefore, we have \[{P}_{v}^{\text{rec}}\,{\propto}\,{G}_{v}{P}_{\theta}^{\text{inc}}{\cos}^{2}{\varphi}_{o}, \quad {P}_{h}^{\text{rec}}\,{\propto}\,{G}_{h}{P}_{\theta}^{\text{inc}}{\sin}^{2}{\varphi}_{o}{.} \tag{S32} \]
For a matched and ideal isotropic antenna, all the power in the incident plane wave is delivered to the antenna connector: \[{P}_{{v},{\text{iso}}}^{\text{rec}} = {P}_{\theta}^{\text{inc}}{\cos}^{2}{\varphi}_{o}, \quad {P}_{{h},{\text{iso}}}^{\text{rec}} = {P}_{\theta}^{\text{inc}}{\sin}^{2}{\varphi}_{o}{.} \tag{S33} \]
From the definition of antenna gain, it can be readily concluded that the proportionality constant in (S32) should be unity, and hence we have \[{P}_{v}^{\text{rec}} = {G}_{v}{P}_{\theta}^{\text{inc}}{\cos}^{2}{\varphi}_{o}, \quad {P}_{h}^{\text{rec}} = {G}_{h}{P}_{\theta}^{\text{inc}}{\sin}^{2}{\varphi}_{o}. \tag{S34} \]
Similarly, \[{\text{SNR}}_{v} = {P}_{v}^{\text{rec}} / {P}_{v}^{n}, \quad {\text{SNR}}_{h} = {P}_{h}^{\text{rec}} / {P}_{h}^{n}, \tag{S35} \] where \[{P}_{v}^{n} = {P}_{v}^{\text{rec}} \frac{\left\langle{\left\vert{N}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}{\left\langle{\left\vert{S}_{v}{\left({t}\right)}\right\vert}^{2}\right\rangle}, \quad {P}_{h}^{n} = {P}_{h}^{\text{rec}} \frac{\left\langle{\left\vert{N}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle}{\left\langle{\left\vert{S}_{h}{\left({t}\right)}\right\vert}^{2}\right\rangle}. \tag{S36} \]
The equations for the case ${\vec{E}}_{\text{inc}} = {E}_{\varphi}^{\text{inc}}{\hat{\varphi}}$ are very similar and can be obtained by replacing ${\varphi}_{o}$ with ${90} + {\varphi}_{o}$. For the general case where ${\vec{E}}_{\text{inc}} = {E}_{\theta}^{\text{inc}}{\hat{\theta}} + {E}_{\varphi}^{\text{inc}}{\hat{\varphi}}$, we have \begin{align*}{P}_{v}^{\text{rec}} & = {G}_{v}{\left\{{P}_{\theta}^{\text{inc}}{\cos}^{2}{\varphi}_{o} + {P}_{\varphi}^{\text{inc}}{\sin}^{2}{\varphi}_{o}\right\}}, \tag{S37} \\ {P}_{h}^{\text{rec}} & = {G}_{h}{\left\{{P}_{\theta}^{\text{inc}}{\sin}^{2}{\varphi}_{o} + {P}_{\varphi}^{\text{inc}}{\cos}^{2}{\varphi}_{o}\right\}}{.} \tag{S38} \end{align*}
The testing and characterization of millimeter-wave UE is commonly conducted inside CATRs, where an offset parabolic reflector is used for indirect far-field (IFF) measurements [1], [7], [8]. By design, within a cylindrical volume known as the quiet zone (QZ), the collimated wavefront is expected to be planar with uniform amplitude and phase and polarization that depends on that of the feed antenna. UE is placed at the center of the QZ, mounted on a positioning system with at least two axes of freedom (the x- and y-axes in Figure 2). The SS plays the role of the base station (BS) in the network. Depending on the excitation of the feed antenna, the SS can transmit one of two orthogonal polarizations: 1) ${\theta}$ polarization, with an E-field parallel to the ${y}$-axis and transmit power of ${P}_{\theta}^{\text{SS}}$, and 2) ${\varphi}$ polarization, with an E-field parallel to the ${x}$-axis and transmit power of ${P}_{\varphi}^{\text{SS}}$.
Figure 2. The IFF test setup.
Radiated measurements of a device under test (DUT) depend not only on the AoA or departure of the plane waves but also on the angle between the polarization axis of the Tx and DUT (the polarization mismatch angle). Tests conducted in wireless communication labs are HW characterization tests and intended to capture the capabilities of the HW for compliance purposes. While polarization mismatch is unavoidable in the field, it is not related to HW design and should be avoided in the lab. Knowledge of the radiated characteristics of UE under polarization match conditions should be enough to predict the performance in the presence of polarization mismatch. As shown in the following, polarization mismatch can be avoided if the test setup is capable of simultaneously generating two orthogonal polarizations. This, of course, requires two SSs, which is not economically tenable in wireless communication labs. Instead, the orthogonal polarizations are generated one at a time, and the results are then combined to account for any processing gain. This test method strikes a balance among cost, time, and accuracy.
To use a familiar analogy, the length of a football field can be accurately and quickly measured using a laser distance measurement tool. But laser tools are expensive. The same measurement can be conducted with a tape measure and by counting the number of segments needed to cover the length. The distance can then be calculated by multiplying the number of segments by the length of the tape. Regardless of the tool, an expensive laser or a cheap tape measure, the results are expected to be reasonably accurate. Furthermore, the equation used with a tape measure should correctly reflect the length of the tape. We do not count the number of segments for a 9-ft-long tape but multiply the number of segments by 10 because it is easier. EIS measurements are no different. The correct formula should be used with two-step measurements to correctly predict the performance of UE.
The calibration of the path loss from the SS to the QZ is conducted using a standard gain horn (SGH) antenna, which can also receive the E-field in either ${\theta}$ or ${\varphi}$ polarization. If ${P}_{\theta}^{\text{inc}}$ and ${P}_{\varphi}^{\text{inc}}$ represent the power in the ${\theta}$- and ${\varphi}$-polarized uniform plane waves in the QZ, respectively, we have \[{P}_{\theta}^{\text{inc}} = {L}_{\theta}{P}_{\theta}^{\text{SS}}, \quad {P}_{\varphi}^{\text{inc}} = {L}_{\varphi}{P}_{\varphi}^{\text{SS}}, \tag{3} \] where ${L}_{\theta}$ and ${L}_{\varphi}$ represent the path loss from the SS to the center of the QZ for the ${\theta}$- and ${\varphi}$-polarized incident plane waves. The standard specifies several different options for the initial orientation of the UE as the default test condition (see [1, Appendix C]), e.g., with the front of the UE pointing toward the reflector [Figure 3(a)]. If ${\hat{r}}$ is the unit vector perpendicular to the front of the UE, the default test condition can be defined as ${\hat{r}} = {\hat{z}}$. As the UE rotates, so does ${\hat{r}}$. The link angle refers to ${\left({\theta}_{l},{\varphi}_{l}\right)}$ corresponding to ${\hat{r}}$ obtained from rotating ${\hat{z}}$ [Figure 3(b)]. For example, if the UE is rotated first by ${\alpha}$ around the ${x}$-axis, followed by a rotation of ${\beta}$ around the ${y}$-axis, the corresponding link angle becomes \[{\cos}\,{\theta}_{l} = {\cos}\,{\alpha}{\cos}\,{\beta},{\tan}\,{\varphi}_{l} = {-}{\tan}\,{\alpha} / {\sin}\,{\beta}{.} \tag{4} \]
Figure 3. (a) The reference test condition. (b) The mismatch among the polarization axes of the UE and SS. (c) The UE rotation around the y-axis.
UE with a dual-polarized feed network can also receive two orthogonal polarizations: ${v}$ and ${h}$. Regardless of the UE orientation, in the far-field and in the direction of the reflector (along the ${z}$-axis), ${E}_{v}$ and ${E}_{h}$, i.e., the far-field electric field components, will always be in the ${xy}$-plane [Figure 3(a)], albeit not necessarily parallel to the ${x}$- and ${y}$-axes [Figure 3(c)]. The angular relationship between ${\left({\hat{v}},{\hat{h}}\right)}$ and ${\left({\hat{x}},{\hat{y}}\right)}$ is represented by azimuthal angle ${\varphi}_{o}$ in the reference coordinate system, with UE at the origin. Misalignment among the polarization axes of the UE and feed antenna depends on the orientation of the UE and changes by rotating it, but it always remains in the ${xy}$-plane. To understand this better, consider an arbitrarily oriented half-wave dipole antenna with sinusoidal current distribution: \[{I}{\left({\ell}\right)} = {I}_{o}{\cos}\,{\left({k}{\ell}\right)} \quad {\text{for}} \quad {\left\vert{\ell}\right\vert}\,{≤}\,{\lambda} / {4}. \tag{5} \]
Let ${\hat{\ell}}$ represent the unit vector along the axis of the dipole. The far-field component of the electric field due to this dipole is given by [10] \[{\vec{E}}_{ff}^{\hat{\ell}} = {-}{j}{\omega}{\vec{A}}{-}{\left({-}{j}{\omega}{\vec{A}}\,{\cdot}\,{\hat{r}}\right)}{\hat{r}} = {-}{j}{\omega}{\left({A}_{\theta}{\hat{\theta}} + {A}_{\varphi}{\hat{\varphi}}\right)}, \tag{6} \] where ${k} = {2}{\pi} / {\lambda}$ and ${\vec{A}}$ is the vector potential given by \[{\vec{A}} = {\hat{\ell}}{\mu} \frac{{e}^{{-}{jkr}}}{{4}{\pi}{r}}\,{\int}{I}{\left({\ell}\right)}{e}^{{jk}{\ell}{\left({\hat{r}}\,{\cdot}\,{\hat{\ell}}\right)}}{d}{\ell} = {\hat{\ell}}{\mu}{I}_{o} \frac{{e}^{{-}{jkr}}}{{2}{\pi}{kr}} \frac{{\cos}{\left(\frac{\pi}{2}{\hat{r}}\,{\cdot}\,{\hat{\ell}}\right)}}{{1}{-}{\left({\hat{r}}\,{\cdot}\,{\hat{\ell}}\right)}^{2}}{.} \tag{7} \]
Here, ${\vec{E}}_{ff}^{\hat{\ell}}$ can further be simplified as \[{\vec{E}}_{ff}^{\hat{\ell}} = {E}_{o}{E}_{p}{\left\{{\left({\hat{\ell}}\,{\cdot}\,{\hat{\theta}}\right)}{\hat{\theta}} + {\left({\hat{\ell}}\,{\cdot}\,{\hat{\varphi}}\right)}{\hat{\varphi}}\right\}}, \tag{8} \] where \[{E}_{o} = {-}{jZ}_{o}{I}_{o} \frac{{e}^{-{jkr}}}{{2}{\pi}{r}}, \quad {E}_{p} = \frac{{\cos}{\left(\frac{\pi}{2}{\hat{r}}\,{\cdot}\,{\hat{\ell}}\right)}}{{1}{-}{\left({\hat{r}}\,{\cdot}\,{\hat{\ell}}\right)}^{2}}, \tag{9} \]ula> in which ${Z}_{o}$ is the free-space impedance.
Table 1 summarizes ${E}_{p}$, ${\hat{\ell}}\,{\cdot}\,{\hat{\theta}}$, and ${\hat{\ell}}\,{\cdot}\,{\hat{\varphi}}$ for diploes oriented along axes in cartesian coordinates. In the direction of reflector and along the z-axis, we have \[{\vec{E}}_{ff}^{\hat{x}} = {E}_{o}{\hat{x}}, \quad {\vec{E}}_{ff}^{\hat{y}} = {E}_{o}{\hat{y}}, \quad {\vec{E}}_{ff}^{\hat{z}} = {0}{.} \tag{10} \]
Table 1. The far-field pattern of a half-wave dipole.
Dipoles along the x- and y-axes are obtained by rotating the dipole along the z-axis by ${\pm}$90° around the y- and x-axes, respectively. Whereas ${\varphi}_{o}$ is undefined for a dipole along the z-axis, it is 0 and 90° for dipoles along the x- and y-axes, respectively. But in all three cases, radiation in the far-field is dominated by transverse, i.e., ${\theta}$ and ${\varphi}$, components, which in the direction of the reflector, are always in the xy-plane. While this may seem obvious, a review of RAN4 implies that it may not have been so for some authors.
For UE with the reference test condition in 3(a) (i.e., with the front of the UE facing the reflector) and for the antenna array on the front of the UE, it can be easily shown that ${\varphi}_{o}$ corresponding to the link angle in (4) is \[{\tan}\,{\varphi}_{o}{\left({\theta}_{l},{\varphi}_{l}\right)} = \frac{{\tan}\,{\varphi}_{o}^{\text{ref}}\,{\cdot}\,{\cos}\,{\alpha}}{{\cos}\,{\beta} + {\tan}\,{\varphi}_{o}^{\text{ref}}\,{\cdot}\,{\sin}\,{\alpha}\,{\cdot}\,{\sin}\,{\beta}}, \tag{11} \] where ${\varphi}_{o}^{\text{ref}}$ is ${\varphi}_{o}$ for the front array at the reference test condition and ${\varphi}_{o}{\left({\theta}_{l},{\varphi}_{l}\right)}$ is obtained by applying rotation matrices to the vector ${\hat{x}}{\cos}{\varphi}_{o}^{\text{ref}} + {\hat{y}}{\sin}\,{\varphi}_{o}^{\text{ref}}$. If ${\varphi}_{o}$ is somehow determined at any link angle, it can be calculated for any other UE orientation by a similar method.
Since the UE beam peak direction is unknown, measuring the EIS requires measuring the spherical sensitivity pattern of the UE through a 3D EIS scan [1], [11]. Polarization mismatch between the UE and feed antenna depends not only on the UE’s design but also its orientation and is unavoidable unless the test setup generates both orthogonal polarizations simultaneously. Otherwise, the spherical sensitivity pattern will be weighted as a function of the mismatch angle. Equation (1) is how the polarization mismatch is accounted for in the case of EIS measurement.
It is important to note that the current text in the standard uses ${\theta}$ and ${\varphi}$ to designate the polarization axes of the UE and feed antenna, implying that the tests are conducted under a polarization match condition. The problem of polarization mismatch between the UE and feed antenna was widely discussed in several RAN4 meetings, but as shown here, it was not handled properly. The IFF test setup and method for measuring the full spherical sensitivity pattern of the UE were introduced in this section. The following section offers a detailed derivation of the EIS equation.
The 3GPP defines the EIS as “sensitivity for an isotropic directivity device equivalent to the sensitivity of the discussed device exposed to an incoming wave from a defined AoA†[12]. The authors of [4] offer a similar definition for${\text{EIS}}_{v}{\left({\theta},{\varphi}\right)} / {\text{EIS}}_{h}{\left({\theta},{\varphi}\right)}$ as “power available from an ideal isotropic, ${v} / {h}$-polarized antenna generated by the ${v} / {h}$-polarized plane wave incident from direction ${\left({\theta},{\varphi}\right)}$ which, when incident on the [equipment under test], yields the threshold of sensitivity performance.â€
Using the equivalent circuit representation of a matched receiving antenna (Figure 4), the average power of an incident plane wave and the average power delivered to the antenna connector are related through (S37) and (S38) in “User Equipment Received Power.†For the case where ${v}\,{\parallel}\,{\theta}$ and ${h}\,{\parallel}\,{\varphi}$, we have \[{P}_{v}^{\text{rec}} = {G}_{v}{P}_{v}^{\text{inc}}, \quad {P}_{h}^{\text{rec}} = {G}_{h}{P}_{h}^{\text{inc}}, \tag{12} \]
Figure 4. The equivalent circuit representation of the matched receiving antenna (a) v-polarized chain and (b) h-polarized chain.
where ${G}_{v} / {G}_{h}$ is the gain of the receiving antenna in the direction of arrival of the plane wave for ${v} / {h}$ polarization, respectively.
Figure 5 demonstrates how a microwave switch (a coaxial connector with a switch, also commonly known as an antenna connector) can be used for performing conducted and radiated measurements. In Figure 5(a), the probe is removed from the switch. This enables the switch to remain closed and for the antenna terminals to be connected to the UE Rx/Tx port. In Figure 5(b), inserting the probe into the switch opens the switch and disconnects the antenna. The coaxial probe, and hence the SS, are now connected to the UE Rx/Tx port, and conducted measurements can be performed.
Figure 5. The UE Tx/Rx port connected to (a) the UE antenna system and (b) the SS.
Consider the following thought experiments. First, UE is illuminated by a ${v}$-polarizaed plane wave, and ${P}_{v}^{\text{inc}}$ is measured at the threshold of sensitivity. This is what was earlier defined as ${\text{EIS}}_{v}$. Suppose we also have a way of measuring the power delivered to the UE Rx/Tx port. Let ${P}_{{S},{v}}$ be this power. From (12) we know that \[{P}_{{S},{v}} = {G}_{v}{\text{EIS}}_{v}{.} \tag{13} \]
Assuming that the UE antenna system is matched and lossless, ${P}_{{S},{v}}$ is, in fact, the conducted sensitivity of the ${v}$-polarized receive chain. In other words, (12) shows how radiated and conducted measurements are related. The tests are also repeated by an ${h}$-polarized plane wave: \[{P}_{{S},{h}} = {G}_{h}{\text{EIS}}_{h}{.} \tag{14} \]
If UE has diversity antennas and Rxs, the conducted sensitivity is still defined as the DL signal level at each antenna connector. Figure 6 illustrates the test setup for UE with two antenna connectors [13]. Since (13) and (14) establish a one-to-one correspondence between the conducted signal level and power of incident plane wave for each receive chain, it is only natural to define the EIS as the average power of the incident plane wave for each polarization. This is in accordance with the definition in [4] and the only one applicable regardless of whether the UE can receive one or two polarizations.
Figure 6. The test setup for UE with two antenna connectors [13].
Next, apply the ${v}$- and ${h}$-polarized plane waves simultaneously, making sure that${P}_{v}^{\text{inc}} = {P}_{h}^{\text{inc}}$, and measure this common value where the threshold of sensitivity is achieved at the output of the MRC combiner. This is, indeed, the total EIS of the UE and achieved by combining signals from both receive chains: \[{P}_{{S},{v}}^{\text{rec}} = {G}_{v}{\text{EIS}}, \quad {P}_{{S},{h}}^{\text{rec}} = {G}_{h}{\text{EIS}}, \tag{15} \] where ${P}_{{S},{v}}^{\text{rec}}$ and ${P}_{{S},{h}}^{\text{rec}}$ are measured at the UE Rx/Tx port corresponding to the ${v}$ and ${h}$ polarizations, respectively. Finally, use the setup in Figure 6 to measure the combined conducted sensitivity of the UE with two receive chains, i.e., ${P}_{S}$. Using (S27), we know that \[{\text{SNR}}_{\text{th}} = \frac{{P}_{S}}{{P}_{v}^{n}} + \frac{{P}_{S}}{{P}_{h}^{n}}{.} \tag{16} \]
Unless ${G}_{v} = {G}_{h}$, ${P}_{{S},{v}}^{\text{rec}}\,{≠}\,{P}_{{S},{h}}^{\text{rec}}\,{≠}\,{P}_{S}$. From (S35) we also know that \[{\text{SNR}}_{\text{th}} = \frac{{P}_{{S},{v}}}{{P}_{v}^{n}} = \frac{{P}_{{S},{h}}}{{P}_{h}^{n}}, \tag{17} \] where ${\text{SNR}}_{\text{th}}$ corresponds to the threshold of sensitivity and ${P}_{v}^{n} / {P}_{h}^{n}$ is the input referred noise. Using (S27), we can write a similar equation for EIS measurement: \[{\text{SNR}}_{\text{th}} = \frac{{P}_{{S},{v}}^{\text{rec}}}{{P}_{v}^{n}} + \frac{{P}_{{S},{h}}^{\text{rec}}}{{P}_{h}^{n}}{.} \tag{18} \]
Replacing for ${P}_{v}^{n}$ and ${P}_{h}^{n}$ from (17) in (16), we can write \[\frac{1}{{P}_{S}} = \frac{1}{{P}_{{S},{v}}} + \frac{1}{{P}_{{S},{h}}}{.} \tag{19} \]
In turn, we can replace for ${P}_{{S},{v}}$ and ${P}_{{S},{h}}$ from (13) and (14) in (17) and from (15) in (18) to obtain \begin{align*}{\text{SNR}}_{\text{th}} & = \frac{{G}_{v}{\text{EIS}}_{v}}{{P}_{v}^{n}} = \frac{{G}_{h}{\text{EIS}}_{h}}{{P}_{h}^{n}}, \tag{20} \\ {\text{SNR}}_{\text{th}} & = \frac{{G}_{v}\text{EIS}}{{P}_{v}^{n}} + \frac{{G}_{h}\text{EIS}}{{P}_{h}^{n}}{.} \tag{21} \end{align*}
By replacing for ${P}_{v}^{n}$ and ${P}_{h}^{n}$ from (20) in (21), we have \[\frac{1}{\text{EIS}} = \frac{1}{{\text{EIS}}_{v}} + \frac{1}{{\text{EIS}}_{h}}{.} \tag{22} \]
The work in [4] contains a similar derivation, but it assumes ${P}_{{S},{v}}^{\text{rec}} = {P}_{{S},{h}}^{\text{rec}} = {P}_{S}$, which is true only if ${G}_{v} = {G}_{h}$.
Note that (19) and (22) are derived from (18), which, in turn, is derived by applying MRC coefficients to the UE receive chains. Equation (19) shows how the dual-chain conducted sensitivity of UE with two receive chains can be obtained by measuring the conducted sensitivity of each chain. Similarly, (22) shows how the total EIS of dual-polarization UE can be obtained by measuring ${\text{EIS}}_{v}$ and ${\text{EIS}}_{h}$ separately. But what if the test setup is capable only of supplying two orthogonal polarizations that are not necessarily aligned with ${v}$ and $\begin{gathered}{\text{h}?}\end{gathered}$ For a ${\theta}$ -polarized incident plane wave with an average power of ${EIS}_{\theta}$ at the threshold of sensitivity, (15) takes the following form: \[{P}_{{S},{v},{\theta}}^{\text{rec}} = {G}_{v}{\text{EIS}}_{\theta}{\cos}^{2}{\varphi}_{o}, \quad {P}_{{S},{h},{\theta}}^{\text{rec}} = {G}_{h}{\text{EIS}}_{\theta}{\sin}^{2}{\varphi}_{o} \tag{23} \] where ${\varphi}_{o}$ is defined in Figure S1.
With UE at the threshold of sensitivity, ${P}_{{S},{v},{\theta}}^{\text{rec}}$ is the power delivered to the ${v}$-polarization antenna connector due to an incident ${\theta}$-polarized plane wave; ${P}_{{S},{h},{\theta}}^{\text{rec}}$ is defined similarly. For a ${\varphi}$-polarized incident plane wave with an average power of ${\text{EIS}}_{\varphi}$ at the threshold of sensitivity, we have \[{P}_{{S},{v},{\varphi}}^{\text{rec}} = {G}_{v}{\text{EIS}}_{\varphi}{\sin}^{2}{\varphi}_{o}, \quad {P}_{{S},{h},{\varphi}}^{\text{rec}} = {G}_{h}{\text{EIS}}_{\varphi}{\cos}^{2}{\varphi}_{o}. \tag{24} \]
Using (S27), we can also write \[{\text{SNR}}_{\text{th}} = \frac{{P}_{{S},{v},{\theta}}^{\text{rec}}}{{P}_{v}^{n}} + \frac{{P}_{{S},{h},{\theta}}^{\text{rec}}}{{P}_{h}^{n}} = \frac{{P}_{{S},{v},\varphi}^{\text{rec}}}{{P}_{v}^{n}} + \frac{{P}_{{S},{h},{\varphi}}^{\text{rec}}}{{P}_{h}^{n}}{.} \tag{25} \]
Replacing for ${P}_{{S},{v},{\theta}}^{\text{rec}}$, ${P}_{{S},{h},{\theta}}^{\text{rec}}$, ${P}_{{S},{v},{\varphi}}^{\text{rec}}$, and ${P}_{{S},{h},{\varphi}}^{\text{rec}}$ from (23) and (24), we have \begin{align*}{\text{SNR}}_{\text{th}} & = \frac{{G}_{v}{\text{EIS}}_{\theta}{\cos}^{2}{\varphi}_{o}}{{P}_{v}^{n}} + \frac{{G}_{h}{\text{EIS}}_{\theta}{\sin}^{2}{\varphi}_{o}}{{P}_{h}^{n}} \\ & = \frac{{G}_{v}{\text{EIS}}_{\varphi}{\sin}^{2}{\varphi}_{o}}{{P}_{v}^{n}} + \frac{{G}_{h}{\text{EIS}}_{\varphi}{\cos}^{2}{\varphi}_{o}}{{P}_{h}^{n}}, \tag{26} \end{align*} which can be readily rearranged to \begin{align*}{\frac{{\text{SNR}}_{\text{th}}}{{\text{EIS}}_{\theta}}} & = \frac{{G}_{v}{\cos}^{2}{\varphi}_{o}}{{P}_{v}^{n}} + \frac{{G}_{h}{\sin}^{2}{\varphi}_{o}}{{P}_{h}^{n}}, \tag{27} \\ \frac{{\text{SNR}}_{\text{th}}}{{\text{EIS}}_{\varphi}} & = \frac{{G}_{v}{\sin}^{2}{\varphi}_{o}}{{P}_{v}^{n}} + \frac{{G}_{h}{\cos}^{2}{\varphi}_{o}}{{P}_{h}^{n}}{.} \tag{28} \end{align*}
Solving (27) and (28) for ${\left[{1} / {P}_{v}^{n}\,{1} / {P}_{h}^{n}\right]}$ and replacing it in (21) gives us (1): \[{\frac{1}{\text{EIS}}} = \frac{1}{{\text{EIS}}_{\theta}} + \frac{1}{{\text{EIS}}_{\varphi}}{.} \tag{29} \]
Equation (29) shows how the total EIS of a dual-polarized UE can be obtained by measuring the EIS for any two orthogonal but otherwise arbitrary polarizations. The same equation is applicable even for UE with a single active polarization at any given time. If only the ${v}$ polarization is active, ${P}_{{S},{h},{\theta}}^{\text{rec}} = {P}_{{S},{h},{\varphi}}^{\text{rec}} = {0}$, and (23)–(28) become \begin{align*}{P}_{{S},{v},{\theta}}^{\text{rec}} & = {G}_{v}{\text{EIS}}_{\theta}{\cos}^{2}{\varphi}_{o}, \quad {P}_{{S},{v},{\varphi}}^{\text{rec}} = {G}_{v}{\text{EIS}}_{\varphi}{\sin}^{2}{\varphi}_{o}, \tag{30} \\ {\text{SNR}}_{\text{th}} & = \frac{{P}_{{S},{v},{\theta}}^{\text{rec}}}{{P}_{v}^{n}} = \frac{{P}_{{S},{v},{\varphi}}^{\text{rec}}}{{P}_{v}^{n}}, \tag{31} \\ {\frac{{\text{SNR}}_{\text{th}}}{{\text{EIS}}_{\theta}}} & = \frac{{G}_{v}{\cos}^{2}{\varphi}_{o}}{{P}_{v}^{n}}, \quad {\frac{{\text{SNR}}_{\text{th}}}{{\text{EIS}}_{\varphi}}} = \frac{{G}_{v}{\sin}^{2}{\varphi}_{o}}{{P}_{v}^{n}}. \tag{32} \end{align*}
Note that for any given array, if only the ${v}$ polarization is active, then ${\text{EIS}} = {\text{EIS}}_{v}$, and from (20) or (21) we have \[{\text{SNR}}_{\text{th}} = \frac{{G}_{v}{\text{EIS}}}{{P}_{v}^{n}} \tag{33} \]
Equations (32) and (33) directly lead us to the same equation as (29). To achieve proper spherical coverage, UE is usually equipped with more than one antenna array, but only one of them is active at any given time. During EIS measurements, as UE is rotated, the active array is selected using some type of modem algorithm. It is important to note that for (29) to be valid, at any given link angle, measurements for ${\text{EIS}}_{\theta}$ and ${\text{EIS}}_{\varphi}$ should come from the same active array. In other words, ${\varphi}_{o}$ in (23) and (24) must belong to the same array.
Equation (29) also provides the blueprint for measuring the spherical coverage patterns of ${\text{EIS}}_{v}$ and ${\text{EIS}}_{h}$. By disabling the ${h}$-polarized receive chain and measuring the spherical coverage patterns of ${\text{EIS}}_{\theta}$ and ${\text{EIS}}_{\varphi}$, ${\text{EIS}}_{v}$ can be calculated using (29). A similar procedure can be repeated for ${\text{EIS}}_{h}$. In the case that both orthogonal polarizations are present simultaneously, (23) and (24) take the following form: \begin{align*}{P}_{{S},{v}}^{\text{rec}} & = {P}_{{S},{v},{\theta}}^{\text{rec}} + {P}_{{S},{v},{\varphi}}^{\text{rec}} = {G}_{v}{P}_{v}^{\text{inc}}, \tag{34} \\ {P}_{{S},{h}}^{\text{rec}} & = {P}_{{S},{h},{\theta}}^{\text{rec}} + {P}_{{S},{h},{\varphi}}^{\text{rec}} = {G}_{h}{P}_{h}^{\text{inc}}, \tag{35} \end{align*} where ${P}_{v}^{\text{inc}} = {\text{EIS}}_{\theta}{\cos}^{2}{\varphi}_{o} + {\text{EIS}}_{\varphi}{\sin}^{2}{\varphi}_{o}$ and ${P}_{h}^{\text{inc}} = {\text{EIS}}_{\theta}{\sin}^{2}$${\varphi}_{o} + {\text{EIS}}_{\varphi}{\cos}^{2}{\varphi}_{o}$. Since ${\text{EIS}}_{v} / {\text{EIS}}_{h}$ are ${P}_{v}^{\text{inc}} / {P}_{h}^{\text{inc}}$ at the threshold of sensitivity, the only choice to make the EIS independent of ${\varphi}_{o}$ is ${\text{EIS}}_{\theta} = {\text{EIS}}_{\varphi}$, which gives us the same result as (15).
There are two important takeaways from this section. If any two orthogonal polarizations can be applied simultaneously, the measurements are free from polarization mismatch. If we can apply only one polarization at a time, by properly combining the results from these two measurements, we can remove the effects of polarization mismatch and achieve the same results with reasonable accuracy. The key is that the formula that is used should mimic a modem processing algorithm and not be arbitrary. While this test method is not perfect, as, e.g., an MRC gain of three can be achieved only if no cross-polarized components are present, it offers a tradeoff between accuracy and cost.
As mentioned, depending on the excitation of the feed antenna, an SS can transmit one of two orthogonal polarizations. For each polarization, the path loss from the SS to the center of the QZ is measured using an SGH antenna with a known efficiency and gain [1]. For example, for a large reflector with no edge effects, the power received by the SGH can be derived as [1] \begin{align*}{P}_{\theta}^{\text{rec}} & = {G}_{\theta}^{\text{sgh}}{G}_{\theta}^{\text{feed}}{\left(\frac{\lambda}{{2}{\pi}{d}_{\text{focal}}}\right)}^{2}{P}_{\theta}^{\text{SS}}, \tag{36} \\ {P}_{\varphi}^{\text{rec}} & = {G}_{\varphi}^{\text{sgh}}{G}_{\varphi}^{\text{feed}}{\left(\frac{\lambda}{{2}{\pi}{d}_{\text{focal}}}\right)}^{2}{P}_{\varphi}^{\text{SS}}, \tag{37} \end{align*} where it is assumed that the SGH aperture is pointing toward the reflector and that the SGH can transmit and receive with the same two orthogonal polarizations as the feed antenna. Here, ${d}_{\text{focal}},{\lambda}$, ${G}_{\theta}^{\text{sgh}} / {G}_{\varphi}^{\text{sgh}}$, and ${G}_{\theta}^{\text{feed}} / {G}_{\varphi}^{\text{feed}}$ are the focal length of the reflector, wavelength, and gain of the SGH and feed antennas, respectively; ${P}_{\theta}^{\text{SS}} / {P}_{\varphi}^{\text{SS}}$ is the power applied to the feed antenna by the SS. Since ${G}_{\theta}^{\text{sgh}}$ and ${G}_{\varphi}^{\text{sgh}}$ are known, the path loss from the feed antenna to the QZ can be calculated as \[{L}_{\theta} = {G}_{\theta}^{\text{feed}}{\left(\frac{\lambda}{{2}{\pi}{d}_{\text{focal}}}\right)}^{2}, \quad {L}_{\varphi} = {G}_{\varphi}^{\text{feed}}{\left(\frac{\lambda}{{2}{\pi}{d}_{\text{focal}}}\right)}^{2}. \tag{38} \]
In reality, ${L}_{\theta}$ and ${L}_{\varphi}$ can be accurately obtained only from measurements and should also account for cable losses (e.g., between the feed antenna and SS). For discussion, we assume that there are no cable losses. Note that \[{P}_{\theta}^{\text{inc}} = {L}_{\theta}{P}_{\theta}^{\text{SS}}, \quad {P}_{\varphi}^{\text{inc}} = {L}_{\varphi}{P}_{\varphi}^{\text{SS}}. \tag{39} \]
To measure ${\text{EIS}}_{\theta}$, ${P}_{\varphi}^{\text{SS}}$ is set to zero, and ${P}_{\theta}^{\text{SS}}$ is adjusted until the threshold of sensitivity is achieved. This process is repeated for ${\text{EIS}}_{\varphi}$. Let us call these values ${P}_{{\text{EIS}},{\theta}}^{\text{SS}}$ and ${P}_{{\text{EIS}},{\varphi}}^{\text{SS}}$, respectively. The tester can use (39) to calculate ${\text{EIS}}_{\theta}$ and ${\text{EIS}}_{\varphi}$: \[{\text{EIS}}_{\theta} = {L}_{\theta}{P}_{{\text{EIS}},{\theta}}^{\text{SS}}, \quad {EIS}_{\varphi} = {L}_{\varphi}{P}_{{\text{EIS}},{\varphi}}^{\text{SS}}. \tag{40} \]
This is the two-step method that was alluded to earlier. Since ${\varphi}_{o}$ is generally unknown, there is no way for the test operator to know ${\text{EIS}}_{v}$ or ${\text{EIS}}_{h}$. Nonetheless, the total EIS can still be calculated using (1). Many RAN4 papers disregard the fact that ${L}_{\theta}$ and ${L}_{\varphi}$ are measured under a polarization match condition and that ${\text{EIS}}_{v}$ and ${\text{EIS}}_{h}$ are generally unknown [16]. This is discussed in more detail in the final section.
Spatial diversity, multiple antenna techniques, and combining methods are extensively covered in the communication literature. Here, only certain aspects of the topics related to wave propagation in a wireless environment are revisited. For any given polarization, the large-scale path loss from a BS to UE is the same for all UE antennas. In other words, in the absence of fading, the average power of an incident plane wave with any polarization is the same on all UE antennas. By adding more antennas or polarization feeds and their corresponding receive chains to UE, we are simply capturing more of the radio wave that already exists in the vicinity of the UE. But diversity reception is about more than just picking up more DL signal. Otherwise, increasing BS transmit power would achieve the same goal. Primary and diversity antennas should be separated by enough distance that they experience independent fading.
At millimeter-wave frequencies, where beamforming is commonly used, polarization diversity and dual-feed antenna arrays are utilized instead. Nonetheless, the core concept remains the same. For diversity techniques to work optimally, the channels from a BS to primary and diversity antennas or from a BS to each orthogonal polarization, i.e., ${P}_{\theta}^{\text{inc}}$ and ${P}_{\varphi}^{\text{inc}}$, should experience uncorrelated fading. For planning a network, we need individual spatial and temporal statistics of ${P}_{\theta}^{\text{inc}}$ and ${P}_{\varphi}^{\text{inc}}$. Therefore, it makes more sense to define UE sensitivity as the DL signal level per antenna connector or the average power of the incident plane wave per orthogonal polarization.
Figure 7, which is from [14], shows the relative bit error probability (BEP) for different numbers of receive antennas versus the SNR in a fading channel. It is assumed that there is only a single transmit antenna. Furthermore, the BEP is normalized to one (0 dB) for each scenario. The array gain is the same as increasing the transmit power by ${10}\,{\cdot}\,{\log}_{10}{N}_{r}$ dB. As an example, consider two UE designs. ${\text{UE}}_{1}$ has only ${v}$ polarization, and ${\text{EIS}} = {\text{EIS}}_{v} = {-}{88}{\text{ dBm}}$, and ${\text{UE}}_{2}$ has both ${v}$ and ${h}$ polarizations, and ${\text{EIS}}_{v} = {\text{EIS}}_{h}{-}{85}{\text{ dBm}}$ or ${EIS} = {-}{88}{\text{ dBm}}$. From Figure 7, we can see that for achieving a BEP of ${10}^{{-}{1}}$ in a fading channel, ${\text{UE}}_{1}$ needs an SNR of 10 dB versus only 5 dB for ${\text{UE}}_{2}$. Note that both report the same EIS if tested in the lab. Diversity is about more than just reporting a single number. Diversity order matters just as much. In turn, received signals processed by modems depend on ${P}_{v}^{\text{inc}}$ and ${P}_{h}^{\text{inc}}$. Modem processing gain applies to ${\text{SNR}}_{v}$ and ${\text{SNR}}_{h}$. This is an important distinction that is discussed in the following section.
Figure 7. The relative bit error probability (BEP) curves for different numbers of receive antennas [14].
Equation (1) had been used for more than a decade to calculate the UE spherical receive pattern before a factor of two was added to it in 2018 [5], [15]. Since some of the arguments presented in [5] and [15] are used to justify more consequential changes, they are discussed in more detail here. Note that all 3GPP-related documents can be downloaded from www.3gpp.org at no cost, simply by searching for the document number.
The fundamental problem with [15] is its failure to recognize the purpose of testing EIS in two steps and the importance of combining the results by using (1). This is an issue that is observed throughout the article. The plots in that article that claim to show EIS dependency on ${\varphi}_{o}$, the polarization mismatch angle, actually include only the contribution from one of the two orthogonal polarizations (Figure 8). These plots are reproduced in Figure 9 for the same three UE configurations as [15]: 1) dual-polarized UE with ${\text{EIS}}_{v} = {\text{EIS}}_{h} + {3}{\text{ dB}}$, 2) single-polarized UE with only ${\text{EIS}}_{v}$ or ${\text{EIS}}_{h}$, and 3) dual-polarized UE with ${\text{EIS}}_{v} = {\text{EIS}}_{h}$. ${\text{EIS}}_{\theta}$, ${\text{EIS}}_{\varphi}$, and ${\text{EIS}}$ calculated using (1) are plotted in Figure 9. Calculation details can be found in “Calculation of EIS From v and h Component Polarizations.†As can be seen, EIS is independent of ${\varphi}_{o}$.
Figure 8. The calculated EIS plots from [15]: (a) dual-polarized UE, with ${\text{EIS}}_{v} = {\text{EIS}}_{h} + {3}{\text{ dB}}$; (b) single-polarized UE; and (c) dual-polarized UE, with ${\text{EIS}}_{v} = {\text{EIS}}_{h}$.
Figure 9. The calculated EIS, ${\text{EIS}}_{\theta}$, and ${\text{EIS}}_{\varphi}$ for (a) dual-polarized UE, with ${\text{EIS}}_{v} = {\text{EIS}}_{h} + {3}{\text{ dB}}$; (b) single-polarized UE; and (c) dual-polarized UE, with ${\text{EIS}}_{v} = {\text{EIS}}_{h}$.
The work in [15] presents four options for reporting EIS from measurements: Equations (1) and (2), ${\text{EIS}}_{M} = {\text{MAX}}{\left({\text{EIS}}_{\theta}, {\text{EIS}}_{\varphi}\right)}$, and ${\text{EIS}}_{\text{MAX}} = {\text{MAX}}_{\text{polarization}}{\left({\text{EIS}}\right)}$, which is described as “EIS for any link angle should be reported as the worst (maximum) EIS over all potential DL polarization angles.†In other words, the authors seem to be under the impression that the only way to capture the EIS without polarization mismatch is to measure it for all possible ${\varphi}_{o}$ and that (1) and (2) are some types of empirical formulas. In choosing (2) over (1), they argue that ${\text{EIS}}_{\theta}$ and ${\text{EIS}}_{\varphi}$ already contain the MRC gain and that using (1) accounts for it twice, unnecessarily overestimating UE performance. As evidence, they point out that the EIS using (1) is better than ${\min}{\left({\text{EIS}}_{v},{\text{EIS}}_{h}\right)}$ and hence does not have a physical basis, a flawed argument reiterated in [16].
Equations (23) and (24) do show MRC gain during ${\text{EIS}}_{\theta}$ and ${\text{EIS}}_{\varphi}$ measurements, respectively. However, the MRC gain is applied to ${P}_{v}^{\text{inc}}$ and ${P}_{h}^{\text{inc}}$ and not ${P}_{\theta}^{\text{inc}}$ and ${P}_{\varphi}^{\text{inc}}$, which are the only measurable quantities. During ${\text{EIS}}_{\theta}$ measurement, ${P}_{v}^{\text{inc}}$ and ${P}_{h}^{\text{inc}}$ are ${\text{EIS}}_{\theta}{\cos}^{2}{\varphi}_{o}$ and ${\text{EIS}}_{\theta}{\sin}^{2}{\varphi}_{o}$, respectively. We know that ${\text{EIS}}_{\theta} = {L}_{\theta}{P}_{{\text{EIS}},{\theta}}^{\text{SS}}$. But ${\varphi}_{o}$, and therefore ${P}_{v}^{\text{inc}}$ and ${P}_{h}^{\text{inc}}$, are unknown. But most importantly, the authors fail to address the most obvious case with ${\varphi}_{o} = {l}{(}{\pi} / {2}{)}$ for an integer ${l}$. When this happens, ${\theta}$ and ${\varphi}$ are aligned with ${v}$ and ${h}$. The receive chain for ${v} / {h}$ polarization can receive only ${v} / {h}$-polarized plane waves. The output of the receive chain for a cross-polarized incident plane wave is noise. For MRC gain to materialize, two copies of the signal with uncorrelated interference are needed. In each of these cases, there is only one copy. So how could MRC possibly improve the output SNR? This is the same as claiming that the diversity gain in Figure 6 is materialized even if the splitter/combiner is removed and the signal is applied only to the UE Tx/Rx port and not the Rx-only port.
The work in [5] offers an even a more curious explanation for the necessity of adding a factor of two to (1). The authors point out that the TRS for FR1 is calculated using ${\text{TRS}} = {4}{\pi} / {\oint}{\left({\text{EIS}}_{\theta}^{{-}{1}} + {\text{EIS}}_{\varphi}^{{-}{1}}\right)}{d}{\Omega}$ and claim that the EIS is defined as the integrand of the TRS. Since the TRS is the amount of DL power on each UE antenna connector, they argue that this definition of the EIS is not suitable for FR2 because FR2 UE does not have physical antenna connectors. The statement that the TRS is the amount of DL power on each UE antenna connector is attributed to [4]. In fact, [4] asserts only that the best achievable TRS for UE with a perfectly matched and 100% efficient isotropic antenna is its conducted sensitivity. Since the antenna is matched and lossless, there is no return loss and conversion to heat, and all the power in the incident plane waves is delivered to the antenna connector.
The reference to the antenna connector in this context is where the antenna connects to the rest of the circuitry, which all antennas should have. The article neither explains what role the antenna connector plays in OTA measurements nor how it could affect modem processing of the signals from receive chains. The TRS definition does apply to FR2 UE. There is no difference between OTA test methods for FR1 or FR2. It is worth mentioning that there is no requirement that in an FR1 UE, the primary and diversity antennas have the same polarization. Furthermore, FR1 predicts up to four receive antennas for some bands [6]. Does this mean we should add a factor of four to (1)? The authors of [17] and [18] more or less recycle arguments from [5], without providing any convincing argument. Those of [19], [20], [21], [22] argue for restoring (1) but only for UE with a single active polarization and without offering anything more than just an opinion.
At this point, the reader may wonder whether the absence or presence of a factor of two in the EIS equation really matters. The introduction of a factor of two, in a way, changed the purpose of the two-step test method by ignoring the importance of the combining equation. Similar arguments were later used to change the equation for the UE spherical transmit pattern and peak effective isotropic radiated power (EIRP) [23]. It can be easily shown that \begin{align*}{\text{EIRP}}_{\theta} & = {\text{EIRP}}_{v}{\cos}^{2}{\varphi}_{o} + {\text{EIRP}}_{h}{\sin}^{2}{\varphi}_{o}, \tag{41} \\ {\text{EIRP}}_{\varphi} & = {\text{EIRP}}_{v}{\sin}^{2}{\varphi}_{o} + {\text{EIRP}}_{h}{\cos}^{2}{\varphi}_{o}{.} \tag{42} \end{align*}
The legacy equation used ${\text{EIRP}} = {\text{EIRP}}_{\theta} + {\text{EIRP}}_{\varphi}$, which gives the correct total EIRP of UE as ${\text{EIRP}}_{v} + {\text{EIRP}}_{h}$. Currently, the EIRP is calculated as ${\max}{\left({\text{EIRP}}_{\theta},{\text{EIRP}}_{\varphi}\right)}$ (this was a compromise solution to avoid forfeiting 3 dB as the result of the averaging equation). Even if ${\text{EIRP}}_{v} = {\text{EIRP}}_{h}$ for dual-polarized UE, this method reports an EIRP that is 3 dB lower. The problem is even worse for single-polarized UE, as the spherical transmit pattern is weighted as a function of ${\varphi}_{o}$. Given the fact that U.S. Federal Communications Commission-mandated limit applies to the total EIRP [24], this reporting method creates the potential of running afoul of the regulation.
The maximum Tx power and Rx sensitivity, or in the case of 5G FR2 UE, the peak EIRP and minimum EIS, are two of the most important factors in deciding the edge of coverage of a cellular BS. While planning a cellular network is not as simple as covering it with hexagonal-shaped cells, there is no doubt that UE with a better EIRP and EIS can travel farther from the BS. A smaller cell size means a shorter distance between adjacent cell towers, which is not always a good thing. While it means a smaller path loss and stronger signal at the cell edge, it can also lead to more intercell interference, more frequent handoffs, higher deployment costs, and shorter battery life. To cut a long story short, 5G millimeter-wave standards deserve a closer look from the 3GPP to address the existing discrepancies and prevent future ones. You can drive a car with a missing wheel nut but not for long.
This article showed that polarization mismatch in the OTA testing of wireless UE is unavoidable unless the test system can support two orthogonal polarizations simultaneously. Due to the high cost of simultaneously generating two orthogonal polarizations, an alternative test method has been adopted by the 3GPP, where the measurements are conducted in two steps, in which only one of the two orthogonal polarizations is present. The article showed that in this case, it is important to use to the correct equation to combine the results from the two steps. Mathematical details were provided for the case of EIS measurement with MRC in the modem. The method can be applied to other test cases, but it is important to analyze each case with the pertinent modem processing algorithm. Finally, the article tried to highlight how misunderstanding the two-step test method has led to inaccurate reporting of some performance indicators and emphasized the importance of prompt action on this matter.
Amirreza Khajehnasiri (amirreza.khajehnasiri@gmail.com) is with InnoPhase, San Diego, California, 92121, USA. 92121. Previously, he was a radio-frequency systems engineer at Qualcomm.
[1] “NR; Study on test methods,†3GPP TS 38.810. V16.0.0 (September 2018) and V16.1.0 (January 2019).
[2] “User equipment (UE)/mobile station (MS) over the air (OTA) antenna performance; conformance testing,†3GPP TS 34.114, Oct. 5, 2016.
[3] “User equipment (UE) over the air (OTA) performance; conformance testing,†3GPP TS 37.544, Dec. 19, 2018.
[4] CTIA Test Plan for Wireless Device Over-the-Air Performance. (v. 3.8.1). (v.3.8.2, April 2019) by CTIA (Cellular Telecommunications Industry Association) https://api.ctia.org/wp-content/uploads/2019/04/CTIA_OTA_Test_Plan_3_8_2.pdf
[5] “On OTA EIS metric in FR2,†3GPP R4-1815626.
[6] “NR; user equipment (UE) radio transmission and reception,†3GPP, TS 38.101-1.
[7] IEEE Standard Test Procedures for Antennas, IEEE Standard 149-1979.R2008, Oct. 2003.
[8] R. C. Johnson and R. J. Poinsett, “Compact antenna range techniques,†Rome Air Development Center, Griffiss AFB, Rome, NY, Contract, AF 30(602)-3594, Final Rep. RADC-TR-66-15, Apr. 1966.
[9] W.C. Jakes, Ed., Microwave Mobile Communications. Hoboken, NJ, USA: Wiley, 1974, ch. 3.
[10] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 3rd ed. Hoboken, NJ, USA: Wiley, 2013.
[11] “Measurements of user equipment (UE) radio performances for LTE/UMTS terminals; total radiated power (TRP) and total radiated sensitivity (TRS) test methodology,†3GPP TS 37.902, Jul. 9, 2018.
[12] “NR; User equipment (UE) radio transmission and reception; part 2: range 2 standalone,†3GPP TS 38.101-2, Jan. 9, 2021.
[13] “Evolved universal terrestrial radio access (E-UTRA) and evolved packet core (EPC); common test environments for user equipment (UE) conformance testing,†3GPP TS 36.508.
[14] A. Ghosh, J. Zhang, J. G. Andrews, and R. Muhamed, Fundamentals of LTE. Boston, MA, USA: Pearson, Apr. 11, 2021.
[15] “On OTA EIS metric in FR2,†3GPP R4-1813370.
[16] “Discussion on EIS calculation,†3GPP R4-1900260.
[17] “On FR2 EIS,†3GPP R4-1900671.
[18] “On FR2 EIS,†3GPP R4-1902846.
[19] “On EIS test metric for FR2,†3GPP R4-1906378.
[20] “On EIS test metric,†3GPP R4-1912420.
[21] “Testability_EIS,†3GPP R4-1913941.
[22] “On EIS test metric,†3GPP R4-1915390.
[23] “EIRP measurement for polarization mismatch,†3GPP R4-2010129.
[24] Telecommunication, eCFR 47, section 30.202, 2021.
[25] M. Schwartz, W. R. Bennett, and S. Srein, Communication Systems and Techniques. Piscataway, NJ, USA: Wiley-IEEE Press, 1996.
Digital Object Identifier 10.1109/MAP.2022.3169393
