Dustin Brown, Yahya Rahmat-Samii
IMAGE LICENSED BY INGRAM PUBLISHING
The error vector magnitude (EVM) is a comprehensive measure of the amplitude and phase distortion of digitally modulated signals and a potential tool for characterizing the performance of active beamforming arrays, especially at millimeter-wave frequencies, because of the high level of integration and limited spacing between the radiating elements. A review of the EVM measurement process and fundamental principles for time-domain demodulation with a vector signal analyzer (VSA) and spectral correlation with a vector network analyzer (VNA) was presented in [1]. This article demonstrates how researchers have modeled and measured impairments from antenna and phased array signal transmission to characterize their influence on the EVM of single and multicarrier digital signals. It presents examples of simulation models and measurements of the distortion from ultrawideband (UWB) passive antenna transmission, power amplifier (PA) nonlinearities, and phased array beam squint and beam scan intersymbol interference (ISI). A brief survey of over-the-air (OTA) test ranges for EVM analysis is also provided. This overview of state-of-the-art EVM simulation and measurement methods is intended to help interested readers understand and enhance the EVM performance analysis methods applied to antennas and phased arrays in this new era of millimeter-wave communications.
This article presents a complete survey of the recent literature related to the modeling, simulation, and measurement of EVM resulting from antenna and phased array transmission of digitally modulated signals. As described in [1], measurements of the far-field radiated power distribution as a function of beam scan angle with a single tone stimulus are no longer sufficient for characterizing phased array performance because the received signal quality depends not only on the signal-to-noise ratio (SNR) but also on the phased array linearity over the complete bandwidth of the applied signal. The EVM is dependent on both and is more closely related to the data throughput rate, the metric of utmost concern in millimeter-wave communication systems [2]. When measured OTA, EVM accounts for the frequency response of the embedded transceivers and array elements to a modulated signal with frequency spectrum ${X}{\left({f}_{n}\right)}$ as shown in (1), where ${Y}{\left({f}_{n}\right)}$ is the received far-field signal spectrum and ${E}{\left({f}_{n}\right)}$ is the equalization filter for N tones spanning the occupied bandwidth ${f}_{1}\ldots{f}_{N}$ of ${X}{\left({f}_{n}\right)}$: \[{\text{EVM}} = \frac{\sqrt{\mathop{\sum}\limits_{{n} = {1}}^{N}{\left\vert{{X}{\left({f}_{n}\right)}{-}{E}{\left({f}_{n}\right)}{Y}{\left({f}_{n}\right)}}\right\vert^{2}}}}{\sqrt{\mathop{\sum}\limits_{{n} = {1}}^{N}{\left\vert{{X}{\left({f}_{n}\right)}}\right\vert^{2}}}}{.} \tag{1} \]
OTA measurements of EVM are usually required for millimeter-wave phased arrays because of their compact designs and limited spacing between the radiating elements. Since they are sensitive to both the linear and nonlinear characteristics of the phased array components as well as the statistical properties of the modulated test signal, they have received increasing consideration as a figure of merit for 5G and beyond [3]. The EVM definitions and measurement methodologies described in [1] provide a context for the survey of EVM simulations and measurements of antennas and phased arrays presented in the sections “EVM Simulations: Antennas and Phased Arrays” and “EVM Measurements: Far Field, Indirect Far Field, and Near Field” of this article. The section “Conclusions and Future Challenges” summarizes these EVM research contributions and identifies areas for future improvement in modeling and measurements.
This section reviews techniques for modeling and simulating the EVM resulting from the amplitude and phase distortion of passive antennas (in the section “Passive Antennas”) and from the superposition of the radiated fields from a large number of passive or active phased array element channels (in the section “Phased Arrays”).
Signal transmission by a passive antenna results in linear distortion when the amplitude response ${\left\vert{H}{{\left({f}\right)}}\right\vert}$ is not constant for all frequency components of the applied voltage signal ${X}{\left({f}\right)}$ and when the phase response ${\angle}{H}{\left({f}\right)}$ deviates from the negative linear slope ${-}{\omega}{t}_{d}$, as shown in [1]. Some researchers have modeled far-field antenna links as linear time-invariant (LTI) systems to simulate this linear distortion and estimate its effects on the EVM of single-carrier quadrature amplitude modulation (QAM) signals. A model of 16-QAM signal transmission by the 28-GHz microstrip patch antenna from [4] was demonstrated by the authors in [5]. A block diagram of the link model is presented in Figure 1(a), and the effect of antenna gain variation across the signal bandwidth of baseband signals ${x}_{i}{\left({t}\right)}$ and ${x}_{q}{\left({t}\right)}$ is shown in Figure 1(b). The amplitude response of the antenna ${\left\vert{H}_{\text{tx}}{\left({f}\right)}\right\vert} = {\left\vert{\vec{E}}_{\text{rad}}{\left({\theta}_{t},{\phi}_{t},{d}\right)}\right\vert} / {\left\vert{X}{\left({f}\right)}\right\vert}$ to a Gaussian pulse excitation voltage ${X}{\left({f}\right)}$ was acquired at the boresight far-field position ${\left({0},{0},{1}{\text{ m}}\right)}$ via full-wave simulation in CST Microwave Studio, and the signal reception was modeled as that of a lossless copolarized receiving antenna with ${\left\vert{H}_{\text{rx}}{\left({f}\right)}\right\vert} = {1}$ for all frequencies. The EVM of a 16-QAM signal distorted by microstrip patch amplitude response ${\left\vert{H}_{\text{tx}}{\left({f}\right)}\right\vert}$ was simulated as a function of the symbol rate and carrier frequency offset ${\Delta}{f}$ from the patch resonance and shown to increase with both.
Figure 1. A far-field signal transmission model for determining the EVM from the amplitude distortion of a 28-GHz microstrip patch antenna [5]. (a) Block diagram of the MATLAB communication system model with antenna transfer function ${H}{(}{f}{)}$ derived from full-wave simulations. (b) 16-QAM in-phase (left) and quadrature (right) input (blue) and output (green) signal amplitudes and ${\Delta}{I}$, ${\Delta}{Q}$ error components (red), input and output signal spectrum (top left), and antenna realized gain ${G}_{r}$ (top right, blue), transfer function ${H}{(}{f}{)}$ (top right, red), and S11 (top right, green). Tx: transmitter; Rx: receiver.
A method of simulating the EVM of a far-field link between nearly identical antennas, based on finite-impulse response (FIR) filter models derived from two port S-parameter measurements, was demonstrated in [6], [7], and [8]. In [6], the inverse fast Fourier transform (IFFT) of the antenna link S-parameters measured in an anechoic test range was used to synthesize FIR filter models that were imported into a Simulink model of a 16-QAM communication system to determine how the antenna gain and group delay variations across the signal bandwidth influenced unequalized EVM and bit error rate (BER) performance. See Figure 2. The same approach was also used to compare the simulated EVM of far-field antenna links for different propagation environments [7], frequency bands [6], and antenna orientations [8]. For antennas with low loss and negligible nonlinear phase, S-parameters can also be estimated from equivalent circuit models that are meticulously designed and optimized to match the measured ${S}_{11}$ [6], [8].
Figure 2. EVM simulation for far-field antenna link based on FIR filter models matching measured two-port S-parameters [6]: (a) block diagram of the digital communication system modeling procedure. (b) FIR filter, (c) antenna link simulation models, and (d) constellation diagrams of scattered symbols from the simulated transmission of 16-QAM signals at 2 GHz (left) and 6 GHz (right). MER: modulation error rate.
Full-wave simulations can be used to determine the radiation characteristics of antenna designs without making the kinds of simplifying assumptions required for the synthesis of a circuit model. The authors of [8] used CST Microwave Studio to simulate the S-parameters of two wideband printed monopoles separated by 50 cm, but they also acknowledged that a lack of computational resources for adaptive meshing could limit the accuracy of such electrically large simulations. One potential solution was presented in [9]. The grid impulse response (GIR) of a finite difference time-domain (FDTD) solver was obtained via simulations with delta function excitation signals and subsequently convolved with a 16-QAM signal to estimate the EVM of a link between two identical 90-GHz dipole antennas as a function of the separation distance, both in free space and over an infinite ground plane. The advantage of this method is that the FDTD simulation used to acquire the GIR can be performed much faster than one in which a modulated source signal is used. The development of computationally efficient full-wave simulation schemes such as these could improve the accuracy and efficiency of estimating the UWB amplitude and phase response from far-field antenna links. While linear distortion from this frequency response can generally be removed via equalization [1], the collective distortion effects of antennas, phase shifters, and nonlinear components embedded in phased arrays must be predicted using more advanced simulation models. The next section focuses on the design and application of such models by representative researchers for far-field EVM analysis.
Fully characterizing the simultaneous transmission and reception of wideband modulated signals by a large number of array elements, radio frequency (RF) transceiver nonlinearities, and mutual coupling for various beam states would require the use of highly advanced phased array simulation models. As a result, most researchers develop simpler models that account for some, but not all, of these EVM effects.
One approach to estimating the EVM of digital beamforming arrays is to develop an analytical expression in which the amplitude and phase errors are modeled as random variables [10], [11]. In [10], the EVM was calculated from (2), where N is the number of antennas, ${W}_{mn}$ is the weighting coefficient of the mth beam, ${A}_{mn}$ is the spatial response function (3), ${\sigma}_{\delta}^{2}$ and ${\sigma}_{\psi}^{2}$ are the variances of the amplitude and phase errors $\left({{1} + {\delta}_{n}}\right){e}^{{\psi}_{n}}$ of each element n, and ${G}_{0} = {\Sigma}_{{n} = {1}}^{N}{W}_{mn}{A}_{mn}$ is the sum of the element excitations for m beam states. In (3), $\left({{\rho}_{n},{\varphi}_{n}}\right)$ represent the cylindrical coordinates of element n, and $\left({{\theta}_{0},{\varphi}_{0}}\right)$ are the beamsteering angles. The array geometry and EVM for the Gaussian random amplitude and phase variables ${\delta}_{n}$ and ${\psi}_{n}$ with different ranges are shown in Figure 3. \begin{align*}{\text{EVM}} & = \sqrt{\mathop{\sum}\limits_{{n} = {1}}^{N}{\left\vert{{W}_{mn}{A}_{n}}\right\vert^{2}\frac{\left({{\sigma}_{\delta}^{2} + {\sigma}_{\psi}^{2}}\right)}{{G}_{0}^{2}} + \frac{1}{\text{SNR}}}} \tag{2} \\ {A}_{n} & = {\exp}\left\{{{-}{j}\frac{{2}{\pi}}{\lambda}{\rho}_{n}\sin{\theta}_{0}\cos\left({{\varphi}_{n}{-}{\varphi}_{0}}\right)}\right\}{.} \tag{3} \end{align*}
Figure 3. A digital beamforming array EVM simulation based on the random amplitude and phase errors ${\delta}_{n}$ and ${\psi}_{n}$ in (2) and (3) [10]: (a) array geometry and element ${n}$ cylindrical coordinates ${(}{\rho}_{n},{\phi}_{n}{)}$ and (b) EVM simulation results for amplitude (left) and phase (right) error ranges with ${\text{SNR}} = {23}\,{\text{dB}}$. RMS: root mean square.
In [11], an analytical expression with random variables was used to predict how different numbers and distributions of damaged array elements would influence the EVM. In [12], in-phase/quadrature (IQ) amplitude and phase errors modeled as random variables were used to estimate the EVM of a 64-QAM signal transmitted and received by four-element phased arrays. Modeling the amplitude and phase errors of array elements as random variables is a relatively simple approach that may yield some insights on statistical correlations of the far-field EVM performance, but a more accurate EVM characterization requires a modeling of amplitude and phase error that is consistent with the power and frequency response of distortion-dominating components, particularly the beamforming integrated circuit (IC) PAs.
One technique of incorporating PA nonlinearity effects in phased array EVM simulations is to use the cubic polynomial model for amplitude-to-amplitude (AM–AM) and amplitude-to-phase (AM–PM) modulation as a function of the input signal power ${P}_{\text{in}}$, as shown in Figure 4(a). In [13], a linear gain with random amplitude and phase error was applied to each array element channel, and AM–AM and AM–PM modulation was subsequently applied to each signal based on its power ${P}_{\text{in}}$. The EVM from a 64-element array transmitting a 64-QAM signal at 5 dB backoff from the output 1-dB compression point ${\left({P}_{\text{in}} = {-}{16}\,{\text{dB}}\right)}$ is plotted in Figure 4(b) as a function of the root mean square (RMS) gain and phase error, and the EVM is plotted in Figure 4(c) at the same backoff level with an RMS gain error of 1 dB and a phase error of 10° as a function of the number of array elements. The results demonstrate that the EVM is more strongly correlated with channel gain error than phase error since the AM–AM and AM–PM distortion depend on gain, and the phase error averages out in the summation of received signals from a large number of elements [13]. This is highlighted in Figure 4(c) by the convergence of 60 EVM simulations to the same average value as the number of array elements increases. However, the authors acknowledged that phase errors will reduce array gain, thereby decreasing the SNR and increasing the EVM. Since the array element patterns and geometry are not defined in this baseband phased array model, it excludes this effect on the EVM [13]. Beam scanning to angle ${\left({\theta}_{s},{\varphi}_{s}\right)}$ could also be implemented by applying a progressive phase shift to the array element channels and adding time delays before the ideal combiner to compensate for the free-space propagation of signals.
Figure 4. MATLAB model of 64-QAM signal transmission by ${8}\,{\times}\,{8}$ phased array and simulated EVM results [13]: (a) array model and memoryless PA AM–AM (left) and AM–PM (right) versus ${P}_{\text{in}}$, (b) EVM versus RMS amplitude and phase error for 5-dB backoff ${(}{P}_{\text{in}} = {-}{16}\,{\text{dB}}{)}$, and (c) EVM versus number of elements for 5-dB backoff, 1-dB RMS amplitude error, and 10° RMS phase error.
A similar MATLAB phased array transmission model was implemented in [15] to simulate the effects of PA nonlinearity variation on the far-field EVM of a uniform linear array. The PA AM–AM and AM–PM distortion versus ${P}_{\text{in}}$ was based on a lookup table from measurements of a 13-GHz CMOS PA, and raised-cosine filtering was applied to the 100-MHz 64-QAM test signal. Monte Carlo simulations demonstrated that the EVM is reduced slightly with an increase in the number of array elements and random variation in PA nonlinearity when digital predistortion (DPD) is not applied, and that this improvement is significantly better when DPD is applied. The authors of [15] therefore suggest that it might be possible to reinforce the DPD and lower the EVM by intentionally biasing the PA transistors in phased array element channels to generate a larger variation. In [16], a simulation of PA nonlinearity variation based on a Gaussian distribution of the input 1-dB compression point showed that the adjacent channel power ratio, another measure of wideband signal distortion, also decreases with increasing numbers of radiating elements in phased arrays.
While the array model from [15] includes element gain and spacing, it excludes mutual coupling and PA loading effects, which will change the PA nonlinearity and EVM performance in beam scanning. Researchers are starting to develop models of mutual coupling effects by deriving mathematical representations or combining independent simulations of PA active load-pull data with active S-parameters [17], [18], [19], [20], [21]. In [20], a mathematical model of a dual-input PA based on a Volterra series was designed to include the nonlinearity, mutual coupling, and wideband impedance mismatch effects. PA model coefficients for each array element channel were obtained from load-pull measurements and array element S-parameters. In [21], an algorithm was designed to use load-pull and S-parameter data to estimate PA nonlinearity from mutual coupling effects in phased array beam scanning. The nonlinearity of radiated far fields was characterized by the received power and the AM–AM and AM–PM distortion. To the authors’ knowledge, PA load modulation in phased array transmission has not yet been modeled and simulated for far-field EVM estimation.
In addition to these PA contributions to the far-field EVM, researchers have modeled and simulated other phased array effects, such as large array beam scan ISI, carrier aggregation, beam squint, and embedded element frequency response. In [22], a MATLAB model of phased array signal transmission was used to demonstrate that a linear equalizer with a sufficient number of channel taps could compensate for a variation in array element signal time delays caused by beam scanning, regardless of QAM modulation order, beam scan angle, and symbol rate. PA nonlinearity effects were excluded from this model to isolate the effect of beam scan signal delays on the EVM, and root-raised cosine filters were included at both the transmit and receive ends. The beam scanning was modeled by applying a true time delay instead of a phase shift to each array element, so beam squint effects were excluded. Beam squint is a form of amplitude distortion caused by variation in scanned beam direction for frequency components of a wideband signal [14], [22], [23]. In [23], beam squint was simulated using a phased array model similar to that of [22], and the EVM of a 64-QAM signal transmitted at ${f}_{c} = {30}{\text{ GHz}}$ was shown to increase with the number of array elements, beam scan angle, and symbol rate. However, in [13], [22], and [23], all of the array elements were modeled as isotropic radiators with a constant-amplitude frequency response across the signal bandwidth. In [14], the authors simulated phased array amplitude distortion caused by both the radiating element frequency response and beam squint. The input voltage to far-field transfer function ${\vec{H}}_{mn}{\left({f},{\theta},{\phi}\right)}$ was acquired for a central microstrip patch element in a 39-GHz ${8}\,{\times}\,{8}$ array via full-wave simulation and used to estimate the wideband radiated fields of the phased array at beam scan angles in the two principle planes. The simulated EVM of a 100-Mbaud 16-QAM signal transmitted at different scan angles and received in the far field was dominated by the array element frequency response for most test cases in which the phased array had fewer than 1,024 elements and the beam scan angles were less than 45°. The phased array signal transmission model and EVM results for the E- and H-plane scans are shown in Figure 5. The phased array’s far fields were calculated using (4), (5a), and (5b), where ${X}_{mn}{\left({f}\right)}$ is the applied voltage signal spectrum for element $\left({{m},{n}}\right)$, ${\vec{H}}_{mn}{\left({f},{\theta},{\phi}\right)}$ is the embedded element transfer function, and ${\psi}_{x}$ and ${\psi}_{y}$ are the applied phase terms for beam scan angle${\left({\theta}_{s},{\phi}_{s}\right)}$ with wavenumbers ${k} = {2}{\pi} / {\lambda}$ and ${k}_{c} = {2}{\pi}{/}{\lambda}_{c}$ and an array element spacing of d. \[{\vec{E}}_{\text{array}} = \mathop{\sum}\limits_{m=1}^{M}\mathop{\sum}\limits_{n=1}^{N}{X}_{mn}{\left({f}\right)}{\vec{H}}_{mn}\left({f},{\theta},{\phi}\right){e}^{{j}{[}{(}{m-1}{)}\Psi{x} + {(}{n-1}{)}\Psi{y}{]}} \tag{4} \] \begin{align*}{\psi}_{x} & = {kd}{\sin}\,{\theta}{\cos}\,{\phi}{-}{k}_{c}{d}{\sin}\,{\theta}_{s}{\cos}\,{\phi}_{s} \tag{5a} \\ {\psi}_{y} & = {kd}{\sin}\,{\theta}{\sin}\,{\phi}{-}{k}_{c}{d}{\sin}\,{\theta}_{s}{\sin}\,{\phi}_{s}{.} \tag{5b} \end{align*}
Figure 5. A phased array model demonstrating the effects of beam squint and embedded element amplitude response on far-field EVM [14]. (a) A 16-QAM signal transmission model for phased array. (b) Field pattern of central embedded element of ${8}\,{\times}\,{8}$ microstrip patch array. (c) and (d) EVM results for the E- and H-plane beam scan angles, respectively, with ${N}$-element rectangular phased arrays.
The combined EVM effects of carrier aggregation and PA nonlinearity for 400-Mbaud 64-QAM signals transmitted by phased arrays were demonstrated with a MATLAB-based simulation in [24]. The EVM effects of varying beamwidths and positions of 28-GHz phased arrays transmitting and receiving an 800-MHz orthogonal frequency-division multiplexing (OFDM) signal in an indoor-to-outdoor multipath channel were estimated via RF circuit and ray tracing cosimulation in [25]. As researchers challenge themselves to develop more accurate and efficient EVM simulation models for antennas and phased arrays, they will have to validate these new models with OTA measurements performed in controlled test environments.
The large number, limited size and spacing, and high level of integration of radiating elements with beamforming ICs in large millimeter-wave phased arrays necessitate the use of OTA techniques for measurements of modulated signal quality. This section highlights OTA measurement systems that have been used by representative researchers to perform EVM measurements and enhance their precision and versatility in characterizing the performance of millimeter-wave phased arrays.
Far-field anechoic test ranges have traditionally been used to measure the radiated electric field magnitude patterns of antennas using continuous-wave signals. However, large increases in signal bandwidth and decreases in array element size and spacing at millimeter-wave frequencies have generated interest in characterizing the wideband effects of antennas and arrays on system-level performance metrics, such as EVM and BER. Controlled OTA test environments are therefore being adapted for these measurements. In [26], the EVM pattern of a 3.5-GHz probe-fed microstrip patch antenna was measured in an anechoic test range. This patch was rotated and received a 50-MHz-wide ${\pi} / {4}$ differential quadrature phase-shift keying (QPSK) signal transmitted by an identical patch along the broadside direction. In [27], another EVM pattern measurement was taken for a 32-element 28-GHz phased array [28] transmitting an 800-MHz QPSK signal. The EVM pattern was shown to be strongly correlated with the inverted radiated far-field pattern, demonstrating that the EVM measured at far-field distances is dominated by the SNR when the array amplifiers are not driven into saturation. The EVM versus effective isotropic radiated power (EIRP) of the modulated test signal applied to the common port of the phased array takes the form of the plot shown in Figure 6(b). At a lower EIRP, the EVM is dominated by the low SNR resulting from the large free-space path loss between antennas and dynamic range limitations of the receiver. This is shown in the representative OTA measurement configuration for the 39-GHz ${8}\,{\times}\,{8}$ phased array of [13] in Figure 6(a). At a higher EIRP, the EVM is dominated by the nonlinear distortion of the phased array PAs. At a median EIRP, the EVM is limited by test instrument imperfections, such as arbitrary waveform generator (AWG) noise and local oscillator (LO) phase noise [13]. In [29], an OTA measurement system similar to that shown in Figure 6(a) was used to measure the EVM of an analog beamforming front-end module with an integrated ${4}\,{\times}\,{1}$ linear array. For the 64-QAM test signal, EVM components from independent error sources were estimated to isolate the device under test (DUT)-based EVM degradation from that of the rest of the test system. Linear distortion in the array link was removed by VSA equalization, and nonlinear distortion from the PAs was limited by operating at a 14-dB backoff, such that the authors could approximate the EVM as a root sum square of components. The EVM from the module and OTA measurement system errors were estimated at 1.23% and 1.07%, respectively. This underscores the difficulty in performing far-field EVM measurements with adequate precision.
Figure 6. Far-field EVM measurement system for a 39-GHz ${8}\,{\times}\,{8}$ phased array and probe separated by D = 1m [13]. (a) Block diagram of measurement system with AWG, external mixer, and a real-time scope for signal demodulation. (b) EVM versus EIRP plot demonstrating 1) SNR-, 2) test instrument-, and 3) PA linearity-dominated regions. IF: intermediate frequency; UCSD: University of California San Diego; Cal: calibration; SLF: spatial loss factor.
Limited dynamic range in the VSA receiver and embedded signal generator errors can create an EVM comparable with that of the DUT in demodulation-based measurements. As a result, compact antenna test ranges (CATRs) are generally preferable. CATRs utilize a parabolic reflector to transform the spherical plane wave emanating from the feed antenna to a plane wave at the antenna under test (AUT) position. This indirect far-field method results in smaller test ranges with a lower spatial path loss than that of direct far-field ranges, and it generates a cylindrical quiet zone in the AUT vicinity, where the amplitude and phase variation are minimal [2], [31], [32], [33]. For example, at 28 GHz, a quiet zone of 27 cm with a 2-dB amplitude taper can be accommodated within a CATR with a maximum dimension of 2 m, but the same quiet zone requires a far-field test range dimension ${R}\,{\geq}\,{2}{D}^{2} / {\lambda} = {13.6}\,{\text{m}}$ and 26 dB of additional path loss [32]. Consequently, CATRs provide a significant improvement in dynamic range that can be leveraged for more precise EVM measurements, especially for large millimeter-wave beamforming arrays. In [2], ${S}_{21}$ for a 26-GHz massive multiple-input, multiple-output base station array in a CATR was simulated and used to scale each frequency component of a 400-MHz OFDM test signal. The EVM of the CATR link was calculated for the ${S}_{21}$-scaled signal after performing a linear equalization to evaluate the CATR’s wideband performance. The influence of AUT gain variation and reflector edge diffraction on CATR links becomes more significant as the signal bandwidth increases. Simulation tools such as these can help engineers examine tradeoffs in the CATR reflector design and define the bandwidth and AUT beamforming measurement limitations [2]. Another type of indirect far-field measurement system is the plane wave converter (PWC), in which a large array of probe antennas is used to synthesize a spherical quiet zone around the AUT. In [32] and [34], a PWC with 158 wideband Vivaldi antennas was used to measure the EVM of antennas transmitting 100-MHz OFDM signals at 2.4 GHz. At frequencies below 6 GHz, CATR reflectors are prohibitively large and expensive, which makes PWCs an attractive alternative for OTA EVM measurements in these lower frequency bands. However, bandwidth limits imposed by PWC arrays [34] and the large expense of installing a large number of probes [33] at millimeter-wave frequencies means that they are less likely to be used for EVM measurements at these frequencies. CATRs provide a convenient and controlled OTA test environment for instantaneous EVM measurements at millimeter-wave frequencies, and their limitations are primarily based on the reflector design: it must be large enough to synthesize a quiet zone that fully encloses the AUT, and edge treatments must limit quiet zone amplitude and phase ripple across the test signal bandwidth. In [30], an F9650A CATR and a PNA-X network analyzer provided by Keysight Technologies were used to measure the EVM of a 16-QAM test signal transmitted by individual elements of the ${8}\,{\times}\,{8}$ phased array presented in [13]. The measurement system and EVM results as a function of array elevation angle and source power at the 39-GHz carrier frequency are highlighted in Figure 7.
Figure 7. (a) Indirect far-field EVM measurement system with F9650A millimeter-wave CATR and PNA-X network analyzer, (b) ${8}\,{\times}\,{8}$ millimeter-wave phased array [13], (c) CATR reflector and feed-horn enclosure, and (d) EVM results for individual array elements transmitting 400-Mbaud 16-QAM signals from [30]. VSG: vector signal generator.
Planar near-field (PNF) measurement systems are particularly well equipped for determining the far-field radiation characteristics of large apertures and planar phased arrays with broadside radiation, such as silicon millimeter-wave phased arrays built for base stations and satellites [23], [36], [37]. The AUT size is not constrained by the dimensions of a quiet zone as it is with a CATR, and the AUT remains stationary during probe scanning [38]. The smaller test volume reduces spatial path loss as well. This has motivated some researchers to investigate the feasibility of applying near-to-far-field transformation techniques to the analysis of modulated signal quality metrics, such as EVM and BER [32], [39]. The numerous challenges associated with this endeavor have been outlined in the recent literature. First, the PNF-to-far-field transformation process is implemented by calculating a 2D FFT of the near fields sampled at probe positions with a minimum separation of ${\Delta}{x}\,{\leq}\,{\lambda} / {2}$ and ${\Delta}{y}\,{\leq}\,{\lambda} / {2}$ at discrete wavelengths ${\lambda}$ [32], [40]. Because the PNF-to-far-field transformation is predicated on an FFT computation for a plane wave spectrum that is unique for each frequency and PNF distribution, a large number of FFT computations may be required to acquire wideband radiated far fields of large phased arrays transmitting multicarrier modulated signals, such as OFDM signals [31], [41], and additional spatial FFTs would be required for each beam [42]. Probe positioning errors and temperature variations during the PNF scan could reduce the accuracy of far fields derived from PNF-to-far-field transforms as well [31], [39]. For system-embedded millimeter-wave phased arrays that lack ports with direct access to the feed point, a separate stationary probe must be included to retrieve phase during the PNF scan [31], [40]. Finally, SNR differences between PNF probe positions and far-field positions would have to be determined, which is challenging because of the noise figure dependence on PNF position [32].
Despite these challenges, researchers have started to develop and patent novel techniques for estimating the far-field EVM from wideband PNF measurements of modulated test signals, periodically transmitted by the AUT during each near-field probe acquisition [35]. Two representations of the EVM measurement algorithm from [35] are shown in Figures 8 and 9. In Figure 8, near-field sampled signals are downconverted to an intermediate frequency (IF) and digitized, interpolated, and upconverted to acquire digital RF signals for the sampled near fields at each PNF probe position. Propagation to the far-field angle ${\left({\theta},{\phi}\right)}$ is modeled by time-advancing each digitized near-field signal by $\Delta{t}$ in (6), where $\left({{x},{y}}\right)$ is the PNF probe coordinate and ${c}_{0}$ is the speed of light. \[{\Delta}{t} = \frac{\left[{{(}{x}{\cos}\,{\phi} + {y}{\sin}\,{\phi}{)}{\sin}\,{\theta}}\right]}{{c}_{0}}{.} \tag{6} \]
Figure 8. A patented procedure for far-field EVM estimation from wideband near-field-to-far-field transformation [35] based on time-advancing the digitized near-field waveforms using (6). ADC: analog-to-digital converter; DSP: digital signal processor; LPF: low-pass filter.
Figure 9. A patented procedure for far-field EVM estimation from wideband near-field-to-far-field transformation [35] based on Fourier transforming multiple near-field waveform time segments and frequency bins.
The superposition of the time-advanced digital RF waveforms represents the far-field signal, which is downconverted, filtered, and decimated to calculate the far-field EVM. In Figure 9, digitized IF near fields from each near-field probe position are partitioned into multiple smaller time segments that are Fourier transformed to the frequency domain. Multiple spatial Fourier transforms are then performed for a collection of frequency bins spanning the modulated signal bandwidth to acquire wideband far fields for all observation angles ${\left({\theta},{\phi}\right)}$. These are subsequently transformed back to time-domain segments and stitched together to calculate the far-field EVM. The patent holders recommended that the near-field segment time durations T satisfy ${\Delta}{\omega}_{\text{bin}}{T}\,{\geq}\,{50}$ and that the frequency bins ${\Delta}{f}_{\text{bin}}$ satisfy ${\Delta}{f}_{\text{bin}}\,{\leq}\,{0.01}\,{\ast}\,{f}_{c}$ to the nearest power of two to facilitate fast and efficient FFT computations.
Some researchers have even suggested that the EVM measured directly in the near-field region of phased arrays may not differ significantly from the EVM measured in the far-field region [43], [44]. In [43], it was shown that the closer proximity of the receive probe to the phased array surface in near-field measurements yields larger variations in the measured group delay because of larger differences in the element path lengths and phase. Despite this, the change in the measured EVM of a 100-MHz 64-QAM signal transmitted by a 16-element array was demonstrated to be negligible as the probe distance varied from one to ten wavelengths. In [44], EVM simulation of a four-element linear array transmitting 16- and 64-QAM signals showed that changing the distance between the receive probe and array along the array’s normal axis did not change the EVM when the array’s steering matrix was periodically updated from channel estimates. The early stages of research on the adaptation and application of near-field-to-far-field transformation techniques for far-field EVM estimation and studies on the relationship between near-field and far-field EVM demonstrate potential for meaningful EVM analysis from near-field data [45].
This article presents a detailed survey of methods that various researchers have used to simulate and measure the EVM that results from antenna and phased array transmission of IQ-modulated signals. Far-field links between two passive antennas in free space can be modeled as two-port LTI networks with voltage transfer functions ${H}{\left({f}\right)}$. The linear distortion from the UWB amplitude and phase response of ${H}{\left({f}\right)}$ will affect the far-field EVM results if the received signal is not equalized. Phased array models capable of predicting the far-field EVM are more challenging to construct because of the PA nonlinearity, beam squint, and mutual coupling. While some progress has been made in simulating the EVM from these phased array effects, current models lack a comprehensive characterization of the joint response of the embedded elements and PAs, which is required to accurately predict the EVM as a function of carrier power and beam scan angle. More advanced models must be developed to incorporate these effects without degrading computational speed and efficiency. Additionally, OTA measurement techniques must continue to be modernized for more accurate and cost-effective far-field EVM characterization of millimeter-wave links. CATRs are typically used for testing of millimeter-wave phased arrays, but far-field EVM analysis based on near-field measurements is also being explored. In this era of millimeter-wave communication, EVM models and measurement methods must continue to be improved and applied to a link-level analysis of antennas and phased arrays to ensure adequate signal quality and data throughput.
We thank Prof. Gabriel Rebiez and his research group at the University of California San Diego (UCSD) for permitting us to use their 5G phased array [13] and Keysight Technologies of Santa Rosa, CA for permitting us to use their millimeter-wave CATR (Figure 7) and instruments for OTA EVM measurements.
Dustin Brown (dustinbrown@g.ucla.edu) is currently an application development engineer in the Space and Satellite Industry Solutions Division at Keysight Technologies, Santa Rosa, CA 95403 USA. His research interests include phased array design, modeling, simulation, calibration, and measurement, over-the-air modulation distortion analysis, and near-field antenna test ranges. He is a Member of IEEE.
Yahya Rahmat-Samii (rahmat@ee.ucla.edu) is a distinguished professor and holder of the Northrop-Grumman chair in Electromagnetics at the University of California, Los Angeles, Los Angeles, CA 90095 USA and a member of the U.S. National Academy of Engineering. Access his research at http://www.antlab.ee.ucla.edu/. His awards include the 2019 Ellis Island Medal of Honor and 2011 IEEE Electromagnetics Field Award. He is a Life Fellow of IEEE.
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Digital Object Identifier 10.1109/MAP.2023.3272842