Dustin Brown, Yahya Rahmat-Samii
IMAGE LICENSED BY INGRAM PUBLISHING
Millimeter-wave communication systems will rely on the large-scale deployment of wideband digital modulation schemes and highly integrated phased arrays to significantly expand the number of simultaneous connections and enable multigigabit per second wireless links. The quality of a digitally modulated signal transmitted and received by millimeter-wave phased arrays depends not only on radiated power and directivity, but also the wideband frequency response of phased array elements and linearity of embedded radio-frequency (RF) transceivers. Error vector magnitude (EVM) is a measure of the amplitude and phase distortion in digitally modulated signals that accounts for all of these effects, and it is increasingly used to characterize the system-level performance of over-the-air (OTA) links between antennas and phased arrays. This article examines the merits of adopting EVM as a performance standard for antennas and phased arrays by comparing it with other commonly applied OTA metrics and reviewing EVM measurement methods and concepts applied to time and frequency domain analysis. The sequel to this article [1] will provide a comprehensive survey of the various techniques applied by representative researchers to model, simulate, and measure EVM for OTA links between antennas and phased arrays, and reflect on open challenges for further research.
Improving the performance of mobile wireless communication systems has required us to transition from a congested spectrum at microwave carrier frequencies (300 MHz to 3 GHz) to higher millimeter-wave carrier frequencies (30 GHz to 300 GHz), where tens of gigahertz of additional spectrum are available to support a larger number of connections and enable multigigabit per second data rates at low latencies on the order of a few milliseconds [2], [3], [4]. These advances in wireless network capacity are essential for accommodating the rapidly expanding number of Internet-connected devices around the world and for realizing a wide variety of new applications, such as intelligent transportation systems and smart factories. This transition to a millimeter-wave spectrum has already been initiated through the adoption of 28- and 39-GHz bands for 5G new radio (NR), and it will likely continue on with 6G, as researchers investigate the feasibility of ultrahigh data rate wireless systems at terahertz frequency bands [5], [6], [7].
One of the primary challenges of establishing reliable wireless communication links at millimeter-wave carrier frequencies is the significantly higher signal attenuation at these frequencies, which results from the increased absorption by atmospheric gasses, such as oxygen, and the scattering from precipitation, vegetation, and other objects in the propagation environment. Millimeter-wave wireless links will depend on widespread deployment of high-gain phased arrays that dynamically form directive beams to compensate for this higher OTA path loss [3], [8], [9]. Phased array design engineers can leverage the down-scaled size of radiating elements at millimeter-wave frequencies and state-of-the-art integrated circuit (IC) packaging technologies to create printed circuit board-integrated architectures with digitally controlled amplitude and phase states for thousands of printed antennas [10], [11], [12], [13], [14], [15], [16].
While this emerging class of millimeter-wave phased arrays is becoming an essential component of modern terrestrial and satellite communication systems, it is also challenging the conventional methods of antenna simulation and measurement, in which the radiation and impedance-matching characteristics of passive antennas have been evaluated in isolation from the RF transceivers with which they are later integrated. The reduced size and spacing between the radiating elements and lack of direct access to the RFIC ports precludes installation of RF connectors and consequently the independent measurement of antenna radiation and RF performance characteristics, such as power amplifier linearity or intermodulation distortion [5], [7], [17], [18], [19], [20]. This has increased interest in the adoption of new performance metrics, which can account for the combined response of the transceiver components and radiating elements to the applied voltage signal as a function of the carrier power and beam-scan angle [21]. One such metric that has been applied in performance characterization of transmitting millimeter-wave phased arrays is EVM, a measure of IQ-modulated signal quality that is calculated as the root-mean-square (RMS) value of the vector difference between the transmitted and received symbols in the complex plane. In OTA measurements of these active phased arrays, EVM represents net amplitude and phase distortion from the ICs and antenna elements [21], [22]. It is therefore a function of the signal modulation characteristics and the linear and nonlinear distortion properties of the phased arrays.
Previously published works have provided insightful surveys about indoor and outdoor millimeter-wave channel models [23], [24], [25], millimeter-wave active phased array design topologies [10], [11], [12], [13], [14], [15], [16], and beamforming algorithms [26], [27], [28]. This article is differentiated by its unique focus on EVM as a measure of antenna or phased array influence on far field signal quality, and is organized as follows: The “Link-Level Performance of Millimeter-Wave Phased Arrays” section introduces EVM as a modulated signal quality metric, which is valuable for characterizing the beamforming performance of millimeter-wave phased arrays. The “Overview of EVM Measurement: Signal Modulation, Impairment, and Equalization” section describes the signal generation, distortion, and equalization processes that precede the EVM calculation. The “EVM Characterization: Demodulation and Spectral Correlation” section describes two methods of evaluating EVM through signal processing in the time and frequency domain. The “Conclusions” section concludes the article and introduces related EVM topics to be addressed in [1].
Highly integrated phased arrays will play an essential role in maximizing the reliable range and the data-throughput rates of millimeter-wave communication systems, in part by maximizing radiated power in the direction of intended receivers while simultaneously minimizing it in other directions. However, due to the presence of nonlinear distortion from active transceiver components, such as power amplifiers, they must also minimize this distortion to prevent degradation of received signal quality. Figure 1 shows three broad categories of metrics for phased array performance: radiated power and polarization, beam-scanning speed and versatility, and received signal quality.
Figure 1. Three categories of phased array performance metrics: (a) radiated power, beam pattern, and polarization; (b) angular beam-scan range, loss, and execution speed; and (c) received signal quality. Examples of the (a) co-pol and cross-pol radiation patterns, (b) beam-scan loss, and (c) EVM versus data rate (MBaud) of the 1,024-element Ka-band active phased array from [29] are shown. ACPR: adjacent channel power ratio; BER: bit error rate; EIRP: effective isotropic radiated power; NPR: noise power ratio; SNR: signal-to-noise ratio.
When radiating elements are integrated with beamforming ICs in active phased arrays, radiated power depends on the spatial power combining of array elements and RF power supplied to each element. Effective isotropic radiated power (EIRP) accounts for both, as shown in (1) from [21]: \[{\text{EIRP}} = {P}_{t} + {G}_{t} + {10}\,{\ast}\,{\log}_{10}{(}{N}{)}{-}{L} \tag{1} \] where ${P}_{t}$ is the transmit power per array element, ${G}_{t}$ is the total array directivity (dBi), N is the number of array elements, and L is the loss measured between the power amplifier output and the radiating surface. Increasing EIRP results in higher signal-to-noise ratio (SNR) at a polarization-matched receiver in the far field, resulting in higher data rates over a larger antenna link range. The independent amplitude and phase control of each element or subarray is leveraged to maximize angular beam-scan range and minimize beam-scan loss, both of which can be defined from the EIRP reduction [21]. Phased array engineers must exercise caution and ensure that element spacing and mutual coupling mitigation measures are adequate to prevent grating lobes and scan blindness [30], [31].
While EIRP is a helpful metric for estimating the SNR of a far field antenna link and has been used to characterize the bandwidth and beam-scan range of phased arrays, it has a limited utility for estimating modulated signal quality and data-throughput rate at the receiver. This is because the data-throughput rate depends not only on the SNR but also the wideband signal distortion that is generated from far field transmission of the phased array and not subsequently compensated via equalization. Link-level metrics, such as EVM and bit error rate (BER), which are more directly correlated with the data-throughput rate, supplement traditional antenna metrics, which only account for the radiated power and field polarization properties [21], [32]. EVM is a comprehensive measure of IQ-modulated signal quality that quantifies amplitude and phase distortion from a device under test (DUT). A block diagram of the EVM measurement process is shown in Figure 2. EVM is calculated as the RMS value of the difference between input signal ${x}{(}{t}_{n}{)}$ and demodulated output signal ${y}{(}{t}_{n}{)}$ equalized by the filter impulse response ${e}{(}{t}_{n}{)}$, as highlighted in (2) from [33], where N is the number of data symbols, ${t}_{n}$ are the symbol time instances, and ${\circledast}$ represents convolution: \[{\text{EVM}} = \frac{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{\left\vert{x}{(}{t}_{n}{)}{-}{y}{(}{t}_{n}{)}\,{\circledast}\,{e}{(}{t}_{n}{)}\right\vert}^{2}}}{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{\left\vert{x}{(}{t}_{n}{)}\right\vert}^{2}}}. \tag{2} \]
Figure 2. Block diagram of EVM measurement system. In this single-carrier representation, a quadrature amplitude modulation-modulated (QAM) baseband signal ${x}{(}{t}{)}$ is generated, upconverted, transmitted through a generic DUT, and demodulated to obtain a received signal y(t) with frequency spectrum Y(f), which represents the sum of a linear component H(f)X(f), nonlinear distortion D(f), and noise N(f).
In general, the number of symbols N is a large multiple of signal modulation order M [e.g., ${M} = {16}$ for a 16-quadrature amplitude modulation (QAM) input signal], so normalization in (2) becomes that of the average reference signal power [34]. However, normalization can alternatively be based on the peak reference power [35]. Unlike EVM, BER only permits a simplified pass/fail test of the modulated signal quality [34], [36]. Observation of the error vectors in the complex IQ plane as shown in Figure 2 yields greater insight on the nature of signal impairments caused by the DUT [36], [37]. Power amplifier intermodulation distortion introduces harmonics within the test signal bandwidth that cannot be removed via equalization, so power amplifier nonlinearity typically dominates EVM degradation from active phased arrays at high carrier power levels [32], [38]. However, because the beamforming IC ports are not accessible for conducted testing in highly integrated millimeter-wave phased arrays, EVM must be measured OTA. Figure 1(c) shows the EVM from a 1,024-element phased array [29], plotted as a function of symbol rate for three digital modulation formats. The EVM versus beam-scan range has been identified as a sensible figure of merit for millimeter-wave active arrays that can provide a linearity constraint for setting maximum EIRP [21]. Furthermore, other link-level metrics, such as SNR and BER, can be mathematically derived from EVM [39], [40], [41], [42], [43], and EVM is equivalent to the reciprocal of the noise power ratio (NPR) when specific signal formatting and test conditions are implemented [44], [45]. Measurement noise ${N}{\left({f}\right)}$ and in-band signal distortion ${D}{\left({f}\right)}$ both contribute to EVM, which makes it an insightful performance metric for modern millimeter-wave active phased arrays.
The following section reviews the three stages of the EVM measurement process that are shown in Figure 2 and their influence on EVM. These stages are: 1) generation of a modulated test signal, 2) transmission and distortion of the signal by a DUT, and 3) demodulation and equalization of the measured signal ${y}{(}{t}{)}$. These stages precede EVM calculation using ${x}{(}{t}{)}$, ${y}{(}{t}{)}$, and ${e}{(}{t}{)}$ in (2).
The conventional EVM measurement process first requires the generation of a modulated reference signal ${x}{(}{t}{)}$ that the measured signal ${y}{(}{t}{)}$ is compared with, where the modulation format is selected to maximize data-throughput rate based on channel conditions and the constraints of the communication system protocol. The QAM format is commonly used in digital communication systems. A QAM voltage signal ${v}{(}{t}{)}$ has two components, in-phase ${v}_{i}{(}{t}{)}$ and quadrature ${v}_{q}{(}{t}{)}$, which are modulated by orthogonal carriers as shown in (3) from [35]: \[{v}{(}{t}{)} = {v}_{i}{(}{t}{)}\,{\cos}\,{(}{2}{\pi}{f}_{c}{t}{)} + {v}_{q}{(}{t}{)}\,{\sin}\,{(}{2}{\pi}{f}_{c}{t}{)}. \tag{3} \]
A digital bit sequence is mapped to a discrete set of amplitude states for ${v}_{i}{(}{t}{)}$ and ${v}_{q}{(}{t}{)}$ that will correspond to unique amplitude and phase states of ${v}{(}{t}{)}$. These represent a collection of symbols in the complex IQ plane as shown in Figure 2. The number of unique symbols is modulation order ${M} = {2}^{{N}_{b}}$, where ${N}_{b}$ is the number of bits per symbol. For example, a 16-QAM signal has ${M} = {16}$ symbols, and each symbol corresponds to a unique sequence of ${N}_{b} = {4}$ bits. Allowing ${v}_{i}{(}{t}{)}$ and ${v}_{q}{(}{t}{)}$ to take the form of rectangular pulses, with amplitude states held constant over each symbol period, would result in sharp edges and large bandwidths. Pulse-shaping filters, such as the root-raised cosine, are generally used to reduce bandwidth and adjacent channel power. The impulse response of a root-raised cosine filter has an amplitude of zero at integer multiples of the symbol period, eliminating intersymbol interference (ISI) [9], [37]. Maximum achievable data rate ${R}_{\max}$ is limited by bandwidth B and SNR in accordance with Shannon’s capacity theorem (4) from [9]: \[{R}_{\max} = {B}\,{\log}_{2}{(}{1} + {\text{SNR}}{)}. \tag{4} \]
Increasing bandwidth allows a modulated test signal to make more-refined amplitude and phase transitions at a fixed symbol rate, and enables the use of higher-ordered modulation formats, which maximize the data-throughput rate and spectral efficiency. The maximum permissible modulation order is limited by the SNR, however, because decreasing amplitude and phase differences between the unique symbols increases the probability of error in the demodulated symbols. As a result, the Third Generation Partnership Project (3GPP) has standardized EVM limits for 5G NR based on modulation order [46]. When the received symbols are only corrupted by additive white Gaussian noise, and the number of measured symbols N is very large, the EVM and can be approximated from the SNR as shown in (5) [39], [40], [41], [42]: \[{\text{EVM}}\,{\approx}\,{\sqrt{\frac{1}{\text{SNR}}}}{.} \tag{5} \]
Transmitting a QAM signal with hundreds of megahertz or gigahertz of bandwidth on a single carrier ${f}_{c}$ is not practical, however, because frequency-selective fading from multipath in millimeter-wave channels results in nonuniform amplification of the frequency components and causes ISI that cannot be compensated without complex and expensive equalizers. Instead, multicarrier modulation schemes, such as orthogonal-frequency division multiplexing (OFDM), are applied to transmit parallel streams of QAM symbols on multiple orthogonal subcarriers, each occupying a small fraction of the complete OFDM signal bandwidth. Each of the subcarriers experiences flat-fading with a uniform frequency amplification, which can be compensated with a simple, single-tap equalizer [47]. This resilience against multipath fading and simplified equalization has led to the widespread adoption of OFDM for wideband digital systems [47], [48]. However, OFDM signals also have undesirable properties, such as their large peak-to-average power ratios (PAPR) [47], [48], [49]. Power amplifiers transmitting OFDM signals must have a large dynamic range or operate at a large backoff to limit nonlinear distortion due to this high PAPR. Operating at large backoff reduces power-added efficiency and EIRP, but may be necessary to lower EVM and ensure adequate signal quality [15], [21], [49].
The amplitude and phase of modulated reference symbols ${x}{(}{t}_{n}{)}$ differ from those of the measured symbols ${y}{(}{t}_{n}{)}$ as a result of measurement system noise and various signal impairments. Six types of impairments in communication systems are described in [36] and listed in Table 1, each of which can be categorized as either a linear or nonlinear form of distortion. Because passive antennas behave as linear and time-invariant systems, their wideband response to an applied voltage signal ${x}{(}{t}{)}$ with frequency spectrum ${X}{\left({f}\right)}$ can be modeled as a transfer function ${H}{\left({f}\right)}$ or impulse response ${h}{(}{t}{)}$ [50], [51], [52], [53]. The far field spectrum ${Y}{\left({f}\right)}$ of a transmitting antenna can be represented as shown for the linear DUT in Figure 2: ${Y}{\left({f}\right)} = {H}{\left({f}\right)}{X}{\left({f}\right)}$, where ${H}{\left({f}\right)}$ is the antenna’s normalized vector effective length. Antennas generate linear distortion when transfer function amplitude ${\left\vert{H}{\left({f}\right)}\right\vert}$ and group delay ${t}_{d}$ vary over the channel bandwidth. Nonlinear devices, such as power amplifiers, can generate an additional distortion component ${D}{\left({f}\right)}$ which, like measurement noise ${N}{\left({f}\right)}$, is not linearly correlated with input signal spectrum ${X}{\left({f}\right)}$ [33], [37]. In the absence of external sources of interference, ${D}{\left({f}\right)}$ represents the intermodulation products of ${X}{\left({f}\right)}$ due to nonlinear amplification or mixing in transceiver circuits, as indicated in Table 1. The absorber materials lining the interior of OTA antenna test chambers are designed to minimize reflections, such that multipath effects are negligible. However, in indoor and outdoor propagation environments, reflections and diffractions from electrically large objects cause multiple copies ${Y}_{n}{\left({f}\right)}$ of the transmitted signal to take different physical paths from the transmitter to the receiver, each with its own complex frequency response. The linear distortion from passive antenna transmission and multipath can generally be removed with an equalizer before calculating EVM. Thus, limiting nonlinear distortion is of primary concern.
Table 1. Types of Signal Impairments in Communication Systems [36].
An equalization filter ${E}{\left({f}\right)}\,{\approx}\,{1} / {H}{\left({f}\right)}$ is obtained by estimating the linear channel response ${H}{\left({f}\right)}$ and then nullifying it in either the time domain or frequency domain. Time-domain equalizers, such as the minimum mean-square error and decision feedback equalizers, are implemented in digital signal processing with finite impulse response filters, which consist of a series of time-delay units and complex weighting coefficients, which are periodically updated in response to the dynamic channel conditions [9], [48], [54]. Time-domain equalizers face challenges meeting the demands of wideband millimeter-wave communication systems. In multipath channels having a high delay spread, a large number of channel taps are required to eliminate ISI, which increases the need for memory and processing power [9]. Furthermore, the analog to digital converter sampling rate must increase proportionally with signal data rate, and time delays and multiplications must be executed faster [48].
In frequency domain equalizers, the channel response estimate is generally obtained with the aid of a cyclic prefix, symbols repeated at the beginning and end of each data sequence, and intermittent transmission of pilot subcarriers. Two commonly applied methods are OFDM and single-carrier frequency domain equalization [9], [48]. In general, channel equalization is performed with a receiver that has no awareness of applied input signal ${x}{(}{t}{)}$. Such is the case for the single-channel vector signal analyzers (VSAs) that are used to demodulate the measured signal in some EVM measurements, as described in the “Demodulation” subsection and shown in Figure 3. However, EVM measurements can alternatively be performed with a multichannel vector network analyzer that measures input and output signals simultaneously and leverages awareness of the input signal to perform frequency domain equalization, as described in the “Spectral Correlation” subsection and shown in Figure 4. An ideal equalizer would remove the linear distortion from channel response ${H}{\left({f}\right)}$ entirely, leaving only noise ${N}{\left({f}\right)}$ and nonlinear distortion ${D}{\left({f}\right)}$. If the signal is not equalized, the amplitude and delay distortion shown in Table 1 (with constants K and ${t}_{d}$) remains and results in higher EVM.
Figure 3. EVM far field measurement system based on the demodulation method and (2). An arbitrary waveform generator generates a baseband IQ signal, which is upconverted via the vector signal generator and applied to the antenna under test (AUT) port. A VSA is used to demodulate and equalize the signal y(t) measured by the probe before computing EVM.
Figure 4. EVM far field measurement system based on the spectral correlation method and (6). A VNA is used to synchronously measure the AUT input signal spectrum X(f) and probe output signal spectrum Y(f), compute spectral correlation components, and perform equalization before computing EVM. DFT: discrete Fourier transform.
While EVM has traditionally been determined from the symbol errors measured in the time domain, as shown in (2), it can also be determined from modulation distortion quantities calculated in the frequency domain under certain testing conditions. This section describes the distinctions between these two methods of EVM measurement, demodulation and spectral correlation, including their signal requirements, test configurations, and performance tradeoffs.
The EVM of a DUT representing a two-port network is traditionally defined as a time domain measurement and computed as the RMS value of error vectors representing the difference between complex DUT output signal ${y}{(}{t}{)}$ and input signal ${x}{(}{t}{)}$ at N symbol instances, as shown in (2). It is expressed as either a percentage or ratio on a decibel scale. An example of an error vector within the constellation diagram of a 16-QAM signal is shown in Figure 2. A VSA is typically used to demodulate the measured output signal ${y}{(}{t}{)}$, as shown in Figure 3 [34], [35], [36], [37]. For the case of a single-carrier QAM test signal, the I and Q amplitude pulses corresponding to data symbols are first filtered to reduce the signal bandwidth, resulting in ${x}_{i}{(}{t}{)}$ and ${x}_{q}(\text{t})$ that only pass through the right symbol amplitude states at specific time instances ${t}_{n}$. An IQ modulator generates passband signal ${x}{(}{t}{)}$, which passes through a DUT that introduces noise and distortion, and an IQ demodulator obtains baseband signals ${y}_{i}{(}{t}{)}$ and ${y}_{q}{(}{t}{)}$. These signals are sampled at the symbol rate, filtered to eliminate ISI and out-of-band signal distortion, and normalized by either their peak or RMS amplitudes to compensate for the linear channel gain. Because it lacks direct access to the DUT input signal, the VSA estimates the applied input signal ${x}{(}{t}{)}$ from measured output signal ${y}{(}{t}{)}$, such that the reference symbols ${x}{(}{t}_{n}{)}$ become the ideal ${M}{-}{\text{QAM}}$ symbols nearest to the demodulated symbols ${y}{(}{t}_{n}{)}$ [32], [36]. Time-alignment of input and output signals is performed by a cross-correlation [37], and a uniform phase shift is typically applied to ${y}{(}{t}{)}$ to derotate the constellation [34]. Further corrections, such as compensating for IQ imbalance and frequency offset, can also be implemented [37].
After time-alignment and equalization, EVM is calculated as shown in (2). Thus far, the EVM measurements of millimeter-wave phased arrays reported in the literature have been based on digital demodulation with VSAs, as shown in Figure 3, but demodulation of the wideband modulated signals deployed in millimeter-wave systems is difficult to achieve because digitizing such signals and computing fast Fourier transforms for frequency domain analysis requires significant memory and processing power [33], [34], [38]. Increasing signal bandwidth also raises the noise floor and the minimum distortion EVM component that can be measured [55]. Because the ideal reference signal generated from by the VSA is only an estimate of the DUT input signal ${x}{(}{t}{)}$, vector errors resulting from the signal generator and DUT distortions cannot be distinguished. Therefore, EVM measurements with a VSA should not be considered an accurate characterization of DUT distortion unless the signal generator errors are negligible. Finally, if signal distortion is high enough to yield error vectors spanning multiple symbols, the registered EVM will be erroneously low due to reference estimation from nearest ideal symbols [33], [37]. Consequently, researchers have explored alternative methods of measuring or estimating EVM within the frequency domain.
A method of estimating EVM based on frequency domain analysis of DUT input and output signal spectra has recently been proposed and implemented in VNA-based measurements of nonlinear devices, including power amplifiers [32], [33], mixers [57], and phased arrays [58]. A modulated test signal with specific properties is required to facilitate EVM estimation directly in the frequency domain: The test signal ${x}{(}{t}{)}$ must be periodic and measured over an integer multiple of the period. It must also satisfy the Nyquist sampling condition, such that ${x}{(}{t}{)}$ has a band-limited frequency spectrum ${X}{\left({f}\right)}$, which can be represented as a discrete Fourier series. These conditions enable the application of Parseval’s power theorem to formulate a frequency domain EVM (6) that is equivalent to the time domain (2), where ${E}{(}{f}_{n}{)}$ is the equalization filter [33], [37]: \[{\text{EVM}} = \frac{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{\left\vert{X}{\left({f}_{n}\right)}{-}{E}{\left({f}_{n}\right)}{Y}{\left({f}_{n}\right)}\right\vert}^{2}}}{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{\left\vert{X}{\left({f}_{n}\right)}\right\vert}^{2}}}. \tag{6} \]
The summation of (6) is for N tones spanning the bandwidth of the input signal spectrum ${X}{\left({f}\right)}$, rather than for N symbol times in (2). Using a multichannel VNA to capture both the DUT input signal spectrum ${X}{\left({f}\right)}$ and output signal spectrum ${Y}{\left({f}\right)}$ simultaneously removes the burden of accurate time alignment and resampling that exists for the demodulation process. Figure 4 shows a far field EVM measurement system based on spectral correlation with a VNA. Because the VNA spectrum analyzer measures the signal applied to the DUT ${x}{(}{t}{)}$ directly on one of its receivers rather than estimating the ideal reference symbols from the DUT output signal ${y}{(}{t}{)}$, signal generator errors will not be embedded in the EVM. This, combined with the application of VNA calibration techniques that eliminate RF cables and connector errors, leads to more accurate characterization of the DUT distortion [32], [33], [36], [37]. Moreover, VNAs can acquire the frequency spectrum of wideband signals over multiple coherent acquisitions with a narrowband filter, as long as the test signal period is known, as demonstrated in [59]. The VNA measurement noise floor can therefore be lowered by either reducing the filter bandwidth or applying vector averaging techniques at the expense of longer frequency sweeps [33], [37].
An equalization filter ${E}{\left({f}\right)}\,{\approx}\,{1} / {H}{\left({f}\right)}$ compensating for DUT linear distortion is acquired by decomposing output signal spectrum ${Y}{\left({f}\right)}$ into a part ${H}{\left({f}\right)}$ that is linearly correlated with the input signal spectrum ${X}{\left({f}\right)}$ and a nonlinear part ${D}{\left({f}\right)}$, as shown in Figure 5. In this model, ${H}{\left({f}\right)}$ is the best linear approximation (BLA) of the DUT frequency response and ${D}{\left({f}\right)}$ is an independently distributed random noise source with zero mean and standard deviation ${\sigma}_{D}$ that represents the DUT nonlinear stochastic distortion. A separate nonlinear component ${N}{\left({f}\right)}$ can also be included to represent the additive measurement noise as shown in Figure 2 [39], [60]. The partitioning of linear and nonlinear frequency components is achieved by calculating the following spectral correlation quantities from [33]: ${S}_{XY}$, the cross-spectral density of ${X}{\left({f}\right)}$ and ${Y}{\left({f}\right)}$; ${S}_{XX}$, the power spectral density (PSD) of input signal spectrum ${X}{\left({f}\right)}$; and ${S}_{YY}$, the PSD of output signal spectrum ${Y}{\left({f}\right)}$. Then ${H}{\left({f}\right)} = {S}_{XY}{\left({f}\right)} / {S}_{XX}{\left({f}\right)}$ is the BLA of the DUT frequency response and ${D}{\left({f}\right)} = {Y}{\left({f}\right)}{-}{H}{\left({f}\right)}{X}{\left({f}\right)}$ is the nonlinear distortion spectrum with PSD ${S}_{DD}{\left({f}\right)} = {S}_{YY}{\left({f}\right)}{-}{\left\vert{S}_{XY}{\left({f}\right)}\right\vert}^{2} {\big{/}} {S}_{XX}{\left({f}\right)}$. The EVM is calculated as shown in (7) [38], which is equivalent to (6) with an equalization filter ${E}{\left({f}\right)} = {1} / {H}{\left({f}\right)}$: \[{\text{EVM}} = \frac{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{{S}_{DD}}{\left({f}_{n}\right)}}}{\sqrt{\mathop{\sum}\limits_{{n} = {1}}\limits^{N}{{\left\vert{{H}{\left({f}_{n}\right)}}\right\vert}^{2}}{S}_{XX}{\left({f}_{n}\right)}}}{.} \tag{7} \]
Figure 5. Decomposition of output signal spectrum Y(f) from a power amplifier into linear component H(f)X(f) and nonlinear component D(f) using a PNA-X vector network analyzer [56].
When the modulation format does not require equalization, an unequalized version of EVM can be obtained by letting ${E}{\left({f}\right)} = {Ge}^{{-}{j}{2}{\pi}{f}_{n}{t}_{d}}$, where G is the reciprocal of the average linear gain, and ${t}_{d}$ is the group delay [33]. A BLA model with linear gain ${H}{\left({f}\right)}$, nonlinearity variance ${\sigma}_{D}^{2}{\left({f}\right)}$, and noise variance ${\sigma}_{N}^{2}{\left({f}\right)}$ provides an accurate characterization of the DUT frequency response in the mean-squared sense for test signals with the same PSD and probability density function (PDF) [33], [60], [61]. Periodic, random-phase signals having a sufficient number of tones can thus be designed with a PSD and PDF matching those of a specific modulation format, such as OFDM [55]. This enables standard-independent EVM measurement and the supplementation of compact test signals, which can be generated by extracting a small segment of a longer periodic signal that has fewer tones but approximately the same PSD and PDF, as shown in Figure 6. Compact signals have already been used to emulate the PSD and PDF of OFDM signals in power amplifier EVM simulations and measurements [33], [62]. Spectral correlation therefore presents an attractive alternative to the conventional demodulation method for wideband millimeter-wave measurements.
Figure 6. Compact test signal generation with a PNA-X vector network analyzer. The CTS (blue) is extracted from a longer original test signal (yellow) and filtered to eliminate spectral leakage. The PSD and CCDF of the original test signal and CTS are shown to be approximately the same [56].
This article outlines the advantages of adopting EVM, a measure of digital signal quality comparing the symbols of the input and output signals of a two-port network, as a beamforming performance metric for millimeter-wave phased arrays. A systematic and comprehensive review of fundamental EVM measurement concepts was also presented, including a detailed discussion of a three-stage EVM measurement algorithm and two distinct methodologies for determining the EVM from a DUT. Spectral correlation measurement with a multichannel VNA was identified as a more robust means of characterizing the EVM from DUT distortion than time-domain demodulation measurement with a VSA, since signal generator errors are removed and dynamic range is improved for the testing of wideband signals. In a future issue of IEEE Antennas and Propagation Magazine, [1] will review research contributions related to the modeling and measurement of modulated signal transmission by antennas, power amplifiers, and phased arrays for subsequent EVM analysis. It will also analyze several concrete examples and highlight research opportunities in this fast-growing field of signal-sensitive antenna simulations and 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.2023.3272838