Digital Predistortion of 5G Massive MIMO Wireless Transmitters Based on Indirect Identification of Power Amplifier Behavior With OTA Tests

In this article, we present a novel digital predistortion (DPD) architecture for multiple-input–multiple-output (MIMO) transmitters using a real-time single-channel over-the-air (OTA) data acquisition loop. The proposed feedback data acquisition strategy captures OTA signals from a fixed location and indirectly identifies the nonlinear behavior of all power amplifiers (PAs) in the array, as well as their combined signals in the far-field direction. The DPD can, therefore, be effectively constructed without direct measurement at PA output or at user end. The proposed linearization solution can run in real-time and, thus, does not interfere with data transmission in the MIMO transmitters. It can also achieve robust performance when mutual coupling occurs between antenna elements. Simulation and experimental results demonstrate that the proposed scheme can accurately estimate both PA outputs and far-field main beam data. Excellent linearization performance can be achieved with low complexity hardware implementation and reduced computational complexity.


I. INTRODUCTION
T O ACHIEVE higher system capacity, millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO), featuring wide bandwidth and large number of antennas, is expected to be adopted in 5G communication systems [1]. The very short wavelength of mmWave frequencies is beneficial for massive MIMO, as the physical size of antenna can be reduced significantly and considerable performance improvement can be achieved by using largescale antenna arrays. However, similar to lower frequencies, Manuscript  designing power amplifiers (PAs) at such high frequencies also faces severe challenges because it is not only difficult to maintain both linearity and efficiency at mmWave but also the crosstalk and integration issues in large arrays probably make the design more complicated. On one hand, the complex behavior of PAs and crosstalk between radio frequency (RF) chains require more powerful distortion mitigation methods.
On the other hand, the large number of RF chains in the massive MIMO system puts serious constraints on hardware complexity and the related power consumption. Thus, effective yet realistic linearization techniques are desirable. Digital predistortion (DPD) is one of the most popular PA linearization techniques in modern communication systems [2], [3]. Within the context of MIMO DPD, many DPD models have been proposed to tackle the nonlinearity and crosstalk induced by PAs [4]. In general, these behavioral models can cancel MIMO crosstalk by adding specific cross terms into the model structure [5]- [7]. Nonlinearity caused by mutual coupling effect between different antenna elements was also considered in recent works [8]- [10]. The hardware complexity of such systems is usually very high as they use separate DPD models for each PA. They are also only suitable for fully digital beamforming systems where the digital baseband chain is available for each RF chain. In hybrid beamforming, it is not feasible to use separate DPDs for each PA. Some compromising solutions have been proposed. Reference [11] employed one DPD for each subarray, which minimized average error between the input and output signals of all PAs. The concept of beam-oriented linearization was proposed in [12]- [14]. The target of linearization was the main beam over-the-air (OTA) signals rather than the output of each PA. In this case, the farfield main beam signals can be linearized, but distortions in other directions are not well compensated.
To extract DPD coefficients, a feedback data acquisition path is required. In single-input single-output (SISO) communication systems, there is only one PA in each transmitter, and thus, the PA output signal is usually acquired directly from the output of the PA with a coupler. In massive MIMO systems, the number of RF chains can reach several hundred. Using dedicated feedback paths for each PA separately is not feasible because it will dramatically increase the system implementation complexity. Choi and Jeong [15] proposed a DPD method based on combined feedback, which added This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ all the PAs' outputs to form single feedback. A single-PAfeedback DPD method in [16] constructs a DPD model for only one PA and uses this model, as all PAs' DPD models in the array. Reference [13] proposed to capture the output of different PAs in a time-division manner so that all PAs can share one feedback receiver. Reference [17] put forward a DPD scheme linearizing all PAs in the transmitter with received signals measured over the air by a few observation receivers. Despite numerous efforts devoted to this field, some critical architecture issues persist. In particular, the use of bulky couplers and the large number of dedicated feedback paths are becoming serious obstacles in the integration of larger arrays.
Another issue that has long been neglected to date is the real-time requirement of MIMO DPD systems. In real operation, MIMO DPD systems must have the following two features: 1) support continuous data transmission and 2) support beam steering. The first requirement implies that DPD calibration cannot interrupt data transmission, and the transmitters in actual base stations do not send repeated data sequences. Thus, for time-shared feedback strategies, such as [13], if output data of each PA are captured one after another, they must come from different input sequences. Since the captured data blocks from different PAs cannot be synchronized or combined, such algorithms are not suitable for real-time mobile communication systems. The second requirement affects the DPD approaches based on beam-oriented linearization or OTA data acquisition. In [12] and [13], all PAs in the array, together with precoder, beamformer, and channel effect, are modeled by a single behavioral model, making the coefficients dependent on the beam direction. As a result, when the beam steers to a different direction, the DPD coefficients need to change accordingly. Therefore, model extraction should either execute fast enough to track the beam change or precalculate the coefficients at all directions. Some DPD methods based on OTA data acquisition also suffer from similar problems. For example, [18] and [19] directly linearize signals received by an observation antenna. However, signals received from different directions generally differ in nonlinear characteristics; these schemes are, therefore, valid only if the DPD antenna happens to be in the same direction as that of the user end.
In this article, we propose a novel MIMO DPD architecture based on a real-time single-channel OTA feedback scheme that can be used to identify PA behavior without direct measurements at PA outputs. In the proposed scheme, an external observation path is equipped to receive the feedback signal at a fixed location near the main transmitter. Due to user movements, the data blocks received at the DPD antenna carry time-varying phase information so that they can be used to identify behavioral models of PAs and to extract DPD model coefficients. In our architecture, bulky couplers are eliminated from the feedback path, and only one feedback receiver is required for the whole array. Moreover, our proposed method is suitable for real-time operation, as continuous data transmission and beam steering are both supported. Beyond our earlier publication [20], this article presents a comprehensive theoretical analysis of the proposed solution and explains how it can be feasibly implemented in a real system. It also includes simplified forward modeling algorithms, theoretical analysis of mutual coupling effect in the signal reconstruction algorithm, and the application of our method to full-angle MIMO DPD systems [21]. Practical considerations, as well as more extensive simulation and experimental results, are also presented.
The rest of the article is organized as follows. In Section II, the proposed OTA-based indirect PA identification method is described. The construction of DPD is given in Section III. The simulation and experimental results are given in Sections IV and V, respectively, followed by a conclusion in Section VI.

II. INDIRECT IDENTIFICATION OF MIMO PAS
In the existing DPD schemes of MIMO systems, couplers and attenuators are widely used to acquire the output of PAs in MIMO. Considering a large number of PAs in massive MIMO transmitters, it is unrealistic to deploy multiple couplers. To solve these issues, an OTA signal acquisition technique is proposed. As illustrated in Fig. 1, an external antenna, named DPD antenna, is set beside the transmitter antenna array to acquire the transmitted data from a fixed location. In real-time operation, the phase of the signals radiated from the antenna array changes according to the location of the user equipment (UE) and, thus, multiple blocks of data with different phase combinations of the PA outputs can be received by the DPD antenna. These data can then be used to reconstruct the signals from the PA outputs and to calibrate the DPD to linearize the signal at the user end. Compared with the existing methods, this proposed solution avoids the use of couplers or switches in the transmitter, thereby alleviates insertion loss and greatly decreases hardware implementation cost. Please note that the OTA data acquisition architecture itself shown in Fig. 1 is not new, and it has been presented in the literature, e.g., in [17]. However, how to utilize this architecture to reconstruct the PA outputs and calibrate DPD in real-time has not been presented before, which we will discuss in detail in the following.

A. Signal Model of MIMO Array
In massive MIMO systems, different array configurations, e.g., linear array or 2-D array, may be deployed. These arrays may have different radiation patterns, but they all share the same operation principle, namely, by adjusting amplitude and phase of the transmit signals radiated from multiple antennas to form beams pointing to user directions. Since the channel in the space is linear, the signals received at the user end can be considered as linear combinations of the antenna outputs [1]. For simplicity, we consider a uniform linear array (ULA), shown in Fig. 2, as an example in the following analysis. The derivation for other arrays can be conducted in the same way. The ULA consists of N antenna elements with equal spacing d, and each succeeding RF chain has a β progressive phase increase relative to the preceding one, i.e., the phase shifts in RF chains are 0, β, 2β, . . . , (N − 1)β. The channel effects are considered as phase shifts only in the system model and all signals are expressed in complex baseband equivalent format. We also assume that all the RF chains use the same input data, which is a single-user case here. The multiuser case using multiple data streams can be extended later.
Assuming the input signal is x and the transfer function for the nth PA is H n , the output of nth RF chain at the input of antenna can be expressed as The far-field transmitted signal in the direction angle α is where p n (α) represents phase shifts induced by the channel and can be calculated by where θ is the progressive phase shift in the channel. The farfield transmitted signal is maximized when the phase shifts in phase shifter compensate for that caused in antenna array and channels, which means Therefore, ideally, in the main beam direction, we have Correspondingly, when the signal is received from a different direction, e.g., in the angle of α D , which can be assumed to be the direction of the DPD antenna, the received signal is or it can be expressed as where θ D = kd cos α D .

B. PA Output Reconstruction
In real environment, the phase of transmitted signals changes based on the movement of the UE. For instance, if the UE moves from location A to location B, shown in Fig. 3, the progressive phase shift in the phase shifters will change from β A to β B to ensure that the main beam is pointed to the new location. The output of nth RF chain, thus, will change from Since the DPD antenna is located at the fixed location, it will receive different data blocks with different phase shifts during the operation time. When the UE is at location A, the signal received by the DPD antenna is while at location B, the signal is changed to It is shown that y DPD combines information of all PAs, and the combining weights change with the main beam direction. Because the phase shift in the phase shifters can be known, after multiple movements and measurements, it is possible to use the signals received at the DPD antenna to reconstruct the output of each PA in the input of antenna.
Let us assume that the transmitter steers to a different main beam direction M times and M measurements are performed at the DPD antenna. β m is the progressive phase increase between two adjacent antenna elements in the mth measurement. For input data x, the received data blocks at DPD antenna can be represented by where In (10), Y DPD and β are known, and the output of PA, H[x], can be calculated by using least squares (LS)

C. Real-Time Operation With Forward Modeling
The proposed linear inversion in (11) can reconstruct the PA output, but it cannot be directly applied in real-time operation because it assumes that the same input data are transmitted repeatedly. To resolve this issue, in this article, we propose to extract the behavior of PAs with forward modeling. It is generally safe to assume that the PA characteristics do not change suddenly. In other words, within a reasonably short time period, e.g., seconds or minutes, the PA behavior is relatively stable and the PA model coefficients do not change with the input data or the main beam direction. In this case, we can represent the PA behavior using forward models in the data process, and we can then reconstruct the PA outputs and characterize the DPD in real time even if the input data streams are varying.
Let the mth input data be x m . The received signal at DPD antenna is Assume that the nonlinear behavior of each PA can be modeled by a PA behavioral model, the output of PA can be represented by H n [x m ] = X m c n , where X m includes all basis functions built by input signal x m and c n is the coefficients of nth PA. The received signal can be expressed as If we can observe M input data with different β, (13) can be written into matrix format y DPD = X β c PA (14) where Equation (14) can be solved by using LS, and thus, the PA coefficients can be calculated as Once c PA is known, the output of each PA can be reconstructed.

D. Complexity-Reduced Model Identification
Solving the inverse equation is straightforward, but large matrix operations can be costly. An improved algorithm is developed to reduce the computational complexity.
We can rewrite (13) as If we make the following definition we can solve c βm by If we gather M data blocks, (17) can be written in matrix format where The PA coefficients can be calculated as A large-scale LS problem is transformed into a few smaller ones, so the total complexity compared to the previous algorithm is reduced approximately by a factor of N 2 .

E. Mutual Coupling Consideration
One practical problem caused by massive MIMO systems is mutual coupling between elements in the antenna array. In this part, we prove that the proposed MIMO DPD scheme can achieve robust reconstruction performance under the mutual coupling effect.
In transmitters with low isolation, the effects of mutual coupling on transmitted signals change with the beam-steering angle. Assume that the mutual coupling coefficients are 13) can be rewritten with mutual coupling as Accordingly, (17) can be rewritten as Write (22) into matrix format where Gathering h (m) mc calculated from all data blocks, we could stack the vectors to build matrix H mc column by column. Equation (19) can be rewritten as If we still follow the procedures in (20) to estimate the forward model coefficients, the estimated coefficients are We define the deviation from the ideal coefficients as It reveals that is the LS solution to the problem As is proven in the Appendix All off-diagonal elements are zero, and the diagonal elements only depend on the coupling coefficients and the direction of DPD receiver, which are both constant after the deployment. Therefore, compared with c PA , each element of c PA mc is simply scaled by a scalar. As the scaling factor is constant, it can be easily compensated in the calibration process. Therefore, our proposed algorithm can work robustly even under the situations where the coupling matrix is unknown to the system.

F. Other Practical Considerations 1) DPD Observation Receiver:
The deployment of the proposed method requires a proper choice of the location of DPD observation receiver. Though the theoretical derivation assumes the receiver locates in the far field of the antenna array, this requirement can be greatly relaxed in practice. According to [22], the linear superposition rule for calculating antenna pattern is valid as long as the receiving antenna is in the far-field of the antenna element. Thus, the derivation is still valid if DPD antenna is in far field of antenna element but near field of the whole antenna array. These observations suggest that the DPD antenna can be placed very close to the mmWave TX array, greatly reducing the integration cost of the proposed DPD architecture.
Another issue for DPD observation receiver design concerns the beam-steering problem. In 5G massive MIMO transmitters, the direction of transmitted signals is changed with the location of users, and the radiation pattern of TX array will change accordingly. As a result, the received signal power at the DPD receiver varies with main beam directions. Therefore, a large dynamic range is required for the receiver. However, because the DPD antenna is fixed in a known location, preselection and calibration can be conducted to choose suitable angles to receive the signal. To further relax the dynamic range requirement, variable gain amplifier (VGA) for power normalization can be used at the early stage of the DPD receiver. A lookup table (LUT) can be built to record the antenna radiation pattern and the corresponding complexvalued array gain in the direction of DPD antenna. With the assistance of LUT, the array gain will be normalized by the VGA and subsequently restored in the digital domain.
2) Effect of MIMO Channels: The time variation and noise in channels affect the amplitudes and phase of received signals at UEs, as well as that at DPD antenna. However, the proposed DPD method can work well in such noisy communication systems. On one hand, the channel between the transmitter and DPD observation receiver is fixed, so the potential performance degradation can be greatly alleviated by calibration. On the other hand, even in systems with the high noise level, the proposed DPD method can suppress the influence by building forward models with signals from more directions.
In massive MIMO transmitters, the beam update speed is expected to be fast. It may pose severe problems to the existing DPD methods based on the time-division acquisition of PA output. However, in the proposed scheme, the rapid beam changing actually helps collect signals from more directions, which decreases the impact of channel noise and increases the III. DPD CONSTRUCTION In a DPD system, data reconstruction targets can be chosen according to the aim of linearization. In recent works, the main beam and PA outputs are two common linearization targets. In this section, the application of the proposed MIMO DPD method with different linearization targets will be explained.

A. DPD for Beam-Oriented Linearization
The beam-oriented linearization approach [13] treats all the PAs in the array as a single input and a single combined output system. It uses the far-field user-end signal as the target, and thus, only the signal at the main beam direction is linearized. The proposed MIMO DPD method consists of two main elements, far-field estimation and DPD coefficient extraction, as shown in Fig. 4. The output of each PA can be estimated with the input signal and the forward models extracted as discussed in Section II. Far-field received signals in any direction can be estimated based on the output of each PA and channel information. DPD coefficients are identified with the estimated far-field received signals.

1) Far-Field Distortion Prediction:
The nonlinear behavior of PAs has been modeled by behavioral models. Considering channel effect, far-field main beam signal is the linear combination of transmitted signals at antennas. The regression matrix U can be built using PA input signal u and the forward models extracted in Section II. The estimated main beam signal iŝ Apart from the prediction of far-field main beam distortion, signals received at other direction can also be predicted. In the α direction, the estimated far-field received signals arê 2) Linearization of Far-Field Received Signal: To linearize the main beam, the estimated received signalŷ RX is used as the linearization reference. It is also possible to set an artificial target by adding up signals from more than one directions in order to achieve balanced linearization performance across multiple directions.
The remaining procedures of linearization are the same as the conventional DPD architectures. The proposed method works with any existing DPD models, and both direct learning and indirect learning algorithms can be used to extract the model coefficients. For example, in indirect learning, the regression matrix of postinverse model, Y, can be built by feedingŷ RX to the DPD model, and the DPD coefficients can be solved by where u is the output of DPD, and in the next iteration, the DPD output can be updated as

B. DPD for Full-Angle Linearization
In a MIMO system, each PA typically has different nonlinear characteristics. Therefore, it is not possible to linearize all PAs with only one DPD. The beam-oriented DPD linearizes far-field main beam signals with low hardware cost, but it is unable to remove distortions in other spatial directions. For better distortion cancellation capabilities, full-angle linearization scheme [21] can be employed to fulfill the stringent linearity requirement across all spatial directions in 5G mmWave MIMO transmitters.
To ensure linear response in all directions, all PAs in the array must be linearized simultaneously. Traditionally, fullangle linearization is achieved by linearizing each PA separately. Reference [21] proposed a novel two-step DPD scheme that has much lower complexity and enables the potential application to hybrid beamforming systems.
The method works as follows. In the first step, all but the first PAs are tuned by a tuning box to remove the nonlinear behavior variations between different PAs. After forcing all PAs to have the same characteristics, a common DPD is employed to simultaneously linearize all PAs. The tuning boxes can be implemented in either analog or digital domain, making the approach also applicable to hybrid beamforming. 1) PA Reconstruction: Unlike beam-oriented DPD, fullangle linearization scheme produces separate predistorted signals for each PA. In the proposed improved reconstruction algorithm, however, the PAs are assumed to share the same input signal. To solve this conflict, we propose to build the forward models using the original input signal, instead of the actual input signal of PA. Thus, the output of PA is always modeled by Xc n , where matrix X is built by input signal x, rather than predistorted signal u. In this way, the forward model actually characterizes the cascade system of both DPD and PA, and the estimated PA outputs are still accurate. It is worth noting that all procedures in Section II are still valid, and the only difference is that the input signals used to build forward models are changed.
2) Linearization of Each PA: To update DPD coefficients, all PAs' outputs are required. Based on the reconstruction algorithm, the estimated PA output in the nth RF chain is The common DPD block can be extracted using the standard DPD parameter estimation methods, e.g., LS. Similarly, the tuning box can be extracted by where u n is the output of nth tuning box, and Y n is built by first feeding H n [u] to the common DPD model and then generating the required basis functions. Applying the DPD coefficients, the predistorted signal can be updated as

IV. SIMULATION RESULTS
The performance of the proposed MIMO DPD scheme will be analyzed with beam-oriented linearization and fullangle linearization based on 1 × 8 ULA (a subarray with eight RF chains) and 1 × 32 ULA (a subarray with 32 RF chains) with element spacing λ/2 at 28 GHz. To model PAs, memory polynomials (MP) models [23] with different nonlinear characteristics were used. For DPD, magnitudeselective affine (MSA) model [24] with M = 1 and K = 6 was considered. Input waveforms were 20-MHz long-term evolution (LTE) input signals with peak-to-average power ratio (PAPR) of 6.5 dB.
In MATLAB simulation, the main beam was steered to different directions by adjusting the phase shifts of each simulated RF chain. The DPD antenna was fixed at 120°direction. After capturing a number of data blocks, the forward modeling and DPD model extraction were performed. Afterward, the extracted DPD coefficients were applied to the input signal for the next iteration. To simulate the linearization performance, the estimated and actual PA outputs were compared. Similar comparisons were also made on the far-field signals.

1) 1 × 8 ULA:
To simulate the proposed method, the main beam was steered to 35°, 45°, 55°, 65°, 75°, 85°, 95°, and 105°to produce signals for reconstruction. Fig. 5(a) shows the direction of DPD antenna relative to the main beam in each measurement. Fig. 5(c) and (d) draws a comparison between reconstructed and actual outputs of the first and the fourth PAs, respectively. The corresponding NMSE values are summarized in Table I, showing that the reconstructed PA outputs have very good agreement with actual PA outputs. To this step, the proposed DPD method has obtained the same PA output information as the traditional DPD methods where couplers are used. Based on the forward models, farfield signal at any direction can be estimated. Taking the main beam signal as an example, Fig. 5(b) shows that the estimated far-field main beam signals agree with the actual main beam   signals, and NMSE between them is −37.60 dB. Fig. 6(a) presents the linearity of the main beam signal without and with the proposed DPD method. The proposed DPD method improves the linearity in the desired direction and achieves −52.15/−49.48 dBc ACPR. The detailed linearization performance is summarized in Table II. 2) 1 × 32 ULA: For 1 × 32 ULA, MATLAB simulation results also show very good performance in the reconstruction of PA outputs and far-field main beam signals, as well as the linearization of main beam signals.
Main beam directions used for reconstruction were uniformly distributed within the range from 35°to 160°. Fig. 7(c) and (d) compares the reconstructed and the actual outputs of the two PAs in the array. NMSE values between    Table I. Fig. 7(b) shows that the estimated main beam signals agree with the actual main beam signals and NMSE between them reaches −36.60 dBc. Fig. 8(a) presents the linearity of the main beam signal without and with the proposed DPD method. The proposed DPD method improves the linearity in the desired direction and achieves −50.96/−49.09 dBc ACPR. More details are listed in Table II.

B. Performance of Proposed Method in Full-Angle DPD
A 1 × 8 ULA was simulated to verify the proposed method in full-angle DPD. Radiation pattern and signals  used for reconstruction were the same as that in simulation for the beam-oriented DPD method shown in Fig. 5(a). PA reconstruction procedures in full-angle DPD were also the same as that in beam-oriented DPD. Fig. 9(a) and (b) show good linearity of PA output with the proposed full-angle DPD method. Performance of the main beam signal is confirmed in Fig. 9(c) and (d). The achieved NMSE and ACPR values are presented in Table III.

C. Effect of Mutual Coupling
Effect of mutual coupling between antenna elements was considered in the MATLAB simulation. In the simulation, the coupling effect was set as a constant but unknown factor. A matrix identifying mutual coupling coefficients was multiplied with the PA output signals before transmitted to UE. In the simulation, only the coupling from one adjacent antenna element from each side was considered, and the coupling coefficient was set as 0.5, which means that half of the signals from the adjacent RF chain is coupled.
As listed in Table IV, even with such high coupling factors, the mutual coupling has little influence on the reconstructed PA outputs and far-field main beam signals, compared with the results of ULA 1 × 32 in Table I.

D. Effect of MIMO Channel Noise
The effect of massive MIMO channel condition on the proposed DPD method was analyzed based on the array with 1 × 32 ULA. To model the influence of noise in massive MIMO channel, a random variation was added to the phase and amplitude of received signals at user end and DPD antenna, and the SNR was set as −60 dB. The simulation results in Table IV show that the noise does not affect the performance of the proposed method.

E. Effect of Proposed DPD on Other Directions
In this part, we analyze how the in-band power and out-ofband (OOB) emissions in all different spatial directions behave after applying the proposed DPD. Figs. 10-12 show that the OOB emissions of massive MIMO transmitters generally follow the beam pattern of the array before DPD. OOB emissions are more powerful in the direction of the intended receiver but weaker in other directions.

1) In Beam-Oriented DPD:
After applying the beamoriented DPD method, the OOB emissions are attenuated significantly in the main beam direction as they are set as the linearization targets, as shown in Figs. 10 and 11. Besides, the simulation results also show that the beam-oriented DPD scheme does not fully remove distortions in other directions.  2) In Full-Angle DPD: Fig. 12 shows that the OOB emissions are attenuated in all spatial directions after applying the full-angle DPD method. Compared with beam-oriented DPD, full-angle DPD achieves better overall linearization performance.

V. EXPERIMENTAL RESULTS
To verify the proposed idea, several experiments have been carried out, and the results are presented in this section.

A. Experimental Setup
Due to limited hardware resource available, we only conducted tests on 1×2 ULA and 1×4 ULA cases. The test bench was set up as shown in Fig. 13. For setup of a 1 ×2 ULA, two baseband input signals with bandwidth of 20 MHz and PAPR of 5.5 dB were generated by the software MATLAB in PC and downloaded to the two signal channels provided by a dualchannel signal generator (R&S SMW200A). The two channels can be phase-shifted separately to realize the beamforming operation. In 1×4 ULA test, because of instrument limitations, all RF chains share one baseband channel, i.e., the baseband signal generated by the signal generator was split into four ways by a one-to-four power divider. The same test signal as 1 × 2 ULA tests was used.
In both cases, the baseband signals were upconverted to IF @ 5.5 GHz and fed into the designed RF front end. In this module, all signals were again upconverted to 27 GHz by 10.75-GHz local oscillator (LO) signals generated by a signal generator (Keysight E8267D) with a power divider and fed into corresponding PAs with the average output power of around 14 dBm. PAs not in use were turned off to avoid crosstalk. Next, the Tx chain outputs were fed into the antenna elements to form the desired radiation pattern. In the receiver side, one horn antenna was employed for OTA test and a spectrum analyzer (Keysight N9030A) was utilized to capture the OTA outputs. Both the outputs and the inputs were sent back to the PC for DPD procedures.

B. System Calibration
Hardware impairments that exist in all physical implementations of wireless systems and especially in massive MIMO systems may, in practice, lead to severe performance losses [25]. Therefore, calibration procedures were employed.
The calibration process consisted of two stages, namely, delay calibration and phase calibration. The first stage was intended to cancel the delay mismatch between different channels of the signal generator, while the second stage aimed to align the phase between different RF chains.
In the delay calibration stage, uncorrelated signals were sent to the transmitter. The received OTA signals were time-aligned with the input signals one by one to determine the delay of each baseband channel. After obtaining the delay values, the time delay mismatch can be compensated by delaying the transmitted signals in MATLAB before sending them to the instrument.
In the phase calibration stage, the same data were fed to the RF chains. After obtaining the OTA signal, the output of each PA was individually captured over the air. As time delay has already been calibrated in the previous stage, signals received at OTA antenna and output of each PA can be used to calculate gain and phase difference across different RF chains by LS. Accordingly, gain and phase differences across RF chains can be compensated by either inversely scaling the corresponding transmitted baseband signals or preprocessing the received OTA data before reconstruction algorithm.

C. Measurement Results
In the experimental tests of 1 × 2 ULA, the main beam was steered to different directions by adjusting the phase shifts of the input signal chains. The DPD antenna was fixed at a specific location while receiving signals of different main beam directions. After capturing a number of data blocks, the forward modeling and DPD model extraction were performed. To verify linearization performance, the main beam direction was set to point to the DPD antenna so that the main beam signal can be acquired and evaluated without moving the DPD antenna.
For 1 × 4 ULA, however, due to the lack of mmWave phase shifters, the main beam was fixed at 90°, and beam steering was not available. As all RF chains shared one baseband channel, only beam-oriented DPD was evaluated. To validate the proposed reconstruction algorithm, the location of the receiver was moved to different locations to capture data with different phase information. The receiver was finally moved back to 90°to verify the performance in the main beam direction.  In both cases, the PA output was reconstructed using the proposed algorithm. Based on the estimated PA output, farfield signals at specified direction can be estimated with channel information, as summarized in Table V. Then, they were used to extract DPD coefficients. Afterwards, the extracted coefficients were applied to the input signal for the next iteration.
Individual PA outputs were also captured. Baseband signals were sent by single RF chain separately, and the RF chains were switched on one at a time. Time delay, phase shifts, and power loss due to channel effect can be removed by synchronization and normalization. Note that the captured individual PA outputs were only used for comparison purpose, and they were not used in reconstruction or DPD.
1) Beam-Oriented DPD for 1 × 2 ULA: To build forward models, the main beam is steered to 50°, 80°, 100°, and 130°. The spectral results are demonstrated in Fig. 14(c) and (d), showing good agreement between the reconstructed and actual outputs of each PA. The NMSE of them were −36.93 and −40.04 dB, respectively. As shown in Fig. 14(b), estimated far-field main beam signal also agrees with actual far-field signals with NMSE reaching −40.54 dB.
The performance of the proposed DPD method is illustrated in Fig. 15. It is shown that the proposed MIMO DPD method achieves an ACPR of −51.27/−50.09 dBc. Detailed performance metrics are listed in Table VI. 2) Full-Angle DPD for 1 × 2 ULA: OTA data with the same main beam directions as previous tests were collected. The PA linearization results of the full-angle DPD method are illustrated in Fig. 16(a) and (b) and Table VII. It is shown that DPD successfully linearized both PAs in the array and achieves an ACPR of −49.30/−47.77 and −47.93/−46.83 dBc, respectively. The far-field main beam signal is also linearized, as depicted in Fig. 16(a), with ACPR of −51.71/−50.22 dBc.
3) Beam-Oriented DPD for 1× 4 ULA: The spectral results of PA reconstruction are demonstrated in Fig. 17

VI. CONCLUSION
In this article, a novel DPD architecture with real-time OTA data acquisition for massive MIMO transmitter has been proposed. A single OTA receiver is employed to capture data for indirect PA identification by taking advantage of the beam-steering scenario. According to the simulation and experimental validation, the proposed method can accurately predict the PA output signals and efficiently linearize the mmWave MIMO transmitters with low complexity, proving itself as a viable solution to the linearization of 5G massive MIMO transmitters.
It is worth noting that although only a single user linear array case was demonstrated in the article and a small number of RF chains were used in the experimental test due to limitation of available facility in the lab, the principle of the proposed approach is generally extendable and it can be easily applied to other types of arrays and more complicated multiuser MIMO communication systems. The proposed DPD method is solely based on the assumption that the far-field signals are the linear combinations of the transmitted signals from the RF chains, which is valid in not only linear arrays but also the other types of arrays, such as 2-D arrays. For instance, moving from linear array to 2-D array, we just need to change the radiation angle α in (2) to 2-D. The generalization to multiuser MIMO systems is also possible. For example, in the popular hybrid MIMO system where each user transmits the signal via a separate subarray, the proposed method can be directly applied by processing each subarray one by one. Furthermore, for proof of concept, in this article, some practical effects, e.g., the physical configuration of the antennas and attenuation of the channel, are omitted. Although the actual linearization performance may vary, these effects do not impact the operation principle of the proposed approach.

APPENDIX DERIVATION OF E
The expectation of can be derived as follows. is the solution to (26). Each row of can be solved independently, so without loss of generality, we solve the kth row of , i.e., k . The linear system to be solved becomes Since only one row in has nonzero correlation with H k , E k will only have one nonzero element.
As All off-diagonal elements are zero, and the diagonal elements only depend on the coupling coefficients and the direction of DPD receiver.