Digital Predistortion Combined with Iterative Method for MIMO Transmitters

Abstract

In this article, an auxiliary module is proposed to mitigate the crosstalk effect in multiple input–multiple output (MIMO) transmitters. Other branches do not need considering when linearizing the local branch, which reduces the complexity of the auxiliary module, and some branches can be linearized selectively to make the module application more flexible. In the end, the feasibility of the proposed algorithm is verified through the experimental platform. Experimental results demonstrate that for a 2 × 2 MIMO transmitter with −20 dB nonlinear crosstalk, the proposed DPD improves the adjacent channel power ratio (ACPR) more than 16 dB, and normalized mean square error (NMSE) more than 12.8 dB, which has similar performance to the PH model. The predistorter can be implemented with fewer coefficients by separating the crosstalk from the nonlinearity, enabling the technique to be used in 5G massive MIMO scenarios.

1. Introduction

The goal of the fifth-generation (5G) wireless communication system is to provide higher data rates, system capacity, and lower power consumption to an ever-increasing number of mobile users over a limited radio frequency (RF) spectrum [1]. More complex signal types and larger bandwidths make power amplifiers (PAs) show more nonlinear and memory effects during operation, which deteriorates the communication quality. Therefore, linearization techniques such as digital predistortion (DPD) are utilized to increase the efficiency of PAs. The DPD module is one of the most basic building blocks in wireless communication systems. The efficiency of PAs can be greatly improved by reducing the distortion in the nonlinear region.

DPD applied in the MIMO system is different from the linearization of a single PA. To start with, we need to explain the advantages of the DPD technology. Wireless base stations that do not use crest factor reduction (CFR) or DPD algorithms are generally less efficient and therefore have higher operating and capital equipment costs. Using CFR and DPD algorithms, efficiencies can be greatly improved, resulting in significant reductions in capital expenditures and operating expenses for network operators.

In this article, crosstalk is mainly processed by the iterative method, and the single input–single output (SISO) DPD can be used in MIMO transmitters. This makes the DPD in MIMO transmitters easy to be implemented. In the simulation part, we use the Parallel Hammerstein (PH) [2] and modified canonical piecewise-linear function-based (m-CPWL) model [3] to model and design DPD devices, respectively. Experimental results show that the proposed method can compensate for the nonlinear distortion of PAs effectively and linearize the output signal of 2 × 2 MIMO transmitters with less complexity.

This article is structured as follows: Section 2 introduces general DPD concepts used in MIMO transmitters. Section 3 details the implementation of the proposed method. The simulation result is shown in Section 4. In Section 5, experiments are conducted to verify the compensation effect of the DPD scheme. Section 6 concludes the article.

2. DPD Technique Designed for MIMO Transmitters

The crosstalk effect which can be considered to aggravate the nonlinear of PAs worsens the application of SISO DPD in MIMO systems. It means that the application of linearization technology cannot simply imitate the situation of a single PA.

Due to the existence of the nonlinear crosstalk, the input signal is the addition of the signal of the local branch and the coupling signal of other branches. Considering a 2 × 2 MIMO transmitter:

$$g_1\left(x_1+\alpha \times x_2\right)=y_1,$$

(1)

$$g_2\left(\beta \times x_1+x_2\right)=y_2,$$

(2)

where x₁ and x₂ are the source signals, α and β are the coupling factors, and g₁(.) and g₂(.) are the behavioral modeling function of a single PA. It is observed that y₁ and y₂ are simultaneously affected by all input branches. Therefore, different behavior models are needed to characterize the potential relationship between the output and the input of each branch.

$$g_1\left(x_1+\alpha \times x_2\right)\to f_1\left(x_1,x_2\right),$$

(3)

$$g_2\left(\beta \times x_1+x_2\right)\to f_2\left(x_1,x_2\right).$$

(4)

PH models are known to be very effective in stimulating MIMO systems. However, the higher accuracy of the model is at the cost of high complexity. That is, the PH model requires more coefficients and basis functions than other behavior models such as the crossover memory polynomial model [4] or the augmented crossover memory polynomial model [5]. In 5G large-scale antenna arrays, the increase in antennas will not significantly increase the number of data links in the case that multiple load antennas share a set of data flow. In addition, the horizontal attenuation of the crosstalk becomes larger after adjacent branches [6]. This makes it possible to use the PH model and its extended form to represent the input and output of large-scale antenna arrays.

For 2 × 2 MIMO transmitter systems, the PH model has the following form:

$$\begin{array}{c}{y}_i^{\left(\mathrm{PH}\right)}(n)={\displaystyle \sum _{m=0}^M{\displaystyle \sum _{k_2=0}^{K-1}{\displaystyle \sum _{k_1=0}^{k_2}{b}^{(i)}{}_{m,k_2,k_1,1}{x}_1(n-m){\left|x_1(n-m)\right|}^{k_2-k_1}{\left|x_2(n-m)\right|}^{k_1}}}}\hfill \\ +{\displaystyle \sum _{m=0}^M{\displaystyle \sum _{k_2=0}^{K-1}{\displaystyle \sum _{k_1=0}^{k_2}{b}^{(i)}{}_{m,k_2,k_1,2}{x}_2(n-m){\left|x_2(n-m)\right|}^{k_2-k_1}{\left|x_1(n-m)\right|}^{k_1}}}}\hfill \end{array}$$

(5)

where x₁(n) and x₂(n) are baseband modulated input signals of the ith(i = 1, 2) transmitter’s path. M is the memory depth. K is the nonlinearity order and b⁽ⁱ⁾_m,k2,k1 are the complex coefficients. When the number of input links increases, only the influence of two adjacent branch signals on the local output needs to be considered because the signal from the further branch coupled to the local branch is negligible. The signal coupling from a branch other than the adjacent branch to the considered branch is very small, so even if the factors are not considered, the modeling error will not increase significantly. The advantage of doing so is that it will significantly reduce the complexity of the model and provides convenience for the design of the DPD model.

The 5G communication system has a higher degree of integration, and there may be hundreds of antennas used for transmission and reception. Figure 1 and Figure 2 simply represent a traditional MIMO transmitter and a mMIMO transmitter. The paper [7] emphasizes the importance of the crosstalk occurring after PAs and designs elaborate experiments. In [7], the two-input model is proposed based on ignoring the crosstalk that occurs before PAs. Then, the experimental verification of a four-element transmitter is carried out. It can be seen that the output of a branch is only related to the two input signals of this branch PA. Therefore, even if the number of data links increases significantly, the complexity of the system model is linearly related to the number of input links. The crossover digital predistorter (CO-DPD) in [4] is a DPD scheme applied in MIMO transmitters in the early stage, and its predistorter is based on Volterra polynomials. The decomposed vector rotation (DVR) model [8] makes us more abundant in the choice of behavior models for the PA modeling. In addition, model itself has its own characteristics. Compared with the traditional Volterra model and its simplified forms, the DVR model uses absolute value operations instead of high-order power to characterize the nonlinearity of PAs. The idea of segmentation is also reflected in the DVR model.

Based on the CPWL model proposed in [8,9], the dual-input CPWL model [3] is proposed. Taking the 2 × 2 MIMO transmitter as an example, it has the following form:

$$\begin{array}{c}{y}_i^{\left(CPWL\right)}\left(n\right)=f_0^{\left(i\right)}\left(n\right)+f_{21}^{\left(i\right)}\left(n\right)+f_{22}^{\left(i\right)}\left(n\right)+f_{23}^{\left(i\right)}\left(n\right)+f_{24}^{\left(i\right)}\left(n\right)\hfill \\ +F_0^{\left(i\right)}\left(n\right)+F_{21}^{\left(i\right)}\left(n\right)+F_{22}^{\left(i\right)}\left(n\right)+F_{23}^{\left(i\right)}\left(n\right)+F_{24}^{\left(i\right)}\left(n\right)+F_{25}^{\left(i\right)}\left(n\right)\hfill \end{array}$$

(6)

$$\begin{array}{l}{f}_{{}_0}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{m=0}^M{a}_{m,1}{x}_i\left(n-m\right)}\\ f_{{}_{21}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,21}\left|x_i\left(n-m\right)-{\beta }_k\right|}}{x}_i\left(n-m\right)\left|x_i\left(n\right)\right|\\ f_{{}_{22}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,22}\left|x_i\left(n-m\right)-{\beta }_k\right|}}{x}_i\left(n\right)\\ f_{{}_{23}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,23}\left|x_i\left(n-m\right)-{\beta }_k\right|}}{x}_i\left(n-m\right)\\ f_{{}_{24}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,24}\left|x_i\left(n\right)-{\beta }_k\right|}}{x}_i\left(n-m\right)\end{array},$$

(7)

$$\begin{array}{l}{F}_{{}_0}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{m=0}^M{a}_{m,2}{x}_i^{\text{'}}\left(n-m\right)}\\ F_{{}_{21}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,31}\left|x_i^{\text{'}}\left(n-m\right)-{\beta }_k\right|}}{x}_i^{\text{'}}\left(n-m\right)\left|x_i\left(n\right)\right|\\ F_{{}_{22}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,32}\left|x_i^{\text{'}}\left(n-m\right)-{\beta }_k\right|}}{x}_i\left(n\right)\\ F_{{}_{23}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,33}\left|x_i\left(n-m\right)-{\beta }_k\right|}}{x}_i^{\text{'}}\left(n-m\right)\\ F_{{}_{24}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,34}\left|x_i\left(n\right)-{\beta }_k\right|}}{x}_i^{\text{'}}\left(n-m\right)\\ F_{{}_{25}}^{\left(i\right)}\left(n\right)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^M{a}_{km,35}\left|x_i\left(n\right)-{\beta }_k\right|}}{x}_i^2\left(n\right)x_i^{\prime \ast }\left(n-m\right)\end{array},$$

(8)

where x_i(n) is the baseband modulated input signal of the ith(i = 1, 2) transmitter’s path, $x_i^{\prime }\left(n\right)$ is the incident signal due to antenna crosstalk and mismatch, M is the memory depth, K is the nonlinearity order, and a_m,1, a_m,2, a_km,* are the complex coefficients.

The complexity and power consumption are two issues to be considered in the practical DPD model. It is pointed out in [10] that when the number of digital links in the transmitting branch increases, the energy consumed by CO-DPD, a relatively simple solution to MIMO DPD, exceeds the sum of the energy consumed by all PAs on transmitting and receiving branches, making the application of the DPD method lose the advantage of the low energy consumption compared with the linearization of a single PA.

To enable the SISO method applied to the linearization of MIMO transmitters, we propose our own scheme. By considering the crosstalk and nonlinearity separately, the crosstalk of each branch can be decoupled. The SISO DPD scheme with lower complexity and better robustness can improve the quality of output signals. The proposed scheme has lower complexity, although the linearization result is slightly inferior to the PH DPD.

3. Proposed Method

3.1. Steps of the Implementation

The coupling and mismatch as a kind of linear crosstalk can be compensated at the receiver side of the MIMO link [11]. Therefore, in this article, we mainly consider the influence of the nonlinear crosstalk in MIMO transmitters and propose an interesting solution. By adopting an auxiliary DPD module to the SISO DPD scheme, the crosstalk between different branches can be “decoupled”, and the problem of MIMO DPD can be transformed into the problem of the single PA linearization. Since the single PA linearization technology is very mature, our method can be used in conjunction with many cutting-edge DPD techniques [12,13,14], which provides a new idea for MIMO DPD.

In this section, a 2 × 2 MIMO transmitter is taken as an example for theoretical derivation, and branch 2 is taken as an example to illustrate the steps of the proposed method; $y_2^{\prime }={\left[y_2^{\prime }\left(1\right) y_2^{\prime }\left(2\right)\dots y_2^{\prime }\left(N\right)\right]}^{\mathrm{T}}$ is obtained without the input from other branches, which means

$$f_2\left(x_1,x_2\right){|}_{x_1=0}=y_2^{\prime }.$$

(9)

It should be noted that f₂(x₁,x₂) used here represents the behavior model of the MIMO transmitter, such as the configuration for the PH model and m-CPWL model mentioned in Section 2. We store the signal $y_2^{\prime }$ in LUT1 to help us produce the injection signal x_2,inject, which is produced according to Equation (16) in the iteration phase, and then store x_2,inject in the LUT2. Finally, the SISO DPD scheme is used to improve the performance of DPD by involving the injection signal to reduce the influence of crosstalk. It should be pointed out that changes in other branch signals will affect the compensation effect of injected signals on crosstalk. Therefore, injected signals need to be re-iterated after the generation of DPD signals. The detailed iteration process can be seen in Algorithm 1.

$$f_2\left(x_{1,dpd}+x_{1,inject},x_{2,dpd}+x_{2,inject}\right)=y_{2,dpd}.$$

(10)

It can be seen that all modules can be identified by using lookup tables and the least square estimation in experimental steps. Figure 3 shows the block diagram of the proposed DPD solution, and the algorithm is formally defined and summarized in Figure 4. It should be pointed out that the signals of the auxiliary module and training predistorter are only related to the local branch, so we can selectively linearize the signal of the specific branch. This shows the flexibility of the proposed approach.

Algorithm 1 Algorithm of Proposed Method

1: Obtain the injection signal x1,inject, and x2,inject according to (16). 2: Use SISO method to linearize each branch, respectively. 3: Use the predistorted signal obtained in step 2 to modify the injection signal x1,inject, and x2,inject in step 1. 4: Update the predistorted signal using the method in step 2. 5: Obtain the optimal signal of each branch.

3.2. Conditions for Convergence

Via the steps in Section 3.1, we take branch 2 as an example to theoretically explain how the injected signal reduces the nonlinear crosstalk.

Assuming the ideal signal x_2opt has been found to eliminate the influence of the crosstalk effect, and considering that x_2opt is a signal of finite length and power,

$$x_{2opt}=x_2+x_{2,inject},$$

(11)

$${‖x_{2opt}‖}_{\infty }=B.$$

(12)

In (12), $\Vert {\Vert }_{\infty }$ denotes the supremum norm [15] and B is a constant representing the maximum value of the ideal signal

$$f_2\left(x_1,x_{2opt}\right)=y_2^{\prime }.$$

(13)

It is easily obtained by using (13):

$$\mu x_{2opt}+f_2\left(x_1,x_{2opt}\right)=\mu x_{2opt}+y_2^{\prime }.$$

(14)

Equation 14 is transformed to obtain Equation (15):

$$x_{2opt}=x_{2opt}+\gamma \left(y_2^{\prime }-f_2\left(x_1,x_{2opt}\right)\right),$$

(15)

where γ = 1/μ, and finally we obtain the iterative equation:

$$x_2^{\left(r+1\right)}=x_2^{\left(r\right)}+\gamma \left(y_2^{\prime }-f_2\left(x_1,x_2^{\left(r\right)}\right)\right).$$

(16)

We define the right-hand side of the nonlinear Equation (15) as a function h(·) given as

$$h\left(x_2\right)=x_2+\gamma \left(y_2^{\prime }-f_2\left(x_1,x_2\right)\right).$$

(17)

Then, (15) is rewritten as

$$x_{2opt}=h\left(x_{2opt}\right).$$

(18)

If we prove that h(x₂) is Lipschitz continuous and λ < 1, we can prove the existence of x_2opt and that Equation (15) holds. We now need to prove Equation (19):

$$\left|h\left(x_2\left(a\right)\right)-h\left(x_2\left(b\right)\right)\right|\le \lambda \left|x_2\left(a\right)-x_2\left(b\right)\right|,\lambda \in \left(0,1\right)$$

(19)

where x₂(a) and x₂(b) are the inputs of two arbitrary moments in x₂.

Substituting Equation (17) into the left side of Equation (19) yields:

$$\begin{array}{c}\left|h\left(x_2\left(a\right)\right)-h\left(x_2\left(b\right)\right)\right|=\left|x_2\left(a\right)-x_2\left(b\right)+\gamma \left(f_2\left(x_1,x_2\left(b\right)\right)-f_2\left(x_1,x_2\left(a\right)\right)\right)\right|\hfill \\ =\left|x_2\left(a\right)-x_2\left(b\right)-\gamma \left(f_2\left(x_1,x_2\left(a\right)\right)-f_2\left(x_1,x_2\left(b\right)\right)\right)\right|\hfill \end{array}$$

(20)

From the mean value theorem, it follows that [15,16]

$$f_2\left(x_1,x_2\left(a\right)\right)-f_2\left(x_1,x_2\left(b\right)\right)=f_2^{\prime }\left(x_1,\xi \right)\left(x_2\left(a\right)-x_2\left(b\right)\right),$$

(21)

where $f_2^{\prime }\left(x_1,x_2\right)$ is the derivative of $f_2\left(x_1,x_2\right)$ and ξ is a moment between a and b.

Substituting (21) into (20), the transformation obtains

$$\left|h\left(x_2\left(a\right)\right)-h\left(x_2\left(b\right)\right)\right|=\left|\left(1-\gamma f_2^{\prime }\left(x_1,\xi \right)\right)\left(x_2\left(a\right)-x_2\left(b\right)\right)\right|.$$

(22)

Due to (21), (22), and (12), λ in (19) is indeed non-negative and finite, which completes the proof.

4. Simulation Results

In this section, we verify the feasibility of the algorithm through the simulation analysis. To briefly illustrate our model, we performed a simulation test on one branch and used the PH model with K = 5, M = 2 and the m-CPWL model with K = 5, M = 2 to model the 2 × 2 MIMO transmitter. The signal x_i,inject was obtained after three iterations. The SISO DPD uses the MP model [17] with K = 7, M = 3, and spectrum characteristics are analyzed. The normalized spectrum curve is shown in Figure 4, and the MP model can be written as the following:

$$y(n)={\displaystyle \sum _{k=1}^K{\displaystyle \sum _{m=0}^{M-1}{a}_{km}x(n-m){\left|x(n-m)\right|}^{k-1}}},$$

(23)

where x(n) and y(n) represent the baseband input and output signal of PAs, respectively. K is the nonlinear order. M is the memory depth and a_km are the model coefficients.

Figure 4 shows a good suppression on the out-of-band distortion of signals, providing an extra solution for crosstalk suppression in MIMO systems. The simulation result also shows that for the modeling of the 2 × 2 MIMO transmitter, the proposed method has a better out-of-band inhibition in the PH model than the m-CPWL model.

5. Experimental Verification

5.1. The Experimental Setup

Figure 5 shows the block diagram of the platform setup when nonlinear crosstalk is introduced, and the block diagram of the experimental platform when linear crosstalk is introduced is shown in Figure 6. Agilent MXG N5182A generated a 20 MHz OFDM signal with a PAPR of 7 dB as the test signal in branch 1. Agilent MXG E4438C generated a 20 MHz OFDM signal with a PAPR of 7 dB as the test signal in branch 2. Each OFDM signal had 180,000 data samples with a sampling rate of 122.88 MHz. Two channel signals used in the test were uncorrelated. The coupling factor was set to −20 dB in our experiments. In the experiment of nonlinear crosstalk, we set couplers [18] at PA input. For the experiment of linear crosstalk, we set couplers at the output end of PAs. The PA models used in each branch were ZHL-5W-63-S+ and ZHL-16W-43-S+. The PA output signal was fed into a 40 dB attenuator, and the R&S FSW43 Signal and Spectrum Analyzer was used to receive and sample the output with a sampling rate of 122.88 MHz. The sampled output was then sent back to the personal PC for processing. For different predistortion models, experimental verification was performed by setting different order K and memory depth M. In case 1, we set K = 5, M = 2, and we set K = 7, M = 3 in case 2. The nonlinear crosstalk was introduced in the above two cases. In case 3, linear crosstalk was introduced, and we set K = 5, M = 2. The output power of branch 1 and branch 2 were set to 31.0 dBm and 35.3 dBm in all cases. ACPR and NMSE indicators were used to evaluate the performance of different DPD methods. ACPR is a metric of out-of-band linearization performance. It is the ratio between the power of adjacent channels (left or right or the average of both) and the power of the main channel in the frequency domain [19]. It is expressed as the following:

$$\mathrm{ACPR}=10{\log }_{10}\left(\frac{{\displaystyle \underset{f_c-\frac{B}{2}\pm \mathrm{offset}}{\overset{f_c+\frac{B}{2}\pm \mathrm{offset}}{\int }}P(f)df}}{{\displaystyle \underset{f_c-\frac{B}{2}}{\overset{f_c+\frac{B}{2}}{\int }}P(f)df}}\right),$$

(24)

where f_c represents the carrier frequency (2.3 GHz). B represents the bandwidth (20 MHz), offset represents the frequency offset (20 MHz), and P(f) represents the power spectral density (PSD). NMSE is a metric used to measure the in-band linearization performance of DPD [19]. For linearization, NMSE is calculated in the time domain between the received and transmitted signals. It is expressed as the following:

$$\mathrm{NMSE}=10{\log }_{10}\left(\frac{{\displaystyle \sum _{n=1}^N{\left|y\left(n\right)-x\left(n\right)\right|}^2}}{{\displaystyle \sum _{n=1}^N{\left|x\left(n\right)\right|}^2}}\right),$$

(25)

where y(n) is the measured output and x(n) is the source signal.

5.2. Results Analysis

To facilitate the comparison, we only make a comparison with the PH model in the actual test to reflect advantages of the proposed algorithm.

Figure 7 shows changes in the out-of-band distortion of the output signal of various DPD models. Compared with the PH model, it can be observed that our scheme has fewer coefficients and less flexibility under the condition that the linearization effect is similar.

Table 1. DPD results for different models in 2 × 1 MIMO transmitters for LTE 20 MHz signal in case 1 (K = 5, M = 2).

DPD Technique	Channel 1		Channel 2
	ACPR, dBc	NMSE, dB	ACPR, dBc	NMSE, dB
None	−38.99/−38.74	−16.2872	−32.14/−32.33	−16.5668
PH DPD	−52.53/−56.85	−28.1770	−53.01/−56.58	−31.7422
Proposed DPD	−56.73/−57.13	−32.8119	−51.95/−52.65	−30.7402

,

Table 2. DPD results for different models in 2 × 1 MIMO transmitters for LTE 20 MHz signal in case 2 (K = 7, M = 3).

DPD Technique	Channel 1		Channel 2
	ACPR, dBc	NMSE, dB	ACPR, dBc	NMSE, dB
None	−38.99/−38.74	−16.2872	−32.14/−32.33	−16.5668
PH DPD	−52.28/−53.35	−29.0262	−53.68/−54.57	−29.8215
Proposed DPD	−55.14/−54.73	−33.7672	−51.10/−52.05	−29.3378

and

Table 3. DPD results for different models in 2 × 1 MIMO transmitters for LTE 20 MHz signal in case 3 (K = 5 M = 2).

DPD Technique	Channel 1		Channel 2
	ACPR, dBc	NMSE, dB	ACPR, dBc	NMSE, dB
None	−36.42/−34.54	−21.2948	−30.83/−30.49	−13.9403
PH DPD	−51.36/−50.88	−33.6664	−52.35/−52.27	−37.1289
Proposed DPD	−49.00/−49.02	−33.4946	−50.26/−49.77	−37.0343

summarize clearer comparative data.

In both cases where only nonlinear crosstalk is introduced, the proposed model has almost the same NMSE and ACPR with fewer coefficients than the PH model. In the first case, the coefficient required by the PH model is 2 × 60, while the proposed method only needs 2 × 10. The coefficient required in the predistortion signal generation stage is reduced by 83%, and with the increasing of K and M, the advantage of our method is more obvious. However, in the case where only linear crosstalk is introduced, the performance of our method deteriorates. We will investigate that in our future work.

6. Conclusions

Another way to study DPD in MIMO systems is to deal with the crosstalk and PA nonlinearity, respectively, while previous works mainly dealt with the crosstalk effect jointly with the nonlinearity and memory effect. The behavioral model achieves the purpose of solving them together through an indirect learning structure. Therefore, the direct impact is that with the increasing of input branches, the corresponding behavioral model may involve more coefficients and basis functions to meet the needs of model accuracy. At the same time, as the crosstalk level increases, the compensation effect of some DPD models is extremely limited.

In this article, SISO DPD combined with the iterative method was proposed. In addition, the complexity of the method was mainly reflected in the running stage of the DPD model. The flexibility of the proposed approach was also indicated by selectively linearizing the signals of a particular branch. It can be seen that the complexity is linearly related to the number of input links, proving itself as a promising solution for the linearization of 5G mMIMO transmitters.