Robust Modeling and Linearization of MIMO RF Power Amplifiers for 4 G and 5 G Applications

In this paper, a novel set of orthogonal crossover polynomials for the baseband modelling and linearization of MIMO RF Pas is presented. The proposed solution is applicable to WCDMA and LTE applications. The new modelling approach has considerably reduced the numerical instability problem associated with the conventional polynomial model identification. In order to validate the efficiency and the robustness of the proposed solution, a 2x2 MIMO LDMOS RF power amplifier has been measured modelled and linearized. A comparison with the conventional polynomial MIMO models showed clearly the superiority of the proposed solution in a fixed-point calculation environment such as DSP and FPGA boards. Keywords—MIMO transmitter; RF power amplifiers; orthogonal polynomials; nonlinear transmitters; digital predistortion


INTRODUCTION
Power amplifiers (PAs) are the major source of nonlinearity in communication system that causes spectral regrowth as well as in-band distortion.Accurate modeling of the RF PAs is required which increases the problem size and reduces the numerical stability of the model identification procedure.This problem is more pronounced in multiple input systems.Multiple-input multiple-output (MIMO) transceivers allow high service quality and increase the capacity range of wireless transmission requiring very high-speed data transfer [1]- [3].In fact, the capacity of MIMO transceivers M times the capacity of a single-input-single-output (SISO) equivalent system.Volterra series were developed in [4]- [7] and were intensively used in modelling power amplifiers and DPD.However, the Volterra series involves a great number of coefficients, which increases the complexity of the problem in the case of MIMO systems.
In the literature, several attempts reduce the number of coefficients for PA models.In [8] a dynamic deviation reduction model has been suggested for modelling single input PAs.In [9]- [13], reduced polynomial models have been proposed for modelling MIMO PAs.Saffar et al. [14], addressed the joint mitigation of I/Q modulator impairment and PA nonlinearity in MIMO transmitters through an optimized memory polynomial model.However, all these published models suffer from the increasing numerical instability as the problem size increases.This generally happens when a high nonlinearity order, a high memory depth, or a big number of inputs were considered.
Radio frequency (RF) PAs presents a challenge to the transceiver designers.In fact, designers need to boost two contradictory parameters of the PA such as power efficiency and linearity.Several linearization techniques are proposed in the literature to improve PA"s linearity.Due to its simplicity and efficiency, the digital pre-distortion (DPD) is considered as the most popular linearization technique.Hence, the DPD approach has been intensively used to compensate for the transmitter nonlinearities as in [15]- [17].However, the DPD technique requires an accurate modelling of the PA.In fact, high polynomial order leads generally to an accurate model with high complexity.This generally increases the numerical instability and vulnerability of the model identification procedure [18], [19].Therefore, a tradeoff is always required between the model accuracy and complexity.A good metric for measuring the model accuracy is the time domain normalized mean squared error (NMSE) while the DPD performance can be measured using the frequency domain adjacent channel power ratio (ACPR) of the compensated system [20], [21].However, the model identification procedure is based on the inversion of an observation matrix that has to be well conditioned in order to avoid numerical instability.The condition number is a metric to test the conditioning of this matrix by measuring the linear dependence of its columns [18], and is generally related to the distribution of the input signal envelope, as well as the dimensionality of the problem.In MIMO PAs, the problem size increases drastically with the number of input signals, making the instability issue more pronounced.This is mainly due to the high correlation that can exist between the input data resulting in ill-conditioned observation matrix with a high condition number.In this work, the numerical instability of the model identification procedure is addressed through the development of a complex multi-input orthogonal polynomial model.
The organization of the paper is as follows.In Section.2, we introduce the reference crossover conventional polynomials www.ijacsa.thesai.orgfor modelling the MIMO PAs.In Section 3, we present the derived novel robust orthogonal polynomial model.The experimental validation and performance assessment of the proposed model is given in Section 4. Finally, the conclusion is drawn in Section 5.

II. MODELLING MIMO SYSTEMS USING CONVENTIONAL POLYNOMIALS
In this section, the crossover polynomial model [13] is detailed, and its limitations are discussed for a 2×2 MIMO PA case.The conventional polynomial model is a base band model that has been developed to characterizing nonlinear radio frequency power amplifiers with and without memory effects.The closed form expression that relates the input and output complex envelope signals of the PA is given in [13] such that where j x and i The inversion of the observation matrix H M ΦΦ    is often imperfectly conditioned.Thus, the inversion of such matrix will undergo numerical errors.This problem is more pronounced and leads to erroneous results when the finite precision calculation is used.Define the condition number of a matrix M as:  This implies that, in practice, the model can be unstable.Therefore, reduction of the condition number is highly required to ensure the numerical stability of the model coefficient identification procedure.This can be achieved by substituting the set of conventional basis functions () K x


with an orthogonal set () K x


leading to an observation matrix with lower  .

III. ORTHOGONAL POLYNOMIALS BASED MODEL FOR MIMO RF POWER AMPLIFIERS
In this section, we propose a new set of orthogonal polynomial basis functions for modelling PAs excited with two independent signals.To adhere to the statistics of the widely used communication signals such as WCDMA and LTE, Gaussian distributions have been considered for the complex envelops of the RF inputs.The proposed solution is expected to reduce the condition number of the observation matrix and hence ensure the numerical stability of the model identification procedure.
Considering the memory less case and in an attempt to alleviate the numerical instability problem associated with the inversion of the observation matrix in (6) let"s consider The new data matrix is given by with U is a (2 +2)×(2 +2) kk matrix.
We consider the following requirements for the orthogonality of the suggested basis functions: () k x  and () x  to be orthogonal, the following condition has to be satisfied: Where, E[.] denotes statistical expectation, and (*) stands for complex conjugation.Therefore, finding the appropriate orthogonal polynomial basis functions returns to finding the U matrix such that The resolution of the problem returns to find the elements of the matrix U such that U M U= To find the U matrix elements for the proposed 2×2 MIMO model consider and two independent complexes Gaussian input signals with zero means and variances σ x1 and σ x2 respectively.The matrix can then be given by  where 0 k+1 is a (k+1)×(k+1) sub matrix with zero elements.R 1 and R 2 are two (k+1)×(k+1) symmetric sub-matrices such that for i={1,2} we have  To construct the required set of basis functions the U matrix is proposed to be a (2k+2)×(2k+2) matrix such that 0 k  is a null (k+1)×(k+1) sub matrix.For the proposed 2×2 MIMO model with two independent inputs 1 x and 2 x with zero mean and i x  variances.The u i,lk elements of the matrix U i can be given by [19].
with i={1,2} refers to the inputs of the PA.

A. N×N MIMO Orthogonal Crossover Model
The suggested Crossover orthogonal model can be extended to N×N MIMO transmitters such that: with N is the number of inputs and outputs of the MIMO transmitter.
For MISO transmitters with N inputs and a single output, expression (9) leads to the following matrix form:


In order to convert the observation matrix to a diagonal one that ensures the numerical stability of the model identification procedure, the F matrix is substituted with a Y matrix such that with U is an (N +N)×(N +N) kk orthogonal matrix.U is composed of sub matrices U i placed in the diagonal line with i={1,2,…,N},such that:

B. Numerical Validation
In this paper, we proposed a closed-form expression for orthogonal polynomials to model MIMO Pas excited by RF signals with Gaussian complex envelops such as WCDMA and LTE signals.To demonstrate the efficiency of the proposed set of basis functions in reducing the risk of numerical instability, the condition number of the resulting observation matrix has been calculated.To do so 184,239 independent realizations of a four-channel WCDMA1001 and a four-channel WCDA1111 have been generated as input signals x 1 and x 2 respectively and used to calculate the H  and H  observation matrices for different non-linearity orders K.The "1" refers to an ON channel while the "0" refers to an OFF channel.Fig. 2 shows the condition number for the memory less 2×2 MIMO.

Fig. 2. The condition number of H
 and H  for the 2×2 MIMO case.
Fig. 2 shows clearly that for the conventional model the condition number grows exponentially with the non-linearity order K to reach 10 30 for K=11.However, for the proposed orthogonal model, the condition number increases at a much lower rate without exceeding 100 for the same range of K.
Hitherto, the above-described numerical simulations have proven the remarkable performance of the proposed orthogonal polynomial basis functions for crossover 2×2 MIMO models.However, an experimental validation is required to verify the impact of the numerical stability on the model accuracy.

IV. EXPERIMENTAL VALIDATION
The performances of the proposed set of orthogonal basis functions have been evaluated by modelling and linearizing a 2x2 RF power amplifier.To do so the experimental setup shown in Fig. 3 has been developed.We generated two signals using two vector signal generators (VSG) ESG1 and ESG2 of type E4438C in order to excite the MIMO transmitter.The latter is equivalent to two drivers followed by two class-AB RF PAs and two attenuators.Two couplers have been used to introduce a non-linear cross talk as shown in the figure .A vector signal analyzer (VSA) of type E4440A is then used to collect and analyze one of the two attenuators" outputs through the RF switch.The measurement is performed by connecting the sources via a (GPIB) bus.The I/Q components of the two multi-carrier wideband code division multiple access (WCDMA) signals (WCDMA1001 and WCDMA1111) are generated using Matlab then downloaded to the source with an average power equal to 9 dBm and a bandwidth of 20 MHz.The output signal is down-converted, sampled and digitized at a sampling frequency of (fs= 92.16MHz) for a time window of 2ms leading to a data set of 184,239 samples.The non-linear crosstalk is set to −20 dB

C. Key Performance Metrics
To evaluate the accuracy of the proposed MIMO model in predicting the transmitter output the time domain normalized mean square error (NMSE) has been used.The NMSE [20], [21] is expressed in the logarithmic scale as shown in the following equation: where and are respectively the complex envelopes of the measured and modelled output signals.In the other hand, a frequency domain metric such as the adjacent channel power ration (ACPR) can be used to evaluate the spectral regrowth in the output spectrum as well as the performances of the proposed model based digital predistorter.The ACPR defined as follows [20], [21]:

 
The adjacent channel power ration (ACPR) is measured for the adjacent channels below and above the main carrier in dBc.

D. Forward Model Experimental Results
The importance of a low condition number resides in the fact that the PA model is generally used in the PAs linearization operation.The linearization algorithm runs on a fixed-point processor like DSP or FPGA with a limited number of bits.In fact, the fixed-point processors are efficient with low computation time, cost and power consumption.However, during the modelling process, the model is generally simulated with a floating-point processor.
The nonlinearity order is set to 11 and the memory depths P and Q are set to 2 and 3 respectively.Under these conditions, three different calculation environments have been considered.
 Scenario 2: fixed point calculation with a fraction length of 32 bits.
 Scenario 3: fixed point calculation with a fraction length of 24bits.
As a first test, the coefficients of the conventional and the proposed orthogonal models for the PA under test have been identified in a floating calculation environment.The models" outputs are then compared to the measured PA output and the NMSE have been calculated.Fig. 4(a) shows the NMSE of the two models.The figure reveals that the two models exhibit a comparable accuracy with a pretty similar NMSEs that reached -47.5dB for K = 10.However, the NMSE of the conventional model bounced back to -42dB for K = 11 while the one of the proposed models continued its fall to -48.53dB.In a second test, the same procedure has been repeated using a 64 bits fixed-point processor with fractional lengths of 32 bits and 24 bits.Fig. 4(b) shows The NMSEs for the two models and for the two fraction lengths.The figure demonstrates clearly that the performance of the conventional model deteriorates for the two cases leading to NMSEs diverging to near -3dB for K = 11.However, the proposed orthogonal model maintained its accuracy with NMSEs below -40dB for the whole range of K. Once the coefficients of the model are identified, the PA output can be predicted under different excitation signals.However, in an attempt to linearize the PA and improve its efficiency-linearity compromise, a reverse model can be developed and a digital predistorter (DPD) can be obtained using the indirect learning architecture (ILA).As shown in Fig. 5, the ILA consists of identifying and updating the DPD coefficients using the LS algorithm while exchanging the PA inputs and outputs.The resulting DPD can then linearize the PA and improve the quality of its output spectrum.To validate the effectiveness of the proposed orthogonal model in the linearization of MIMO PAs, a nonlinearity order of 11 and memory depths P and Q of 2 and 3, respectively were considered.The DPD is then developed and applied using the three different scenarios of calculation environment.Tables I, II and III show the adjacent channel power ratios (ACPR) of the 2x2 MIMO PA outputs with and without DPD.The table revealed the high values (above -28dBc) of the ACPR for the different scenarios when the DPD is turned OFF.Under the floating-point environment calculation, the conventional and the proposed orthogonal model based DPDs performed pretty well with an ACPR below -42dBc for the two PA outputs.In addition, the results revealed a better performance of the orthogonal model based DPD with an ACPR as low as -49dBc.
Under the fixed-point calculation environment, the performance of the conventional model-based DPD deteriorates significantly, leading to an ACPR exceeding -19dBc.However, the orthogonal model based DPD maintained its good performance with an ACPR below -41dBc for the two PA outputs and for the two fraction lengths of 24 and 32 bits In Fig. 6 the power spectral densities of the PA outputs has been plotted for the different scenarios where Yi_w/o_DPD are the measured PA outputs when the DPD is turned OFF (Black); Yi_C_fl_DPD are the predicted PA outputs when the conventional model based DPD is applied using a floating point calculation environment (Blue); Yi_O_fl_DPD are the predicted PA outputs when the orthogonal model based DPD is applied using a floating point calculation environment (Red); Yi_C_fx(24b)_DPD are the predicted PA outputs when the conventional model based DPD is applied using a fixed point calculation environment with a fraction length of 24 bits (Orange); Yi_C_fx(32b)_DPD are the predicted PA outputs when the conventional model based DPD is applied using a fixed point calculation environment with a fraction length of 32 bits (Yellow); Yi_O_fx(24b)_DPD are the predicted PA outputs when the orthogonal model based DPD is applied using a fixed point calculation environment with a fraction length of 24 bits (Violet); Yi_O_fx(32b)_DPD are the predicted PA outputs when the orthogonal model based DPD is applied using a fixed point calculation environment with a fraction length of 32 bits (Green), with i = {1,2} denotes the number of outputs.Fig. 6 shows the power spectral densities of the predicted PA outputs for the different scenarios.The measured outputs of the nonlinearity PA, shown in Fig. 6(a)-(b), confirm the spectrum regrowth caused by the PA nonlinearity.In the same time, the figures reveal quite similar and perfect PA linearization when using a floating-point processor (scenario 1) regardless of the DPD model.However, Fig. 6(c)-(d) shows the spectra of the same signals in the cases of scenario 2 and 3.The figures reveal that when using a fixed-point processor, the conventional model-based DPD loses completely its performance and fail to linearize the PA.However, the proposed orthogonal model based DPD maintains its good performances and succeeds to linearize the PA for the 32 bits and 24 bits fraction lengths.

V. CONCLUSION
In this paper, we proposed closed-form expressions for orthogonal polynomials for MIMO PA modelling under RF signals with Gaussian complex envelopes.The numerical and experimental validations have confirmed the robustness and stability of the proposed model identified in fixed-point calculation environments.The proposed orthogonal model based DPD outperformed the conventional model-based DPD in terms of adjacent channel emission reduction in the presence of a nonlinear coupling in the MIMO PA.Due to its simplicity and closed-form expression, the proposed model can be tuned to fit special cases such as massive MIMO PAs where only the coupling between adjacent inputs needs to be considered.

Fig. 1 .
Fig. 1.The condition number of where and are the complex envelope input and output signals respectively and are the model coefficients, are the coefficients of the orthogonal model, P and Q are the memory depths, K is the model nonlinearity, with .

Fig. 6 .
Fig. 6.Frequency domain performance for two channels for 2×2 MIMO transmitter (a) and (b) outputs of DPD for scenario 1, (c) and (d) outputs of DPD for scenario 2 and 3.

TABLE I .
MEASURED ACPR IN DBC FOR THE FIRST SCENARIO