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Efficient Pruning Technique of Memory Polynomial Models Suitable for PA Behavioral Modeling and Digital Predistortion Wenhua Chen, Senior Member, IEEE, Silong Zhang, Student Member, IEEE, You-Jiang Liu, Member, IEEE, Fadhel M. Ghannouchi, Fellow, IEEE, Zhenghe Feng, Fellow, IEEE, and Yuanan Liu, Member, IEEE

Abstract—This paper proposes an error variation ranking (EVR)-based pruning method to reduce the complexity of memory polynomials (MPs) for power amplifier behavioral modeling. During the EVR pruning, the variation of prediction error caused by removing each term is calculated and ranked as a quantification factor to show the term’s importance. The dominant terms are then selected based on their ranking positions among all terms. This method is verified by comparing its results with all other possible selections under the same conditions. When it is used to prune digital predistorters, approximately 74% of the terms in the MP model and 78% of the terms in the 2-D digital-predistortion model can be removed with negligible deterioration of the prediction and linearization performance. Moreover, further discussion is presented to strategize the configuration of MP models based on the EVR pruning results. Index Terms—Basis selection, digital predistortion (DPD), error variation, nonlinear model, power amplifier (PA).

I. INTRODUCTION

P

OLYNOMIAL-BASED models have greatly contributed to solving the problem of describing the behavior of power amplifiers (PAs), including nonlinearities and memory effects. They are also widely used as digital predistorters to linearize

Manuscript received April 01, 2014; revised June 14, 2014 and July 28, 2014; accepted August 10, 2014. This work was supported in part by the National Basic Research Program of China under Grant 2014CB339900, the National Science and Technology Major Project of the Ministry of Science and Technology of China under Grant 2014ZX03003007-008, the National Natural Science Foundation of China under Grant 61201043, and the New Century Excellent Talents in University (NCET). W. Chen and Z. Feng are with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]). S. Zhang was with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. He is now with Smarter Micro Inc., Shanghai 201203, China (e-mail: [email protected]). Y.-J. Liu is with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA 92093 USA (e-mail: [email protected]). F. M. Ghannouchi is with the Intelligent RF Radio Laboratory (iRadio Lab), Department of Electrical and Computer Engineering, Schulish School of Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4 (e-mail: fadhel. [email protected]). Y. Liu is with the School of Electronic Engineering, Beijing University of Posts and Telecommunications, 100876 Beijing, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2014.2351779

PAs, due to their potential for estimation or implementation in real-time systems [1]–[12]. However, the tradeoff between implementation complexity and modeling capacity is still a challenge. In recent years, several types of modern PA structures (e.g., supply modulated PAs [12],[13] and concurrent dual-band PAs [20],[21]) have been developed and are considered promising for next-generation communication systems. These new PA structures have two or more input ports with nonlinear interferences between each other. Correspondingly, several two-to-one mapping models have been proposed. However, they further exacerbate the difference between complexity and performance [10]–[21]. In fact, much research has been published on the simplification of the Volterra series, which is the origin of most polynomial-based models. Most of the methods have been based on the analysis of the physical characteristics of PAs and/or modification of the structure of the original polynomials. Several compact and useful PA models have been obtained, such as memory polynomials (MPs), dynamic deviation reduction (DDR)-based Volterra series, and 2-D digital predistortion (2-D-DPD) model. Although these models have led, in most cases, to good performance, they encompass a relatively high number of coefficients. This has triggered the need and provided the motivation for further reduction in the dimensions of such models without sacrificing performance. Recently, some efforts have been made to prune the models adaptively with captured input and output PA signals. In [22] and [23], terms with small kernels were considered negligible and removed, which effectively pruned the Wiener G-function. However, the Wiener G-function is impractical for field-programmable gate-array (FPGA) implementation, due to the complexity in constructing and implementing its basis functions. In [24]–[26], an adaptive pruning method was used to prune the MP model online. However, since significant multicollinearity effects exist in the MP model (i.e., some basis functions in the model can be approximated by a linear combination of the other basis functions), a stable result using this method cannot be achieved, especially when the number of coefficients in the original model is large. We have presented an iterative pruning method where only the term with the minimum kernel was removed in each iteration [27]. Although it outperformed the previous pruning techniques, it still suffers from inherent instability.

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In this paper, the effects of pruning 1-D and 2-D MP models on the prediction error are analyzed, and it is shown that the accuracy of the pruned model is determined by the magnitudes of the removed kernels and the variance of their estimated results. We then propose an error variation ranking (EVR)-based pruning method that directly uses the variation of the prediction error, thereby alleviating the influence of multicollinearity. This method is proven effective through the comparison of the results of all possible basis function subsets. Its performance for digital predistortion (DPD) in single-band Doherty and concurrent dual-band class-AB PAs clearly demonstrates the method’s capability of reducing the complexity of the model and its implementation. The experimental results also reveal that only a small number of terms dominate the prediction accuracy of the whole model. By observing the dominant groups in different cases, some conclusions are drawn to strategize the configuration of the MP model. This paper is organized as follows. Section II presents the theory for model pruning and introduces some important criteria used in this paper. Section III outlines two basis selection methods, e.g., the all possible (AP) test and the EVR pruning method. Experiment results under both single- and dual-band circumstances are then presented and examined in Section IV. The pruning results are further discussed in Section V. Finally, Section VI discusses the conclusions.

II. THEORY AND CRITERIA FOR MODEL PRUNING

and is a matrix filled with all the basis functions and the input series’ values. Specifically, they are given by

(3) (4) (5) (6) (7) where is the total sample number, and each element in is calculated with (8) Consequently, the LS solution to (2) is computed using (9) where is the Hermitian transpose operator, is the inverse operator, and the hat indicates an estimator. Here, it should be noted that, in the following discussions, the order of the basis functions in vector may be different from (6), as needed. And when it is reordered, the and vectors in (2) are modified consistently. Herein it is can be assumed that the residual error, is independent and identically distributed (i.i.d.) (10)

Usually, an MP model is expressed as

(1) where and represent the input and output series of a tested PA, respectively, is the residual error series, including the noise and all other differences between the PA and MP model, is the kernel of the MP and the coefficient of the corresponding term, and and are the nonlinear order and memory depth of the model, respectively. This model considers not only the present terms, but also the lagged terms and their power, which makes it efficient in describing nonlinear behavior with weak and fading memory effects. It is used as a representative model for the PA in the following descriptions. An important advantage of MP models is that they are linear combinations of the basis functions, which means that regression analysis and the least squares (LS) method can be applied to identify the coefficients of the model. Normally, (1) can be denoted in matrix form as (2) where and are the vectors of outputs and error series, respectively, is the vector of the coefficients to be determined,

which means they are in line with the Gaussian distribution with a zero expectation and variance. This is an ideal assumption in order to simplify the analytical derivation to draw essential conclusions. The main function of the model is the prediction of the output of the PA as (11) where is a row vector generated with the basis functions of a certain sample of the test signal. It can be proven that the prediction mean square error (MSEP) of can be expressed as

(12) where is the expectation. Assuming that the selected bases are put in the first columns of matrix and the removed bases are in the other columns, the total number of bases in the full set is . Two selection matrices are then defined as (13)

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and (14) where and are -dimension and -dimension identity matrices, respectively, and, and are zero matrices. When the model is pruned, it is equivalent to solving an LS problem, with a solution of

is actually the same as , and where is the probability density distribution of , which is decided by the statistic properties of the test signal. Therefore, with the same input, the NMSE should obey the same law indicated in (20) as the MSEP, but more comprehensively. The variation of the NMSE caused by pruning can be obtained by

(15) where the asterisk denotes the estimator is based on the selected basis subset after the pruning. As proven in [33], the MSEP of the pruned prediction, is

(16) where is a transform of . Comparing (12) and (16), the variation of the MSEP caused by the pruning is

(23) Equation (23) indicates that two factors, in addition to the property of the test signal, influence the variation of the prediction error when some of the bases are removed. The first factor is related to the magnitudes of the corresponding kernels. When deleted terms, , are associated with significant kernels, their removal greatly deteriorates the NMSE of the model. is significant is the The singular case when the variance of second factor, which is usually induced by a large and/or multicollinearity. A large suggests that the model identification leads to a bad model for the PA, while multicollinearity is a common source of instability in identification of polynomial models, such as the MP model. Another criterion used in this paper is the condition number of , which is used to evaluate the degree of multicollinearity, given by (24)

(17) Since the kernels of the deleted terms are (18) and the variance of

is (19)

it can be concluded that (20) varies as the input signal differs, i.e., it becomes time dependent when the test signal changes with time. Therefore, it is not an appropriate criterion to evaluate the overall prediction precision in practice. Another popular criterion is the normalized mean square error (NMSE) between the predicted and measured output signals, which is given by (21) The NMSE considers the joint effect of the prediction error for each sample and the distribution of the test signal. When the time series is long enough, the NMSE can be regarded as a normalized average of MSEP, which means (22)

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where and are the largest and smallest eigenvalues of , respectively. A large condition number usually implies considerable multicollinearity, therefore, the model is unstable with a large estimate variance and a prediction error [32]. Moreover, a small condition number often leads to fast convergence in most iterative LS algorithms, such as the conjugate gradient method [28]. Therefore, it should be taken as a rough rate when the LS method is used. The final criterion is the number of terms, which is used as an indicator of a model’s implementation complexity, as they are positively related to each other in the usual implementations, e.g., lookup tables (LUTs) or multiplier structures [11]. III. EVR PRUNING METHOD There are many strategies to prune an MP model based on data analysis [31], [32] such as the AP test, stepwise methods, and optimal methods, the feasibility of which are dependent on conditions. In this section, the AP test is introduced and compared with our proposed EVR-based pruning method. If the number of terms in the original model, , is not overly large, it may be realistic to try AP models to find the best combination. There should be sweeps if all sub-models are tested, or, there should be sweeps if terms are selected, i.e., the combinations of bases are considered and the orders do not matter. The best fitting model can be selected with this method.

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Fig. 2. Setup of the platform of the single-band measurements.

3) Ranking and Pruning: The bases in can then be ranked in descending order in line with the significance of its corresponding element in , denoting its importance in the whole model, and moved back to . Afterward, is organized as Fig. 1. Flowchart of the EVR pruning method.

However, when becomes sufficiently large, a huge number of computations are needed. The EVR pruning method is, therefore, proposed with the objective of identifying better models with a moderate amount of computations. The process is illustrated in Fig. 1, and the details of each step are described in the following sections. 1) Initialization: During the initialization, four vectors are defined and denoted as , , , and , respectively. is the vector for all remaining bases, is the buffer, is a vector for all pruned bases, and is a vector for the NMSE variation caused by removing each basis, corresponding to the elements in . Obviously, and complement each other: when a basis is pruned away from , it is moved to . and are both initialized with all potential bases

where denotes the MP basis with nonlinear order and memory depth , whose basis function is as in (8). Both and are defined as blank vectors. 2) One-by-One Pruning: After initialization, the bases in are removed one by one in order, until there is only one base in . In each iteration, the last basis in is moved to as (end) end where (end) is the last basis of vector , and end is all bases in , except the last one. The two models built with the bases in and are then estimated, and the difference of their NMSEs is calculated and added to vector

where and are the NMSEs of the model estimated with and , respectively. At the end of each loop, is buffered to ,

where is the index of the th basis in after ranking. Finally, the first bases of are selected as a tradeoff between complexity and performance. 4) Supplementary Comments: • It is worth noting that the importance of the first basis in the original is not quantified in the process: it is kept as the most important one in the ranked . Therefore, it is strongly recommended to initialize it with , for the PA is generally expected to be a linear component. • All NMSEs in this algorithm should be calculated in a linear unit, although they are given in decibels in the following sections. • Since the sweeping procedure needs only iterations, the computation is much simpler than the AP test. • Since significant multicollinearity may exist in the original model, the ranking results are different if is initialized in a different order. Therefore, the best results can be missed if is not well ordered. Our experience shows that sometimes the ranking algorithm should be performed once or twice more in order to optimize the results, i.e., buffering in , emptying and , and running the algorithm again without initialization. IV. EXPERIMENTS A. Comparison of EVR Pruning Method and AP Test The EVR pruning method was evaluated with a group of signals captured from the RF/DSP platform as Figs. 2 and 3 [29]. The device-under-test (DUT) is denoted as PA 1, and the test signal as Signal 1, as presented in Table I. The DUT was operated at a constant average output power of 48 dBm, corresponding to the output power back-off of 6.8 dB, and a drain efficiency of 41.73%. The dimensions of the original model were determined using a general sweep method described in [34]. Finally, it was set with a nonlinear order ( ) of 7 and a memory depth ( ) of 4, therefore, the total number of terms ( ) was 35. During the experiment, 10 000 samples were used to fit the model, and another 10 000 samples were used to calculate the

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TABLE II COMPARISON OF EVR PRUNING RESULT AND AP TEST RESULT

Fig. 3. Photograph of the test bench with PA1, which is a Doherty PA with two LDMOS transistors.

TABLE III NUMBER OF EVALUATIONS IN EVR PRUNING METHOD AND AP TEST WHEN 25 OF 35 TERMS ARE PRUNED OUT OF THE MP MODEL (10 TERMS REMAINED)

TABLE I PAs AND SIGNALS USED IN THE EXPERIMENTS

LDMOS: laterally diffused metal oxide semiconductor CW: continuous wave LTE: long term evolution PAPR: peak-to-average power ratio WCDMA: wideband code division multiple access

NMSE of the prediction. Since there are at most 35 terms in the model, a time series of 10 000 samples could guarantee enough degrees of freedom to make reliable parameter estimations [32]. In the first step, the MP model was pruned using both the EVR pruning method and the AP test, resulting in different downsized models. Their corresponding NMSEs are listed in Table II, as well as the ranking positions of the pruning results of all possibilities. It is shown that the EVR pruning results were very close to the best ones of the AP test, which proves that the EVR technique is an efficient method to prune the polynomial models used in PA modeling. The number of evaluations in EVR pruning and AP test, when the model is pruned from 35 terms to 10 terms, is shown in Table III. Apparently, EVR pruning needs much less evaluations than AP test so it is much faster. Moreover, EVR pruning method provides the flexibility of the selection of usable bases, for the result after ranking is an ordered vector with all available bases, meaning that a basis can be added or deleted considering the available hardware resources.

Fig. 4. Performance of the EVR pruning method on the forward MP model.

B. Evaluations of EVR Pruning on MP Model Fig. 4 displays the curves of the NMSE and condition number versus the number of selected bases corresponding to different MP model sizes. It can be seen that, when the number of selected terms changed from 35 to 1, the condition number first decreased exponentially and even more so in the end. On the other hand, the NMSE barely changed in the beginning, but deteriorated dramatically at last. In other words, a small number of terms dominated the prediction precision of the whole model. For example, when 10 of the 35 bases were chosen, the condition number decreased almost by half, but the NMSE deteriorated by only 0.78 dB. More discussion is presented in Section IV-C. One of the most important applications of PA behavioral models is as digital predistorters so it is necessary to evaluate the performance of the EVR pruning method in DPD, i.e., to

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Fig. 7. Setup of the platform of the dual-band measurements.

Fig. 5. Performance of the EVR pruning method on an inverse MP model.

TABLE IV DPD PERFORMANCE OF MP AND PRUNED MP

is an extension of the 1-D MP model and has a greater need for simplification. The dual-band platform was the setup shown in Fig. 7, which was the same as [14]. The DUT was PA 2 [30] in Table I. The test signals on the two bands were signals 1 and 2, respectively, as also listed in Table I. The PA delivered 30.3and 30.5-dBm output power on each band in the operation and equal to 33.4 dBm in total. The 2-D-DPD model is expressed as (25) (26) and are the measured output signals of PA where on the upper and lower bands, respectively, and are the errors series, respectively, and are the nonlinear order and memory depth, respectively, and are the kernels of the model, and and are the basis functions, which are given by

(27) (28) Fig. 6. Measured spectrum of the PA output before and after DPD.

prune the MP model as an inverse model of the PA. Fig. 5 illustrates the pruning process when the EVR pruning method was used to prune MP predistorters. The DPD performance when 9 of the 35 bases were selected is depicted in Table IV and Fig. 6. In the experiment, about 74% of the terms are removed, along with a reduction of the condition number by about 93%. However, the NMSE deteriorates by only by 1.8 dB, and the adjacent channel power ratio (ACPR) almost remained the same. This successfully proves the outstanding performance of the EVR pruning method in DPD. C. Evaluation of EVR Pruning on Dual-Band DPD To further assess the performance of the EVR pruning method, it was used to prune the 2-D-DPD model [14], which

where and are the input signals in the upper and lower bands, respectively. The original model was a 2-D-DPD model with and , and the total number of terms in each band was . Similarly, 10 000 samples of each band were used to estimate the model, and another 10 000 samples were used to calculate the NMSE of the prediction during the experiment. The two bands were operated separately during the pruning. The performance of EVR pruning method on the 2-D-DPD model when used as predistorters of the concurrent dual-band PA is depicted in Fig. 8. The DPD performance with 10 of the 45 bases selected in each band is given in Table V and Fig. 9. On the upper band, 78% of the terms were pruned, along with a reduction in the condition number by about 98.9%. However, the NMSE increased by only 1.3 dB, and the ACPR deteriorated by only 2.28 dB. On the lower band, the condition number decreased by roughly 98.4%, the NMSE increased by only 1.2 dB,

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TABLE V DPD PERFORMANCE OF 2-D-DPD MODEL AND PRUNED 2-D-DPD MODEL

Fig. 8. Performance of the EVR pruning method on 2-D-DPD model as an inverse model of the concurrent dual-band PA. (a) Upper band. (b) Lower band.

and the ACPR deteriorated by only 1.82 dB. Again, the excellent capability of the EVR pruning method in 2-D-DPD simplification has been successfully demonstrated. V. FURTHER DISCUSSION As previously mentioned, only a small number of terms dominate the prediction precision of the whole model, and they can be distinguished using the EVR pruning method. Therefore, some basis selection rules of the PA behavioral models can be summarized by comparing the dominant bases in different cases. The configurations of the four tests are listed in Table VI with the details of the PAs and signals described in Table I. Cases 1–3 correspond to single-band scenarios, where MP models with and were selected as the original models, and Cases 4 and 5 were the two bands in a concurrent dual-band test, which was based on the 2-D-DPD model with and . The dominant groups under each condition are presented in Table VII. The dominant group is defined as the smallest subset selected by the EVR pruning method with deterioration of the NMSE less than 1 dB. The pruning result statistics are shown in Figs. 10 and 11. The following statements can be concluded after careful scrutiny of the results.

Fig. 9. Measured spectrum of the output of concurrent dual-band PA before and after DPD. (a) Upper band. (b) Lower band.

1) The nonlinear order and memory depth requirements are different: e.g., although they were pruned from the same model, cases 1 and 2 had a of 7 and an of 2, and case 3 had a of 4 and an of 1. It is quite possible that the difference can be attributed to the inherent characteristics of the PA since the Doherty PA is much more complex than the class-AB PA. 2) The odd- and even-order terms contribute equally to the nonlinearities of the MP model. Among all the nonlinear terms in Table VII, there are 19 even-order and 17 odd-

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TABLE VI DESCRIPTION OF THE FIVE CASES IN THE COMPARISON

TABLE VII BASIS FUNCTIONS IN THE DOMINANT GROUP UNDER DIFFERENT CONDITIONS Fig. 11. Memory depth and nonlinear order of the terms in the EVR pruning results in different cases.

The expressions of basis functions are given by (8), (27), and (28).

3) Fig. 10 depicts the maximum nonlinear order of the terms corresponding to different memory depths of the pruning results of the different cases. The maximum nonlinear order of the static terms was considerably higher than that of the terms with deep memory effects ( or ). 4) Fig. 11 organizes the terms in the EVR pruning results by and values. As shown, the memory depth corresponding to the linear terms is more significant than that of the nonlinear terms . 5) Combining statements 3) and 4), it can be inferred that the nonideal behavior of the PAs in our tests was mainly due to static nonlinearity, and the deep memory effects were usually linear effects caused by the frequency characteristic of the matching networks of the PA. This is clearly demonstrated in Fig. 8, where the remaining terms are categorized into 4 groups by black lines. Among the selected 49 terms, only 6 had and , 8 had and , and 30 terms had and . This suggests a strategy is needed to configure and construct the MP models in PA modeling, i.e., higher and lower orders are more appropriate for the static and dynamic terms, respectively, when the nonlinear order of each memory depth needs to be chosen separately. VI. CONCLUSION

Fig. 10. Maximum nonlinear order of the terms corresponding to different memory depths of the EVR pruning results in different cases.

order terms. This conclusion is consistent with our previous understanding of the contribution of even orders in nonlinearities [33].

In this paper, the theory of model pruning is investigated based on random and statistical theories, which demonstrated that multicollinearity is a potential factor that influences the pruning effects on prediction precision. In order to overcome this problem, the EVR pruning method is proposed and verified. The NMSE variation caused by removing a term is directly used as the quantification factor of its importance. The model obtained through this easy-to-implement method exhibits comparable performance with the best one that could be achieved by the AP test method with the same number of terms. Experiments show that the computational complexity in DPD for both 1-D and 2-D cases can be successfully decreased. The results of the experiments with this method also reveal that a small number of terms dominate the prediction precision of the whole model. After careful investigation of the dominant groups in different

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cases, conclusions are drawn to strategize the configuration of the MP model for PA behavioral modeling. ACKNOWLEDGMENT The authors would like to thank the reviewers for their excellent suggestions and the quality of their comments. The authors would like to thank A. Kwan and M. Younes, both with the iRadio Lab, University of Calgary, Calgary, AB, Canada, for their technical help in experimental measurements. The authors would also thank X. Chen, Microwave and Antenna Laboratory, Tsinghua University, Beijing, China, for his concurrent dual-band PA prototype. REFERENCES [1] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems. Malabar, FL, USA: Krieger, 2006. [2] S. C. Cripps, RF Power Amplifiers for Wireless Communications, 2nd ed. Norwood, MA, USA: Artech House, 2006. [3] H. Ku and J. S. Kenney, “Behavioral modeling of nonlinear RF power amplifiers considering memory effects,” IEEE Trans. Microw. Theory Techn., vol. 51, no. 12, pp. 2495–2504, Dec. 2003. [4] L. Ding, G. T. Zhou, D. R. Morgan, Z. Ma, J. Kim, and C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, no. 1, pp. 159–165, Jan. 2004. [5] D. R. Morgan, Z. Ma, J. Kim, M. G. Zierdt, and J. Pastalan, “A generalized memory polynomial model for digital predistortion of RF power amplifiers,” IEEE Trans. Signal Process., vol. 54, no. 10, pp. 3852–3860, Oct. 2006. [6] A. Zhu, J. C. Pedro, and T. J. Brazil, “Dynamic deviation reductionbased Volterra behavioral modeling of RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 12, pp. 4323–4332, Dec. 2006. [7] M. Isaksson, D. Wisell, and D. Ronnow, “A comparative analysis of behavioral models for RF power amplifiers,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 1, pp. 348–359, Jan. 2006. [8] S. Boumaiza, M. Helaoui, O. Hammi, T. Liu, and F. M. Ghannouchi, “Systematic and adaptive characterizatioin approach for behavior modeling and correction of dynamic nonlinear transmitters,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2203–2211, Dec. 2007. [9] R. N. Braithwaite, “Wide bandwidth adaptive digital predistortion of power amplifiers using reduced order memory correction,” in IEEE MTT-S Int. Microw. Symp. Dig., , Jun. 2008, pp. 1517–1520. [10] A. Zhu, P. J. Draxler, C. Hsia, T. J. Brazil, D. F. Kimball, and P. M. Asbeck, “Digital predistortion for envelope-tracking power amplifiers using decomposed piecewise Volterra series,” IEEE Trans. Microw. Theory Techn., vol. 56, no. 10, pp. 2237–2247, Oct. 2008. [11] C. Quindroit, N. Naraharisetti, P. Roblin, S. Gheitanchi, V. Mauer, and M. Fitton, “FPGA implementation of orthogonal 2-D digital predistortion system for concurrent dual-band power amplifiers based on time-division multiplexing,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 12, pp. 4591–4599, Dec. 2013. [12] H. Cao, H. M. Nemati, A. S. Tehrani, T. Eriksson, J. Grahn, and C. Fager, “Linearization of efficiency-optimized dynamic load modulation transmitter architectures,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 4, pp. 873–881, Apr. 2010. [13] F. H. Raab, B. E. Sigmon, R. G. Myers, and R. M. Jackson, “L-band transmitter using Kahn EER technique,” IEEE Trans. Microw. Theory Techn., vol. 46, no. 12, pp. 2220–2225, Dec. 1998. [14] S. A. Bassam, M. Helaoui, and F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual band transmitters,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2547–2553, Oct. 2011. [15] S. A. Bassam, A. Kwan, W. Chen, M. Helaoui, and F. M. Ghannouchi, “Subsampling feedback loop applicable to concurrent dual-band linearization architecture,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 6, pp. 1990–1999, Jun. 2012. [16] Y. J. Liu, J. Zhou, W. Chen, B. Zhou, and F. M. Ghannouchi, “Lowcomplexity 2-D behavioral model for concurrent dual-band power amplifiers,” Electron. Lett., vol. 48, no. 11, pp. 620–621, May 2012.

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[17] Y. J. Liu, W. Chen, B. Zhou, J. Zhou, and F. M. Ghannouchi, “2D augmented Hammerstein model for concurrent dual-band power amplifiers,” Electron. Lett., vol. 48, no. 19, pp. 1214–1216, Sep. 2012. [18] Y. J. Liu, W. Chen, J. Zhou, B. Zhou, and F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Techn., vol. 61, no. 1, pp. 281–290, Jan. 2013. [19] S. Zhang, W. Chen, Y. Liu, and F. M. Ghannouchi, “A 2-D timemisalignment tolerant memory polynomials predistorter for concurrent dual-band power amplifiers,” IEEE Microw. Wireless Compon. Lett., vol. 23, no. 9, pp. 501–503, Sep. 2013. [20] W. Chen, S. A. Bassam, X. Li, Y. Liu, K. Rawat, M. Helaoui, F. M. Ghannouchi, and Z. Feng, “Design and linearization of concurrent dual-band Doherty power amplifier with frequency-dependent power ranges,” IEEE Trans. Microw. Theory Techn., vol. 59, no. 10, pp. 2537–2546, Oct. 2011. [21] W. Chen, S. Zhang, Y. Liu, Y. Liu, and F. M. Ghannouchi, “A concurrent dual-band uneven Doherty power amplifier with frequency-dependent input power division,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 2, pp. 552–561, Feb. 2014. [22] F. Mkadem, S. Boumaiza, J. Staudinger, and J. Wood, “systematic pruning of Volterra series using Wiener G-functionals for power amplifier and predistorter modeling,” in IEEE Eur. Microw. Integ. Circuits Conf., Oct. 2011, pp. 482–485. [23] F. Mkadem, D. Y. T. Wu, and S. Boumaiza, “Wiener G-functionals for nonlinear power amplifier digital predistortion,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2012, pp. 1–3. [24] X. Yu and H. Jiang, “Digital predistortion using adaptive basis functions,” in Proc. IEEE Asia–Pacific Microw. Conf., Dec. 2010, pp. 13–16. [25] H. Jiang, X. Yu, and P. A. Wilford, “Digital predistortion using stochastic conjugate gradient method,” IEEE Trans. Broadcast., vol. 58, no. 1, pp. 114–124, Mar. 2012. [26] X. Yu and H. Jiang, “Digital predistortion using adaptive basis functions,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 60, no. 12, pp. 3317–3327, Dec. 2013. [27] S. Zhang, W. Chen, F. M. Ghannouchi, and Y. Chen, “An iterative pruning of 2-D digital preditortion model based on normalized polynomial terms,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2013, pp. 1–4. [28] D. Cai and F. Bai, Modern Scientific Computing. Beijing, China: Sci. Press, 2010. [29] M. Helaoui, S. Boumaiza, A. Ghazel, and F. M. Ghannouchi, “On the RF/DSP design for efficiency of OFDM transmitters,” IEEE Trans. Microw. Theory Techn., vol. 53, no. 7, pp. 2355–2361, Jul. 2005. [30] X. Chen, W. Chen, F. M. Ghannouchi, and Z. Feng, “A novel design method of concurrent dual-band power amplifiers including impedance tuning at inter-band modulation frequencies,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2013, pp. 1–4. [31] R. R. Hocking, “A biometrics invited paper. The analysis and selection of variables in linear regression,” Biometrics, vol. 32, no. 1, pp. 1–49, Mar. 1976. [32] S. Chatterjee and A. Hadi, Regression Analysis by Example, 4th ed. New York, NY, USA: Wiley, 2006. [33] D. M. Allen, “Mean square error of prediction as a criterion for selecting variables,” Technometrics, vol. 13, no. 3, pp. 469–475, Aug. 1971. [34] L. Ding and G. Zhou, “Effects of even-order nonlinear terms on power amplifier modeling and predistortion linearization,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 156–162, Jan. 2004. [35] O. Hammi, M. Younes, and A. Kwan, “Performance-driven dimension estimation of memory polynomial behavioral models for wireless transmitters and power amplifiers,” Prog. Electromagn. Res. C, vol. 12, pp. 173–189, 2010. Wenhua Chen (S’03–M’07–SM’11) received the B.S. degree in microwave engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2001, and the Ph.D. degree in electronic engineering from Tsinghua University, Beijing, China, in 2006. From 2010 to 2011, he was a Postdoctoral Fellow with the Intelligent RF Radio Laboratory (iRadio Lab), University of Calgary. He is currently an Associate Professor with the Department of Electronic Engineering, Tsinghua University, Beijing, China. He has authored or coauthored over 100 journal and conference papers. He

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holds two U.S. patents. His main research interests include power-efficiency enhancement for wireless transmitters, PA predistortion, and smart antennas. Dr. Chen is as an associate editor for the International Journal of Microwave and Wireless Technology and a guest lead editor for the International Journal of Antenna and Propagation. He was the corecipient of the Student Paper Award of the 2010 Asia–Pacific Microwave Conference (APMC).

Silong Zhang (S’12) received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 2011 and 2014, respectively. From January 2013 to June 2013, he was an Intern Graduate Student with the iRadio Laboratory, University of Calgary, Calgary, AB, Canada. From July 2012 to May 2014, he was an Intern Student with the Measurement Research Laboratory, Agilent Technology Inc., Beijing, China. He is currently an RF Design Engineer with Smarter Micro Inc., Shanghai, China. His main research interests include PA design for mobile communication and nonlinear system design.

You-Jiang Liu (M’14) received the B.S. and Ph.D. degrees in engineering physics from Tsinghua University (THU), Beijing, China, in 2008 and 2013, respectively. From October 2011 to April 2012, he was a Visiting Student with the Intelligent Radio Laboratory (iRadio Lab), Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. He was one of the most active and influential researchers in dual-band digital predistortion (DPD) techniques for concurrent dual-band transmitters. He is currently a Post-Doctoral Researcher with the High Speed Device Group (HSDG), Department of Electrical and Computer Engineering, University of California at San Diego (UCSD), La Jolla, CA, USA. His recent research at UCSD is focusing on signal generation and processing for envelope tracking (ET) and outphasing, future super-broadband/multi-band DPD with slower ADCs/DACs, frequency quadrupling transmitter architectures for millimeter-wave communications, and long-term evolution (LTE) receive band noise (RxBN) cancellation. He has authored or coauthored over 20 journal and conference papers. Dr. Liu is a member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the IEEE Communications Society. He also actively serves as a reviewer for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, and Circuits, Systems and Signal Processing.

Fadhel M. Ghannouchi (S’84–M’88–SM’93–F’07) is currently a Professor, iCORE/Canada Research Chair, and Director of the iRadio Laboratory, Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. Until 2005, he was with the École Polytechnique de Montréal, where he taught microwave theory and techniques and RF communications systems since 1984. He has held several invited positions at several academic and research institutions in Europe, North America, and Japan. He has provided consulting

services to a number of microwave and wireless communications companies. His research interests are in the areas of RF and wireless communications, nonlinear modeling of microwave devices and communications systems, design of power- and spectrum-efficient microwave amplification systems, and design of software-defined radio (SDR) systems for wireless and satellite communications applications. His research has led to over 500 refereed publications. He holds seven U.S. patents and seven patent applications. Dr Ghannouchi is a Fellow of the Institution of Engineering and Technology (IET). He is an IEEE Microwave Theory and Techniques Soceity (IEEE MTT-S) Distinguish Microwaves Lecturer.

Zhenghe Feng (SM’92–F’11) received the B.S. degree in radio and electronics from Tsinghua University, Beijing, China, in 1970. Since 1970, he has been with Tsinghua University, as an Assistant, Lecturer, Associate Professor, and Full Professor. His main research areas include numerical techniques and computational electromagnetics, RF and microwave circuits and antenna, wireless communications, smart antennas, and spatial temporal signal processing.

Yuanan Liu (M’93) received the B.E., M.Eng., and Ph.D. degrees in electrical engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1984, 1989, and 1992, respectively. In 1984, he joined the 26th Institute of Electronic Ministry of China, to develop the inertia navigating system. In 1992, he began his first post-doctoral position with the Electromagnetic Compatibility (EMC) Laboratory, Beijing University of Posts and Telecommunications (BUPT), Beijing, China. In 1995, he started his second post-doctoral position with the Broadband Mobile Laboratory, Department of System and Computer Engineering, Carleton University, Ottawa, ON, Canada. Since July, 1997, as a Professor, he has been with the Wireless Communication Center, College of Telecommunication Engineering, BUPT, where he is involved in the development of next-generation cellular systems, wireless local area networks (WLANs), Bluetooth application for data transmission, EMC design strategies for high-speed digital systems, and electromagnetic interference (EMI) and electromagnetic susceptibility (EMS) measuring sites with low cost and high performance. He is interested in smart antennas for high-capacity mobile signal processing techniques in fading environments, EMC for high-speed digital systems, intersymbol interference (ISI) suppression, orthogonal frequency division multiplexing (OFDM), and multicarrier system design. Dr. Liu is a senior member of the Electronic Institute of China.