96 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 28, NO. 2, FEBRUARY 2018 Generalized Memory Polynomial Model Dimension Selection Using Particle Swarm Optimization A. Abdelhafiz , Graduate Student Member, IEEE , L. Behjat, Member, IEEE, and F. M. Ghannouchi, Fellow, IEEE Abstract— This letter presents a new method which uses particle swarm optimization and the Akaike information criterion for determining the dimensions of nonlinear amplifier behavioral models, applied to the generalized memory polynomial model. Determining the size of this model has always been a challenge as it depends on eight parameters, and the proposed method provides a fast and efficient solution which can discover the smallest possible model at a very short amount of time. Index Terms— Akaike information criterion (AIC), behavioral modeling, digital predistortion (DPD), generalized memory poly- nomial (GMP), particle swarm optimization (PSO), simulated annealing (SA). I. I NTRODUCTION I N THE area of nonlinear power amplifier (PA) and transmitter modeling, compensation, and digital predistor- tion (DPD), Volterra-based polynomial models are often used due to their strong performance [1]. When the signals are of narrow bandwidth, more basic models such as the memory polynomial model (MPM) can be used but for multicarrier and wideband signals, more complex models such as the generalized MPM (GMPM) are needed [2]. However, GMPM’s size is determined by a set of eight different parameters as opposed to 2 for MPM. As a result, there is a need to dedicate some effort to find the model size. Typically (as in [3]), a fixed initial guess for each of the eight parameters is used, and then the parameters are sequentially swept one by one to arrive at the best value [1], which consumes a large amount of time and requires going through a substantial number of combinations. Another issue is that while the models found this way might be the best in terms of modeling error, there is no guarantee that the model size is optimized as the only criterion typically used here is the modeling error [2]. This could result in larger- than-needed models being used, which results in inefficient memory usage during implementation. To address these issues, we are proposing an efficient process combining the evolutionary computing method of particle swarm optimization (PSO) with the Akaike informa- tion criterion (AIC), to develop a fast and efficient way to determine the best possible model size. We show that our Manuscript received September 8, 2017; accepted December 7, 2017. Date of publication January 3, 2018; date of current version February 12, 2018. (Corresponding author: A. Abdelhafiz.) The authors are with the Department of Electrical and Computer Engineer- ing, University of Calgary, Calgary, AB T2N1N4, Canada (e-mail: ahbabdel@ ucalgary.ca). 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/LMWC.2017.2783847 proposed model achieves a good tradeoff between performance and complexity. The results obtained by the modeling and linearization of an envelope tracking PA in this letter support the value of the proposed method, and show its superiority to traditional in terms of performance and speed. The method proposed in this letter finds models in less than 1/25th of the time needed by the traditional sweep method, while obtaining models of smaller size and almost equal performance to those obtained through sweeps. The results obtained by the modeling and linearization of two different PAs in this letter support the value of the proposed method, and show its superiority to traditional sweeps and the simulated annealing (SA) method in terms of performance and speed. II. MOTIVATION AND PROBLEM STATEMENT A. Generalized Memory Polynomial The GMPM is a very powerful behavioral model used to linearize strongly nonlinear PAs. The output of this model y GMP (n) as a function of its input x (n) [2] y GMP (n) = K a  k=0 N a -1  l =0 a kl x (n - l )|x (n - l )| k + K b  k=0 N b -1  l =0 M b -1  m=0 b klm x (n - l )|x (n - l - m)| k + K c  k=0 N c -1  l =0 M c -1  m=0 c klm x (n - l )|x (n - l +m)| k (1) where N a , N b , and N c are the memory depths of each of the branches of the model. Similarly, K a , K b , and K c are the nonlinearity orders of each of the branches of the model. M b and M c are the memory depths of the lagging and leading branches of the model, respectively. a kl , b klm and c klm are the coefficients of the zero lag, lagging, and leading branches of the model, respectively. There are two issues associated with the use of this model. 1) To use this model, an 8-D sweep would need to be performed, which is very time consuming and compu- tationally expensive. 2) Even if the model parameter set p = [ K a K b K c N a N b N c M b M c ] is found using the sweep in the first step, there is no guarantee that this set provides the best performance at the smallest model size since the sweeping process relies on using the error as a metric. In this letter, AIC and the PSO algorithm are combined to finds the parameter set which provides the best tradeoff between 1531-1309 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.