IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. XX, NO. X, JUNE 2018 1 A Doubly Orthogonal Matching Pursuit Algorithm for Sparse Predistortion of Power Amplifiers Juan A. Becerra, Student Member, IEEE, Mar´ıa J. Madero-Ayora, Member, IEEE, Javier Reina-Tosina, Senior Member, IEEE, Carlos Crespo-Cadenas, Senior Member, IEEE, Javier Garc´ıa-Fr´ıas, Senior Member, IEEE, Gonzalo Arce, Fellow, IEEE Abstract—This letter presents a new method for the digital pre- distortion of power amplifiers (PAs) based on sparse behavioral models. The Gram-Schmidt orthogonalization is synergistically integrated into the orthogonal matching pursuit algorithm to decorrelate the selected model regressors against the components still to be selected. Experiments conducted on a test bench based on a GaN PA driven by a 15-MHz orthogonal frequency division multiplexing signal were conducted in order to validate the algo- rithm. Experimental results in a digital predistortion application and a comparison with other state-of-the-art algorithms highlight the enhancement of its pruning capabilities, reducing the number of coefficients while maintaining the performance. Index Terms—Behavioral modeling, compressive-sensing, dig- ital predistortion, orthogonal matching pursuit, power amplifier. I. I NTRODUCTION T HE evolution of wireless communication systems is pushing the design of power amplifiers (PAs) towards challenging constraints in terms of linearity and efficiency. The power range where the PA is more efficient is also where nonlinearity occurs, leading to the need of techniques as digital predistortion (DPD) to mitigate this inconvenience. The pruning of behavioral models is being extensively researched by the community, coming along with a wide set of techniques that range from the application of the principal component analysis (PCA) method [1] to matching pursuit algorithms like the compressive sampling matching pursuit (CoSaMP) [2] and the orthogonal matching pursuit (OMP) [3]. Compressed-sensing techniques are characterized by their simplicity and flexibility, due to the low computational com- plexity of the greedy algorithms these techniques are based on. Greedy algorithms do not perform any a priori decision about the selection of components, becoming suitable for the pruning of Volterra series. OMP [4] is a greedy algorithm that makes a hard decision based on a local optimal criterion whereby the estimated output of the model is always orthogonal to the residual [5], orthogonalization from which OMP takes its name. Since OMP selects one component of the model Manuscript received April 19, 2018; accepted May 17, 2018. (Correspond- ing author: Juan A. Becerra.) J. A. Becerra, J. Garc´ıa-Fr´ıas and G. Arce are with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716 USA (e-mail: becerra@udel.edu; jgf@udel.edu, arce@udel.edu). J.A. Becerra, M.J. Madero-Ayora, J. Reina-Tosina and C. Crespo-Cadenas are with the Departamento de Teor´ıa de la Se˜ nal y Comunicaciones, Escuela T´ ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Sevilla, 41092, Spain (e-mail: jabecerra@us.es, mjmadero@us.es, jreina@us.es, ccrespo@us.es). at each iteration, a pseudoinverse of the data matrix with a number of columns that is equal to the iteration value has to be performed. The work in [6] proposed a simplified sparse parameter identification resulting in a lower computational complexity and in [7], a greedy algorithm ensures that the selected element has maximum energy in every iteration. The estimation of Volterra coefficients is intricate since the basis functionals of the Volterra series are highly correlated. This correlation leads to a large condition number in the model matrices, implying that the equations system is ill-conditioned, affecting the least-squares (LS) solution. A doubly orthogonal matching pursuit (DOMP) algorithm is proposed in this letter to enhance the selection of coefficients in a sparse parameter identification of the model. The main contribution of this work is the addition of the Gram-Schmidt orthogonalization at one step of the OMP algorithm, decorre- lating the selected regressors and those still to be selected. The remainder of this letter is organized as follows. First, Section II introduces the details of model selection and derives the DOMP algorithm. DPD experimental design and results are detailed in Section III. Finally, Section IV summarizes the main results and concludes the paper. II. THE DOUBLY ORTHOGONAL MATCHING PURSUIT The recovery of a high-dimensional vector from a set of measured observations arises in many different fields. The measurement process equation models a linear relation be- tween the Volterra kernel vector h ∈ C n and the system output y ∈ C m , y = X · h + w, (1) through the measurement matrix X ∈ C m×n , which contains one component of the model in each of its columns. The vector w represents the measurement noise. Greedy algorithms aim at recovering the Volterra kernel vector through an iterative approach following the ‘ 1 -norm minimization. In this section, the DOMP algorithm is devel- oped. Its pseudocode instructions are shown in Algorithm 1. A. Initialization The algorithm returns a support set, S, whose elements are sorted in decreasing impact over the output. The initial state of the support set is empty, S (0) = ∅, since no components have been added to it yet. Prior to the algorithm iterations, the matrix Z (0) = X is defined. This matrix will be used to keep