# Volterra Black-Box Models Identification Methods: Direct Collocation vs. Least Squares  
[PERSON]  
1 Applied Mathematics Department, Melentiev Energy Systems Institute, Siberian Branch of Russian Academy of Sciences, Hkutsk 664003, Russia 12 Industrial  
mathematical approach to the description of nonlinear distortions, we will describe below. However, let us consider the general statement of the problem of digital correction (DPD) with the following structure of the model of correction as shown in Figure 2.  
Here, \(F_{DPD}(. )\) is a nonlinear operator reflecting the essence of nonlinear correction-- imagine it as some function dependent on parameters \(\tilde{\mathcal{W}}=[w_{1},\ldots,w_{p}]^{T},\tilde{\mathcal{W}}\in\mathbb{C}^{p}\). \(F_{PA}(. )\) is a nonlinear operator identified with a nonlinear device which generates some complex vector \(\vec{Y}=[y_{1},\ldots,y_{n}]^{T},\vec{Y}\in\mathbb{C}^{n}\) and also defines some vector from a complex field of numbers \(\vec{Y}_{d}=[y_{d,1},\ldots,y_{d,n}]^{T},\vec{Y}_{d}\in\mathbb{C}^{n}\) on which the operator \(F_{DPD}(. )\) depends. Under the error \(E\in\mathbb{C}^{n}\) we will understand the difference between vectors \(Y\) and \(Y_{d}\)  
\[\vec{E}=\vec{Y}_{d}-\vec{Y}.\]  
Then we can formulate the requirements for the definition of parameters \(\vec{W}\) as follows: \(\vec{\omega}=\arg\min_{\mathcal{W}}\lVert E\rVert^{2}\), where \(\lVert.\rVert\) is Euclidean norm. Considering \(Y=F_{PA}(F_{DPD}(Y_{d}))\), the above introduced expression can be rewritten as  
\[\vec{\omega}=\arg\min_{W}\lVert Y_{d}-F_{PA}(F_{DPD}(Y_{d}))\rVert^{2}.\]  
This equation will be task of DPD (Digital Predistortion). Here, we can highlight several important sub-tasks, which in themselves are quite complex both theoretically and computationally:  
Figure 1: Power spectrum density. Figure 2: Digital correction scheme. 1. Since we have formulated, in fact, the problem of approximation of a function, we need to derive the analytical regression dependence \(F_{DPD}(. )\) on the parameters \(\vec{W}\). How this function is defined will depend on the quality of the correction of nonlinear distortions; 2. The procedure of searching for the parameters \(\vec{W}\) is a classical optimization problem, which is a linear or nonlinear regression with respect to the parameters \(\vec{W}\). Finding efficient methods of convex or non-convex optimization is one of the major difficulties in this problem; 3. Compression of a function \(F_{DPD}(. )\), i.e., reducing its computational complexity. One of the methods to solve the problem (a) for the DPD task is the Volterra functional series. And it is also the conventional tool to characterize the complex nonlinear dynamics in various fields including the radiophysics, mechanical engineering, electronic and electrical engineering, energy sciences (here, readers may refer, e.g., to review [1] or [2]). Volterra series are widely employed to represent the input-output relationship of nonlinear dynamical systems with memory. Volterra power series are among the best-understood nonlinear system representations in signal processing. Such an integral functional series (also called Frechet-Volterra series)  
\[ \begin{split} y(t)=F(x(t)):=\int\limits_{0}^{t}K_{1}(s)x(t-s)\, ds+\int\limits_{0}^{t}\int\limits_{0}^{t}K_{2}(s_{1},s_{2})x(t-s_{1})x(t-s_{2})\, ds_{1}ds_{2}+\ldots\\ +\int\limits_{0}^{t}\int\limits_{0}^{t}\cdots\int\limits_{0}^{t}K _{n}(s_{1},s_{2},\ldots,s_{n})x(t-s_{1})x(t-s_{2})\ldots x(t-s_{n})\,ds_{1}ds_ {2}\ldots ds_{n}+\ldots\ t\in[0,T]\end{split}  \tag{1}\]  
was proposed by [PERSON] for a continuous nonlinear dynamical systems representation [3; 4]. Here, readers may also refer to overview [5] and monograph [6] for more details on relevant Lyapunov-Liechtenstein operator and Lyapunov-Schmidt methods in the theory of non-linear equations. The role of a reproducing kernel Hilbert space in the development of a unifying view of the Volterra theory and polynomial kernel regression is presented in [7].