# Volterra Black-Box Models Identification Methods: Direct Collocation vs. Least Squares
In (1), \(x(t)\) is the input signal and \(y(t)\) is the output of a single-input-single-output (SISO) nonlinear system and \(K_{n}(s_{1},s_{2},\ldots,s_{n})\) are the multidimensional Volterra kernels (or transfer functions) to be identified based on nonlinear system's response \(y(t)\) as a reaction on input \(x(t)\) (Figure 3). It is to be noted that for the basic case \(n=1\), we have a conventional Finite Impulse Response (FIR) linear model which is optimal in the least-squares sense. The Frechet theorem [3] generalizes the famous Weierstrass approximation theorem which characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. Power series (1) characterize the stationary dynamical systems. Stationarity here means that a transfer function does not vary during the transient process as \(t\in[0,T]\). More general power series (2) models nonstationary dynamics when transfer functions depend explicitly on time \(t\)  
\[ \begin{split} y(t)=\int\limits_{0}^{t}K_{1}(t,s)x(s)\,ds+\int \limits_{0}^{t}\int\limits_{0}^{t}K_{2}(t,s_{1},s_{2})x(s_{1})x(s_{2})\,ds_{1 }ds_{2}+\ldots\\ +\int\limits_{0}^{t}\int\limits_{0}^{t}\cdots\int\limits_{0}^{t}K _{n}(t,s_{1},s_{2},\ldots,s_{n})x(s_{1})x(s_{2})\ldots x(s_{n})\,ds_{1}ds_{2} \ldots ds_{n}+\ldots\ t\in[0,T].\end{split}  \tag{2}\]  
Figure 3: Behavioral modeling of the black box system. The Volterra series is an essential tool for the mathematical modeling of the nonlinear dynamical systems appearing in the digital pre-distortion (DPD) iterative process [8]. DPD as we described before is an important part of the digital signal processing algorithms used in transmitters and receivers. Particularly for short-distance applications where the limitations of the transceiver are more significant. DPD is used to improve performance by compensating for the imperfect response of transmitter components, e.g., in [9] the frequency selective DPD was proposed. In [9] the Volterra series model structure consists of a basic linear part and a partial-band pre-compensation part, moreover, a generalized indirect learning architecture is employed to extract the coefficients. Several methods have been studied for DPD, with Volterra series-based methods being popular due to their ease of implementation and the straightforward interpretation of their nonlinear terms. The key issue with Volterra series is the curse of dimension: as the order of the series increases, the number of terms involved in the expansion grows exponentially, making it computationally demanding. On the other hand, estimating the functional coefficients (Volterra kernels) of the Volterra integral functional series can be challenging. It is often considered in its discrete form and requires a significant amount of data and complex optimization algorithms to find the best fit for the model coefficients. An alternative approach based on problem reduction to multi-dimensional integral equations solution [10; 11] requires a special probe signal design. In present paper, the alternative approach for the identification of Volterra kernels is proposed using the direct collocation method. The results are compared with the conventional least squares method (LSM) widely employed for the Volterra series identification problem in the telecommunication domain. The rest of the paper is structured as follows: The subsequent section provides the problem statement. Section 3 focuses on the collocation method. Section 4 carries out computational experiments with LSM, while Section 5 discusses concluding remarks and future work.