Efficient Identification And Inversion Of Complex-Valued Wiener Systems Using B-Spline Neural Networks.pdf

The provided text appears to be an excerpt from a technical paper or document discussing complex-valued (CV) systems and their identification using neural networks, specifically focusing on Wiener systems. Here's a breakdown of the key concepts:

### Key Concepts

1. Wiener System:
- A type of system consisting of two cascaded subsystems:
1. A linear filter representing memory effects.
2. A nonlinear memoryless function.

2. Linear Filter:
- Described by a transfer function \( H(z) \).
- Characterized by coefficients \( h = [h1, h2, \ldots, hL]^T \), with the assumption that \( h0 = 1 \).

3. Nonlinear Function (\( \Psi(\cdot) \)):
- A memoryless function mapping complex inputs to complex outputs.
- Assumed to be a one-to-one mapping.

4. Complex-Valued B-spline Neural Network:
- Used to model the nonlinear function \( \Psi(\cdot) \).
- Involves univariate B-spline basis functions for both real and imaginary parts of the input.

5. Identification and Inversion:
- The goal is to identify the parameters of the Wiener system (i.e., the linear filter coefficients and the nonlinear function) using input-output data.
- Additionally, develop an algorithm to invert the identified Wiener system.

### Technical Details

- B-spline Basis Functions:
- Defined by a polynomial order \( Po \) and a knot vector.
- The knot vector includes internal knots within the input range and external knots outside this range.

- Complex Numbers Representation:
- Complex numbers are represented in both rectangular form (\( x = xR + jx_I \)) and polar form (\( x = |x| \cdot e^{j\angle x} \)).

### Application

The document discusses using these concepts for designing digital predistorters, which are used to compensate for nonlinearities in power-efficient transmitters in broadband communication systems.

This approach leverages the efficiency of B-spline neural networks and the Gauss-Newton algorithm for both identification and inversion tasks, aiming to improve system performance by accurately modeling and compensating for nonlinear effects.