Troublesome distortions often occur in communication systems. For a wide class of systems such distortions can be computed with the help of Volterra series. Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented. The n-fold Fourier transform G n of the nth Volterra kernel plays an important role in the analysis. Methods of computing G n from the system equations are described and several special systems are considered. When the G n are known, items of interest regarding the output can be obtained by substituting the G n in general formulas derived from the Volterra series representation. These items include expressions for the output harmonics, when the input is the sum of two or three sine waves, and the power spectrum and various moments, when the input is Gaussian. Special attention is paid to the case in which the Volterra series consists of only the linear and quadratic terms.

The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs

Analytical modeling of communication receivers to account for their nonlinear response to multiple input signals is discussed. The method is based on the application of the Wiener-Volterra analysis of nonlinear functionals. The derived analytical relations were embodied in a computer program which provides nonlinear transfer functions of large circuits specified by their parameters. This method was applied to the prediction of behavior of communication receivers in the presence of interference. Examples illustrate the method and demonstrate its validity in the small-signal region.

Analysis of nonlinear systems with multiple inputs

https://www.dtc.umn.edu/s/resources/gg-se.pdf

A bibliography on nonlinear system identification

In this study, digital higher-order spectral analysis and frequency-domain Volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time-invariant systems up to third order. The primary focus is on consideration of random excitation of nonlinear systems and, thus, the approach makes extensive use of higher-order spectral analysis to determine the frequency-domain Volterra kernels, which correspond to linear, quadratic, and cubic transfer functions. Although the Volterra model is nonlinear in terms of its input, it is linear in terms of its unknown transfer functions. Thus, a least squares approach is used to determine the optimal (in a least squares sense) set of linear, quadratic, and cubic transfer functions. Of particular practical note, is the fact that the approach of this paper is valid for non-Gaussian, as well as Gaussian, random excitation. It may also be utilized for multitone inputs. The complexity of the problem addressed in this paper arises from two principal causes: (1) the necessity to work in a 3D frequency space to describe cubically nonlinear systems, and (2) the necessity to characterize the non-Gaussian random excitation by computing higher-order spectral moments up to sixth order. A detailed description of the approach used to determine the nonlinear transfer functions, including considerations necessary for digital implementation, is presented. >

Application of higher order spectral analysis to cubically nonlinear system identification

Bibliography on higher-order statistics

This paper presents the noise loading analysis of a third-order nonlinear system characterized by Volterra kernels. The analysis is of practical interest since intermodulation distortion is often a limiting consideration in the design of solid-state long-haul broad-band frequency-division multiplexed transmission systems, and since speech on such systems is best represented by zero mean Gaussian noise. The analysis technique is amenable to digital computer calculation; as such, quantitive results are readily obtained. The analysis was applied to a single-stage transistor amplifier that was represented by a frequency-dependent nonlinear model. An excellent agreement between measured and calculated results was obtained. Portions of the program written for this example are readily applicable to the calculation of intermodulation distortion for any thirdorder nonlinear system (e.g., amplifier, repeater, or cascade of repeaters and cable sections) given the appropriate Volterra kernels.

S. Narayanan

Papers

Noise Loading Analysis of a Third-Order Nonlinear System with Memory