# Higher-order sinusoidal input describing function

## Definition

The higher-order sinusoidal input describing functions (HOSIDF) were first introduced[1] by dr. ir. P.W.J.M. Nuij. The HOSIDFs are an extension of the sinusoidal input describing function[2] which describe the response (gain and phase) of a system at harmonics of the base frequency of a sinusoidal input signal. The HOSIDFs bear an intuitive resemblance to the classical frequency response function and define the periodic output of a stable, causal, time invariant nonlinear system to a sinusoidal input signal:

${\displaystyle u(t)=\gamma \sin(\omega _{0}t+\varphi _{0})}$

This output is denoted by ${\displaystyle y(t)}$ and consists of harmonics of the input frequency:

${\displaystyle y(t)=\sum \limits _{k=0}^{K}|H_{k}(\omega _{0},\gamma )|\gamma ^{k}\cos {\big (}k(\omega _{0}t+\varphi _{0})+\angle H_{k}(\omega _{0},\gamma ){\big )}}$

Defining the single sided spectra of the input and output as ${\displaystyle U(\omega )}$ and ${\displaystyle Y(\omega )}$, such that ${\displaystyle |U(\omega _{0})|=\gamma }$ yields the definition of the k-th order HOSIDF:

${\displaystyle H_{k}(\omega _{0},\gamma )={\frac {Y(k\omega _{0},\gamma )}{U^{k}(\omega _{0},\gamma )}}}$