# The Harmonic Analysis of Kernel Functions

## TL;DR

This paper introduces a harmonic analysis approach to kernel functions in Bayesian system identification, aiding kernel design and reducing computational complexity.

## Contribution

It presents a novel harmonic analysis method for non-stationary Gaussian process kernels, improving kernel design and computational efficiency.

## Key findings

- Harmonic analysis provides insights into kernel structure.
- The approach enables efficient kernel approximation.
- Results suggest improved computational performance.

## Abstract

Kernel-based methods have been recently introduced for linear system identification as an alternative to parametric prediction error methods. Adopting the Bayesian perspective, the impulse response is modeled as a non-stationary Gaussian process with zero mean and with a certain kernel (i.e. covariance) function. Choosing the kernel is one of the most challenging and important issues. In the present paper we introduce the harmonic analysis of this non-stationary process, and argue that this is an important tool which helps in designing such kernel. Furthermore, this analysis suggests also an effective way to approximate the kernel, which allows to reduce the computational burden of the identification procedure.