# On cross-correlogram IRF's estimators of two-output systems in spaces of continuous functions

## TL;DR

This paper develops estimators for the unknown impulse response component of a two-output linear system using cross-correlation, proving their asymptotic normality and enabling confidence interval construction in continuous function spaces.

## Contribution

It introduces a novel estimation method for unknown IRFs in two-output systems using cross-correlation with Wiener process inputs, establishing asymptotic properties in continuous function spaces.

## Key findings

- Establishes weak asymptotic normality of estimators.
- Constructs confidence intervals in spaces of continuous functions.
- Employs Gaussian process techniques for analysis.

## Abstract

In this paper, single input--double output linear time-invariant systems are studied. Both components of system's impulse response function (IRF) are supposed to be real-valued and square-integrable. One component is unknown while the second one is controlled. The problem is to estimate the unknown component after observations of the other component. We apply cross-correlating of the outputs given that the input is a standard Wiener process. Weak asymptotic normality of appropriately centred estimators in spaces of continuous functions is proved. This enables us to construct confidence intervals in these spaces. Our results employ techniques related to Gaussian processes and bilinear forms of Gaussian processes.