Quantifying uncertainties in the input–output identification of Flame Transfer Functions

The Finite Impulse Response model of a flame subject to acoustic forcing can be identified from numerical simulations. It is often subsequently used to obtain the frequency domain Flame Transfer Function (FTF). Its estimation from a finite time series introduces uncertainty in the model coefficients, affecting the prediction of the system response when implemented in a thermoacoustic network model. Quantifying how this uncertainty affects the identified FTF is commonly achieved by repeatedly sampling from the distribution of the model coefficients to obtain numerous model realizations and computing their Fourier transform. In the present work, we instead provide the exact mathematical connection between the uncertainty in the time and frequency domain, and give the sampling distributions for the gain and phase of the transfer function. Confidence intervals can then be associated with each predicted FTF value from a single time series. Moreover, by setting a permissible range in the gain and phase of the FTF, the appropriate time series length of the simulation can be determined on the fly. Novelty and Significance Statement In this paper, a novel approach to quantify uncertainties in Flame Transfer Functions (FTFs), which are crucial for predicting thermoacoustic instabilities in combustion systems, is introduced. This research provides an exact mathematical connection between uncertainties in the time domain impulse response and their impact on FTF gain and phase in the frequency domain. We derive sampling distributions for the gain and phase of the FTF, which enables the assignment of confidence intervals to the Bode representation of the FTF. This advancement helps determine the necessary simulation duration to achieve a desired uncertainty level, improving the reliability and efficiency of thermoacoustic predictions.