A Spectral Framework for Identifying Frequency-Dependent Instrumental Sensitivity

Measured signals are shaped not only by the physical system under investigation but also by the measurement apparatus, whose transfer characteristics are often only approximately known and unevenly distributed over the spectrum. This paper develops a mathematical framework for identifying where a measured signal is most sensitive to an unknown measurement response. The instrument is described by a family of real rational transfer functions with known structural form but uncertain parameters. For a fixed reference signal, the admissible measured-signal ensemble's covariance matrix is formed and eigendecomposed; its dominant directions of instrument-induced variability, projected onto the frequency axis, give a frequency-resolved map of instrumental sensitivity. The framework is diagnostic: it does not estimate the instrument's parameters, reduce the physical signal, or solve an inverse problem, and its integration into inverse analysis is left for future work. The approach is illustrated on simulated photoacoustic signals for three microphone classes. Seven dominant modes explain 98.8% of the variance and reveal a narrow, low-redundancy spectral band that coincides with the regions found by a correlation analysis of the 400-dimensional amplitude-phase representation. A previously validated regression model recovers the instrument parameters with 97.27% accuracy from seven selected features (98.84% from all 400).