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An analysis of accuracy of time-constant spectrum extraction from thermal transients has been performed. Numerical calculations based on analytical models and finite element method simulations have been used in order to obtain the thermal transients. Simple geometries have been used such that analytical expressions for their time-constant spectrums are known. Results show that a large error in the time-constant spectrum is obtained for very small rms error (<;1 mK) in the thermal transient. The estimation problem is ill-conditioned. Moreover, the differential structure function shows a low accuracy identifying stacked structures. The initial part of the differential structure function shows numerical oscillations and the final part has an asymptotic behavior to infinity that has been identified as an artifact related to errors in the time-constant spectrum estimation. Peak identification from the differential structure function heavily depends on an accurate determination of the time-constant spectrum. The limited spectral resolution and dynamic range of the differential structure function are a direct consequence of the time-constant spectrum imprecision.