Bias errors in estimating frequency response and coherence functions from truncated transient signals

An estimation bias error is known to exist when spectra are estimated from truncated responses to transient signals. Expressions for this bias error are developed for zeroth, first and second order systems excited by a burst of white noise, with varying degrees of truncation of the response signal. A series of experiments confirms the theoretical results. For the zeroth order system the effects of truncation are identical for both the frequency response and coherence functions, with a bias error proportional to the fraction of output signal truncation. For first and second order systems, it is shown that the frequency response functions can be accurately estimated with 5% provided that one and three time constants, respectively, of the response signal are captured after the excitation has ceased. On the other hand, the bias of the output autospectrum and the coherence function are much more sensitive to the degree of truncation. Estimating these functions within 5% may require capturing seven or more time constants after the excitation has ceased. Upper bound error limits are developed from the bias functions to help establish data capture requirements when analyzing real systems under varying degrees of signal truncation. It is shown that transient excitation requires significantly shorter capture time per record than continuous excitation to provide the same degree of estimation accuracy.