Complex Time-Response Using Time-Delay Spectrometry
The time response, like the frequency response, is a complex signal character- ized by its magnitude and phase. The magnitude is a square root of the sum of the squares of the real terms (impulse response) and the imaginary terms (doublet response). Due to the insight of Richard C. Heyser the magnitude is now widely accepted as an accurate measure of the temporal distribution of an impulse exciting signal in a system. The phase of the time response is the arc tangent of the quotient of the doublet and the impulse response. The phase introduced by the system, at each discrete delay time, is reflected in the parti- tioning of the signal between the impulse and doublet responses. While the phase of the transfer function or frequency response is routinely examined, the phase of the time response has not been studied. Time-delay spectrometry (TDS) provides an excellent measurement technique to determine the complex time response of a system. In an attempt to understand the significance of the magnitude and phase of the time response we present a mathematical der- ivation of how TDS, and specifically the TEF analyzer, can be used to determine the complex time response. The discussion is illustrated with theoretical plots at various stages in the TEF analyzer using a new tutorial software program. We derive how the use of a quadratic chirp introduces a translation in data collection space, which is a function of the delay time and any time or frequen- cy offsets used in the experiment. Since a translation in data collection space is equivalent to a phase shift in the Fourier space, namely, the time response, this artifact of the experiment can be removed by multiplying the complex Fourier transform of the IF output by a functional-form phase shift (FFPS), which is a complex exponential whose argument is negative phase shift. We introduce the concept of an asymmetrical window in the Fourier transform of the IF output to minimize spurious wings on the impulse response and also to minimize further deweighting of the low frequencies in the linearly swept bandwidth. Once the Fourier transform is correctly phase shifted, it is then possible to calculate the impulse response, the doublet response, and the phase. The magnitude is clearly not affected. We also describe the effect of the frequency variation of the magnitude and phase of the transfer function on the peak shape of the impulse and doublet responses. Theoretical predictions are supported by postprocessing experimental data collected with the Techron TEF analyzer on a variety of systems.
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