Complex Time-Response Measurements Using Time-Delay Spectrometry, Part 1
The time response, like the frequency response, is a complex signal characterized by its magnitude and phase. The magnitude is the square root of the sum of the squares of its real (impulse response) and imaginary (doublet response) terms. Due to the insight of Richard C. Heyser, the magnitude is now widely accepted as an accurate measure of the temporal distribution of an impulsive 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 partitioning of the signal between the impulse an doublet response. 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 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 derivation 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 space which is a function of the delay time and any time or frequency 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 the negative phase shift. Once the Fourier transform is correctly phase shifted, it is then possible to calculate the impulse response, doublet response, and phase. The magnitude is clearly not the magnitude and phase of the transfer function on the peak shape of the impulse and doublet response. Theoretical predictions are supported by post-processing experimental data collected with the Techron TEF analyzer on a variety of systems.
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