The Analytic Impulse and the Energy-Time Curve: The Debate Continues
The analytic impulse is used as a complex-excitation signal to produce the energy-time curve (ETC) of a system. The ETC, usually displayed on a wide-dynamic-range log scale, is the envelope of the system's impulse response and is loosely related to the energy decay in the system. Additional information is presented, using heuristic arguments and simulations, to show that: 1) The ETC is acausal in the same sense that the time response of a theoretical zero-phase filter is acausal; 2) a time-derivative-based complex-excitation signal (rather than Hilbert-transform-based signal) does not work to extract the envelope of a system's impulse response; and 3) even though the ETC is a good general approximation of the energy decay in a system, it does not predict details of the decay such as exact timing and roll-off behavior. The intent here is not to present any radical new information in this debate, but to explain and clarify some of the concepts.
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