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Time-frequency distributions that evaluate the signal's energy content both in the time and frequency domains are indispensable signal processing tools, especially, for nonstationary signals. Various short-time energy computation schemes are used in practice, including the mean squared amplitude and Teager-Kaiser energy approaches. Herein, we focus primarily on the short- and medium-term properties of these two energy estimation schemes, as well as, on their performance in the presence of additive noise. To facilitate this analysis and generalize the approach, we use a harmonic noise model to approximate the noise component. The error analysis is conducted both in the continuous- and discrete-time domains, deriving similar conclusions. The estimation errors are measured in terms of normalized deviations from the expected signal energy and are shown to greatly depend on both the signals' spectral content and the analysis window length. When medium- and long-term analysis windows are employed, the Teager-Kaiser energy operator is proven superior to the common squared energy operator, provided that the spectral content of the noise is more lowpass than the corresponding signal content, and vice versa. However, for shorter window lengths, the Teager-Kaiser operator always outperforms the squared energy operator. The theoretical results are experimentally verified for synthetic signals. Finally, the performance of the proposed energy operators is evaluated for short-term analysis of noisy speech signals and the implications for speech processing applications are outlined.