In noise and vibration analysis, as well as in many other engineering applications, it may be necessary to extract or analyze signals with time-varying frequency components. Examples include start-up and shut-down of rotating machinery, transient structural vibrations, vehicle passing noise, and speech analysis. Both Short-Time Fourier Transforms (STFT), representing a set of non-causal filters of constant bandwidth, and Wavelet Transforms, representing a set of non-causal filters of constant Q or constant percent bandwidth, have been used for such Joint Time Frequency Analysis (JTFA). In the present work, an arbitrary swept frequency signal is approximated locally, in time, by a linearized frequency sweep. We show that an optimal time window can be identified which, at a given frequency, is inversely proportional to the square root of the instantaneous rate of change of frequency. We find that the constant bandwidth of the STFT and the constant-Q of the Wavelet transform represent extreme cases which are each optimal for certain types of signals. In between the two extremes there lies a continuous range of variation of the effective bandwidth with frequency. Many important types of signals require analysis window variation in this range between STFT and Wavelet analysis. The paper concludes with some simple rules for optimizing the variation of the analysis window with frequency for various types of signals.

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