The Uncertainty Principle for the Short-Time Fourier Transform and Wavelet Transform
Leon W. Cohen · Birkhäuser Boston eBooks · 2001
We obtain both a global and local uncertainty principle for the short-time Fourier transform. Explicit expressions are obtained for the uncertainty product in terms of the uncertainty product of the signal and window. Although the uncertainty product of the spectrogram is not the sum of the uncertainty products of the window and signal, nonetheless, we show that the global uncertainty product has to be greater than one. We also derive local uncertainty relations which indicate how the local spread in time and frequency are related at a particlar time–frequency point. In addition, we obtain general results for the time, scale, and frequency moments of the scalogram in terms of the moments of the signal and mother wavelet. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.