Uncertainty Principle and Spectogram with PyLab
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The Fourier transform does not give any information on the time at which a frequency component occurs. One approach which can give information on the time resolution of the spectrum is the Short Time Fourier Transform (STFT). Here, a moving window is applied to the signal and the Fourier transform is applied to the signal within the window as the window is moved [Ref]. The choice of window is very important with respect to the performance of the STFT in practice. Since the STFT is simply applying the Fourier transform to pieces of the time series of interest, a drawback of the STFT is that it will not be able to resolve events if they happen to appear within the width of the window. In this case, the lack of time resolution of the Fourier transform is present. In general, one cannot achieve simultaneous time and frequency resolution because of the Heisenberg uncertainty principle. In the field of particle physics, an elementary particle does not have precise position and momentum. The better one determines the position of the particle, the less precisely is known at that time, and vice versa. For signal processing, this rule translates into the fact that a signal does not simultaneously have a precise location in time and precise frequency [Ref].
The library PyLab provides the function specgram(...) to compute the spectrogram of a signal using the STFT. The following script uses that function to show the spectrogram of a signal with different windows size:
from scipy.io.wavfile import read,write from pylab import plot,show,subplot,specgram # Open the Homer Simpson voice: "Ummm, Crumbled up cookie things." # from http://www.thesoundarchive.com/simpsons/homer/mcrumble.wav rate,data = read('mcrumble.wav') # reading subplot(411) plot(range(len(data)),data) subplot(412) # NFFT is the number of data points used in each block for the FFT # and noverlap is the number of points of overlap between blocks specgram(data, NFFT=128, noverlap=0) # small window subplot(413) specgram(data, NFFT=512, noverlap=0) subplot(414) specgram(data, NFFT=1024, noverlap=0) # big window show()
This image is the result:
The picture shows that changing the number of data points used for each Fourier transform block, the spectrogram loses definition in frequency or in the time.
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