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MIT2_017JF09_p43

MIT2_017JF09_p43 - 43 SPECTRAL ANALYSIS TO FIND A HIDDEN...

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43 SPECTRAL ANALYSIS TO FIND A HIDDEN MESSAGE 180 43 Spectral Analysis to Find a Hidden Message As you know, the fast Fourier transform ( fft() in MATLAB), turns time-domain signals x into frequency-domain equivalents X . Here you will analyze a given signal from the site and apply several of the common treatments so as to get clean power spectral densities. 1. First, download the x data file from the site; it is called computeSpectra.dat . This signal x looks like a mess at first because of the noise, but we will tease out some specific information about X . Make a plot of the data versus time, assuming the time step is 0.0001 seconds. What features do you see if you zoom in on x , if any? It’s an evidently narrowband signal with content around 1.6kHz (10krad/s), but prob- ably with a lot of noise. Overall it is very hard to see anything beyond this. 2. Use the FFT to make the transformed signal X . Plot the magnitude of X versus frequency; recall that the frequency vector corresponding to X goes from zero to the roughly the sampling rate (use the instructions from your previous homework). In your plot, only show the frequency range near 10000 rad/s where there is significant energy, and remember the 2 /n scaling that is needed to put FFT peaks into the same units as x . What do you see? See the top plot of the psd’s. There are about fifteen peaks at frequencies slightly above 10 krad/s. 3. Treatment 1: Windowing. The Fourier transform operation on a finite-length signal x assumes periodicity, and so any discontinuity between the last and the first points in x is a feature that will be accounted for in X . In general we don’t want this effect, because it muddies up the water and makes it harder to see the details in X that we are interested in. For this and some other reasons, it is common to multiply x by a windowing function, before taking the transform.
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