You should work through the tutorial. You may wish to review the output of my Matlab program

There are seven distinct exercises. For the first few exercises the input data can be easily generated. But as you proceed generating the input time series can, in itself, be time consuming. To this end I provide a simple ascii data file for each exercise. I suggest that you simply proceed these data files.

• We start with a pure tone. Generate a pure tone: y(k)=cos(2*pi*k*f0/N) where N=1024 and k=0,1..,N-1. Note for reasons explained in the class the amplitude of the Fourier transform should be normalized by N/2. (1) f0=32. data Plot the power spectrum. You will note that high value of 1/4 at bin 33 (for Matlab convention; this means that the frequency is really 32) and bin N-31 (for Matlab convention). Connect this finding to what you learnt.
• f0=100.5. data You will find that the peak is not longer a delta-like function. That is because your signal has finite temporal width and so you are seeing the effect of a window function.
• Now set f0=924. data A frequency above N/2 is "aliased" as explained in the class. Look at the power spectrum. Change f0=930 and note that the spike in the power spectrum is going to lower index.
• We now add Gaussian noise with zero mean and unit variance, yn=y+noise. data The time series looks quite ratty but in the power spectrum you can easily see the spike at the correct frequency.
• Now we consider a simple square wave. data You will note that the power specturm has many spikes. If you look carefully you will find the spikes are at f0, 3*f0, 5*f0. The square wave is apparently made up of many cosines.
• Finally, we will randomly sample the pure tone [discussed in (1)] above. data I used the Matlab "rand" function which generates a random number uniformly distributed between 0 and <1. If this variate was between 0.8 and 1 then I selected the sample but not otherwise. We can still see the spike!
• Consider a satellite in low earth orbit (e.g. Hubble). For most sources, you can observe only half the time (the Earth will occult the source the remaining half). data This last exercise shows that value of observing a tone with Earth gaps. Notice the "sidebands" (in physics this is called as beat frequency).