In addition to our notes on the web, here are some helpful references for this problem set:
Davis : pp 217 - 276 (for discussion of Fourier/spectral analysis)
Recipes: chapters 12 and 13 (a lot of detail, but a good resource)
Jenkins and Watts : the classic text in time series analysis (see book list)
1. Power Spectral Analysis. Download the data file freqanal.dat , which is a time series which runs from t=0 to 10 seconds, in steps of .005 seconds. It contains three frequency components and some noise.
a) Using "spectrum" and "specplot", do a power spectrum analysis, identify the three frequencies, and the relative power associated with each. Try different binning numbers (64, 128, 256) and identify what is happening. Hint: you can do this by sorting the output of the "spectrum" command in decreasing order, and taking the first several elements. See also section 3 of lecture 7, and/or the lecture 7 demo m-file, and/or Chapter 6.
b) Now add to the data an additional frequency component of frequency 110 hz. You do this by something like
t=0:.005:10;y=freqanal';f=110;y=y + 3*sin(2*pi*f*t);
and do the spectrum. Where does the new peak appear? Repeat the step again with f=160; Where does this new peak appear? Now do it with f=146. Explain your results.
2. Unevenly Spaced Data. Download the data files tu.dat (independent data) and fu.dat (dependent data). These are unevenly spaced samples in the time domain. Plot a histogram of the logarithm (base 10) of the sampling intervals. What is the average nyquist frequency for the data set? Do a spectrum using Lomb's method, and by downloading the file lomb.m and identify the dominant frequencies (to the nearest hertz), and the probability of them being real (i.e. not random noise). Hint: your maximum frequency is going to be some reasonable multiple (say a factor of 2-3) of your average nyquist frequency. How do the dominant frequencies you identify from the spectral analysis compare to the average nyquist frequency?