Part $1.$ Download the file called 'data$\_$final.xlsx'and read the data into MATLAB as a variable $($name it whatever you want$).$ The first column is a time$-$axis $($units of seconds$).$ The next three columns are some measurements of a dimensionless variable at each time point.

a$.$ Plot all the data on a single plot and label everything clearly.

b$.$ Using a cross$-$correlation function, figure out the time$-$shift between the data in column $2$ and column $3.$ Make a plot that illustrates what you did. $($note: since the number of datapoints in each set is odd, the tau $=0$ point will be exactly in the center of the correlation function$)$

c$.$ Do the same thing as you did in $$b$$ but now comparing column $2$ with column $4.$

d$.$ Do the same thing again but now comparing column $3$ to column $4.$

e$.$ What would happen if you switched the order of your datasets in the cross$-$correlation functions?

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Part $2.$ Re$-$use the data from part $1.$

a$.$ Compute the cross$-$spectral density functions for the same pairs of data you compared in problem $1.$ Plot the absolute value of the cross$-$spectral density functions on a single plot. Use a semilog$-$y scale. Also, since the number of datapoints is odd, your zero frequency point should be exactly in the middle of the spectrum.

b$.$ Zoom in on the central region around the peak in the your plot from part a $($roughly $-5$ to $5$

radians per second$).$ Keep your plot on a semiolog$-$y scale. What type of function describes the kind of decay you see in this region?

c$.$ Find the time$-$shift between the data in column $2$ and the data in column $3$ using the crossspectral density functions. I recommend plotting the inverse tangent of the ratio of the imaginary part to the real part of the cross$-$spectral density function. Zoom in on the region around zero$-$frequency. By analyzing this central linear region, you can determine the time$-$shift using the formula phi $=$ omega $*$ tau $,$ where tau is the time shift.

d$.$ Do the same thing as you did in $$c$$ but now comparing column $2$ with column $4.$

e$.$ Do the same thing again but now comparing column $3$ to column $4.$

f$.$ Do your time$-$shifts agree with part $1?($hint: if they don$$t you did something wrong$).$