Problem $2(35$ pts$).$ Re$-$use the data from problem $1.$

a$.(7$ pts $)$ 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$.(5$ pts $)$ 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$.(10$ pts$\backslash ()\backslash )$ 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 $\backslash (\backslash$phi$=\backslash$omega $\backslash$tau $\backslash ),$ where $\backslash (\backslash$tau $\backslash )$ is the time shift.

c$.(5$ pts $)$ Do the same thing as you did in $"$c$"$ but now comparing column $2$ with column $4.$

d$.(5$ pts $)$ Do the same thing again but now comparing column $3$ to column $4.$

e$.(3$ pts$)$ Do your time$-$shifts agree with part $1?($hint: if they don't you did something wrong$).$

answer code must be in MATLAB only