$8\mathrm{.}1$ Use the built$-$in MATLAB function $\backslash ($ f $\backslash )$ ft to re$-$calculate the Discrete

$2$ pts Fourier coefficients for the same four points you used in $7\mathrm{.}5.$ Write out on paper exactly how you used $\backslash (\backslash$mathrm$\{$f$\}$ f $\backslash$mathrm$\{$f$\}\backslash )$ :

i$.$ what vectors you created as input to the fft function,

ii$.$ how you called fft $,$

iii. exactly what your MATLAB output from fft was $($I want you to write out all four complex values from MATLAB$),$

iv$.$ and finally, exactly how you used that output to get all the relevant $\backslash ($ A$\_\{$k$\}\backslash )$ and $\backslash ($ B$\_\{$k$\}\backslash ).$

Hint: There should be $\backslash (\backslash$boldsymbol$\{$N O$\}\backslash )$ difference between the coefficients calculated in $7\mathrm{.}5$ and $8\mathrm{.}1.$ If there are, you're either having difficulty calculating the coefficients correctly by$-$hand, or else misinterpreting the output of $\backslash ($ f $\backslash$mathrm$\{$ft$\}\backslash ).$ Go back and fix the problem. A good idea to confirm you're using fft correctly is to maybe first try it on the known $4-$pt example from class.

The whole point of part $8\mathrm{.}1$ is to make sure you fully understand how to use and interpret the built$-$in fft command before moving on to the final $($more complicated$)$ MATLAB questions in $8\mathrm{.}2,$ where you don't know the $"$by$-$hand" coefficients.

Now, let's use the MATLAB fft command to do a Discrete Fourier Transform using many more points! We can then compare that to the Analytic Fourier Transform and discuss any possible aliasing errors.