Consider the vector \(\xvec=\left[\begin{array}{r} 103 \\ 94 \\ 91 \\ 92 \\ 103 \\ 105 \\ 105 \\ 108 \\ \end{array}\right] \text{.}\)

  1. In the Sage cell below is a copy of the change of basis matrices that define the Fourier transform. Find the Fourier coefficients of \(\xvec\text{.}\)

  2. We will now form the vector \(\yvec\text{,}\) which is an approximation of \(\xvec\text{.}\) To do this, round all the Fourier coefficients of \(\xvec\) to the nearest integer to obtain \(\coords{\yvec}{\ccal}\text{.}\) If a coefficient has an absolute value less than one, set it equal to zero. Now find the vector \(\yvec\) and compare this approximation to \(\xvec\text{.}\) What is the error in this approximation?

  3. Repeat the last part of this problem, but set the rounded Fourier coefficients to zero if they have an absolute value less than five. Use it to create a second approximation of \(\xvec\text{.}\) What is the error in this approximation?

  4. Compare the number of nonzero Fourier coefficients that you have in the two approximations and compare the accuracy of the approximations. Using a few sentences, discuss the comparisons that you find.