Question #$2$: $($Frequency Filtering$)$

This question will study a high$-$pass filter in the frequency domain and apply it to a sum of sinusoids signal. You will compare this result with the convolution in the time domain. Add all code into skeleton ee$13135\_$lab$05\_$skeleton.m from Canvas. Include all code $($and functions$)$ in this one file so that everything is published to a single PDF$.$

$($a$)$ Create a new function $H=$ FreqResponse $(b,w)$ that outputs the following frequency response of an FIR system:

$H(e^{\text{j}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}(w)})={\displaystyle \underset{k=0}{\overset{M}{}}}{b}_k{e}^{-\text{j}\text{w}\text{i}\text{d}\text{e}\text{h}\text{a}\text{t}(w)k}.$

where the input $b$ is a vector of filter coefficients, input $w$ is a vector of angular frequencies, and output $H$ is the complex$-$valued frequency response. Include this function at the end of the skeleton file.

$($b$)$ Let the filter coefficients be $\{b_k\}=\{1,0,-1,0,1\},$ generate the frequency response using the function FreqResponse. Plot the magnitude and phase responses. Use Question #$1$ as a guide.

$($c$)$ Use FreqResponse to determine the frequency response at widehat$(w)=0,$widehat$(w)=\frac{}{2},$ and widehat$(w)=9\frac{}{10}.$ Use the disp function to display each result.

$($d$)$ Given the following input signal:

$x[n]=1+2cos((\frac{}{2})n)-cos((9\frac{}{10})n+\frac{}{4}),$

compute the output signal $y[n].$ Use subplot and stem to plot $x[n]$ and $y[n]$ side$-$by$-$side for a range of $n$ containing $1$ fundamental period of $x[n].$ Label the horizontal axis "Samples" and vertical axis $"x[n]"$ or $"$y$[$n$]".$

$($e$)$ Using the conv function, compute the convolution between the input signal $x[n]($for a range of $n$ containing $1$ fundamental period$)$ and coefficients $\{b_k\}.$ Call this result $z[n].$ Use subplot and stem to plot $x[n]$ and $z[n]$ side$-$by$-$side for a range of $n$ containing $1$ fundamental period of $x[n].$ Label the horizontal axis "Samples" and vertical axis $"$x$[$n$]"$ or $"$z$[$n$]".$

$($f$)$ Answer in your comments: How does $z[n]$ compare to $y[n]?$ Explain the cause for any differences.

$1$