$(1)(10$ marks$)$ Consider a discrete$-$time system with finite impulse response $h[n]=\{1_{,}2,3,4\}.$ To this system, we apply the input sequence $x[n],$ defined as follows:

$x[n]=\{\begin{array}{cc}2,& 0n4\\ 1,& ,& n9\\ -2,& 10,& n14\\ 3,& 15,& n19\end{array}$

$($i$)$ Using MATLAB, find the convolution $y[n]=h[n]**x[n]$

$($ii$)$ We now want to compute the convolution of part $($i$)$ using circular convolution and the overlap and save method. Proceed as follows:

$($a$)$ Find input blocks $x_i[n],i=1,2,3,4,5,$ with $N=8.$

$($b$)$ Using MATLAB, compute the FFT $x_i[k]$ of each input block.

$($c$)$ Using MATLAB, find the corresponding output blocks

$Y_i[k]=H[k]x_i[k],i=1,2,3,4,5$

where $H[k]$ is the $8-$point FFT of $h[n].$

$($d$)$ Using MATLAB, find the sequence

$y_i[n]=$IFFT$\{Y_i[k]\},i=1,2,3,4,5.$

e$)$ Using the result of part $($d$),$ find the output $y[n]=h[n]**x[n]$ and compare to the result of part $($i$).$

Hint: You may use the following MATLAB commands:

The command conv $(x,y)$ computes the convolution between vectors x and y $.$

The commands $".*"\text{'}$ computes the product between two sequences of the same length.

The command $fft(x)$ computes the DFT of the vectors $x.$

The command $fft(x,N)$ computes the N $-$point DFT of the vectors x $($i$.$e$.$ with the vector $x$ padded with zeros as needed$)$

The commands ifft $(x)$ and computes the inverse DFT of the vectors X $.$