All code must be in Matlab. All code must be attached. Absolutely no use of ChatGPT or other AI tools allowed.

The project includes coding in the following three problems in Matlab. All problems are Matlab coding problems. Each problem carries equal marks.

Use of inbuilt functions or taking shortcut approaches to coding can lead to a lower grade for the project. You must follow the coding style and approaches discussed in the class.

The difference equation of a linear time$-$invariant $($LTI$)$ system is given by

$y(n)=x(n)+0\mathrm{.}5x(n-1)-0\mathrm{.}25x(n-2)+0\mathrm{.}125x(n-3)+0\mathrm{.}33y(n-1)$

The system is excited by the unit impulse function $x(n)=2(n-5).$ Find the corresponding output $y(n)=h(n)$ for $128$ time samples. Plot $h(n).$ Use the calculated $h(n)$ to find the output $y_2(n)=h_2(n)$ corresponding to the new input $x_2(n)=2(n-5)+2(n-6)$ without using the above difference equation $($hint: use the previously calculated response $h(n)$ to directly find $h_2(n)$ using properties of LTI systems. Plot $h_2(n).$

$2.$ The difference equation of a LTI system is

$y(n)=x(n)+x(n-1)$

If the corresponding unit impulse response is $h(n)$ find using Matlab the impulse response $h_{10}(n)$ of $10$ cascaded units of the above LTI system. Cascading refers to the connection of the output of the k$-$th LTI system to the input of the $k+1-$th LTI system. Also find the FFT of both $h(n)$ and $h_{10}(n)$ in both magnitude and phase. You may use the inbuilt $fft()$ function in Matlab for this part of the project.

$3.$ Find using Matlab the $1616$ square discrete Fourier transform $($DFT$)$ matrix for $16-$point input signal vectors. Display the real and imaginary parts of DFT matrix separately. An input signal is defined as

$x(n)=cos({}_{1}n)+cos({}_{2}n)$

where ${}_1=2*\frac{3}{16}$ and ${}_2=2*\frac{5}{16}.$ Find the DFT of $x(n)$ using your DFT matrix by writing a matrix$-$vector product. Plot it$.$ Then find the FFT of $x(n)$ and plot it$.$ Use magntiude plots only.