Task $1$

a$.$ A digital signal $x(n)$ is defined as $x(n)=\{1,1,1,1,\frac{1}{2},\frac{1}{2}\}$

Calculate the signal $y(n)=x(n-1)(n-3)$ by direct method, represent the signal in sequence form and sketch the signal.

b$.$ By performing combination of time shifting and time scaling of $x(n)$ how $y(n)$ can be generated?

Task $2$:

$2.$

a$)$ Compute the $8-$point DFT of the sequence.

$x[n]=\{1,2,3,4,4,3,2,1\}$

Using Decimation in Time $($FFT$-$DIT$).$ Follow the signal flow graphs corresponding to the FFT$-$DIT method and annotate all intermediate quantities on the diagrams.

b$)$ Explain the relation between Discrete Time Fourier Transform $($DTFT$)$ and Discrete Time Fourier Transform $($DFT$).$

Task $3$:

$3.$ For the given LTI system,

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

a$)$ Determine the system transfer function and the impulse response, $h(n)$ of the system.

b$)$ Determine the step response of the system if $x(n)=u(n).$

c$)$ Plot the pole$-$zero plot and analyze the stability of the system

d$)$ Explain in detail the implications of the stability analysis and how it affects the system's behavior for different types of inputs $($e$.$g$.,$ step, impulse, sinusoidal$).$ Discuss the role of pole locations in determining transient and steady$-$state responses.

Task $4$

$4.$ For the given LTI system, whose input signal,

$x(n)=\{\begin{array}{ccc}0& \text{f}\text{o}\text{r}& n1\end{array}$

$1$ for $2n5,$

$0,$ otherwise and

$h(n)=(n+2)+(n+1)+(n)+(n-1)$

a$)$ Obtain the system output using Tabulation convolution method.

b$)$ Critically analyze the system with the input, output and impulse response of the systems