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$3$. Marks will be lost for resembling/copied work.$\newline$$4$. Put names of the group members on the cover page. Only the Group Leader to email a single copy of the work to linusaloo$88$@gmail.com.$\newline$ASSIGNMENT I$\newline$$1$a) Briefly discuss the Discrete Fourier Transform (DFT) and derive its pair back and forth defining equations.$\newline$b) Perform the circular and linear convolution of the following sequences using the DFT techniques.$\newline$$$\begin{array}{l} X_{1}(n)=\{1,3,1,4\} \\ X_{2}(n)=\{1,4,2,3\} \\ \end{array}$$$\newline$$2$. An analogue filter has the transfer function, given by:$\newline$$$\mathrm{T}(\mathrm{s})=\frac{12(\mathrm{~s}+2)}{(\mathrm{s}+1)(\mathrm{s}+2)}$$$\newline$Convert this to its discrete equivalent by using the impulse-invariant transform. The sampling frequency is $15 \mathrm{~Hz}$.$\newline$$3$. Given a second-order transfer function $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$. Perform the filter realizations and write the difference equations using the cascade form via the firstorder sections.$\newline$$4$. Find the convolution of the following two sequences:$\newline$$$x(n)=\left\{\begin{array}{ll} 1 & n=0,1,2 \\ 0 & \text { otherwise } \end{array} \text { and } h(n)=\left\{\begin{array}{ll} 0 & n=0 \\ 1 & n=1,2 \\ 0 & \text { otherwise } \end{array}\right.\right.$$$\newline$i. Using the direct evaluation method$\newline$ii. Using tabular and graphical methods.$\newline$$5$ a) Describe the steps of designing digital filters.$\newline$b) Given the Fourier transform of sequences $\{x(n)\}$ and $\{h(n)\}$ are $\left\{x\left(e^{j \omega}\right)\right\}$ and, $\left\{H\left(e^{j \omega}\right)\right\}$ respectively, derive the Fourier transform of:$\newline$i. the delayed sequence $\left\{x_{k}(n)\right\}$ in terms of $\left\{x\left(e^{j \omega}\right)\right\}$, where $x_{k}(n)=x(n-k)$$\newline$ii. the sequence $\{y(n)\}$ in terms of $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$0$ and $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$1$ where $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$2$$\newline$$1$$\newline$ASSIGNMENT II$\newline$$1$. A linear shift invariant system is described by the difference equation$\newline$$$y(n)-\frac{3}{4} y(n-1)+\frac{1}{8} y(n-2)=x(n)+x(n-1)$$$\newline$with $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$3$ and $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$4$.$\newline$Find$\newline$i. the natural response of the system$\newline$ii. the forced response of the system for a step input and$\newline$iii. the frequency response of the system.$\newline$$2$. Compute the $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$5$-point DFT of $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$6$.$\newline$$3$. Given a sequence $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$7$ for $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$8$, where $H(z)=\frac{0.5\left(1-z^{-2}\right)}{1+1.3 z^{-1}+0.3 z^{-2}}$$9$, and $\{x(n)\}$$0$, Evaluate its DFT X(k).$\newline$$4$. Find the digital network in direct and transposed form for the system described by the difference equation.$\newline$$$y(n)=2 x(n)+0.3 x(n-1)+0.5 x(n-2)-0.7 y(n-1)-0.9 y(n-2)$$$\newline$$5$. Consider the sequence $\{x(n)\}$$1$. Calculate the Fast Fourier Transform (FFT).$\newline$END