1.a. Compute N-point DFT of a sequence $x(n)=\frac{1}{2}+\frac{1}{2} \cos \left(\frac{2 \pi}{N}\left(n-\frac{N}{2}\right)\right)$

1.b. Compute 4-point circular convolution of the sequence using time domain and frequency domain.

2.a. Obtain the relationship between DFT and z-transform

2.b. Let x(n) be a real sequence of length N and its N-point DFT is X(K), show that

i. $\mathrm{X}(\mathrm{N}-\mathrm{K})=\mathrm{X}^{*}(\mathrm{K})$

ii. $\mathrm{X}(0)$ is real.

iii. If N is even, then $\mathrm{X}\left(\frac{\mathrm{N}}{2}\right)$ is real

3.a. Let x(n) be a finite length sequence with using properties of DFT, f‌ind the DFT of the followings:

i. $\mathrm{x}_{1}(\mathrm{n})=\mathrm{e}^{\mathrm{j} \frac{\pi}{2} \mathrm{n}} \times(\mathrm{n})$

ii. $x_{2}(n)=\left\{\cos \frac{\pi}{2} n\right\} x(n)$

3.b. Find the response of an LTI system with an impulse response h(n) = {3, 2,1} for the input x(n) = {2, −1, −1,−2, −3, 5, 6, −1, 2, 0, 2,1 } using overlap add method. Use 8-point circular convolution.

4.a. State and prove the,

i. Modulation property.

ii. Circular time shift property.

4.b. Consider a finite duration sequence

i. Find the sequence, $y(n)$ with 6 point $D F T$ is $y(K)=W_{2}^{k} X(K)$

ii. Determine the sequence $y(n)$ with 6 -point DFT $y(K)=\operatorname{Real}[X(K)]$

5.a. Develop the radix - 2 Decimation in frequency FFT algorithm tor N = 8 and draw the signal flow graph.

5.b. What is Goertzel algorithm and obtain the direct form − ll realization?

6.a. Let‘ x(n) be the 8-point sequence of Compute the DFT of the sequence using DIT FFT algorithm.

6.a. What is Chirp-Signals and mention the applications of Chirp-Z-transform?

6.c. A designer is having a number of 8-point FFT chips. Show explicitly how he should interconnect three chips in order to compute a 24-point DFT.

7.a. Design a digital low pass Butterworth Filter using bilinear transformation to meet the following specifications:

$-3 \mathrm{dB} \leq\left|\mathrm{H}\left(\mathrm{e}^{\mathrm{j} \omega}\right)\right| \leq-1 \mathrm{dB}$ for $0 \leq \omega \leq 0.5 \pi$

$\left|\mathrm{H}\left(\mathrm{e}^{\mathrm{j} \omega}\right)\right| \leq-10 \mathrm{dB}$ for $0.7 \pi \leq \omega \leq \pi$

7.b. Obtain the parallel form of realization of a system difference equation,

$y(n)=0.75 y(n-1)-0.125 y(n-2)+6 x(n)+7 x(n-1)+x(n-2)$

8.a. Convert the analog f‌ilter with system function,

$H_{a}(s)=\frac{s+0.1}{(s+0.1)^{2}+9}$ into a digital IIR f‌ilter by means of the impulse invariance method.

8.b. Obtain the DF-l and cascade form of realization of the system function,

$H(z)=\frac{1+\frac{1}{3} z^{-1}}{\left(1-\frac{1}{5} z^{-1}\right)\left(1-\frac{3}{4} z^{-1}+\frac{1}{8} z^{-2}\right)}$.

9.a. Obtain the linear phase realization of FIR filter with impulse reponse,

$h(n)=\delta(n)-\frac{1}{2} \delta(n-1)+\frac{1}{4} \delta(n-2)+\frac{1}{4} \delta(n-3)-\frac{1}{2} \delta(n-4)+\delta(n-5)$

9.b. What are the advantages and disadvantages of the window technique for designing FIR f‌ilter?

9.c. A low pass filter is to be designed with the following desired frequency response:

$\mathrm{H}_{\mathrm{d}}\left(\mathrm{e}^{\mathrm{j\omega}}\right)=\left\{\begin{array}{cc}{\mathrm{e}^{-\mathrm{j} 2 \omega},} & {|\omega|\lt\frac{\pi}{4}} \\ {0,} & {\frac{\pi}{4}\lt|\omega|\lt\pi}\end{array}\right.$

Determine the filter coefficients $\mathrm{h}_{\mathrm{d}}(\mathrm{n})$ and $\mathrm{h}(\mathrm{n})$ if $\omega(\mathrm{n})$ is a rectangular window defined as,

$\omega_{R}(n)=\left\{\begin{array}{ll}{1,} & {0 \leq n \leq 4} \\ {0,} & {\text { Otherwise }}\end{array}\right.$.

10.a. The desired frequency response of a low pass filter is given by, $H_{d}\left(e^{j \omega}\right)=$ $\left\{\begin{array}{cc}{\mathrm{e}^{-j \omega,},} & {|\omega|\lt\frac{3 \pi}{4}} \\ {0,} & {\frac{3 \pi}{4}\lt|\omega|\lt\pi}\end{array}\right.$. Determine the frequency response of the FIR filter if Hamming window is used with N = 7.

10.b. Realize an FIR filter with impulse response h(n) given by,

$h(n)=\left(\frac{1}{2}\right)^{n}[u(n)-u(n-4)]$ using direct form.