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INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1(a) Sketch the even and odd part of the signals shown in Fig. Q1(a).
:!mage
6 M
1(b) For the signal x(t) and y(t) shown in Fig. Q1(b) sketch the signals:
i) x(t+1) - y(t)
ii) x(t) × y(t-1).
:!mage
6 M
1(c) Determine whether the system described by the following input/output relationship is i) memory less ii) causal iii) time invariant iv) linear. \begin {align*} &i)y(t)=x(2-t)\\ &ii)y[n]=\sum ^{\infty}_{k=0}2^k x[n-k] \end{align*}
8 M

2(a) Compute the following convolutions :
i) y(t) = e-2t u(t-2) * {u(t-2) -u (t-12)}
ii) y[n] = αn {u[n] - u[n-6]} * 2{u[n] - u[n-15]}.
14 M
Prove the following
2(b)(i) x(t) * δ(t-t0 = x(t-t0
3 M
2(b)(ii) $$x[n]*u[n]\sum ^n_{k=-\infty}x[k].$$
3 M

3(a) Identify whether the systems described by the following impulse are memory-less, causal and stable.
i) h(t) = 3δ(t-2) + 5δ(t-5)
ii) h[n] = 2nu[-n]
iii) h[n] = (½)n δ[n].
9 M
3(b) Find the natural response and the forced response of the system described by the following differential equation : $$\dfrac{\mathrm{d} ^2y(t)}{\mathrm{d} t^2}-4y(t)=\dfrac{\mathrm{d} }{\mathrm{d} t}x(t), \ \text{if y(0)=1 and}\ \dfrac{\mathrm{d} }{\mathrm{d} t}\ y(t)|_{t=0}=-1.$$
8 M
3(c) Write the difference equation for the system depicted in Fig. Q.3(c).
:!mage
3 M

4(a) State and prove the Parseval's relation for the Fourier series representation of discrete time periodic signals.
6 M
4(b) i) Find the DTFS of the signal x(t) = sin [5πn] + cos [7πn]
ii) Find the FS of the signal shown in Fig. Q4(b)(ii).
:!mage
8 M
4(c) If the FS representation of periodic signal x(t) is : where $$\omega _0=\dfrac{2\pi}{t}$$ then find the FS of y(t) without computing x(t) : \begin{align*} &i)y(t)=x(t+2)\\ &ii)y(t)=\dfrac{\mathrm{d} }{\mathrm{d} t}x(t). \end{align*}
6 M

5(a) i) Compute the DIFT of x[n] = (½)n u[n+2] + (½)n u[n-2]
ii) Find FT of the signal shown in Fig. Q5(a)(ii).
:!mage
10 M
5(b) Find inverse of the following x(jω) :\begin{align*} &i)x(j\omega)=\dfrac{j\omega}{(j\omega)^2+6j\omega+8}\\ &ii)x(j\omega)=j.\dfrac{\mathrm{d} }{\mathrm{d} \omega}\dfrac{e^{3j\omega}}{2+j\omega}. \end{align*}
10 M

6(a) Determine output of the LTI system whose I/P and the impulse response is given as :
i) x(t) = e-2t u(t) and h(t) - e-3t u(t).
ii) x[n] = (⅓)n u[n] and h[n] = δ[n-4].
8 M
6(b) Find the Fourier transform of the signal x(t) = cos ω0t where $$\dfrac{2\pi}{T}$$ and T the period of the signal.
4 M
6(c) State the sampling theorem and briefly explain how to practically reconstruct the signal.
8 M

7(a) State and prove differentiation in z-domain property of z-transforms.
6 M
7(b) Use property of z-transform to compute x(z) of :
i) x[n] = n sin (πn/2) u[-n]
ii) x[n] = (n-2) (½)n u[n-2].
6 M
7(c) Find the inverse z-transforms of \begin {align*} &i) x(z)=\frac{z^2-2z}{\left ( z^2+\frac{3}{2}z-1 \right )}\ \frac{1}{2}<|z|<2\\ &ii) x(z)=\frac{z^3}{\left ( z-\frac{1}{2} \right )}\ |z|>\frac{1}{2}. \end{align*}
8 M

8(a) Determine the impulse response of the following transfer function if :
i) The system is causal
ii) The system is stable
iii) The system is stable and causal at the same time : $$H(z)=\dfrac{3z^2-z}{(z-2)\left ( z+\dfrac{1}{2} \right )}.$$
8 M
8(b) Use unilateral z - transform to determine the forced response and the natural response of the systems described by: $$y[n]-\dfrac{1}{4y}[n-1]-\dfrac{1}{8y}[n-2]=x[n]+x[n-1]$$ where y[-1] = 1 and y[-2] = 1 with I/P x[n] = 3n u[n].
12 M

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