1(a)
Sketch the even and odd part of the signals shown in Fig. Q1(a).

:!mage

:!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

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

ii) y[n] = α

i) y(t) = e

^{-2t}u(t-2) * {u(t-2) -u (t-12)}ii) y[n] = α

^{n}{u[n] - u[n-6]} * 2{u[n] - u[n-15]}.
14 M

Prove the following

2(b)(i)
x(t) * δ(t-t

_{0}= x(t-t_{0}
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] = 2

iii) h[n] = (½)

i) h(t) = 3δ(t-2) + 5δ(t-5)

ii) h[n] = 2

^{n}u[-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

:!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

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)

ii) Find FT of the signal shown in Fig. Q5(a)(ii).

:!mage

^{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

ii) x[n] = (⅓)

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 ω

_{0}t 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)

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 : \( H(z)=\dfrac{3z^2-z}{(z-2)\left ( z+\dfrac{1}{2} \right )}.\)

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] = 3

^{n}u[n].
12 M

More question papers from Signals & Systems