1 (a)
Define signal and system with example. And briefly explain operations performed on independent variable of the signal.

6 M

1 (b)
Determine whether the following signal is energy signal or power signal and calculate its energy of power \[ x(t)=rect \left ( \dfrac {t}{t_o} \right ) \cos \omega_o t. \]

4 M

1 (c)
Find the whether the following system is stable, memory less, linear and time invariant? y(t)=sin[x(t+2)]

4 M

1 (d)
Two signals x(t) and g(t) as shown in Fig Q1(d). Express the signals x(t) in terms of g(t).

Fig.1 d (i)

Fig. 1 d (ii)

6 M

2 (a)
Given the signal x(t) as shown in Fig Q2(a). Sketch the following: \[ i) \ x(-2t+3) \\ ii) \ x\left ( \dfrac {1}{2}-2 \right ) \]

4 M

2 (b)
For a DT LTI system to be stable show that \[ S\overset {\Delta}= \sum^{k=+\infty}_{k=-\infty} |h(K)|<\infty \]

5 M

2 (c)
Two discrete time LTI systems are connected in cascade as shown in Fig Q2(c). Determine the unit sample response of this cascade connection.

6 M

2 (d)
Find convolution of 2 finite duration sequences,

h(n)=a

x(n)=b

i) when a?b

ii) when a=b

h(n)=a

^{n}u(n) for all n andx(n)=b

^{n}u(n) for all ni) when a?b

ii) when a=b

5 M

3 (a)
Determine the LTI system characterized by impulse response. \[ i) \ h(n)=n\left ( \dfrac {1}{2} \right )^n u(n) \\ ii) h(t)=e^{-t}u(t+100) \] Stable and causal.

6 M

3 (b)
Find the forced response of the following system; \[ y(n)- \dfrac {1}{4}y(n-1)- \dfrac {1}{8} y(n-2)= x(n)+x(n-1) \ for \ x(n)= \left ( \dfrac {1}{8} \right )^n u(n) \]

8 M

3 (c)
Draw direct form II implementation for the system described by the following equation and indicate number of delay elements, adders, multipliers.

y(n)-0.25y(n-1)-0.125y(n-2)-x(n)-x(n-2)=0

y(n)-0.25y(n-1)-0.125y(n-2)-x(n)-x(n-2)=0

6 M

4 (a)
Prove the following properties of DTFS:

i) Convolution in time ii) Modulation theorem

i) Convolution in time ii) Modulation theorem

6 M

4 (b)
Determine the complex exponential Fourier series for periodic rectangular pulse train shown in Fig Q4(b). Plot its magnitude and phase spectrum.

8 M

4 (c)
Determine the DTFS representation for the signal \[ x(n)=\cos \left ( \dfrac {n\pi}{3}\] Plot the spectrum of x(n).

6 M

5 (a)
State and prove the following properties of DTFT:

(i) Parseval's theorem ii) Linearity

(i) Parseval's theorem ii) Linearity

6 M

5 (b)
Find the DTFT on the signals shown, \[ i) \ x(n) = \left ( \dfrac {1}{4} \right )^n u(n+4) \\ ii) x(n)= u(n) \]

8 M

5 (c)
Find the inverse Fourier transform of the rectangular shown below

6 M

6 (a)
Consider the continuous time LTI system described by, \[ \dfrac {d}{dt} y(t)+2y(t)=x(t) \] Using FT, find the output y(t) to each of the following input signals.

i) x(t)=e

ii) x(t)=u(t)

i) x(t)=e

^{-t}u(t)ii) x(t)=u(t)

8 M

6 (b)
Find the Nyquist rate and Nyquist interval for each of the following signals:

i) x(t)=sin c

ii) x(t)=2 sin c(50t) sin(5000?t)

i) x(t)=sin c

^{2}(200t)ii) x(t)=2 sin c(50t) sin(5000?t)

6 M

6 (c)
An LTI system is described by \[ H(f)=\dfrac {4} {2+j2\pi f} \] find its response y(t) if the input is x(t)=u(t).

6 M

7 (a)
Define ROC and list its properties.

4 M

7 (b)
State and prove time reversal property of z-transform.

4 M

7 (c)
Determine the inverse z-transform of \[ x(z)=\dfrac {1}{(1+z^{-1})(1-z^{-1})^2} ; ROC;|z|>1 \]

6 M

7 (d)
Determine z-transform and ROC of \[ x(n)= \left ( \dfrac {1}{3} \right )^n \sin \left ( \dfrac {\pi}{4}n \right )u(n) \]

6 M

8 (a)
A causal, stable discrete time system is defined by \[ y(n)= \dfrac {5}{6} y(n-1)-\dfrac {1}{6}y(n-2)+x(n)-2x(n-1) Determine :

i) System function H(z) and magnitude response at zero frequency

ii) Impulse response of the system.

iii) Output y(n) for x(n)=?(n)- 1/3 ?(n-1)

i) System function H(z) and magnitude response at zero frequency

ii) Impulse response of the system.

iii) Output y(n) for x(n)=?(n)- 1/3 ?(n-1)

12 M

8 (b)
Solve the following difference equation for the given initial conditions and input, \[ y(n)-\dfrac {1}{9} y(n-2)=x(n-1) \\ with \ y(-1)=0, y(-2)=1 \ and \ x(n)=3u(n) \]

8 M

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