VTU Signals & Systems - June 2014 Exam Question Paper

VTU Electronics and Communication Engineering (Semester 4)
Signals & Systems
June 2014

Total marks: --
Total time: --

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) Determine the even and odd part of the signal x(t) shown in Fig. Q1(a).

6 M

1 (b) The signal x₁(t) and x₂(t) are shown in Fig 1(b). Sketch the following signals:
i) x₁(t) + x₂(t)
ii) x₁(t).x₂(t)
iii) x₁(1/2)
iv) x₂(2t)
v) x₂(t) x₁(t)

8 M

1 (c) Check whether each of the following signals is periodic or not. If periodic determine its fundamental period:

i) x (n) = \cos (2 n) i i) x (n) = (- 1)^{n} i i i) x (n) = \cos (\frac{π}{8} n^{2})

$i) \ x(n)= \cos (2n) \\ ii) \ x(n)= (-1)^n \\ iii) \ x(n)=\cos \left ( \dfrac {\pi}{8} n^2 \right )$

6 M

2 (a) Perform the convolution of the following signals shown in Fig. Q2(a) and also sketch the o/p signal y(t).

8 M

2 (b) Compare the convolution sum of
x(n)-?ⁿ[u(n)-u(n-8)], |?|<1 and h(n)=u(n)-u(n-5).

8 M

2 (c) Compute the convolution of two sequences
x₁(n)={1, 2, 3} and x₂(n)={1, 2, 3, 4}

4 M

3 (a) Check the followings are stable, causal and memoryless;
i) h(t)=e^-tu(t+100)
ii) h(t)=e^-4|t|
iii) h(n)=2u(n)-2u(n-2)
iv) h(n)=?(n)+sin(n?).

8 M

3 (b) Find the total response of the system given by

\frac{d^{2} y (t)}{d t^{2}} + 3 \frac{d y (t)}{d t} + 2 y (t) = 2 x (t) w i t h y (0) = - 1, \frac{\frac{d y (t)}{d t}}{t = 0} = 1 a n d i n p u t x (t) - \cos t u (r)

$\dfrac {d^2y(t)}{dt^2}+3 \dfrac {dy (t)}{dt}+2y(t)=2x(t) \ with \ y(0)=-1, \dfrac {\frac {dy(t)}{dt}}{t=0}=1 \ and \ input \ x(t)-\cos t\ u(r)$

7 M

3 (c) Find the difference equation corresponding to the block diagram shown in Fig Q3(c).

5 M

4 (a)

I f x (n) \overset{D T F S}{\leftrightarrow} X (k) a n d y (n) \overset{D T F S}{\leftrightarrow} Y (k), t h e n p r o v e t h a t x (n) . y (n) \overset{D T F S}{\leftrightarrow} X (k) ⊛ Y (k) .

$\displaystyle If \ x(n)\overset{DTFS}{\leftrightarrow} X(k) \ and \ y(n) \overset{DTFS}{\leftrightarrow} Y(k), \ then \ prove \ that \ x(n).y(n) \overset{DTFS}{\leftrightarrow} X(k)\circledast Y(k).$

7 M

4 (b) Obtain the DTFS coefficients of

x (n) \cos (\frac{6 π}{13} n + \frac{π}{6}) .

$x(n)\cos \left ( \dfrac {6\pi}{13}n + \dfrac {\pi}{6} \right ).$ Draw the magnitude and phase spectrum.

6 M

4 (c) Determine the time domain corresponding to the following spectra shown in Fig Q4(c)

7 M

5 (a) Let F{x₁(t)}=x₁(j?) and F{x₂(t)}=x₂(j?) then prove that

L e t F {x_{1} (t)} = x_{1} (j Ω) a n d F {x_{2} (t)} = x_{2} (j Ω) t h e n p r o v e t h a t F {x_{1} (t) . x_{2} (t)} = \frac{1}{2 π} \int_{λ = - \infty}^{\infty} x_{1} (j λ) x_{2} (j Ω - λ) d λ

$Let \ F\{x_1(t)\}=x_1(j\Omega) \ and \ F\{ x_2 (t)\}=x_2(j\Omega) \\ then \ prove \ that \\ F\{ x_1(t). x_2(t)\}= \dfrac {1}{2\pi} \int^{\infty}_{\lambda=-\infty} x_1 (j\lambda)x_2 (j\Omega-\lambda)d\lambda$

7 M

5 (b) Find the Fourier transform of the signal x(t) shown in Fig Q5(b)

6 M

5 (c) Find the inverse Fourier transform of

X (j w) = \frac{j w}{(2 + j w)^{2}}

$X(jw)= \dfrac {jw}{(2+jw)^2}$ using properties.

7 M

6 (a) Draw the frequency response of the system described by the impulse response h(t)-?(t)-2e^-2t u(t).

7 M

6 (b) Find the Fourier transform of the periodic impulse train

δ_{T o} (t) = \sum_{k = - \infty}^{\infty} δ (t - k T o)

$\delta_{To}(t)= \sum^{\infty}_{k=-\infty}\delta (t-kTo)$ and draw the spectrum.

8 M

6 (c) A signal x(t)=cos(10?t)+3cos(20?t) is ideally sampled with sampling period Ts. Find the Nyquist rate.

5 M

7 (a) Determine Z-transform of the following DTS and also find the ROC:

i) x (n) = {0.8}^{n} u (n - 1) i i) x (n) = - u (- n + 1) + {(\frac{1}{2})}^{n} u (n)

$i) \ x(n)=0.8^n u(n-1) \\ ii) \ x(n)=-u(-n+1)+ \left ( \dfrac {1}{2} \right )^n u(n)$

8 M

7 (b)

I f x (n) \overset{z}{\leftrightarrow} X (z), w i t h R O C = R t h e n p r o v e t h a t n . x (n) \overset{z}{\leftrightarrow} - z \frac{X (z)}{d z} w i t h R O C - R .

$If \ x(n) \overset{z}{\leftrightarrow}X(z), \ with \ ROC=R \ then \ prove \ that \ n.x(n) \overset{z}{\leftrightarrow}-z\dfrac {X(z)}{dz} \ with \ ROC-R.$

6 M

7 (c) Determine the inverse Z-transform of the function

X (z) = \frac{3 z^{2} + 2 z + 1}{z^{2} + 3 z + 2}

$X(z)= \dfrac {3z^2+2z+1}{z^2+3z+2}$

6 M

8 (a) Determine the impulse response of the sequence describe by y(n)-2y(n-1)+y(n-2)=x(n)+3x(n-3).

8 M

8 (b) Solve the following difference equation using unilateral z-transform.

y (n) - \frac{3}{2} y (n - 1) + \frac{1}{2} y (n - 2) = x (n), f o r n \geq 0, w i t h i n i t i a l c o n d i t i o n s y (- 1) = 4, y (- 2) = 10, a n d i / p x (n) = {(\frac{1}{4})}^{n} u (n)

$y(n) - \dfrac {3}{2} y(n-1)+ \dfrac {1}{2}y(n-2)=x(n), \ for \ n\ge 0, \ with \ initial \ conditions \ y(-1)=4, y(-2)=10, \ and \ i/p \ x(n)= \left ( \dfrac {1}{4} \right )^n u(n)$

8 M

8 (c) Define stability and causality with respect to z-transform.

4 M

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