STUPIDSID
Solve the following:
1 (a)
Show that x(t)*δ(t-to)=x(t-to)
4 M
1 (b)
Obtain the linear convolution of given signals. Also sketch the result. \[
\begin{align*}
x(t)&=1 & \text{for }0\le t \le 1 \ &=0 & \text{elsewhere} \ \ \ \ \ \ \
h(t) &=1 &\text{for }0
4 M
1 (c)
Find the z-transform of the signal \[ x(n) = \left ( \dfrac {1}{2} \right )^n \cos(w_0n) \ u(n). \] Specify its ROC.
4 M
1 (d)
State and explain Dirichlets conditions for the existence of continuous time Fourier series.
4 M
1 (e)
Find the Fourier transform of the signal \[ x(t) = \dfrac {d}{dt}[(e^{-3t}u(t))* (e^{-2t}u(t-2))] \]
4 M
2 (a)
Find if the following sequences are periodic or not. If yes find its fundamental period.
i) x1(n)=ej(π/4)n
ii) x2(n)=3 sin (1/8)n
i) x1(n)=ej(π/4)n
ii) x2(n)=3 sin (1/8)n
6 M
2 (b)
Plot the following sequences:
i) x1(n) = (-2)n u(n)
ii) x2(n)=2+u(t-4)+u(t)
iii) x3(n)=2n u(-n-1)
i) x1(n) = (-2)n u(n)
ii) x2(n)=2+u(t-4)+u(t)
iii) x3(n)=2n u(-n-1)
10 M
2 (c)
Find bilateral z-transform of the signal
x(n)=9 δ(n+2)+3δ(n+1)-4δ(n)+3δ(n-2)+4δ(n-4).
x(n)=9 δ(n+2)+3δ(n+1)-4δ(n)+3δ(n-2)+4δ(n-4).
4 M
3 (a)
Solve the difference equation \[ y(n) - \dfrac {1}{9} y(n-2) = x (n-1) \] with y(-1)=0, y(-2)=1, x(-1)=0 and x(n)=3 u(n).
10 M
3 (b)
Classify the following system for memory, linearity, causality, time variance and stability.
y(n)=a x(n)-b x(n-1).
y(n)=a x(n)-b x(n-1).
5 M
3 (c)
Find x(t) corresponding to FT. \[ x(jw) = \dfrac {-jw}{(jw)^2 + 3jw+2} \]
5 M
4 (a)
Determine complex exponential Fourier series for the signal x(t) shown below:
10 M
4 (b)
Determine z-transform of following function \[ i) \ x(n) = \left ( \dfrac {2}{3} \right )^n u(n+2) \
ii) \ x(n) = n \left ( \dfrac {5}{8} \right )^n u(n) \
iii) \ x(n)=(0.6)^n u(n)*(0.9)^n u(n) \]
10 M
5 (a)
Find Laplace transform of x(t)=te-3t u(t). Prove the property used.
5 M
5 (b)
Find Fourier transform of SINC function.
5 M
5 (c)
Find the inverse Laplace transform of \[ x(s) = \dfrac {-3}{(s+2)(s-1)} \] If the ROC is
i) -2 < Re(s) <1
ii) Re(s)>1
iii) Re(s)<-2
i) -2 < Re(s) <1
ii) Re(s)>1
iii) Re(s)<-2
10 M
6 (a)
If x(t)↔ x(w) is Fourier transform pair then prove that x(t)↔2π x(-w).
5 M
6 (b)
Find the initial and final values of the signal \[ x(z) = \dfrac {(z-3)z}{(z-1)(z-0.4)} \]
5 M
6 (c)
Find inverse z-Transform of \[ x(z) = \dfrac {1- \frac {1}{2}z^{-1}}{1+ \frac {3}{4}z^{-1}+ \frac {1}{8}z^{-2}} |z|>\dfrac {1}{2} \]
10 M
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