STUPIDSID
1 (a)
For each of the following systems
i) y(t)=x(t-2)+x(2-t)
ii) y(n)=nx(n)
Determine which of properties "memoryless", "time invariant", "linear", "Causal" holds and justify your answer.
i) y(t)=x(t-2)+x(2-t)
ii) y(n)=nx(n)
Determine which of properties "memoryless", "time invariant", "linear", "Causal" holds and justify your answer.
7 M
1 (b)
Using the convolution integral to find the response y(t) of the LTI system with impulse response h(t)=e-βtu(t) to the input x(t)=e-αtu(t) for α=β and α ≠ β.
7 M
2 (a)
Determine the Fourier transform of each of the following signals: \[ i) x(t) = [e^{-at}\cos \omega_0 t] u(t), \ a>0 \\ ii) \ x[n]= \left ( \frac {1}{2} \right )^{-n} u[-n-1] \]
7 M
Answer any one question from Q2 (b) & Q2 (c)
2 (b)
Determine the Fourier series representations for the signal x(t) shown in figure below.
7 M
2 (c)
Let x(t) be a periodic signal whose Fourier series coefficients are \[ a_k = \left\{\begin{matrix}
2, &k=0 \\ j \left (\frac {1}{2} \right ) |k|
& otherwise
\end{matrix}\right. \] Use Fourier series properties to answer the following questions:
(a) Is x(t) real?
(b) Is x(t) even?
(c) Is \( \dfrac {dx(t)} {dt} \) even?
(a) Is x(t) real?
(b) Is x(t) even?
(c) Is \( \dfrac {dx(t)} {dt} \) even?
7 M
Answer any two question from Q3 (a), (b) & Q3 (c), (d)
3 (a)
Consider a causal and stable LTI system S whose input x[n] and output y[n] are related through the second-order difference equation \[ y[n] - \dfrac {1}{6} y [n-1]- \dfrac {1}{6} y[n-2]=x[n] \] i) Determine the frequency response H[ejw] for the system S.
ii) Determine the impulse response h[n] for the system S.
ii) Determine the impulse response h[n] for the system S.
7 M
3 (b)
State and prove the following properties of the Fourier transform.
i) Time Shifting
ii) Time Scaling.
i) Time Shifting
ii) Time Scaling.
7 M
3 (c)
Determine the z-transform for the following sequence. Sketch the pole-zero plot and indicate the ROC. Indicate whether or not the Fourier transform of the sequence exists. \[ i) \ \delta [n+5] \\ ii) \ \left ( \dfrac {1}{4} \right )^n \ u[3-n] \]
7 M
3 (d)
Determine the Laplace transform and the associated region of convergence and pole zero plot for each of the following functions of time:
i) x(t) = e-2t u(t)+e-3t u(t)
ii) x(t)=δ(t)+u(t).
i) x(t) = e-2t u(t)+e-3t u(t)
ii) x(t)=δ(t)+u(t).
7 M
Answer any two question from Q4 (a), (b) & Q4 (c), (b)
4 (a)
Using the long division method, determine the sequence that goes with the following z-transform: \[ x[z] = \dfrac {1- \left (\frac {1}{2} \right )z^{-1}}{1+ \left ( \frac {1}{2} \right )z^{-1}} \] and x[n] is right sided.
7 M
4 (b)
Explain with example the properties and importance of LTI Systems.
7 M
4 (c)
Consider a causal LTI system whose input x[n] and output y[n] are related by the difference equation \[ y[n] = \dfrac {1}{4} y[n-1]+x[n] \] Determine y[n] if x[n]=δ[n-1].
7 M
4 (d)
Using the Partial fraction method, determine the sequence that goes with the following z-transforms: \[ X(z) =\dfrac {3}{z- \frac {1}{4} - \frac {1}{8}z^{-1}} \]
7 M
Answer any two question from Q5(a), (b) & Q5 (c), (d)
5 (a)
List the properties of the region of convergence (ROC) for the z-Transform.
7 M
5 (b)
Consider the signal \[ x[n]\left\{\begin{matrix}
\left ( \dfrac {1}{3} \right )^n \cos \left ( \dfrac {\pi}{4}n \right ), & n\le 0\\0 & n > 0 \end{matrix}\right. \] Determine the poles and ROC for X[z].
7 M
5 (c)
Compute and plot the convolution y[n]=x[n]*h[n] where \[ x[n]= \left\{\begin{matrix} 1,&3\le n \le 8 \\ 0,
& otherwise \end{matrix}\right. \ \ and \\ h[n]= \left\{\begin{matrix} 1,& 4 \le n \le 15 \\0, &otherwise
\end{matrix}\right. \]
7 M
5 (d)
Determine whether or not each of the following signals is periodic. If the signal is periodic, determine its fundamental period. \[ i) \ x(t) = \left [ \cos \left (2t-\dfrac {\pi}{3} \right) \right ]^2 \\ ii) \ x[n]=\cos \left ( n^2 \dfrac {\pi}{8} \right ) \]
7 M
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