21. A half-wave rectified sinusoidal waveform has a peak voltage of 10 V. Its average value and the peak value of the fundamental component are respectively given by
22. Two sequences [a, b, c] and [A, B, C] are related as,
\[\left[ {\begin{array}{*{20}{c}}
A \\
B \\
C
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1&1&1 \\
1&{W_3^{ - 1}}&{W_3^{ - 2}} \\
1&{W_3^{ - 2}}&{W_3^{ - 4}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
a \\
b \\
c
\end{array}} \right]\] where, $${W_3} = {e^{i\frac{{2\pi }}{3}}}.$$
If another sequence [p, q, r] is derived as,
\[\left[ {\begin{array}{*{20}{c}}
a \\
b \\
c
\end{array}} \right] = \] \[\left[ {\begin{array}{*{20}{c}}
1&1&1 \\
1&{W_3^1}&{W_3^2} \\
1&{W_3^2}&{W_3^4}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
1&0&0 \\
0&{W_3^2}&0 \\
0&0&{W_3^4}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{A/3} \\
{B/3} \\
{C/3}
\end{array}} \right]\]
then the relationship between the sequences [p, q, r] and [a, b, c] is
\[\left[ {\begin{array}{*{20}{c}} A \\ B \\ C \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{W_3^{ - 1}}&{W_3^{ - 2}} \\ 1&{W_3^{ - 2}}&{W_3^{ - 4}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} a \\ b \\ c \end{array}} \right]\] where, $${W_3} = {e^{i\frac{{2\pi }}{3}}}.$$
If another sequence [p, q, r] is derived as,
\[\left[ {\begin{array}{*{20}{c}} a \\ b \\ c \end{array}} \right] = \] \[\left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{W_3^1}&{W_3^2} \\ 1&{W_3^2}&{W_3^4} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{W_3^2}&0 \\ 0&0&{W_3^4} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {A/3} \\ {B/3} \\ {C/3} \end{array}} \right]\]
then the relationship between the sequences [p, q, r] and [a, b, c] is
23. The signal $$\cos \left( {10\pi t + \frac{\pi }{4}} \right)$$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $$\left( {\frac{{\sin \left( {\pi t} \right)}}{{\pi \tau }}} \right)\cos \left( {40\pi t - \frac{\pi }{2}} \right).$$ The filter output is
24. The voltage across an impedance in a network is V(s) = Z(s). I(s), where V(s), Z(s) and I(s) are the Laplace transform of the corresponding time functions v(t), z(t) and i(t). The voltage v(t) is
25. The z-transform X[z] of a sequence x[n] is given by $$X\left[ z \right] = {{0.5} \over {1 - 2{z^{ - 1}}}}.$$ It is given that the region of convergence of X[z] includes the unit circle. The value of x[0] is
26. A linear time invariant system has an impulse response est, for t > 0. If initial conditions are 0 and the input is e3t, the output for t > 0 is
27. The 4-point discrete Fourier Transform (DFT) of a discrete time sequence {1, 0, 2, 3} is
28. Let x(t) and y(t) (with Fourier transforms X(f) and Y(f) respectively) be related as shown in the figure. Then Y(f) is
29. If $$sL\left[ {f\left( t \right)} \right] = {\omega \over {\left( {{s^2} + {\omega ^2}} \right)}},$$ then the value of $$\mathop {\lim }\limits_{t \to \infty } f\left( t \right)$$
30. The transfer function of a system is given by $$H\left( s \right) = {1 \over {{s^2}\left( {s - 2} \right)}}.$$ The impulse response of the syste is
(* denotes convolution, and u(t) is unit step function)
(* denotes convolution, and u(t) is unit step function)
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