31. The Laplace transform of a continuous-time signal x(t) is $$X\left( s \right) = {{5 - s} \over {{s^2} - s - 2}}.$$ If the Fourier transform of this signal exists, then x(t) is
32. A network consisting of a finite number of linear resistor (R), inducer (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form
$$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t} \right),{\rm{were}}\,{a_k} \ne 0,} \,{\omega _0} \ne 0.$$
The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?
$$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t} \right),{\rm{were}}\,{a_k} \ne 0,} \,{\omega _0} \ne 0.$$
The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?
33. The complex envelope of the bandpass signal $$x\left( t \right) = \sqrt 2 \left( {{{\sin \left( {{{\pi t} \over 5}} \right)} \over {{{\pi t} \over 5}}}} \right)\sin \left( {\pi t - {\pi \over 4}} \right),$$ centered about $$f = {1 \over 2}Hz,$$ is
34. A signal x(t) has a Fourier transform X(ω). If x(t) is a real and odd function of t, then X(ω) is
35. The power in the signal
$$s\left( t \right) = 8\cos \left( {20\pi t - {\pi \over 2}} \right) + 4\,\sin \left( {15\pi t} \right)$$ is
$$s\left( t \right) = 8\cos \left( {20\pi t - {\pi \over 2}} \right) + 4\,\sin \left( {15\pi t} \right)$$ is
36. A 5-point sequence x[n] is given as
x[-3] = 1, x[-2] = 1, x[-1] = 0, x[0] = 5, x[1] = 1.
Let X(ejω) denote the discrete-time Fourier transform of x[n]. The value of $$\int\limits_{ - \pi }^\pi {X\left( {{e^{j\omega }}} \right)} d\omega $$
x[-3] = 1, x[-2] = 1, x[-1] = 0, x[0] = 5, x[1] = 1.
Let X(ejω) denote the discrete-time Fourier transform of x[n]. The value of $$\int\limits_{ - \pi }^\pi {X\left( {{e^{j\omega }}} \right)} d\omega $$
37. The Fourier series representation of an impulse train denoted by
$$s\left( t \right) = \sum\limits_{n = - \infty }^\infty {\delta \left( {t - n{T_0}} \right)} \,{\rm{is}}\,{\rm{given}}\,{\rm{by}}$$
$$s\left( t \right) = \sum\limits_{n = - \infty }^\infty {\delta \left( {t - n{T_0}} \right)} \,{\rm{is}}\,{\rm{given}}\,{\rm{by}}$$
38. Let Y(s) be the unit-step response of a causal system having a transfer function
$$G\left( s \right) = {{3 - s} \over {\left( {s + 1} \right)\left( {s + 3} \right)}}$$
That is, $$Y\left( s \right) = {{G\left( s \right)} \over s}.$$ The forced response of the system is
$$G\left( s \right) = {{3 - s} \over {\left( {s + 1} \right)\left( {s + 3} \right)}}$$
That is, $$Y\left( s \right) = {{G\left( s \right)} \over s}.$$ The forced response of the system is
39. Consider the system shown in the figure below. The transfer function $$\frac{{Y\left( z \right)}}{{X\left( z \right)}}$$ of the system is
40. The impulse response of an LTI system can be obtained by
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