Signal processing

61. The unit impulse response of a linear time invariant system is the unit step function u(t). For t > 0, the response of the system to an excitation e-at u(t), a > 0 will be

ae-at
$$left( {{1 over a}} ight)left( {1 - {e^{ - at}}} ight)$$
a(1 - e-at)
1 - e-at

Detailed SolutionThe unit impulse response of a linear time invariant system is the unit step function u(t). For t > 0, the response of the system to an excitation e-at u(t), a > 0 will be

62. The unilateral Laplace transform of f(t) is $${1 \over {{s^2} + s + 1}}.$$ Which one of the following is the unilateral Laplace transform of g(t) = t.f(t)?

$${{ - s} over {{{left( {{s^2} + s + 1} ight)}^2}}}$$
$${{ - left( {2s + 1} ight)} over {{{left( {{s^2} + s + 1} ight)}^2}}}$$
$${s over {{{left( {{s^2} + s + 1} ight)}^2}}}$$
$${{2s + 1} over {{{left( {{s^2} + s + 1} ight)}^2}}}$$

Detailed SolutionThe unilateral Laplace transform of f(t) is $${1 \over {{s^2} + s + 1}}.$$ Which one of the following is the unilateral Laplace transform of g(t) = t.f(t)?

63. A 1 kHz sinusoidal signal is ideally sampled at 1500 samples/sec and the sampled signal is passed through an ideal low-pass filter with cutoff frequency 800 Hz. The output signal has the frequency

Zero Hz
0.75 kHz
0.5 kHz
0.25 kHz

Detailed SolutionA 1 kHz sinusoidal signal is ideally sampled at 1500 samples/sec and the sampled signal is passed through an ideal low-pass filter with cutoff frequency 800 Hz. The output signal has the frequency

68. Let y[n] denote the convolution of h[n] and g[n], where h[n] = $${\left( {\frac{1}{2}} \right)^n}$$ u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = $$\frac{1}{2},$$ then g[1] equals

0
$${1 over 2}$$
1
$${3 over 2}$$

Detailed SolutionLet y[n] denote the convolution of h[n] and g[n], where h[n] = $${\left( {\frac{1}{2}} \right)^n}$$ u[n] and g[n] is a causal sequence. If y[0] = 1 and y[1] = $$\frac{1}{2},$$ then g[1] equals

69. The impulse response h[n] of a linear time-invariant system is given by h[n] = u[n + 3] + u[n – 2] – 2u[n – 7], where u[n] is the unit step sequence. The above system is

Stable but not causal
Stable and causal
Causal but unstable
Unstable and not causal

Detailed SolutionThe impulse response h[n] of a linear time-invariant system is given by h[n] = u[n + 3] + u[n – 2] – 2u[n – 7], where u[n] is the unit step sequence. The above system is

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