Signal processing

41. The Laplace transform of the causal periodic square wave of period T shown in the figure below is It

$$Fleft( s ight) = rac{1}{{1 + {e^{ - rac{{sT}}{2}}}}}$$
$$Fleft( s ight) = rac{1}{{sleft( {1 + {e^{ - rac{{sT}}{2}}}} ight)}}$$
$$Fleft( s ight) = rac{1}{{sleft( {1 - {e^{ - rac{{sT}}{2}}}} ight)}}$$
$$Fleft( s ight) = rac{1}{{1 - {e^{ - sT}}}}$$

Detailed SolutionThe Laplace transform of the causal periodic square wave of period T shown in the figure below is It

42. A function is given by f(t) = sin2t + cos 2t. Which of the following is true?

$$f$$ has frequency components at 0 and $${1 over {2pi }}Hz$$
$$f$$ has frequency components at 0 and $${1 over pi }Hz$$
$$f$$ has frequency components at $${1 over {2pi }}$$ and $${1 over pi }Hz$$
$$f$$ has frequency components at 0, $${1 over {2pi }}$$ and $${1 over pi }Hz$$

Detailed SolutionA function is given by f(t) = sin2t + cos 2t. Which of the following is true?

43. Letx(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, P3, P4. Properties P1 : Linear but NOT time-invariant P2 : Time-invariant but NOT linear P3 : Linear and time-invariant Relations R1 : y(t) = t2x(t) R2 : y(t) = t |x(t)| R3 : y(t) = |x(t)| R4 : y(t) = x(t – 5)

(P1, R1), (P2, R3), (P3, R4)
(P1, R2), (P2, P3), (P3, R4)
(P1, R3), (P2, R1), (P3, R2)
(P1, R1), (P2, R2), (P3, R3)

Detailed SolutionLetx(t) be the input and y(t) be the output of a continuous time system. Match the system properties P1, P2 and P3 with system relations R1, R2, P3, P4. Properties P1 : Linear but NOT time-invariant P2 : Time-invariant but NOT linear P3 : Linear and time-invariant Relations R1 : y(t) = t2x(t) R2 : y(t) = t |x(t)| R3 : y(t) = |x(t)| R4 : y(t) = x(t – 5)

44. If G(f) represents the Fourier transform of a signal g(t) which is real and odd symmetric in time, then

G(f) is complex
G(f) is imaginary
G(f) is real
G(f) is real and non-negative

Detailed SolutionIf G(f) represents the Fourier transform of a signal g(t) which is real and odd symmetric in time, then

45. 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

-0.5
0
0.25
0.5

Detailed SolutionThe 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

46. A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. y(t) + 5y(t) = u(t) When y(0) = 1 and u(t) is a unit step function, y(t) is

0.2 + 0.8e-5t
0.2 - 0.2e-5t
0.8 + 0.2e-5t
0.8 - 0.8e-5t

Detailed SolutionA system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system. y(t) + 5y(t) = u(t) When y(0) = 1 and u(t) is a unit step function, y(t) is

47. Consider a system whose input r and output y are related by the equation $$y\left( t \right) = \int\limits_{ – \infty }^\infty {x\left( {t – \tau } \right)} h\left( {2\tau } \right)d\tau $$ Where h(t) is shown in the graph Which of the following four properties are possessed by the system? BIBO: Bounded input gives a bounded output Causal: The system is causal. LP : The system is low pass. LTI: The system is linear and time-invariant.

Causal, LP
BIBO, LTI
BIBO, Causal, LTI
LP, LTI

Detailed SolutionConsider a system whose input r and output y are related by the equation $$y\left( t \right) = \int\limits_{ – \infty }^\infty {x\left( {t – \tau } \right)} h\left( {2\tau } \right)d\tau $$ Where h(t) is shown in the graph Which of the following four properties are possessed by the system? BIBO: Bounded input gives a bounded output Causal: The system is causal. LP : The system is low pass. LTI: The system is linear and time-invariant.

48. If the impulse response of a discrete-time system is h[n] = -5nu[- n – 1], then the system function H(z) is equal to

$${{ - z} over {z - 5}}$$ and the system is stable
$${z over {z - 5}}$$ and the system is stable
$${{ - z} over {z - 5}}$$ and the system is unstable
$${z over {z - 5}}$$ and the system is unstable

Detailed SolutionIf the impulse response of a discrete-time system is h[n] = -5nu[- n – 1], then the system function H(z) is equal to

49. Suppose x[n] is an absolutely summable discrete- time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ±2j. Which one of the following statements is TRUE for the signal x[n]?

It is a finite duration signal
It is a causal signal
It is a non-causal signal
It is a periodic signal

Detailed SolutionSuppose x[n] is an absolutely summable discrete- time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ±2j. Which one of the following statements is TRUE for the signal x[n]?

50. The ACF of a rectangular pulse of duration T is

A rectangular pulse of duration T
A rectangular pulse of duration 2T
A triangular pulse of duration T
A triangular pulse of duration 2T

Detailed SolutionThe ACF of a rectangular pulse of duration T is

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