Consider the following statements for continuous-time linear time invariant (LTI) systems. 1. There is no bounded input bounded output (BIBO) stable system with a pole in the right half of the complex plane. 2. There is no causal and BIBO stable system with a pole in the right half of the complex plane. Which one among the following is correct?

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.

A

Causal, LP

B

BIBO, LTI

C

BIBO, Causal, LTI

D

LP, LTI

The transfer function of a discrete time LTI system is given by
$$H\left( z \right) = {{2 - {3 \over 4}{z^{ - 1}}} \over {1 - {3 \over 4}{z^{ - 1}} + {1 \over 8}{z^{ - 2}}}}$$
Consider the following statements:
S₁ : The system is stable and causal for
$$ROC:\left| z \right| > {1 \over 2}$$
S₂ : The system is stable but not causal for
$$ROC:\left| z \right| S₃ : The system is neither stable nor causal for
$$ROC:{1 \over 4} Which one of the following statements is valid?

A

Both S1 and S2 are true

B

Both S2 and S3 true

C

Both S1 and S3 are true

D

S1, S2 and S3 are all true

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)

A

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

B

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

C

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

D

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

A stable linear time invariant (LTI) system has a transfer function $$H\left( s \right) = {1 \over {{s^2} + s - 6}}.$$ To make this system causal it needs to be cascaded with another LTI system having a transfer function H₁(s). A correct choice for H₁(s) among the following options is

A

s + 3

B

s - 2

C

s - 6

D

s + 1

The impulse response functions of four linear systems S₁, S₂, S₃, S₄ are given respectively by
$${h_1}\left( t \right) = 1,$$ $${h_2}\left( t \right) = u\left( t \right),$$ $${h_3}\left( t \right) = \frac{{u\left( t \right)}}{{t + 1}},$$ $${h_4}\left( t \right) = {e^{ - 3t}}u\left( t \right)$$
Where u(t) is the unit step function. Which of these systems is time invariant, causal, and stable?

A

S1

B

S2

C

S3

D

S4

Let h(t) denote the impulse response of a causal system with transfer function $${1 \over {s + 1}}.$$ Consider the following three statements:
S₁ : The system is stable.
S₂ : $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}$$ independent of t for t > 0.
S₃ : A non-causal system with the same transfer function is stable.
For the above system,

A

Only S1 and S2 are true

B

Only S2 and S3 are true

C

Only S1 and S3 are true

D

S1, S2 and S3 are true

A linear time invariant system is said to be BIBO stable if and only if the ROC of the system function _____________

A

Includes unit circle

B

Excludes unit circle

C

Is an unit circle

D

None of the mentioned

Input x(t) and output y(t) of an LTI system are related by the differential equation y"(t) - y'(t) - 6y(t) = x(t). If the system is neither causal nor stable, the impulse response h(t) of the system is

A

$${1 \over 5}{e^{3t}}u\left( { - t} \right) + {1 \over 5}{e^{ - 2t}}u\left( { - t} \right)$$

B

$$ - {1 \over 5}{e^{3t}}u\left( { - t} \right) + {1 \over 5}{e^{ - 2t}}u\left( { - t} \right)$$

C

$${1 \over 5}{e^{3t}}u\left( { - t} \right) - {1 \over 5}{e^{ - 2t}}u\left( t \right)$$

D

$$ - {1 \over 5}{e^{3t}}u\left( { - t} \right) - {1 \over 5}{e^{ - 2t}}u\left( t \right)$$

Consider the following statements regarding a linear discrete-time system: H (z) = z2+1/(z+0.5)(z-0.5) 1. The system is stable 2. The initial value of h(0) of the impulse response is -4 3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency Which of these statements are correct?

A

1,2 and 3

B

1 and 2

C

1 and 3

D

2 and 3

The impulse response and the excitation function of a linear time invariant causal system are shown in figure (a) and (b) respectively. The output of the system at t = 2 sec is equal to

A

0

B

$$\frac{1}{2}$$

C

$$\frac{3}{2}$$

D

1