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?

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

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)

An electronic device rearranges numbers step-by-step in a particular order according to set of rules. The device stops when the final result is obtained.
Input: 84 15 35 04 18 96 62 08
Step 1: 96 84 15 35 04 18 62 08
Step 2: 96 84 62 15 35 04 18 08
Step 3: 96 84 62 35 15 04 18 08
Step 4: 96 84 62 35 18 15 04 08
Step 5: 96 84 62 35 18 15 08 04
And Step 5 is the last step for this input. Now, find out an appropriate step in the following question following the above rule. Question:
Input: 16 09 24 28 15 04
What will be the third step for this input ?

A

28 24 15 16 04 09

B

28 24 16 15 09 04

C

24 28 15 16 09 04

D

28 24 16 15 04 09

An electronic device when fed with the numbers, rearranges them in a particular order following certain rules. The following is a step-by-step process of rearrangement for the given input of numbers. Input :- 85 16 36 04 19 97 63 09 Step I :- 97 85 16 36 04 19 63 09 Step II :- 97 85 63 16 36 04 19 09 Step III :- 97 85 63 36 16 04 19 09 Step IV :- 97 85 63 36 19 16 04 09 Step V :- 97 85 63 36 19 16 09 04 (for the given input step V is the last step). Which of the following will be the last step for the given input ? Input :- 03 31 43 22 11 09

A

IV

B

V

C

VI

D

None of these

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

The impulse response h(t) of a linear time-invariant continuous time system is described by h(t) = exp(αt)u(t) + exp(βt)u(-t), where u(t) denotes the unit step function, and α and β are real constants. This system is stable if

A

α is positive and β is positive

B

α is negative and β is negative

C

α is positive and β is negative

D

α is negative and β is positive

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?

A

Both 1 and 2 are true

B

Both 1 and 2 are not true

C

Only 1 is true

D

Only 2 is true

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

A

ae-at

B

$$\left( {{1 \over a}} \right)\left( {1 - {e^{ - at}}} \right)$$

C

a(1 - e-at)

D

1 - e-at

Assertion (A): It is observed that step function is first derivative of a ramp function and impulse function is first derivative of a step function. Reason (R): From the derived time response expression it is concluded that the output time response also follows the same sequence as that of input functions.

A

Both A and R are true and R is the correct explanation of A

B

Both A and R are true but R is not correct explanation of A

C

Both A is True but R is false

D

Both A is False but R is true