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QUESTION: 1

Is the system y(t) = Rx(t), where R is a arbitrary constant, a memoryless system?

Solution:

The output of the system depends on the input of the system at the same time instant. Hence, the system has to be memoryless.

QUESTION: 2

Does the following discrete system have the parameter of memory, y[n] = x[n-1] + x[n] ?

Solution:

y[n] depends upon x[n-1], i.e at the earlier time instant, thus forcing the system to have memory.

QUESTION: 3

y[t]= ∫x[t],t ranges from 0 to t. Is the system a memoryless one?

Solution:

While evaluating the integral, it becomes imperative to know the values of x[t] from 0 to t, thus making the system requiring memory.

QUESTION: 4

y(t) = sin(x(t-1)) : Comment on its memory aspects.

Solution:

The output at any time t = A, requires knowing the input at an earlier time, t = A – 1, hence making the system require memory aspects.

QUESTION: 5

Construct the inverse system of y(t) = 2x(t)

Solution:

Now, y(t) = 2x(t) => x(t) = 0.5*y(t)

Thus, reversing x(t) <-> y(t), we obtain the inverse system: y(t) = 0.5x(t)

QUESTION: 6

y(t) = x^{2(t)}. Is y(t) = sqrt(x(t)) the inverse of the first system?

Solution:

We cannot determine the sign of the input from the second function, thus, the output doesn’t replicate the input. Thus, the second function is not an inverse of the first one.

QUESTION: 7

Comment on the causality of y[n] = x[-n].

Solution:

For positive time, the system may seem to be causal. However, for negative time, the output depends on time at a positive sign, thus being in the future, enforcing non causality.

QUESTION: 8

y(t) = x(t-2) + x(2-t). Comment on its causality:

Solution:

For a time instant existing between 0 and 1, it would depend on the input at a time in the future as well, hence being non causal.

QUESTION: 9

Comment on the causality of y[n] = n*x[n].

Solution:

For positive time, the system may seem to be causal. For negative time, the output depends on the same time instant, thus making it causal.

QUESTION: 10

Comment on the linearity of y[n] = n*x[n].

Solution:

The function obeys the scaling/homogeneity property, but doesn’t obey the additivity property, thus not being linear.

QUESTION: 11

What is the following type of system called? y[n] = x[n] + y[n-1].

Solution:

If we write for n-1, n-2, .. we will obtain y[n] = x[n] + x[n-1] + x[n-2] …,

thus obtaining an adder system.

QUESTION: 12

Which of the following systems is linear?

Solution:

Only d satisfies both the scaling and the additivity properties.

QUESTION: 13

Which of the following systems is stable?

Solution:

Stability implies that a bounded input should give a bounded output. In a, b, d there are regions of x, for which y reaches infinity/negative infinity. Thus the sin function always stays between -1 and 1, and is hence stable.

QUESTION: 14

Which of the following systems is time invariant?

Solution:

In each of b and c there is a negative sign of t involved, which means a backward shift of t-0 in time, would mean a forward shift in each of them. In option a twice of t leads to time variant. However, only in d, the backward shift will remain as backward, and undiminished.

QUESTION: 15

Which property of delta function indicates the equality between the area under the product of function with shifted impulse and the value of function located at unit impulse instant ?

Solution:

QUESTION: 16

State whether the differentiator system is a stable system or not.

Solution:

The derivative of a function can be unbounded at some bounded inputs, like tan(x) at x=pi/2, hence the differentiator system is unstable in general, when the input is not specified.

QUESTION: 17

Which of the following systems is memoryless?

Solution:

A system possessing no memory has its output depending upon the input at the same time instant, which is prevalent only in option b.

QUESTION: 18

For what value of k, will the following system be time invariant?y(t) = x(t) + x(kt) – x(2t) + x(t-1)

Solution:

A system possessing no memory has its output depending upon the input at the same time instant, which is prevalent only in option b.

QUESTION: 19

State if the following system is periodic or not. y(t) = sin(√(2)*x(t))

Solution:

The function y = sin(nx) is periodic only for rational ‘n’.

QUESTION: 20

State whether the following system is periodic or not. y(t) = log(sin(x(t)).

Solution:

Sin x is a periodic function, but log x is not a periodic function. Thus y is log t, where t= sin x, thus y is not periodic.

QUESTION: 21

Comment on the linearity of y[n] = n*x[n].

Solution:

The function obeys the scaling/homogeneity property, but doesn’t obey the additivity property, thus not being linear.

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