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QUESTION: 1

How is the discrete time impulse function defined in terms of the step function?

Solution:

Using the definition of the Heaviside function, we can come to this conclusion

QUESTION: 2

What is the definition of the delta function in time space intuitively?

Solution:

Arises from the definition of the delta function. There is a clear difference between just the functional value and the impulse area of the delta function.

QUESTION: 3

Is it practically possible for us to provide a perfect impulse to a system?

Solution:

The spread of the impulse can never be restricted to a single point in time, and thus, we cannot achieve a perfect impulse.

QUESTION: 4

The convolution of a discrete time system with a delta function gives

Solution:

The integral reduces to the the integral calculated at a single point, determined by the centre of the delta function.

QUESTION: 5

Find the value of 2sgn(0)d[0] + d[1] + d[45], where sgn(x) is the signum function.

Solution:

sgn(0)=0, and d[n] = 0 for all n not equal to zero. Hence the sum reduces to zero.

QUESTION: 6

Where h*x denotes h convolved with x, x[n]*d[n-90] reduces to

Solution:

The function gets shifted by the center of the delta function during convolution.

QUESTION: 7

Where h*x denotes h convolved with x, find the value of d[n]*d[n-1].

Solution:

Using the corollary, if we take d[n] to be the ‘x’ function, it will be shifted by -1 when convolved with d[n-1], thus rendering d[n-1].

QUESTION: 8

How is the continuous time impulse function defined in terms of the step function?

Solution:

Using the definition of the Heaviside function, we can come to this conclusion.

QUESTION: 9

In which of the following useful signals, is the bilateral Laplace Transform different from the unilateral Laplace Transform?

Solution:

The bilateral LT is different from the aspect that the integral is applied for the entire time axis, but the unilateral LT is applied only for the positive time axis. Hence, the u(t) [unit step function] differs in that aspect and hence can be used to differentiate the same.

QUESTION: 10

What is the relation between the unit impulse function and the unit ramp function?

Solution:

Now, d = du/dt and u = dr/dt. Hence, we obtain the above answer.

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