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QUESTION: 1

The function y[n] = sin(x[n]) is periodic

Solution:

‘y’ will be periodic only if x attains the same value after some time, T. However, if x is a one-one discrete function, it may not be possible for some x[n].

QUESTION: 2

What is the time period of the function x[n] = exp(jwn)?

Solution:

Using Euler’s rule, exp(2pi*n) = 1 for all integer n. Thus, the answer can be derived.

QUESTION: 3

What is the nature of the following function: y[n] = y[n-1] + x[n]?

Solution:

If the above recursive definition is repeated for all n, starting from 1,2.. then y[n] will be the sum of all x[n] ranging from 1 to n, making it an accumulator system.

QUESTION: 4

A time invariant system is a system whose output

Solution:

A time invariant system’s output should be directly related to the time of the output. There should be no scaling, i.e. y(t) = f(x(t)).

QUESTION: 5

Is the function y[n] = x[n-1] – x[n-4] memoryless?

Solution:

Since the function needs to store what it was at a time 4 units and 1 unit before the present time, it needs memory.

QUESTION: 6

Is the function y[n] = x[n-1] – x[n-56] causal?

Solution:

As the value of the function depends solely on the value of the input at a time presently and/or in the past, it is a causal system.

QUESTION: 7

Is the function y[n] = y[n-1] + x[n] stable in nature?

Solution:

It is BIBO stable in nature, i.e. bounded input-bounded output stable.

QUESTION: 8

If n tends to infinity, is the accumulator function a stable one?

Solution:

The system would be unstable, as the output will grow out of bound at the maximally worst possible case.

QUESTION: 9

We define y[n] = nx[n] – (n-1)x[n]. Now, z[n] = z[n-1] + y[n], is z[n] stable?

Solution:

As we take the sum of y[n], terms cancel out and deem z[n] to be BIBO stable.

QUESTION: 10

We define y[n] = nx[n] – (n-1)x[n]. Now, z[n] = z[n-1] + y[n]. Is z[n] a causal system?

Solution:

As the value of the function depends solely on the value of the input at a time presently and/or in the past, it is a causal system.

QUESTION: 11

What is the value of d[0], such that d[n] is the unit impulse function?

Solution:

The unit impulse function has value 1 at n = 0 and zero everywhere else.

QUESTION: 12

What is the value of u[1], where u[n] is the unit step function?

Solution:

The unit step function u[n] = 1 for all n>=0, hence u[1] = 1.

QUESTION: 13

Evaluate the following function in terms of t: {sum from -1 to infinity:d[n]}/{Integral from 0 to t: u(t)}

Solution:

The numerator evaluates to 1, and the denominator is t, hence the answer is 1/t.

QUESTION: 14

Evaluate the following function in terms of t: {integral from 0 to t}{Integral from -inf to inf}d(t)

Solution:

The first integral is 1, and the overall integral evaluates to t.

QUESTION: 15

The fundamental period of exp(jwt) is

Solution:

The function assumes the same value after t+2pi/w, hence the period would be 2pi/w.

QUESTION: 16

Find the magnitude of exp(jwt). Find the boundness of sin(t) and cos(t).

Solution:

The sin(t)and cos(t) can be found using Euler’s rule.

QUESTION: 17

Find the value of {sum from -inf to inf} exp(jwn)*d[n].

Solution:

The sum will exist only for n = 0, for which the product will be 1.

QUESTION: 18

Compute d[n]d[n-1] + d[n-1]d[n-2] for n = 0, 1, 2.

Solution:

Only one of the values can be one at a time, others will be forced to zero, due to the delta function.

QUESTION: 19

Defining u(t), r(t) and s(t) in their standard ways, are their derivatives defined at t = 0?

Solution:

None of the derivatives are defined at t=0.

QUESTION: 20

Which is the correct Euler expression?

Solution:

Euler rule: exp(jt) = cos(t) + jsin(t).

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