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QUESTION: 1

Solution:

QUESTION: 2

X(z) = ln(1 + z^{-1}), |z| > 0

Solution:

QUESTION: 3

If z-transform is given by

X(z) = cos(z^{-3}), |z| > 0

The value of x[12] is

Solution:

QUESTION: 4

X(z) of a system is specified by a pole zero pattern in fig.

Solution:

All gives the same z transform with different ROC. So all are the solution.

QUESTION: 5

Consider three different signal

fig.shows the three different region. Choose the correct option for the ROC of signal

R_{1} , R_{2} , R_{3}

Solution:

x_{1}[n] is right-sided signal

QUESTION: 6

Given

For three different ROC consider there different solution of signal x[n] :

Solution:

QUESTION: 7

X(z) has poles at z =1/2 and z =-1.If x [1] = 1 x [-1] = 1, and the ROC includes the point z = 34. The time signal x[n] is

Solution:

Since the ROC includes the z = 3/4, ROC is

QUESTION: 8

x[n] is right-sided, X (z) has a signal pole, and x[0] = 2, x[2] = 1/2. x[n] is

Solution:

QUESTION: 9

The z-transform function of a stable system is given as

The impuse response h[n] is

Solution:

h[n] is stable, so ROC includes |z| = 1

QUESTION: 10

Let x[n] = δ[n-2] + δ[n+2] The unilateral z-transform is

Solution:

QUESTION: 11

The unilateral z-transform of signal x[n] = u[ n + 4 ] is

Solution:

QUESTION: 12

The z-transform of a signal x[n] is given by

If X (z) converges on the unit circle, x[n] is

Solution:

Since X(z) converges on |z| = 1. So ROC must include this circle.

QUESTION: 13

The transfer function of a system is given as

The h[n] is

Solution:

So system is both stable and causal. ROC includes z = 1.

QUESTION: 14

The transfer function of a system is given as

Consider the two statements

Statement(i) : System is causal and stable.

Statement(ii) : Inverse system is causal and stable.

The correct option is

Solution:

For this system and inverse system all poles are inside |z| = 1. So both system are both causal and stable.

QUESTION: 15

The impulse response of a system is given by

For this system two statement are

Statement (i): System is causal and stable

Statement (ii): Inverse system is causal and stable.

The correct option is

Solution:

Pole of this system are inside |z| = 1. So the system is stable and causal.

For the inverse system not all pole are inside |z| = 1. So inverse system is not stable and causal

QUESTION: 16

The system

is stable if

Solution:

QUESTION: 17

Consider the following three systems

The equivalent system are

Solution:

So y_{1 }and y_{2} are equivalent.

QUESTION: 18

The z-transform of a causal system is given as

The x[0] is

Solution:

QUESTION: 19

The z-transform of a anti causal system is

The value of x[0] is

Solution:

QUESTION: 20

Given the z-transforms

The limit of x[ ∞] is

Solution:

The function has poles at z = 1,3/4. Thus final value theorem applies.

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