Test: The Z-Transform

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

Consider three different signal

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

R₁  , R₂ , R₃

Given

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

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

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

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

The impuse response h[n] is

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

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

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

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

The transfer function of a system is given as

The h[n] is

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

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

The system

is stable if

Consider the following three systems

The equivalent system are

The z-transform of a causal system is given as

The x[0] is

The z-transform of a anti causal system is

The value of x[0] is

Given the z-transforms

The limit of x[ ∞] is