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If ztransform is given by
X(z) = cos(z^{3}), z > 0
The value of x[12] is
X(z) of a system is specified by a pole zero pattern in fig.
All gives the same z transform with different ROC. So all are the solution.
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}
x_{1}[n] is rightsided signal
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
Since the ROC includes the z = 3/4, ROC is
x[n] is rightsided, X (z) has a signal pole, and x[0] = 2, x[2] = 1/2. x[n] is
The ztransform function of a stable system is given as
The impuse response h[n] is
h[n] is stable, so ROC includes z = 1
The ztransform of a signal x[n] is given by
If X (z) converges on the unit circle, x[n] is
Since X(z) converges on z = 1. So ROC must include this circle.
So system is both stable and causal. ROC includes z = 1.
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
For this system and inverse system all poles are inside z = 1. So both system are both causal and stable.
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
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
Consider the following three systems
The equivalent system are
So y_{1 }and y_{2} are equivalent.
The ztransform of a anti causal system is
The value of x[0] is
The function has poles at z = 1,3/4. Thus final value theorem applies.
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