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The Z-Transform X(z) of a discrete time signal x(n) is defined as:
Explanation: The z-transform of a real discrete time sequence x(n) is defined as a power of ‘z’ which is equal to , where ‘z’ is a complex variable.
What is the set of all values of z for which X(z) attains a finite value?
Explanation: Since X(z) is a infinite power series, it is defined only at few values of z. The set of all values of z where X(z) converges to a finite value is called as Radius of Convergence(ROC).
What is the z-transform of the finite duration signal
Explanation: We know that, for a given signal x(n) the z-transform is defined as
Substitute the values of n from -2 to 3 and the corresponding signal values in the above formula
We get, X(z) = 2z2 + 4z + 5 +7z-1 + z-3.
What is the ROC of the signal x(n)=δ(n-k),k>0?
Explanation: We know that, the z-transform of a signal x(n) is
Given x(n)= δ(n-k)=1 at n=k
From the above equation, X(z) is defined at all values of z except at z=0 for k>0.
So ROC is defined as Entire z-plane, except at z=0.
What is the z-transform of the signal x(n)=(0.5)nu(n)?
Explanation: For a given signal x(n), its z-transform
Which of the following series has an ROC as mentioned below?
Let x(n)= αnu(n)
Find the Z-transform of δ(n + 3).
Given x(n) = δ(n + 3)
We know that δ(n + 3) =
What is the ROC of the z-transform of the signal x(n)= anu(n)+bnu(-n-1)?
Explanation: We know that,
ROC of z-transform of a<sup>n</sup>u(n) is |z|>|a|. ROC of z-transform of b<sup>n</sup>u(-n-1) is |z|<|b|. By combining both the ROC's we get the ROC of z-transform of the signal x(n) as |a|<|z|<|b|
What is the ROC of z-transform of finite duration anti-causal sequence?
Explanation: Let us an example of anti causal sequence whose z-transform will be in the form X(z)=1+z+z2 which has a finite value at all values of ‘z’ except at z=∞.So, ROC of an anti-causal sequence is entire z-plane except at z=∞.
What is the ROC of z-transform of an two sided infinite sequence?
Explanation: Let us plot the graph of z-transform of any two sided sequence which looks as follows.
From the above graph, we can state that the ROC of a two sided sequence will be of the form r2 < |z| < r1.
The z-transform of a sequence x(n) which is given as is known as:
Explanation: The entire timing sequence is divided into two parts n=0 to ∞ and n=-∞ to 0.
Since the z-transform of the signal given in the questions contains both the parts, it is called as Bi-lateral z-transform.
What is the ROC of the system function H(z) if the discrete time LTI system is BIBO stable?
Explanation: A discrete time LTI is BIBO stable, if and only if its impulse response h(n) is absolutely summable. That is,
The ROC of z-transform of any signal cannot contain poles.
Explanation: Since the value of z-transform tends to infinity, the ROC of the z-transform does not contain poles.
Is the discrete time LTI system with impulse response h(n)=an(n) (|a| < 1) BIBO stable?
Given h(n)= a<sup>n</sup>(n) (|a|<1) The z-transform of h(n) is H(z)=z/(z-a),ROC is |z|>|a| If |a|<1, then the ROC contains the unit circle. So, the system is BIBO stable
What is the ROC of a causal infinite length sequence?
Explanation: The ROC of causal infinite sequence is of form |z|>r1 where r1 is largest magnitude of poles.