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The ZTransform X(z) of a discrete time signal x(n) is defined as:
Explanation: The ztransform 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).
Explanation: We know that, for a given signal x(n) the ztransform 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) = 2z^{2} + 4z + 5 +7z^{1} + z^{3}.
Explanation: We know that, the ztransform of a signal x(n) is
Given x(n)= δ(nk)=1 at n=k
=> X(z)=z^{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 zplane, except at z=0.
Explanation: For a given signal x(n), its ztransform
Explanation:
Let x(n)= α^{n}u(n)
Given x(n) = δ(n + 3)
We know that δ(n + 3) =
What is the ROC of the ztransform of the signal x(n)= a^{n}u(n)+b^{n}u(n1)?
Explanation: We know that,
ROC of ztransform of a<sup>n</sup>u(n) is z>a. ROC of ztransform of b<sup>n</sup>u(n1) is z<b. By combining both the ROC's we get the ROC of ztransform of the signal x(n) as a<z<b
What is the ROC of ztransform of finite duration anticausal sequence?
Explanation: Let us an example of anti causal sequence whose ztransform will be in the form X(z)=1+z+z^{2} which has a finite value at all values of ‘z’ except at z=∞.So, ROC of an anticausal sequence is entire zplane except at z=∞.
What is the ROC of ztransform of an two sided infinite sequence?
Explanation: Let us plot the graph of ztransform 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 ztransform 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 ztransform of the signal given in the questions contains both the parts, it is called as Bilateral ztransform.
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,
Explanation: Since the value of ztransform tends to infinity, the ROC of the ztransform does not contain poles.
Is the discrete time LTI system with impulse response h(n)=a^{n}(n) (a < 1) BIBO stable?
Explanation:
Given h(n)= a<sup>n</sup>(n) (a<1) The ztransform of h(n) is H(z)=z/(za),ROC is z>a If a<1, then the ROC contains the unit circle. So, the system is BIBO stable
Explanation: The ROC of causal infinite sequence is of form z>r1 where r1 is largest magnitude of poles.
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