The properties of ROC
The ROC can be identified from X(z) and limited knowledge of the characteristics of x[n]. The relationship between the ROC and the characteristics of the time-domain signal is used to find inverse z-transforms.
It is to be noted that the ROC cannot contain any poles. This is because the ROC is defined as the set of all z for which the z-transform converges. Hence, X(z) must be finite for all z in the ROC. If d is a pole, then |X(d)|= infinity and the z-transform does not converge at the pole. Thus, the pole cannot lie in the region of convergence.
Second, the ROC for a finite-duration signal the entire z-plane, except possibly z = 0 or |z|= infinite (or both). To see this, suppose x[n] is nonzero only on the n1 <= n <= n2, We have
X(z)= Σx[n] z^n.
This sum will converge, provided that each of its terms is finite. If a signal has any non-zero causal components (n2 > 0), then the expression for X(z) will have a term involving z ^ – 1, and thus the ROC cannot include z = 0 If a signal is noncausal, a power of (n1<0), then the expression for X(z) will have a term involving a power of z1 and thus the ROC cannot include |z|= infinity. Conversely if n2 <= 0 then the ROC will include z = 0, If a signal has no nonzero noncausal components (n1 >= 0), then the ROC will include |z|= infinity. This line of reasoning also indicates that x[n] = cδ [n] is the only signal whose ROC is the entire z-plane.
Now consider infinite-duration signals. The condition for convergence is |X(z)| <= infinity.
This sum will converge, provided that each of its terms is finite.
Now consider infinite-duration signals. We may thus write |X(z)|<z|
|X(z)|=| sum n=- infty ^ infty x(n]z^ -n | <= sum n=- infty |x[n]z^ -n | = sum n=- infty ^ infty |x[n]||z|^ -n
The second line follows from the fact that the magnitude of a sum of a complex number is less than or equal to the sum of the individual magnitudes. We obtain the third line by ing that the magnitude of a product equals the product of the magnitudes. Splitting the finite sum into negative- and positive-time portions.
We can note that |X(z)|<= I – (z)+I + (z) . If both I – (z) and I, (z) are finite, then X(2) needs to be finite, too. This clearly requires that |x[n]| be bounded in some way Suppose we can bound |x[n]| by finding the smallest positive constants A… and r + such that
|x[n]|<= A – (r – )^ n n < 0
and
|x[n]|<= A + (r + )^ n , n >= 0
This sum converges if and only if |z|>r. Hence, if r * <|z|<r. , then in both the conditions, it will converge. Note that if, then there are no values of |X(z)| z for which convergence is guaranteed. Now define a left-sided signal as a signal for which x[n] = 0 gal as a signal for which x[n] = for , and a two-sided signal as a signal that has n < 0 n = 0, an * g ^ (1/2) * tan d * e
Infinite duration in both the positive and negative directions. Then, for signals that satisfy the exponential bounds of the given equations. We can draw the following conclusions :
The ROC of a right-sided signal is of the form |z|>r x(n).
The ROC of a left-sided signal is of the form |z|<r x(n).
We are often interested in the behavior of a signal either before or after a given time instant. With such signals, the magnitude of one or more poles determines the region of convergence boundaries. and r. Suppose we have a right-sided signal x[n] = alpha ^ n * u * [n] where a is, in general complex. The z-transform of x[n] has a pole at z = o and the ROCis|z| > |alpha| . Hence, the ROC is the region of the z-plane with a radius greater than the radius of the pole. Likewise, x[n] is the left-sided signal x[n]= alpha^ prime prime u[-n-1] , then the region of convergences is |z| < |alpha| the response of the 2-plane with radius less than the radius of the pole. If a signal consists of a sum of exponentials, then the region of convergence is the intersection of the region of convergences associated with each term; it will have a radius greater than that of the pole of the largest radius associated with right-sided terms and a radius less than that of the pole of smallest radius associated with left-sided terms.
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