Signals and Systems For Dummies
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The ZT doesn’t converge for all sequences. When it does converge, it’s only over a region of the z-plane. The values in the z-plane for which the ZT converges are known as the region of convergence (ROC).

Convergence of the ZT requires that

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The right side of this equation shows that x[n]r-n is absolutely summable (the sum of all terms |x[n]rn| is less than infinity). This condition is consistent with the absolute summability condition for the DTFT to converge to a continuous function of

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Convergence depends only on |z| = r, so if the series converges for z = z1, then the ROC also contains the circle |z| = |z1|. In this case, the general ROC is an annular region in the z-plane, as shown. If the ROC contains the unit circle, the DTFT exists because the DTFT is the ZT evaluated on the unit circle.

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The ROC has important implications when you’re working with the ZT, especially the two-sided ZT. When the ZT produces a rational function, for instance, the roots of the denominator polynomial are related to the ROC. And for LTI systems having a rational ZT, the ROC is related to a system’s bounded-input bounded-output (BIBO) stability. The uniqueness of the ZT is also ensured by the ROC.

Consider the right-sided sequence

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The term right-sided means that the sequence is 0 for

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and the nonzero values extend from n0 to infinity. The value of n0 may be positive or negative.

To find X(z) and the ROC, follow these steps:

  1. Reference the definition to determine the sum:

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  2. Find the condition for convergence by summing the infinite geometric series:

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    Thus, the ROC is |z| > |a|.

  3. To find the sum of Step 1 in closed form, use the finite geometric series sum formula:

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    The geometric series convergence condition corresponds with the ROC.

Consider the left-sided sequence xb[n] = –anu[–n–1]. The term left-sided means that the sequence is 0 for

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To find X(z) and the ROC, write the definition and the infinite geometric series:

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If the sum looks unfamiliar in its present form, you can change variables in the sum, an action known as re-indexing the sum, by letting m = –n, then in the limits

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About This Article

This article is from the book:

About the book author:

Mark Wickert, PhD, is a Professor of Electrical and Computer Engineering at the University of Colorado, Colorado Springs. He is a member of the IEEE and is doing real signals and systems problem solving as a consultant with local industry.

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