What is the ROC for X1(z)?
Understand the Problem
The question is asking about the Region of Convergence (ROC) for a function X1(z), which usually appears in the context of signal processing or control theory. The ROC is important in determining the convergence properties of a series or transform related to the function.
Answer
The ROC for X1(z) is where the z-transform converges.
The ROC for a given z-transform, such as X1(z), is the set of values of z for which the z-transform converges. It defines the region in the z-plane where the corresponding sequence representation exists.
Answer for screen readers
The ROC for a given z-transform, such as X1(z), is the set of values of z for which the z-transform converges. It defines the region in the z-plane where the corresponding sequence representation exists.
More Information
The region of convergence (ROC) is crucial as it determines whether a z-transform provides a meaningful and unique representation of the sequence it transforms. Its definition depends on the sequence behavior - distinct types, such as stable or causal, have specific ROC characteristics.
Tips
A common mistake is confusing the ROC with the range of the z-transform itself. The ROC is about convergence, not the values the transform takes.
Sources
- 12.6: Region of Convergence for the Z-Transform - eng.libretexts.org
- Table 3: Properties of the z-Transform - Henry D. Pfister - pfister.ee.duke.edu
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