Question
Download Solution PDFLet u|n| be the unit-step signal and \(x\left[ n \right] = {\left( {\frac{1}{2}} \right)^n}u\left[ n \right] + {\left( { - \frac{1}{3}} \right)^n}u\left[ n \right]\). The region of convergence of z-transform of x[n] is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
\(Z.T[{a^n}u\left( n \right)] = \frac{1}{{1 - a{z^{ - 1}}}}\)
ROC IzI > IaI
Analysis:
\(x\left[ n \right] = {\left( {\frac{1}{2}} \right)^n}u\left[ n \right] + {\left( { - \frac{1}{3}} \right)^n}u\left[ n \right]\)
\(X\left( z \right) = \frac{1}{{1 - \frac{1}{2}{z^{ - 1}}}} + \frac{1}{{1 + \frac{1}{3}{z^{ - 1}}}}\)
Individual ROCs are IzI > (\(\frac{1}{2}\)) and IzI > (\(\frac{1}{3}\))
For write ROC for the causal system by default.
Hence common ROC for the causal system will be:
IzI > (\(\frac{1}{2}\))
Last updated on Jul 2, 2025
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