Question

Consider the infinite-length, discrete-time sequence \(x[n]=0.9^{|n|}\), where \(n\) is an integer. The region of convergence of its Z-transform \(X(z)\) is given by:
(Note: \(z\) is a complex variable)

• A. \(|z|>0.9\)
• B. \(|z|<0.9\)
• C. \(0.9<|z|<1/0.9\)
• D. \(\{z\ \text{such that}\ |z|<0.9\}\cup\{z\ \text{such that}\ |z|>1/0.9\}\)

Hint:

Split the two-sided exponential sequence into a right-sided and a left-sided part, find the ROC of each, and intersect them.

Answer 1

The Correct Option is C

Step 1: Split the two-sided sequence into a right-sided and a left-sided part.
Since \(x[n]=0.9^{|n|}\), write \(x[n]=x_1[n]+x_2[n]\), where
\[ x_1[n]=0.9^{n}u[n] \]
is the causal (right-sided) part, and
\[ x_2[n]=0.9^{-n}u[-n-1] \]
is the anticausal (left-sided) part for \(n\leq-1\). At \(n=0\), \(x_2[0]=0\) and \(x_1[0]=1=x[0]\), so the decomposition is exact.
Step 2: Find the Z-transform and ROC of the causal part.
\[ X_1(z)=\sum_{n=0}^{\infty}0.9^{n}z^{-n}=\frac{1}{1-0.9z^{-1}} \]
This geometric series converges when \(|0.9z^{-1}|<1\), that is,
\[ |z|>0.9 \]
Step 3: Find the Z-transform and ROC of the anticausal part.
Writing \(n=-m\) for \(m\geq1\),
\[ X_2(z)=\sum_{n=-\infty}^{-1}0.9^{-n}z^{-n}=\sum_{m=1}^{\infty}(0.9z)^{m} \]
This geometric series converges when \(|0.9z|<1\), that is,
\[ |z|<\frac{1}{0.9} \]
Step 4: Combine the two regions.
Since \(x[n]=x_1[n]+x_2[n]\), the transform \(X(z)\) exists only where BOTH series converge at the same time, so the overall ROC is the intersection of \(|z|>0.9\) and \(|z|<1/0.9\), which gives the ring (annulus)
\[ 0.9<|z|<\frac{1}{0.9} \]
Because \(0.9<1/0.9\), this annulus is non-empty, exactly as expected for a genuinely two-sided decaying sequence.
Step 5: Rule out the other options.
Option (A), \(|z|>0.9\), is only the ROC of the causal half of the signal by itself, not the full two-sided sequence. Option (B), \(|z|<0.9\), is not even the ROC of either half. Option (D) takes a UNION of the two half-plane conditions rather than an intersection; a union would allow points where only one of the two series converges, but the actual \(X(z)\) requires both parts of the real, physically existing two-sided sequence to converge simultaneously, so a union is not correct here.
Step 6: Final Answer.
\[ \boxed{0.9<|z|<1/0.9} \]