# Find the Z-transform of x(n) = a^|n|; |a|<1.

+1 vote
Find the Z-transform of x(n) = a^|n|; |a|<1.

(a) $\frac{z}{z-a} – \frac{z}{z-(1/a)}$

(b) $\frac{z}{z-(1/a)} – \frac{z}{z-a}$

(c) $\frac{z}{z-a} + \frac{z}{z-(1/a)}$

(d) $\frac{1}{z-a} – \frac{1}{z-(1/a)}$

The question was posed to me in unit test.

I'd like to ask this question from The Z-Transform topic in section Z-Transform and Digital Filtering of Signals and Systems

## 1 Answer

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by (42.1k points)
The correct answer is (a) $\frac{z}{z-a} – \frac{z}{z-(1/a)}$

Best explanation: a^^|n| = a^n u(n) + a^-n u(-n-1) = a^n u(n) + $(\frac{1}{a})^n$ u(-n-1)

Z[a^|n|] = Z[a^n u(n)] + Z[$(\frac{1}{a})^n$ u(-n-1)] = $\frac{z}{z-a} – \frac{z}{z-(1/a)}$.

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