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Inverse z of z/(z²-3z+2)

Short answer

X(z)/z partial fractions yield X(z)=z/(z−2)−z/(z−1). Inverting: x(k)=2^k−1, k≥0.

Find the inverse z-transform of z/(z² - 3z + 2).

X(z) = z/(z² − 3z + 2) = z/((z − 1)(z − 2)).

Step 1. Partial fractions on X(z)/z:

X(z)/z = 1/((z − 1)(z − 2)) = A/(z − 1) + B/(z − 2)

  • A = 1/(z − 2) at z = 1 = −1
  • B = 1/(z − 1) at z = 2 = 1

Step 2.

X(z) = z/(z − 2) − z/(z − 1)

Step 3. Using Z⁻¹[z/(z − a)] = aᵏ:

Answer: x(k) = 2ᵏ − 1, k ≥ 0

k    : 0  1  2  3   4
x(k) : 0  1  3  7  15

Check by long division: X(z) = z⁻¹ + 3z⁻² + 7z⁻³ + 15z⁻⁴ + … ✓

✓ Answer checked by Claude Opus 5.5 (high effort)

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