z-transform of aᵏ cos kπ

Short answer

cos(kπ)=(-1)^k, so a^k cos kπ = (-a)^k. Using Z[b^k]=z/(z-b) gives Z[a^k cos kπ] = z/(z+a), |z|>|a|.

Find the z-transform of aᵏ cos kπ for k ≥ 0.

Simplify first. cos kπ = (−1)ᵏ, so

aᵏ cos kπ = (−a)ᵏ.

Use Z[bᵏ] = z/(z − b) with b = −a:

Answer: Z[aᵏ cos kπ] = z/(z + a), for |z| > |a|.

Cross-check with the general formula. Z[aᵏ cos kθ] = z(z − a cos θ)/(z² − 2az cos θ + a²). Put θ = π (cos π = −1, sin π = 0):

z (z + a) / (z² + 2az + a²) = z (z + a) / (z + a)² = z/(z + a)  ✓

✓ Answer checked by Claude Opus 5.5 (high effort)

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