- 2075 Baisakh · 1 mark
Define the z-transform
Short answer
The z-transform of a sequence x(k) is X(z)=Σ_{k=−∞}^{∞} x(k)z^{−k}, with complex z. For causal sequences, one-sided sum from k=0. For sampled signals, x(k)=x(kT). Example: a^k u(k) ↔ z/(z−a).
Define z-transform.
The z-transform of a discrete-time sequence x(k) (or of the samples x(kT) of a signal x(t)) is the function
X(z) = Z[x(k)] = Σ (k = −∞ to ∞) x(k) z⁻ᵏ
where z is a complex variable. For a causal sequence (x(k) = 0 for k < 0), the one-sided transform is X(z) = Σ (k = 0 to ∞) x(k) z⁻ᵏ. For a sampled signal, x(k) is replaced by x(kT), with T the sampling period.
Example: for x(k) = aᵏ, k ≥ 0, X(z) = Σ (a/z)ᵏ = z/(z − a), |z| > |a|.
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