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Table of Common z-Transforms

Module by: Melissa Selik, Richard Baraniuk. E-mail the authors

Summary: Lists the z-transform and region of convergence (ROC) for several common discrete-time signals.

Note: You are viewing an old version of this document. The latest version is available here.

The table below provides a number of unilateral and bilateral z-transforms. The table also specifies the region of convergence.

note:

The notation for z z found in the table below may differ from that found in other tables. For example, the basic z-transform of u⁢n u n can be written as either of the following two expressions, which are equivalent:
zz−1=11−z-1 z z 1 1 1 z -1
(1)

Table 1
Signal Z-Transform ROC
δ⁢n−k δ n k z−k z k All⁢z All z
u⁢n u n zz−1 z z 1 |z|>1 z 1
−u⁢(−n)−1 u n 1 zz−1 z z 1 |z|<1 z 1
n⁢u⁢n n u n zz−12 z z 1 2 |z|>1 z 1
n2⁢u⁢n n 2 u n z⁢(z+1)z−13 z z 1 z 1 3 |z|>1 z 1
n3⁢u⁢n n 3 u n z⁢(z2+4⁢z+1)z−14 z z 2 4 z 1 z 1 4 |z|>1 z 1
(−αn)⁢u⁢(−n)−1 α n u n 1 zz−α z z α |z|<|α| z α
αn⁢u⁢n α n u n zz−α z z α |z|>|α| z α
n⁢αn⁢u⁢n n α n u n α⁢zz−α2 α z z α 2 |z|>|α| z α
n2⁢αn⁢u⁢n n 2 α n u n α⁢z⁢(z+α)z−α3 α z z α z α 3 |z|>|α| z α
∏ k =1mn−k+1αm⁢m!⁢αn⁢u⁢n k 1 m n k 1 α m m α n u n zz−αm+1 z z α m 1
γn⁢cosα⁢n⁢u⁢n γ n α n u n z⁢(z−γ⁢cosα)z2−(2⁢γ⁢cosα)⁢z+γ2 z z γ α z 2 2 γ α z γ 2 |z|>|γ| z γ
γn⁢sinα⁢n⁢u⁢n γ n α n u n z⁢γ⁢sinαz2−(2⁢γ⁢cosα)⁢z+γ2 z γ α z 2 2 γ α z γ 2 |z|>|γ| z γ

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