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Table of Common z-Transforms

Module by: Melissa Selik, Richard Baraniuk

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

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 un 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)
Signal Z-Transform ROC
δn-k δ n k z-k z k All z All z
un 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
nun n u n zz-12 z z 1 2 |z|>1 z 1
n2un n 2 u n zz+1z-13 z z 1 z 1 3 |z|>1 z 1
n3un n 3 u n zz2+4z+1z-14 z z 2 4 z 1 z 1 4 |z|>1 z 1
-αnu-n-1 α n u n 1 zz-α z z α |z|<|α| z α
αnun α n u n zz-α z z α |z|>|α| z α
nαnun n α n u n αzz-α2 α z z α 2 |z|>|α| z α
n2αnun n 2 α n u n αzz+αz-α3 α z z α z α 3 |z|>|α| z α
∏k=1mn-k+1αmm!αnun k 1 m n k 1 α m m α n u n zz-αm+1 z z α m 1
γncosαnun γ n α n u n zz-γcosαz2-2γcosαz+γ2 z z γ α z 2 2 γ α z γ 2 |z|>|γ| z γ
γnsinαnun γ n α n u n zγsinαz2-2γcosαz+γ2 z γ α z 2 2 γ α z γ 2 |z|>|γ| z γ

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