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Question 9.5: z transform using the differentiation property Using the z-d......

z transform using the differentiation property

Using the z-domain differentiation property, show that the z transform of n\mathbf{u}[n]\mathbf{is}{\frac{z}{(z-1)^{2}}},\ \mathbf{|z|\gt }.

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Start with

\mathbf{u}[n]\overset{\mathcal Z}{\longleftrightarrow }{\frac{z}{z-1}},\;\;|z|\gt 1.

Then, using the z-domain differentiation property,

-n\,u[n]\overset{\mathcal Z}{\longleftrightarrow }z\frac{d}{d z}\left\lgroup\frac{ z}{z-1}\right\rgroup =-\frac{z}{(z-1)^{2}},~|z|\gt 1

or

nu[n]\overset{\mathcal Z}{\longleftrightarrow }{\frac{z}{(z-1)^{2}}},~|z|\gt 1.

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