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Homework Statement
Using the definition of the z-transform, show that if [tex]X(z)[/tex] is the z-transform of [tex]x(n) = x_{R}(n) +jx_{I}(n)[/tex], then:
[tex]Z\{x^{*}(n)\}=X^{*}(z^{*})[/tex]
Homework Equations
z-tranform definition:
[tex]Z\{x(n)\}=X(z)=\sum x(n)z^{-n}[/tex]
The Attempt at a Solution
[tex]x(n) = x_{R}(n) + jx_{I}(n) \Longrightarrow x^{*}(n) = x_{R}(n) - jx_{I}(n)[/tex]
[tex]Z\{x^{*}(n)\}=Z\{x_{R}(n) - jx_{I}(n)\}=\sum x^{*}(n)z^{-n}[/tex]
[tex]=\sum [x_{R}(n) - jx_{I}(n)]z^{-n}[/tex]
[tex]=\sum [x_{R}(n)z^{-n} - jx_{I}(n)z^{-n}][/tex][tex]=\sum x_{R}(n)z^{-n} - j \sum x_{I}(n)z^{-n}[/tex]
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