$X(z) = 1 / ((z-1)^2(z-3))$ where ROC is $|z|>3$.
I've put the equation into partial fractions then I am not sure on how to proceed further with the first term. Any help is appreciated.
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Sign up to join this community$X(z) = 1 / ((z-1)^2(z-3))$ where ROC is $|z|>3$.
I've put the equation into partial fractions then I am not sure on how to proceed further with the first term. Any help is appreciated.
$$X(z)=\frac{1}{(z-1)^2(z-3)}=\frac{1/4}{(z-3)}-\frac{1/4}{z-1}-\frac{1/2}{(z-1)^2},\,|z|>3$$
$$X(z)=0.25z^{-1}\frac{z}{z-3}-0.25z^{-1}\frac{z}{z-1}-0.5z^{-1}\left(\frac{z}{(z-1)^2}\right)$$
Note that:
Hence,
$$x[n]=(0.25)3^{n-1}u[n-1]-(0.25)u[n-1]-(0.5)(n-1)u[n-1]$$