Question 9.2: Inverse z transforms Find the inverse z transforms of (a) X(......
Inverse z transforms
Find the inverse z transforms of
(a) \mathbf{X}(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},\ \ 0.5\lt |z|\lt 2
(b) \mathrm{X}(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},\;|z|\gt 2
(c) X(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},{\big|}z{\big|}\lt 0.5
(a) Right-sided signals have ROCs that are outside a circle and left-sided signals have ROCs that are inside a circle. Therefore, using
\alpha^{n}\,\mathbf{u}[n]\overset{\mathcal Z}{\longleftrightarrow } {\frac{z}{z-\alpha}}={\frac{1}{1-\alpha z^{-1}}},\;\left|z\right|\gt \left|\alpha\right|
and
-\alpha^{n}\mathbf{u}[-n-1]\overset{\mathcal Z}{\longleftrightarrow } {\frac{z}{z-\alpha}}={\frac{1}{1-\alpha z^{-1}}},\;|z|\lt |\alpha|
we get
(0.5)^{n}\,u[n]-(-(-2)^{n}\,u[-n-1])\overset{\mathcal Z}{\longleftrightarrow } X(z)=\frac{z}{z-0.5}-\frac{z}{z+2},\ 0.5\lt |z|\lt 2
or
(0.5)^{n}\,\mathrm{u}[n]+(-2)^{n}\,\mathrm{u}[-n-1]\overset{\mathcal Z}{\longleftrightarrow } \mathrm{X}(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},\;\;0.5\lt |z|\lt 2
(b) In this case both signals are right sided.
[(0.5)^{n}-(-2)^{n}]{u}[n]\overset{\mathcal Z}{\longleftrightarrow } {X}(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},\ \mathbf{{|z|}} >2
(c) In this case both signals are left sided.
-[(0.5)^{n}-(-2)^{n}]\mathbf{u}[-n-1]\overset{\mathcal Z}{\longleftrightarrow } \mathbf{X}(z)={\frac{z}{z-0.5}}-{\frac{z}{z+2}},\;|z|\lt 0.5