In this topic, you study the Z-Transform Properties as Linearity, Time Scaling, Time Shifting, Multiplication by an Exponential Sequence, Differentiation in z-domain, Time Reversal, and Conjugate symmetry.
Linearity
if
\[a{x_1}[n]{\text{ }} \leftrightarrow a{X_1}[z]{\text{ }}\]
\[b{x_2}[n]{\text{ }} \leftrightarrow b{X_2}[z]{\text{ }}\]
Then
\[a{x_1}[n]{\text{ }}+b{x_2}[n]{\text{ }} \leftrightarrow a{X_1}[z]{\text{ }} + b{X_2}[z]{\text{ }}\]
Multiplication by an exponential Sequence (Scaling in z-domain)
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[z]{\text{ }}\]
Then
\[{a^n}x\left[ n \right] = X({a^{ – 1}}z)\]
Time Shifting
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[z]{\text{ }}\]
Then
\[x\left[ {n – {n_0}} \right] \leftrightarrow {z^{ – {n_0}}}X\left( z \right)\]
Differentiation in z-domain
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[z]{\text{ }}\]
Then
\[nx[n] \leftrightarrow -{\text{ }} z\frac{{dX(z)}}{{dz}}\]
Time Reversal
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[z]{\text{ }}\]
Then
\[x[ – n] \leftrightarrow X\left( {\,\frac{1}{z}} \right)\]
Conjugate Symmetry
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[z]{\text{ }}\]
Then
\[{x^ * }[n] \leftrightarrow {X^ * }({z^ * })\]