Properties of z- transform:
Linearity:
$$ ax(n)+by(n) \longleftrightarrow aX(Z)+bY(Z) $$
Time Shifting:
$$ x(n−m) \longleftrightarrow z^{−m}X(Z) $$
Scaling in Z domain:
$$ a^n\cdot x(n) \longleftrightarrow X \left(\frac Za\right) $$
Time Reversal:
$$ x(−n) \longleftrightarrow X \left(\frac 1Z\right) $$
Differentiation in Z-Domain:
$$ n^kx(n) \longleftrightarrow [-1]^k z^k \frac {d^k X(Z)}{dZ^k} $$
Convolution:
$$ x(n)\ast y(n) \longleftrightarrow X(Z)\cdot Y(Z) $$
Correlation:
$$ x(n) \otimes y(n) \longleftrightarrow X(Z)\cdot Y(Z^{−1}) $$
Example:
Find the Z-Transform of the convolution of the two sequences x1(n) = 3 δ(n) + 2 δ(n-1), x2(n) = 2 δ(n) – δ(n-1)
Solution:
Applying Z-transform on the two sequences,
$$ X_1(z) = 3+ 2 Z^{-1} $$
$$ X_2(z) = 2- Z^{-1} $$
Therefore we get,
$$ X(z)= X_1(z)X_2(z) = (3+ 2 Z^{-1})(2- Z^{-1}) $$
$$ X(z)= 6+ Z^{-1} -2 Z^{-2} $$