Z-Transforms (ZT)

Analysis of continuous time LTI systems can be done using z-transforms. It is a powerful mathematical tool to convert differential equations into algebraic equations.

The bilateral (two sided) z-transform of a discrete time signal x(n) is given as

$Z.T[x(n)] = X(Z) = Sigma_{n = -infty}^{infty} x(n)z^{-n} $

The unilateral (one sided) z-transform of a discrete time signal x(n) is given as

$Z.T[x(n)] = X(Z) = Sigma_{n = 0}^{infty} x(n)z^{-n} $

Z-transform may exist for some signals for which Discrete Time Fourier Transform (DTFT) does not exist.

Concept of Z-Transform and Inverse Z-Transform

Z-transform of a discrete time signal x(n) can be represented with X(Z), and it is defined as

$X(Z) = Sigma_{n=- infty }^ {infty} x(n)z^{-n} ,...,...,(1)$

If $Z = re^{jomega}$ then equation 1 becomes

$X(re^{jomega}) = Sigma_{n=- infty}^{infty} x(n)[re^{j omega} ]^{-n}$

$= Sigma_{n=- infty}^{infty} x(n)[r^{-n} ] e^{-j omega n}$

$X(re^{j omega} ) = X(Z) = F.T[x(n)r^{-n}] ,...,...,(2) $

The above equation represents the relation between Fourier transform and Z-transform.

$ X(Z) |_{z=e^{j omega}} = F.T [x(n)]. $

Inverse Z-transform

$X(re^{j omega}) = F.T[x(n)r^{-n}] $

$x(n)r^{-n} = F.T^{-1}[X(re^{j omega}]$

$x(n) = r^n,F.T^{-1}[X(re^{j omega} )]$

$= r^n {1 over 2pi} int X(re{^j omega} )e^{j omega n} d omega $

$= {1 over 2pi} int X(re{^j omega} )[re^{j omega} ]^n d omega ,...,...,(3)$

Substitute $re^{j omega} = z$.

$dz = jre^{j omega} d omega = jz d omega$

$d omega = {1 over j }z^{-1}dz$

Substitute in equation 3.

$ 3, o , x(n) = {1 over 2pi} int, X(z)z^n {1 over j } z^{-1} dz = {1 over 2pi j} int ,X(z) z^{n-1} dz $