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, to , 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 $
$$X(Z) = sum_{n=- infty }^{infty} ,x(n)z^{-n}$$
$$x(n) = {1 over 2pi j} int, X(z) z^{n-1} dz$$
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