All of these properties of z-transform are applicable for discrete-time signals that have a Z-transform. Meaning these properties of Z-transform apply to any generic signal x(n) for which an X(z) exists. (x(n) X(z)). We will also specify the Region of Convergence of the transform for each of the properties.
Property | Mathematical representation | Exceptions/
ROC |
Linearity | a_{1}x_{1}(n)+a_{2}x_{2}(n) = a_{1}X_{1}(z) + a_{2}X_{2}(z) | At least
ROC_{1}∩ROC_{2} |
Time shifting | x(n-k) z^{-k}X(z) | ROC of x(n-k) |
Scaling | a_{n}x(n) x(z/a) | If r_{1} <|z|< r_{2},
then |a|r_{1}<|z|<|a|r_{2} |
Time reversal | x(-n) x(1/z) | 1/r_{2}<|z|<1/r_{1} |
Differentiation
in Z-domain or Multiplication by n |
n^{k}x(n) [-1]^{k}z^{k} | ROC = All R |
Convolution | x(n)*h(n) x(Z)*h(Z) | At least
ROC_{1}∩ROC_{2} |
Correlation | x(n)⊗y(n) x(Z).y(Z^{-1}) | |
Initial Value theorem
in Z-transform |
x(0) = x(Z) | For a causal
signal x(n) |
Final Value theorem
in Z-transform |
x() = x(Z)(1-Z^{-1}) | For a causal
signal x(n) |
Conjugation | x*(n) x*(Z*) | ROC of x(n) |
Parseval’s
Theorem |
x(n).h*(n) = F(z)F*(z) dz
Parseval’s relation tells us that the energy of a signal is equal to the energy of its Fourier transform. |