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))
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. |