Properties of Z-Transforms

1.Linearity Property:-

x_{1}[n]\leftrightarrow X_{1}(Z) \ \ \ ROC :a_{1}< \left | Z \right | <b_{1}-R_{1}

x_{2}[n]\leftrightarrow X_{2}(Z) \ \ \ ROC :a_{2}< \left | Z \right | <b_{2}-R_{2}

a\ x_{1}[n]+b\ x_{2}[n]\leftrightarrow \ \ ?.

we know that  Bi-lateral Z- Transform of a signal  x[n]  is X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

Z\left \{a\ x_{1}[n]+b\ x_{2}[n]\right \} = \sum_{n=-\infty }^{\infty } \ \ \left \{ a\ x_{1}[n]+b\ x_{2}[n] \right \} Z^{-n}.

Z\left \{a\ x_{1}[n]+b\ x_{2}[n]\right \} = \sum_{n=-\infty }^{\infty } \ \ \left \{ a\ x_{1}[n]\ Z^{-n}+b\ x_{2}[n] \ Z^{-n}\right \}

Z\left \{a\ x_{1}[n]+b\ x_{2}[n]\right \} = \ a\ \sum_{n=-\infty }^{\infty } x_{1}[n]\ Z^{-n}+b \ \sum_{n=-\infty }^{\infty } x_{2}[n] \ Z^{-n}.

Z\left \{a\ x_{1}[n]+b\ x_{2}[n]\right \} = \ a \ X_{1}(Z)+b \ X_{2}(Z) .          ROC: R_{1} \cap R_{2}.

2.Time-shifting Property:-

x[n]\leftrightarrow X(Z) \ \ \ ROC : R

x[n-k]\leftrightarrow \ ?.

we know that  X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

Z\left \{ x[n-k] \right \} = \sum_{n=-\infty }^{\infty } x[n-k] \ Z^{-n}.    Let     n-k=m \Rightarrow m= n+k

Here n is a variable and k is a constant.

Z\left \{ x[n-k] \right \} = \sum_{m\ =-\infty }^{\infty } x[m] \ Z^{-(m+k)} .

Z\left \{ x[n-k] \right \} = Z^{-k} \sum_{m\ =-\infty }^{\infty } x[m] \ Z^{-m}.

Z\left \{ x[n-k] \right \} = Z^{-k} X(Z)

x[n-k]\leftrightarrow \ Z^{-k} X(Z)\ , \ \ ROC:R.

from the above equation x[n-k]  forms Z Transform pair with Z^{-k} X(Z).

3. Scaling  in-Z-domain property:-

x[n]\leftrightarrow X(Z) \ \ \ ROC :r_{1}< \left | Z \right | <r_{2}-R_{1}

a^{n}x[n]\leftrightarrow \ \ \ ? ,

we know that  X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

Z\left \{ a^{n}\ x[n] \right \} = \sum_{n=-\infty }^{\infty }a^{n}\ x[n] \ Z^{-n}.

Z\left \{ a^{n}\ x[n] \right \} = \sum_{n=-\infty }^{\infty }\ x[n] \ (a^{-1}Z)^{-n} .

Z\left \{ a^{n}\ x[n] \right \} = X(a^{-1}Z)

a^{n}x[n]\leftrightarrow \ X(\frac{Z}{a}), \ \ \ if \ a>0.   \ \ ROC \ of \ a^{n}x[n] \ is :\left | a \right |\ r_{1}< \left | Z \right | <\left | a \right |\ r_{2} .

4. Time-reversal property:-

x[n]\leftrightarrow X(Z) \ \ \ ROC :r_{1}< \left | Z \right | <r_{2}-R_{1}

x[-n]\leftrightarrow \ \ ?.

we know that  X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

Z\left \{ x[-n] \right \} = \sum_{n=-\infty }^{\infty } x[-n] \ Z^{-n} .  Let  -n=\ m ,

Z\left \{ x[-n] \right \} = \sum_{m=-\infty }^{\infty } x[m] \ (Z^{-1})^{-m}\ \ \ ROC :r_{1}< \left | Z^{-1} \right | <r_{2}.

x[-n]\leftrightarrow \ X(Z^{-1})\ \ \ ROC :\frac{1}{\left | r_{2} \right |}< \left | Z \right | <\frac{1}{\left | r_{1} \right |}.

from the above equation x[-n]  forms Z Transform pair with X(\frac{1}{Z}).

5. Differentiation in Z-domain:-

x[n]\leftrightarrow X(Z) \ \ \ ROC :r_{1}< \left | Z \right | <r_{2}-R_{1}

n\ x[n]\leftrightarrow \ \ ?.

we know that  X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

\frac{d(X(Z))}{dZ} = \sum_{n=-\infty }^{\infty } x[n] \frac{d (Z^{-n})}{dZ} .

\frac{d(X(Z))}{dZ} = \sum_{n=-\infty }^{\infty } \ -n \ x[n] Z^{(-n-1)} .

\frac{d(X(Z))}{dZ} = -\left  \ Z^{-n}\right ] Z^{-1} .

\frac{d(X(Z))}{dZ} = - Z^{-1} \ Z\left \{ n\ x[n] \right \} .

from the above equation \frac{d(X(Z))}{dZ}  forms Z-Transform pair with n\ x[n]  and the ROC is same as that of the original sequence x[n].

6. Convolution in Time-domain:-

x_{1}[n]\leftrightarrow X_{1}(Z) \ \ \ ROC :a_{1}< \left | Z \right | <b_{1}-R_{1}

x_{2}[n]\leftrightarrow X_{2}(Z) \ \ \ ROC :a_{2}< \left | Z \right | <b_{2}-R_{2}

x_{1}[n]* x_{2}[n]\leftrightarrow \ \ ?.

we know that  X(Z) = \sum_{n=-\infty }^{\infty } x[n] \ Z^{-n} .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = \sum_{n=-\infty }^{\infty }(x_{1}[n] *x_{2}[n]) \ Z^{-n} .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = \sum_{k=-\infty }^{\infty }(x_{1}[k] x_{2}[n-k]) \sum_{n=-\infty }^{\infty }\ Z^{-n} .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = \sum_{k=-\infty }^{\infty }x_{1}[k] \sum_{n=-\infty }^{\infty }x_{2}[n-k] \ Z^{-n} .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = \sum_{k=-\infty }^{\infty }x_{1}[k] \ Z^{-k}\ X_{2}(Z) .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = \left  \ Z^{-k} \right ]\ X_{2}(Z) .

Z\left \{ x_{1}[n] *x_{2}[n]\right \} = X_{1}(Z) \ X_{2}(Z) .

 

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Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.