Z-Transforms Properties

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Z-Transform has following properties:

Linearity Property

If $\,x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

and $\,y(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} Y(Z)$

Then linearity property states that

$a\, x (n) + b\, y (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} a\, X(Z) + b\, Y(Z)$

Time Shifting Property

If $\,x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

Then Time shifting property states that

$x (n-m) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} z^{-m} X(Z)$

Multiplication by Exponential Sequence Property

If $\,x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

Then multiplication by an exponential sequence property states that

$a^n\, . x(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z/a)$

Time Reversal Property

If $\, x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

Then time reversal property states that

$x (-n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(1/Z)$

Differentiation in Z-Domain OR Multiplication by n Property

If $\, x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

Then multiplication by n or differentiation in z-domain property states that

$ n^k x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} [-1]^k z^k{d^k X(Z) \over dZ^K} $

Convolution Property

If $\,x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

and $\,y(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} Y(Z)$

Then convolution property states that

$x(n) * y(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z).Y(Z)$

Correlation Property

If $\,x (n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z)$

and $\,y(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} Y(Z)$

Then correlation property states that

$x(n) \otimes y(n) \stackrel{\mathrm{Z.T}}{\longleftrightarrow} X(Z).Y(Z^{-1})$

Initial Value and Final Value Theorems

Initial value and final value theorems of z-transform are defined for causal signal.

Initial Value Theorem

For a causal signal x(n), the initial value theorem states that

$ x (0) = \lim_{z \to \infty }⁡X(z) $

This is used to find the initial value of the signal without taking inverse z-transform

Final Value Theorem

For a causal signal x(n), the final value theorem states that

$ x ( \infty ) = \lim_{z \to 1} [z-1] ⁡X(z) $

This is used to find the final value of the signal without taking inverse z-transform.

Region of Convergence (ROC) of Z-Transform

The range of variation of z for which z-transform converges is called region of convergence of z-transform.

Properties of ROC of Z-Transforms

• ROC of z-transform is indicated with circle in z-plane.

• ROC does not contain any poles.

• If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is entire z-plane except at z = 0.

• If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-plane except at z = ∞.

• If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. |z| > a.

• If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius a. i.e. |z| < a.

• If x(n) is a finite duration two sided sequence, then the ROC is entire z-plane except at z = 0 & z = ∞.

The concept of ROC can be explained by the following example:

Example 1: Find z-transform and ROC of $a^n u[n] + a^{-}nu[-n-1]$

$Z.T[a^n u[n]] + Z.T[a^{-n}u[-n-1]] = {Z \over Z-a} + {Z \over Z {-1 \over a}}$

$$ ROC: |z| \gt a \quad\quad ROC: |z| \lt {1 \over a} $$

The plot of ROC has two conditions as a > 1 and a < 1, as you do not know a.

In this case, there is no combination ROC.

Here, the combination of ROC is from $a \lt |z| \lt {1 \over a}$

Hence for this problem, z-transform is possible when a < 1.

Causality and Stability

Causality condition for discrete time LTI systems is as follows:

A discrete time LTI system is causal when

• ROC is outside the outermost pole.

• In The transfer function H[Z], the order of numerator cannot be grater than the order of denominator.

Stability Condition for Discrete Time LTI Systems

A discrete time LTI system is stable when

• its system function H[Z] include unit circle |z|=1.

• all poles of the transfer function lay inside the unit circle |z|=1.

Z-Transform of Basic Signals