Z Transform

THE Z TRANSFORM

Z Transform

z Transform is used for the analysis of discrete time signal as well as discrete time system.

Definition of z Transform

z transform of discrete time signal x(n) is given as S

z transform pair

Types of z transform: unilateral and bilateral

(i) Unilateral (or) one sided z transform.

Here the summation exist in the range n = 0 to ∞.i.e. one sided.

(ii) Bilateral z transform: It has both sided summation.

Here the summation exist in between n = -∞ to ∞. i.e. both sided

Region of Convergence (ROC)

ROC is the region where z transform converges. z transform is an infinite power series. This series is not convergent for all values of z.

Significance of ROC

(i) ROC gives an idea about the values of z for which z transform can be calculated.

(ii) ROC is used to determine causality of the system

(iii) ROC is used to determine stability of the system.

Properties of ROC

Property 1:

ROC for a finite duration sequence includes entire z plane except at z = 0, and |z| = ∞.

Proof:

Here x(z) = ∞ for z = 0 and ∞. Hence proved

Property 2:

ROC does not contain any poles

Proof:

The z transform of a" u(n) is calculated as

This function has pole at z = a. But ROC is |z| > |a|

Hence poles does not lie in ROC

Property 3:

ROC is the ring in the z plane centered about origin.

Here |z| is always a circular region centered around origin.

Property 4:

ROC of causal sequence (right hand sided sequence) is in the form z > r

Proof:

Consider right hand sided sequence aⁿ u(n).

It's ROC is |z| > |a|, thus the ROC of right hand sequence is in the form of |z|> r. Here 'r" is radius of circle.

Property 5:

ROC of left sided sequence is in the form |z| < r.

Proof:

Consider left sided sequence -an u(-n-1). Its ROC is |z| < |a|.

Thus the ROC of left sided sequence is inside the circle of radius "r".

Property 6:

ROC of two sided sequence is the concentric ring in z plane.

Property 7:

If x(n) is finite causal sequence, then its

ROC is entire z Plane except z = 0.

Proof:

Consider the causal sequence, x(n) = {1,2,3},

Then its z transform will be . This sequence converges in entire z plane.

Property 8:

ROC of stable LTI system contains unit circle in the z plane.

Property 9:

The ROC is a connected region.

Proof:

Convergence of the sequence exists over certain area rather than discrete points. Hence ROC is a connected region.

Properties of z Transform

Linearity

Proof:

Time Shifting or Translation

Proof:

Scaling in z Domain (or) Multiplication by Exponential

Proof:

Time Reversal

Proof:

Differentiation in z Domain or Multiplication by a Ramp

Proof:

Convolution in Time Domain

Proof:

By interchanging order of summation

Correlation of Two Sequences

Proof:

Correlation of two sequence is given as

Conjugation of a Complex Sequence

Proof:

Parseval's Relation

Proof:

Inverse z transform of X1(z) is

Initial Value Theorem

Proof:

z transform of causal sequence