Z 변환의정의

1

z-transform

Definition

Relationship with DFTF

( ) [ ] n

n

X z x n z

is a complex variable

( ) [ ] n

n

X z x n z

(e ) [ ]ej j n

n

X x n

jz e

The z-transform evaluated on the unit circle corresponds to the Fourier transform

z-transform

Region of convergence (ROC)• The range of for which infinite sum in z-transform converges.

• How to get the ROC?

( ) ( ) [ ] [ ]j n jn n

n n

X z X re x n r e x n r

ROC will consist of a ring in the z-plane.

ROC contains a unit circle. Fourier transform converges.

z-transform

Rational z-transform

( )( )( )

P zX zQ z

Polynomial in z

Pole: the roots of denominator

Zero: the roots of numerator

0 0{ : ( ) 0}z Q z

0 0{ : ( ) 0}z P z

z-transform

Example• Find the z-transform of

• When does the Fourier transform converge?• Draw pole-zero plot with ROC.

[ ] [ ]nx n a u n

Right-sided sequence

z-transform

Example

• When does the Fourier transform converge?• Draw pole-zero plot with ROC.

[ ] [ 1]nx n a u n Left-sided sequence

z-transform

Example1 1[ ] [ ] [ ]2 2

n n

x n u n u n

ROC1 ROC2

{ 1 2}ROC ROC ROC

z-transform

Example1 1[ ] [ ] [ 1]3 2

n n

x n u n u n

Double-sided sequence

z-transform

Example0 1

[ ]0

na n Nx n

else

Hint: the roots of are /

Finite duration sequence

z-transform

Properties of the ROC for the z-transform

ROC properties

Property 1. The ROC will either be of the form 0 ∞.

Property 2.The Fourier transform of x[n] converges absolutely if and only if the ROC of z‐transform includes the unit circle.

Properties of the ROC for the z-transform

Property 3. The ROC cannot contain any poles.

Property 4. If  is a finite‐duration sequence, the ROC is the entire z‐plane except possibly z=0 or z=∞

Properties of the ROC for the z-transform

Property 5. If x[n] is a right‐sided sequence, the ROC extends outward from the outermost finite pole

Property 6. If x[n] is a left‐sided sequence, the ROC extends inward from the innermost nonzero pole

Properties of the ROC for the z-transform

Property 7. If x[n] is a two‐sided sequence, the ROC consists of a ring bounded by the interior and exterior by a pole

Property 8. ROC must be a connected region

Properties of the ROC for the z-transform

Example• Find the ROC of the following double-sided sequence

1 1[ ] [ ] [ 1]2 3

n n

x n u n u n

Properties of the ROC for the z-transform

A sequence cannot be determined only by poles and zeros. ROC should be specified!

Properties of the ROC for the z-transform

z transform of system impulse response h[n]• Causality check

• If h[n] is left-sided or double-sided, the system cannot be causal.• If the system has at least one pole, the ROC of causal systems should

extend outward.

• Stability check• If the system is stable (or equivalently h[n] is absolutely summable and

therefore has Fourier transform), the ROC must include the unit circle.