H AA Signals and Systems Fall 2003 Lecture #22 2 December 2003 Properties of the roc of the z-Transform 2. Inverse z-Transform 3. Examples 4. Properties of the z-Transform 5. System Functions of DT LTI Systems a causality b. Stabil
Signals and Systems Fall 2003 Lecture #22 2 December 2003 1. Properties of the ROC of the z-Transform 2. Inverse z-Transform 3. Examples 4. Properties of the z-Transform 5. System Functions of DT LTI Systems a. Causality b. Stability
The z-Transform rlx(2)=∑a2n=z(rl m=二 ROC=2=relu at which 2(ngr-n< -depends only on r==l, just like the roc in s-plane only depends on Re(s) Last time Unit circle(r=1)in the roc dtfT(e/o)exists rational transforms correspond to signals that are linear combinations of dt exponentials
The z-Transform • Last time: •Unit circle (r = 1) in the ROC ⇒DTFT X(ejω) exists •Rational transforms correspond to signals that are linear combinations of DT exponentials -depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)
Some Intuition on the relation between zT and lt a(tes dt=Ca(t)) lim a(nT)lesz ST\-n lim T T→0 ∑(e n三- The bilateral) z-Transform m←→X(2)=∑ ern z ZanI Can think of z-transform as dt version of Laplace transform with T
Some Intuition on the Relation between z T and LT Can think of z-transform as DT version of Laplace transform with The (Bilateral) z-Transform
More intuition on zT-LT, S-plane-z-plane relationship jw axis in s-plane(s= jw)+2=ejw'l a unit circle in z-plane gn jo-axis z|=1 S-plane Z-plane e LHP RHP " RHPW LHP in s-plane, Re(s)<0===es <l, inside the ==1 circle Special case, Re(s)=-00==0 RHP in s-plane, Res)>0===es> l, outside the = 1 circle Special case,Re(s)=+∞分|=∞ A vertical line in s-plane, Re(s)=constant oes/=constant, a circle in z-plane
More intuition on zT-LT, s-plane - z-plane relationship • LHP in s-plane, Re(s) < 0 ⇒ |z| = | esT| < 1, inside the |z| = 1 circle. Special case, Re(s) = -∞ ⇔ |z| = 0. • RHP in s-plane, Re(s) > 0 ⇒ |z| = | esT| > 1, outside the |z| = 1 circle. Special case, Re(s) = +∞ ⇔ |z| = ∞. • A vertical line in s-plane, Re(s) = constant ⇔ | esT| = constant, a circle in z-plane
Properties of the roCs of z-Transforms (1) The roc of X(z)consists of a ring in the z-plane centered about the origin(equivalent to a vertical strip in the s-plane) gr Z-plane g 2)The roc does not contain any poles(same as in LT
Properties of the ROCs of z-Transforms (1) The ROC of X(z) consists of a ring in the z-plane centered about the origin (equivalent to a vertical strip in the s-plane) (2) The ROC does not contain any poles (same as in LT)