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, 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, 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)
More roc properties (3)Ifxn] is of finite duration, then the roc is the entire z-plane except possibly at z=0 and/or2=oo Why? X() ∑ rnz Examples COunterpart 6mn]←1 ROC all z|ot)←1 roC all s ROC2≠0|0t-T) e-e{s}≠ 6+1]←2ROC2≠0∞o6(+1)←cTe(}≠
More ROC Properties (3) If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly at z = 0 and/or z = ∞. Why? Examples: CT counterpart
ROC Properties Continued (4 )Ifx[n] is a right-Sided sequence, and if ==ro is in the roC, then all finite values of z for which =>ro are also in the roc ●●鲁● ro 1 TTIITTTTTT. converges faster th n r1",r1>r0 ∑ml P99●99
ROC Properties Continued (4) If x[n] is a right-sided sequence, and if | z| = r o is in the ROC, then all finite values of z for which | z| > r o are also in the ROC
Side by side (5)Ifxn] is a left-sided sequence, and if =ro is in the roc, then all finite values of z for which 0<=<ro are also in the roc (6)If x[n] is two-Sided and if==ro is in the roC, then the roc consists of a ring in the z-plane including the circle =r What types of signals do the following roc correspond to? g 9x 77 Z-plane Z-plane z-plane g g g right-sided left-sided two-sided
Side by Side (6) If x[n] is two-sided, and if |z| = ro is in the ROC, then the ROC consists of a ring in the z-plane including the circle |z| = ro. What types of signals do the following ROC correspond to? right-sided left-sided two-sided (5) If x[n] is a left-sided sequence, and if |z| = ro is in the ROC, then all finite values of z for which 0 < |z| < ro are also in the ROC
Example #1 a n=bn, 6>0 XIn]=binl x[n]=binl b 1_b-1y-17
Example #1
Example #I continued X(z) -b2 1b-1x Unit circle z-plan oX Clearly, roc does not exist if b> 1= No z-transform for binl
Example #1 continued Clearly, ROC does not exist if b > 1 ⇒ No z-transform for b|n|