H AA Signals and Systems Fall 2003 Lecture#23 4 December 2003 1. Geometric Evaluation of z-Transforms and dt frequency Responses 2. First-and Second-Order Systems 3. System Function Algebra and block diagrams Unilateral z-Transforms
Signals and Systems Fall 2003 Lecture #23 4 December 2003 1. Geometric Evaluation of z-Transforms and DT Frequency Responses 2. First- and Second-Order Systems 3. System Function Algebra and Block Diagrams 4. Unilateral z-Transforms
Geometric Evaluation of a Rational z-Transform Example #1: X1()=2-a-a first-order zero Example #2: Z-plane A first-order pole 2-a 2(2)= x1(22X2(2)=-∠X1() Z1-a 1(z-) ∠(z1a Example# a3:x(2)=M1(2-0) a e R X()=1Mm= All same as plane ∠X()=∠M+∑∠(2-1)-∑4( i=1
Geometric Evaluation of a Rational z-Transform Example #1: Example #3: Example #2: All same a s in s-plane
Geometric Evaluation of DT Frequency responses First-Order System H(a) z>a one real pole 2- hn=a"un, a< e [/2 7 Unit circle Z-plane a=0.95 a=05 10 =0.95 a=0.5 T H(e)=-,|H(e)= ,∠H(e4)=∠1-∠02=u-∠
Geometric Evaluation of DT Frequency Responses First-Order System — one real pole
Second-Order system Two poles that are a complex conjugate pair(re/=z2) H(2) (2-x1)(2-2)1-(2rcos0)-1+y2-2 0<r<1,0≤6≤ H(eu) /(eju - b )(eju -re e), hIni sin(n +1)0 sin e Clearly, H peaks near (=+0 IH(ejo)I T/2 Unit circle z-plane r=095/ 10 r=095 r=0.75 r=0 2
Second-Order System Two poles that are a complex conjugate pair (z1= rejθ =z2*) Clearly, |H| peaks near ω = ±θ
Demo: dt pole-zero diagrams. frequency response vector diagrams, and impulse-& step-responses Remove FREQUENCY (radians) oddy Markers 1080.60.40.2002040608 Line width. REQUENCY (radians)
Demo: DT pole-zero diagrams, frequency response, vector diagrams, and impulse- & step-responses