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
DTLTI Systems described by lCCdes N M ∑akn-树=∑am- k=0 k=0 Use the time-shift property ∑ak2-Y(2)=∑2X( k:=0 (2)=H(2)(a) H(a) k=0h之一k M k=0 ROC: Depends on Boundary Conditions, left-, right-, or two-sided For Causal Systems RoC is outside the outermost pole
DT LTI Systems Described by LCCDEs ROC: Depends on Boundary Conditions, left-, right-, or two-sided. — Rational Use the time-shift property For Causal Systems ⇒ ROC is outside the outermost pol e
System Function Algebra and block diagrams Feedback System e[n] h,[n] (causal systems) [n] H2(z) negative feedback h2In] configuration H1(z) Example #1: H(2)=X(2)H1(2)H2 yIn] x[n] H(a) z Del /n=un-1+an
Feedback System (causal systems) System Function Algebra and Block Diagrams Example #1: negative feedback configuration z-1 ⇔ D Delay
Example #2: 1-2z H(z) 114 Cascade of two systems yn x[n]
Example #2: — Cascade of two systems
Unilateral z-Transform 2(x) Inez 7=0: Note (1)Ifx[n]=0 for n<0, then 2(2)=X(a (2)UZT of x[n]=BZT of xnjun]= roc always outside a circle and includes =oo For causal ltI systems, H(2)=H(z)
Unilateral z-Transform Note: (1) If x[n] = 0 for n < 0, then (2) UZT of x[n] = BZT of x[n]u[n] ⇒ ROC always outside a circle and includes z = ∞ (3) For causal LTI systems
Properties of Unilateral z-Transform Many properties are analogous to properties of the bZt e. g Convolution property(for x,[n<0]=x2[n<0]=0) 1 m]*2m]+241(z)22(2) But there are important differences. For example, time-shift ym]=xn-1←→y(2)=-1:x-1(z) Derivation Initial condition 0(2)=∑yn2n=∑mn-1z-n=2-1]+∑n-1z-n n =x-1]+2-1∑ elma m=0
• But there are important differences. For example, time-shift Properties of Unilateral z-Transform Many properties are analogous to properties of the BZT e.g. • Convolution property (for x1[n<0] = x2[n<0] = 0) Derivation: Initial condition
