Time-Domain Representations of LTI Systems

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1 2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable descriptios for LTI systems 2.2 Covolutio Sum 1. A arbitrary sigal is expressed as a weighted superpositio of shifted impulses. Discrete-time sigal x[]: 0 x x x x Fig. 2.1 x[] = etire sigal; x[] = specific value of the sigal x[] at time x x x x x x 1

2 Figure 2.1 (p. 99) Graphical example illustratig the represetatio of a sigal x[] as a weighted sum of time-shifted impulses. x x (2.1) 2. Impulse respose of LTI system H: Iput x[] Output: LTI system H Output y[] y H x H x y H x y[] x[ ]H{ [ ]} (2.2) Liearity Liearity 2

3 The system output is a weighted sum of the respose of the system to timeshifted impulses. For time-ivariat system: H{ [ ]} h[ ] (2.3) h[] = H{[]} impulse respose of the LTI system H y[] x[ ]h[ ] (2.4) 3. Covolutio sum: x h x h Covolutio process: Fig Figure 2.2a (p. 100) Illustratio of the covolutio sum. (a) LTI system with impulse respose h[] ad iput x[]. 3

4 Figure 2.2b (p. 101) (b) The decompositio of the iput x[] ito a weighted sum of time-shifted impulses results i a output y[] give by a weighted sum of time-shifted impulse resposes. d Sigals ad Systems_Simo Hayi & Barry Va Vee 4

5 The output associated with the th iput is expressed as: Hx[ ] [ ] x[ ]h[ ] y x h Example 2.1 Multipath Commuicatio Chael: Direct Evaluatio of the Covolutio Sum Cosider the discrete-time LTI system model represetig a two-path propagatio chael described i Sectio If the stregth of the idirect path is a = ½, the 1 y x x 1 2 Lettig x[] = [], we fid that the impulse respose is h 1, 0 1, 2 1 0, otherw ise 5

6 Determie the output of this system i respose to the iput x <Sol.> 2, 0 4, 1 2, 2 0, otherw ise 1. Iput: x Sice time-shifted impulse iput [ ] Iput = 0 for < 0 ad > 0 time-shifted impulse respose output h [ ] 3. Output: y h h h y 0, 0 2, 0 5, 1 0, 2 1, 3 0, 4 6

7 2.3 Covolutio Sum Evaluatio Procedure 1. Covolutio sum: y x h = idepedet variable 2. Defie the itermediate sigal: [ ] x[ ]h[ ] (2.5) is treated as a costat by writig as a subscript o w. 3. Sice h [ ] = h [ ( )] is a reflected (because of ) ad time-shifted (by ) versio of h []. y[] [ ] (2.6) The time shift determies the time at which we evaluate the output of the system. Example 2.2 Covolutio Sum Evaluatio by usig Itermediate Sigal 3 Cosider a system with impulse respose h u 4 Use Eq. (2.6) to determie the output of the system at time = 5, = 5, ad = 10 whe the iput is x [] = u []. 7

8 <Sol.> Fig. 2.3 depicts x[] superimposed o the reflected ad time-shifted impulse respose h[ ] h [ ]: 4 3, h 4 0, otherw ise 2. Itermediate sigal w []: For = 5: w w Eq. (2.6) y[ 5] = 0 For = 5: Eq. (2.6), 0 5 0, otherw ise y For = 10: w 10 Eq. (2.6) y , , otherw ise y

9 Figure 2.3 (p. 103) Evaluatio of Eq. (2.6) i Example 2.2. (a) The iput sigal x[] above the reflected ad time-shifted impulse respose h[ ], depicted as a fuctio of. (b) The product sigal w 5 [] used to evaluate y [ 5]. (c) The product sigal w 5 [] used to evaluate y[5]. (d) The product sigal w 10 [] used to evaluate y[10]. 9

10 w 3 4, 0 0, otherw ise Procedure 2.1: Reflect ad Shift Covolutio Sum Evaluatio 1. Graph both x[] ad h[ ] as a fuctio of the idepedet variable. To determie h[ ], first reflect h[] about = 0 to obtai h[ ]. The shift by. 2. Begi with large ad egative. That is, shift h[ ] to the far left o the time axis. 3. Write the mathematical represetatio for the itermediate sigal w []. 4. Icrease the shift (i.e., move h[ ] toward the right) util the mathematical represetatio for w [] chages. The value of at which the chage occurs defies the ed of the curret iterval ad the begiig of a ew iterval. 5. Let be i the ew iterval. Repeat step 3 ad 4 util all itervals of times shifts ad the correspodig mathematical represetatios for w [] are idetified. This usually implies icreasig to a very large positive umber. 6. For each iterval of time shifts, sum all the values of the correspodig w [] to obtai y[] o that iterval. 10

11 Example 2.3 Movig-Average System: Reflect-ad-shift Covolutio Sum Evaluatio The output y[] of the four-poit movig-average system is related to the iput x[] accordig to the formula 3 1 y x 4 0 The impulse respose h[] of this system is obtaied by lettig x[] = [], which yields 1 h u u 4 4 Fig. 2.4 (a). Determie the output of the system whe the iput is the rectagular pulse defied as 1 st iterval: < 0 x u u 10 Fig. 2.4 (b). 2 d iterval: 0 3 <Sol.> 1. Refer to Fig Five itervals! 3 rd iterval: 3 < st iterval: w [] = 0 4th iterval: 9 < 12 5th iterval: > d iterval: For = 0: w 0 1 / 4, 0 0, otherw ise Fig. 2.4 (c). 11

12 Figure 2.4 (p. 106) Evaluatio of the covolutio sum for Example 2.3. (a) The system impulse respose h[]. (b) The iput sigal x[]. (c) The iput above the reflected ad time-shifted impulse respose h[ ], depicted as a fuctio of. (d) The product sigal w [] for the iterval of shifts 0 3. (e) The product sigal w [] for the iterval of shifts 3 < 9. (f) The product sigal w [] for the iterval of shifts 9 < 12. (g) The output y[]. Sigals ad Systems_Simo Hayi & Barry Va Vee 12

13 For = 1: w 1 1 / 4, 0,1 0, otherw ise For geeral case: 0: w 1 / 4, 0 0, otherw ise Fig. 2.4 (d) rd iterval: 3 < 9 w Fig. 2.4 (e). 5. 4th iterval: 9 < 12 w 1 / 4, 3 0, otherw ise 1 / 4, 3 9 0, otherw ise Fig. 2.4 (f). 6. 5th iterval: > 12 w [] = 0 7. Output: The output of the system o each iterval is obtaied by summig the values of the correspodig w [] accordig to Eq. (2.6). N M c c N M 1 1) For < 0 ad > 12: y[] = 0. 2) For 0 3: y 0 1 / 4 3) For 3 < 9: y 1 / ) For 9 < 12: / y 3 Fig. 2.4 (g) 13

14 Example 2.4 First-order Recursive System: Reflect-ad-shift Covolutio Sum Evaluatio The iput-output relatioship for the first-order recursive system is give by 1 y y x Let the iput be give by x b u 4 We use covolutio to fid the output of this system, assumig that b ad that the system is causal. <Sol.> 1. Impulse respose: h h 1 (2.7) Sice the system is causal, we have h[] = 0 for < 0. For = 0, 1, 2,, we fid that h[0] = 1, h[1] =, h[2] = 2,, or u h 2. Graph of x[] ad h[ ]: Fig. 2.5 (a). x b, 4 0, otherw ise ad h, 0, otherw ise 3. Itervals of time shifts: 1 st iterval: < 4; 2 d iterval: 4 14

15 Figure 2.5a&b (p. 109) Evaluatio of the covolutio sum for Example 2.4. (a) The iput sigal x[] depicted above the reflected ad time-shifted impulse respose h[ ]. (b) The product sigal w [] for 4. 15

16 4. For < 4: w [] = For 4: w b, 4 0, otherw ise Fig. 2.5 (b). 6. Output: 1) For < 4: y[] = 0. 2) For 4: y 4 y b 4 Let m = + 4, the y b m b b m0 m0 Next, we apply the formula for summig a geometric series of + 5 terms to obtai b y m 5 b b b b b b 1 Combiig the solutios for each iterval of time shifts gives the system output: 5 5 b y b 4 0, 4 b Fig. 2.5 (c)., 4 Assumig that = 0.9 ad b =

17 Figure 2.5c (p. 110) (c) The output y[] assumig that p = 0.9 ad b =

18 Example 2.5 Ivestmet Computatio The first-order recursive system is used to describe the value of a ivestmet earig compoud iterest at a fixed rate of r % per period if we set = 1 + (r/100). Let y[] be the value of the ivestmet at the start of period. If there are o deposits or withdrawals, the the value at time is expressed i terms of the value at the previous time as y[] = y[ 1]. Now, suppose x[] is the amout deposited (x[] > 0) or withdraw (x[] < 0) at the start of period. I this case, the value of the amout is expressed by the first-order recursive equatio y y 1 x We use covolutio to fid the value of a ivestmet earig 8 % per year if $1000 is deposited at the start of each year for 10 years ad the $1500 is withdraw at the start each year for 7 years. <Sol.> 1. Predictio: Accout balace to grow for the first 10 year, ad to decrease durig ext 7 years, ad afterwards to cotiue growig. 2. By usig the reflect-ad-shift covolutio sum evaluatio procedure, we ca evaluate y[] = x[] h[], where x[] is depicted i Fig. 2.6 ad h[] = u[] is as show i Example 2.4 with =

19 Figure 2.6 (p. 111) Cash flow ito a ivestmet. Deposits of $1000 are made at the start of each of the first 10 years, while withdrawals of $1500 are made at the start of each of the secod 10 years. 3. Graphs of x[] ad h[ ]: Fig. 2.7(a). 4. Aswer: y , , 17 Figure 2.7e (p. 113) (e) The output y[] represetig the value of the ivestmet immediately after the deposit or withdrawal at the start of year. 19

20 2.4 The Covolutio Itegral 1. A cotiuous-time sigal ca be expressed as a weighted superpositio of time-shifted impulses. x(t) x( ) (t - )d (2.10) - The siftig property of the impulse! 2. Impulse respose of LTI system H: Iput x(t) LTI system Output: H y t H x t H x t d - y(t) x( )H{ (t - )}d (2.10) 3. h(t) = H{ (t)} impulse respose of the LTI system H If the system is also time ivariat, the H{ (t - )} h(t - ) (2.11) y(t) x( )h(t )d (2.12) - Liearity property A time-shifted impulse geerates a time-shifted impulse respose output Fig Output y(t) 20

21 Covolutio itegral: x t h t x h t d 2.5 Covolutio Itegral Evaluatio Procedure 1. Covolutio itegral: y(t) x( )h(t )d (2.13) - 2. Defie the itermediate sigal: t w x h t = idepedet variable, t = costat h (t ) = h ( ( t)) is a reflected ad shifted (by t) versio of h(). 3. Output: y(t) w ( )d (2.14) t - The time shift t determies the time at which we evaluate the output of the system. 21

22 Example 2.6 Reflect-ad-shift Covolutio Evaluatio Give ad h t u t u t 2 as depicted i Fig. 2-10, Evaluate the covolutio itegral y(t) = x(t) h(t). <Sol.> 1. Graph of x() ad h(t ): Fig (a). 2. Itervals of time shifts: Four itervals 1 st iterval: t < 1 2 d iterval: 1 t < 3 3 rd iterval: 3 t < 5 4th iterval: 5 t 3. First iterval of time shifts: t < 1 4. Secod iterval of time shifts: 1 t < 3 w t 1, 1 t 0, otherw ise Fig (b). w t () = 0 Figure 2.10 (p. 117) Iput sigal ad LTI system impulse respose for Example

23 Figure 2.11 (p. 118) Evaluatio of the covolutio itegral for Example 2.6. (a) The iput x() depicted above the reflected ad timeshifted impulse respose. (b) The product sigal w t () for 1 t < 3. (c) The product sigal w t () for 3 t < 5. (d) The system output y(t). t 23

24 5. Third iterval: 3 t < 5 w t 1, t 2 3 0, otherw ise 6. Fourth iterval: 5 t w t () = 0 Fig (c). 7. Covolutio itegral: 1) For t < 1 ad t 5: y(t) = 0 2) For secod iterval 1 t < 3, y(t) = t 1 3) For third iterval 3 t < 5, y(t) = 3 (t 2) y t 0, t 1 t 1, 1 t 3 5 t, 3 t 5 0, t 5 Figure 2.12 (p. 119) RC circuit system with the voltage source x(t) as iput ad the voltage measured across the capacitor y(t), as output. Example 2.7 RC Circuit Output (Exercise) For the RC circuit i Fig. 2.12, assume that the circuit s time costat is RC = 1 sec. Ex shows that the impulse respose of this circuit is h(t) = e t u(t). Use covolutio to determie the capacitor voltage, y(t), resultig from a iput voltage x(t) = u(t) u(t 2). 24

25 <Sol.> RC circuit is LTI system, so y(t) = x(t) h(t). 1. Graph of x() ad h(t ): Fig (a). x 1, 0 2 ad 0, otherw ise 2. Aswer: Fig (d). 0, t 0 t y t 1 e, 0 t 2 2 t e 1 e, t 2 t, t e t h t e u t 0, otherw ise Figure 2.13 (p. 120) Evaluatio of the covolutio itegral for Example 2.7. (a) The iput x() superimposed over the reflected ad time-shifted impulse respose h(t ), depicted as a fuctio of. (b) The product sigal w t () for 0 t < 2. (c) The product sigal w t () for t 2. (d) The system output y(t). 25

26 2.6 Itercoectio of LTI Systems Parallel Coectio of LTI Systems 1. Two LTI systems: Fig. 2.18(a). Figure 2.18 (p. 128) Itercoectio of two LTI systems. (a) Parallel coectio of two systems. (b) Equivalet system. 2. Output: y ( t ) y ( t ) y ( t ) 1 2 x( t ) h ( t ) x( t ) h ( t ) 1 2 y( t) x( ) h ( t ) d x( ) h ( t ) d 1 2 y ( t ) x( ) h ( t ) h ( t ) d 1 2 x( ) h( t ) d x( t ) h( t ) where h(t) = h 1 (t) + h 2 (t) Fig. 2.18(b) 26

27 Distributive property for Cotiuous-time case: x(t) h (t) x(t) h (t) x(t) {h (t) h (t)} (2.15) Distributive property for Discrete-time case: x[] h [] x[] h [] x[] {h [] h []} (2.16) Cascade Coectio of LTI Systems 1. Two LTI systems: Fig. 2.19(a). Figure 2.19 (p. 128) Itercoectio of two LTI systems. (a) Cascade coectio of two systems. (b) Equivalet system. (c) Equivalet system: Iterchage system order. 27

28 2. The output is expressed i terms of z(t) as y(t) z(t) h (t) 2 (2.17) 2 (2.18) - y(t) z( )h (t )d Sice z(t) is the output of the first system, so it ca be expressed as z( ) x( ) h ( ) x( )h ( )d (2.19) Substitutig Eq. (2.19) for z(t) ito Eq. (2.18) gives y( t ) x( v ) h ( v ) h ( t ) dvd 1 2 y(t) x( ) h ( )h (t )d d (2.20) Defie h(t) = h 1 (t) h 2 (t), the Chage of variable = h( t v) h ( ) h ( t v ) d 1 2 (2.21) y(t) x( )h(t )d x(t) h(t) - 3. Associative property for cotiuous-time case: Fig. 2.19(b). 28

29 {x(t) h (t)} h (t) x(t) {h (t) h (t)} (2.22) 4. Commutative property: Write h(t) = h 1 (t) h 2 (t) as the itegral h( t ) h ( ) h ( t ) d 1 2 Chage of variable = t (2.23) - h(t) h (t )h ( )d h (t) h (t) Fig. 2.19(c). Iterchagig the order of the LTI systems i the cascade without affectig the result: x( t ) h ( t ) h ( t ) x( t ) h ( t ) h ( t ), Commutative property for cotiuous-time case: h (t) h (t) h (t) h (t) (2.24) 5. Associative property for discrete-time case: {x[] h []} h [] x[] {h [] h []} (2.25) Commutative property for discrete-time case: h [] h [] h [] h [] (2.26) 29

30 Example 2.11 Equivalet System to Four Itercoected Systems Cosider the itercoectio of four LTI systems, as depicted i Fig The impulse resposes of the systems are h [ ] u[ ], 1 2 ad h [ ] u [ 2] u [ ], h [ ] [ 2], [ ] [ ]. 3 h u 4 Fid the impulse respose h[] of the overall system. <Sol.> 1. Parallel combiatio of h 1 [] ad h 2 []: h 12 [] = h 1 [] + h 2 [] Fig (a). Figure 2.20 (p. 131) Itercoectio of systems for Example

31 Figure 2.21 (p. 131) (a) Reductio of parallel combiatio of LTI systems i upper brach of Fig (b) Reductio of cascade of systems i upper brach of Fig. 2.21(a). (c) Reductio of parallel combiatio of systems i Fig. 2.21(b) to obtai a equivalet system for Fig

32 2. h 12 [] is i series with h 3 []: h 123 [] = h 12 [] h 3 [] h 123 [] = (h 1 [] + h 2 []) h 3 [] 3. h 123 [] is i parallel with h 4 []: h[] = h 123 [] h 4 [] Fig (b). h[ ] ( h [ ] h [ ]) h [ ] h [ ], Fig (c) Thus, substitute the specified forms of h 1 [] ad h 2 [] to obtai h [ ] u [ ] u [ 2] u [ ] 12 u [ 2] Covolvig h 12 [] with h 3 [] gives h [ ] u[ 2] [ 2] 123 u [ ] h [ ] 1 u [ ]. Table 2.1 summarizes the itercoectio properties preseted i this sectio. 32

33 2.7 Relatio Betwee LTI System Properties ad the Impulse Respose Memoryless LTI Systems 1. The output of a discrete-time LTI system: y[ ] h[ ] x[ ] h[ ] x[ ] y[] h[ 2]x[ 2] h[ 1]x[ 1] h[0]x[] h[1]x[ 1] h[2]x[ 2] (2.27) 33

34 2. To be memoryless, y[] must deped oly o x[] ad therefore caot deped o x[ ] for 0. A discrete-time LTI system is memoryless if ad oly if h[ ] c [ ] Cotiuous-time system: 1. Output: y( t) h( ) x( t ) d, c is a arbitrary costat 2. A cotiuous-time LTI system is memoryless if ad oly if h( ) c ( ) c is a arbitrary costat Causal LTI Systems The output of a causal LTI system depeds oly o past or preset values of the iput. Discrete-time system: 1. Covolutio sum: y[ ] h[ 2] x[ 2] h[ 1] x[ 1] h[0] x[ ] h[1] x[ 1] h[2] x[ 2]. 34

35 2. For a discrete-time causal LTI system, h[ ] 0 for 0 3. Covolutio sum i ew form: y[ ] h[ ] x[ ]. 0 Cotiuous-time system: 1. Covolutio itegral: y( t) h( ) x( t ) d. 2. For a cotiuous-time causal LTI system, 3. Covolutio itegral i ew form: y( t) h( ) x( t ) d. 0 h( ) 0 for Stable LTI Systems A system is BIBO stable if the output is guarateed to be bouded for every bouded iput. Discrete-time case: Iput x[ ] M Output: y[ ] M x y 35

36 1. The magitude of output: y[ ] h[ ] x[ ] h[ ] x[ ] y[ ] h[ ] x[ ] a b a b y[ ] h[ ] x[ ] ab a b 2. Assume that the iput is bouded, i.e., x[ ] M x[ ] M x x ad it follows that y[] M h[ ] (2.28) x Hece, the output is bouded, or y[] for all, provided that the impulse respose of the system is absolutely summable. 3. Coditio for impulse respose of a stable discrete-time LTI system: 36

37 h [ ]. Cotiuous-time case: Coditio for impulse respose of a stable cotiuous-time LTI system: h( ) d. Example 2.12 Properties of the First-Order Recursive System The first-order system is described by the differece equatio y[ ] y[ 1] x[ ] ad has the impulse respose h[ ] u[ ] Is this system causal, memoryless, ad BIBO stable? <Sol.> 1. The system is causal, sice h[] = 0 for < The system is ot memoryless, sice h[] 0 for > Stability: Checig whether the impulse respose is absolutely summable? 37

38 h [ ] if ad oly if < Special case: A system ca be ustable eve though the impulse respose has a fiite value. 1. Ideal itegrator: t y(t) x( )d (2.29) Iput: x() = (), the the output is y(t) = h(t) = u(t). h(t) is ot absolutely itegrable Ideal itegrator is ot stable! 2. Ideal accumulator: y[ ] x[ ] Impulse respose: h[] = u[] h[] is ot absolutely summable Ideal accumulator is ot stable! 38

39 2.7.4 Ivertible Systems ad Decovolutio A system is ivertible if the iput to the system ca be recovered from the output except for a costat scale factor. 1. h(t) = impulse respose of LTI system, 2. h iv (t) = impulse respose of LTI iverse system Fig Figure 2.24 (p. 137) Cascade of LTI system with impulse respose h(t) ad iverse system with impulse respose h -1 (t). 3. The process of recoverig x(t) from h(t) x(t) is termed decovolutio. 4. A iverse system performs decovolutio. i v x( t) ( h( t) h ( t)) x( t). iv h( t) h ( t) ( t) (2.30) Cotiuous-time case 5. Discrete-time case: iv (2.31) h[] h [] [] 39

40 Example 2.13 Multipath Commuicatio Chaels: Compesatio by meas of a Iverse System Cosider desigig a discrete-time iverse system to elimiate the distortio associated with multipath propagatio i a data trasmissio problem. Assume that a discrete-time model for a two-path commuicatio chael is y[ ] x[ ] ax[ 1]. Fid a causal iverse system that recovers x[] from y[]. Chec whether this iverse system is stable. <Sol.> 1. Impulse respose: 1, 0 h[ ] a, 1 0, otherw ise 2. The iverse system h iv [] must satisfy h[] h iv [] = []. iv iv h [ ] ah [ 1] [ ]. (2.32) 1) For < 0, we must have h iv [] = 0 i order to obtai a causal iverse system 40

41 2) For = 0, [] = 1, ad eq. (2.32) implies that iv iv h [ ] ah [ 1] 0, (2.33) 3. Sice h iv [0] = 1, Eq. (2.33) implies that h iv [1] = a, h iv [2] = a 2, h iv [3] = a 3, ad so o. The iverse system has the impulse respose iv iv h [ ] ( a) u[ ] iv h [] ah [ 1] 4. To chec for stability, we determie whether h iv [] is absolutely summable, which will be the case if iv h [ ] a is fiite. For a < 1, the system is stable. Table 2.2 summarizes the relatio betwee LTI system properties ad impulse respose characteristics. 41

42 2.8 Step Respose 1. The step respose is defied as the output due to a uit step iput sigal. 2. Discrete-time LTI system: Let h[] = impulse respose ad s[] = step respose. s[ ] h[ ] * u[ ] h[ ] u[ ]. 3. Sice u[ ] = 0 for > ad u[ ] = 1 for, we have 42

43 s[ ] h[ ]. The step respose is the ruig sum of the impulse respose. Cotiuous-time LTI system: t s(t) h( )d (2.34) The step respose s(t) is the ruig itegral of the impulse respose h(t). Express the impulse respose i terms of the step respose as d h[ ] s[ ] s[ 1] ad h( t ) s( t ) dt Example 2.14 RC Circuit: Step Respose The impulse respose of the RC circuit depicted i Fig is t 1 RC h( t ) e u ( t ) RC Fid the step respose of the circuit. <Sol.> Figure 2.12 (p. 119) RC circuit system with the voltage source x(t) as iput ad the voltage measured across the capacitor y(t), as output. 43

44 1. Step respose: t 1 RC s( t ) e u ( ) d. RC s() t 1 RC t 0, t 0 RC e u ( ) d t 0 s() t 1 RC Fig , t 0 t RC e d t 0 0, t 0 t RC 1 e, t 0 0 Figure 2.25 (p. 140) RC circuit step respose for RC = 1 s. 44

45 2.9 Differetial ad Differece Equatio Represetatios of LTI Systems 1. Liear costat-coefficiet differetial equatio: N M d d a y(t) b x(t) (2.35) 0 dt 0 dt 2. Liear costat-coefficiet differece equatio: N M a y[ ] b x[ ] (2.36) 0 0 Iput = x(t), output = y(t) Iput = x[], output = y[] The order of the differetial or differece equatio is (N, M), represetig the umber of eergy storage devices i the system. Ex. RLC circuit depicted i Fig Iput = voltage source x(t), output = loop curret 2. KVL Eq.: d 1 t Ry t L y t y d x t dt C Ofte, N M, ad the order is described usig oly N. 45

46 Figure 2.26 (p. 141) Example of a RLC circuit described by a differetial equatio. 2 1 d d d y t R y t L y t x t N = 2 2 C dt dt dt 46

47 Ex. Secod-order differece equatio: 1 y[] y[ 1] y[ 2] x[] 2 x[ 1] 4 (2.37) N = 2 Differece equatios are easily rearraged to obtai recursive formulas for computig the curret output of the system from the iput sigal ad the past outputs. Ex. Eq. (2.36) ca be rewritte as M N 1 1 y b x a y a 0 0 a0 1 Ex. Cosider computig y[] for 0 from x[] for the secod-order differece equatio (2.37). <Sol.> 1. Eq. (2.37) ca be rewritte as 1 y[] x[] 2 x[ 1] y[ 1] y[ 2] 4 2. Computig y[] for 0: (2.38) 47

48 1 y[0] x[0] 2 x[ 1] y[ 1] y[ 2] (2.39) 4 1 y[1] x[1] 2 x[0] y[0] y[ 1] 4 (2.40) 1 y x x y y y x x y y Iitial coditios: y[ 1] ad y[ 2]. The iitial coditios for Nth-order differece equatio are the N values y N, y N 1,..., y 1, The iitial coditios for Nth-order differetial equatio are the N values 2 N 1 d d d y t y t y t y t,,..., t 0, 2 N 1 dt t 0 dt t 0 dt t 0 48

49 Example 2.15 Recursive Evaluatio of a Differece Equatio Fid the first two output values y[0] ad y[1] for the system described by Eq. (2.38), assumig that the iput is x[] = (1/2) u[] ad the iitial coditios are y[ 1] = 1 ad y[ 2] = 2. <Sol.> 1. Substitute the appropriate values ito Eq. (2.39) to obtai 1 1 y Substitute for y[0] i Eq. (2.40) to fid y

50 2.10 Solvig Differetial ad Differece Equatios Complete solutio: y = y (h) + y (p) y (h) = homogeeous solutio, y (p) = particular solutio The Homogeeous Solutio Cotiuous-time case: N d h 1. Homogeeous differetial equatio: a y t 0 0 dt 2. Homogeeous solutio: (h ) N rt i (2.41) y (t) c e i0 3. Characteristic eq.: i N a r 0 (2.42) 0 Discrete-time case: 1. Homogeeous differetial equatio: 2. Homogeeous solutio: N (h ) y [] c r (2.43) i1 3. Characteristic eq.: i i N 0 N N a r 0 (2.44) 0 h a y Coefficiets c i is determied by iitial coditio 0 Coefficiets c i is determied by iitial coditio 50

51 If a root r j is repeated p times i characteristic eqs., the correspodig solutios are r j t r j t p 1 r j t Cotiuous-time case: e, te,..., t e Discrete-time case: p 1 r, r,..., r j j j Example 2.17 RC Circuit: Homogeeous Solutio The RC circuit depicted i Fig is described by the differetial equatio d y t RC y t x t dt Determie the homogeeous solutio of this equatio. <Sol.> d 1. Homogeeous Eq.: y t RC y t 0 dt 2. Homo. Sol.: h 1 y t c e rt V 1 3. Characteristic eq.: 1 RCr 0 1 r 1 = 1/RC 4. Homogeeous solutio: t h RC V 1 y t c e Figure 2.30 (p. 148) RC circuit. 51

52 Example 2.18 First-Order Recursive System: Homogeeous Solutio Fid the homogeeous solutio for the first-order recursive system described by the differece equatio 1 y y x <Sol.> 1. Homogeeous Eq.: y y 2. Homo. Sol.: h 1 1 y c r Characteristic eq.: r 0 1 r 1 = 4. Homogeeous solutio: h 1 y c The Particular Solutio A particular solutio is usually obtaied by assumig a output of the same geeral form as the iput. Table

53 Example 2.19 First-Order Recursive System (Cotiued): Particular Solutio Fid a particular solutio for the first-order recursive system described by the differece equatio 1 y y x if the iput is x[] = (1/2). <Sol.> 1. Particular solutio form: y [ ] c p 2. Substitutig y (p) [] ad x[] ito the give differece Eq.: p

54 c p CHAPTER c 1 p Both sides of above eq. are multiplied by (1/2) c (1 2 ) 1 p (2.45) 3. Particular solutio: y p Figure 2.30 (p. 148) RC circuit. Example 2.20 RC Circuit (cotiued): Particular Solutio Cosider the RC circuit of Example 2.17 ad depicted i Fig Fid a particular solutio for this system with a iput x(t) = cos( 0 t). <Sol.> d 1. Differetial equatio: y t RC y t x t dt 2. Particular solutio form: ( p ) y ( t ) c cos( t ) c si( t ) Substitutig y (p) (t) ad x(t) = cos( 0 t) ito the give differetial Eq.: 54

55 c cos( t ) c si( t ) RC c si( t ) RC c cos( t ) cos( t) c RC c RC c c Coefficiets c 1 ad c 2 : c 1 RC 1 RC Particular solutio: 0 ad c RC p 1 RC 0 y t cos t 0 si t 0 V RC 1 RC The Complete Solutio Complete solutio: y = y (h) + y (p) y (h) = homogeeous solutio 0 y (p) = particular solutio The procedure for fidig complete solutio of differetial or differece equatios is summarized as follows: 55

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