Signal Processing and Linear Systems-B.P.Lathi copy

2 eq 215 we found the zero state component in example

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Unformatted text preview: tem response are lumped together, they form t he s ystem's n atural r esponse Yn(t) (also known as t he h omogeneous s olution o r c omplementary s olution). T he r emaining p ortion o f t he response consists entirely of noncharacteristic mode terms a nd is called the system's f orced r esponse y¢(t) (also known as the p articular s olution). E quation (2.51b) shows these two components for t he loop current in t he R LC circuit o f Fig. 2.1a. or Q(D) [Yn(t) Q(D)Yn(t) + y¢(t)] + y¢(t). Forced Response ( f. Ai (i = 1, 2, . .. , n ) l. + Q(D)y¢(t) = e (t 2. e (t 3. 4. = P (D)j(t) k cos (wt (t r + a r_lt r - l + ... + a lt + ao) e (t ( = Ai {3e(t {3te(t {3 + 6) + </» + { 3r_ltr-1 + ... + {31t {3 cos (wt P (D)j(t) ({3rtr + (3o)e(t Q(D)Yn(t) = 0 Q (D)y¢(t) = P (D)j(t) (2.52) T he n atural r esponse, being a linear combination of t he s ystem's characteristic modes, has t he s ame form as t hat o f t he z ero-input response; only its a rbitrary c onstants are different. These constants are determined from auxiliary conditions, as explained later. W e shall now discuss a method of determining t he forced response. 2.5-1 I nput j (t) Since y(t) m ust satisfy the B ut Yn(t) is c omposed entirely of characteristic modes. Therefore so t hat T ABLE 2.2 5. T he t otal s ystem response is y(t) = Yn(t) system equation [Eq. (2.1)], 141 Forced Response: T he M ethod o f Undetermined Coefficients I t is a r elatively simple t ask t o determine y¢(t), t he forced response of a n LTIC s ystem, when t he i nput j (t) is such t hat i t yields only a finite number of independent derivatives. I nputs having t he form e (t o r t r fall into this category. For example, e (t h as only one independent derivative; t he r epeated differentiation of e (t yields t he same form as this input; t hat is, e (t. Similarly, t he repeated differentiation o f r t yields only T i ndependent derivatives. T he forced response t o such a n i nput c an be expressed as a linear combination of t he i nput a nd i ts independent derivatives. Consider, for example, t he i nput at 2 + bt +c. T he successive derivatives of this input are 2at + b and 2 a. I n this case, t he i nput has only two independent derivatives. Therefore, t he forced response can be assumed t o be a linear combination o f j (t) a nd i ts two derivatives. T he s uitable form for y¢(t) in this case is, therefore y¢(t) = I ht 2 + {3lt + {30 T he u ndetermined coefficients {30, {31, a nd (32 a re determined by substituting this expression for y¢( t ) in Eq. (2.52) Q(D)y¢(t) = P (D)j(t) a nd t hen e quating coefficients of similar terms on b oth sides of t he resulting expression. N ote: B y definition, y¢(t) c annot have any characteristic mode t~rms. I f a ny term appearing in the right-hand column for t he forced response IS also a characteristic mode of t he system, t he c orrect form o f t he forced response must ~e modified t o t iy¢(t), where i is t he s mallest possible integer t hat c an b e used a nd still can prevent tiy¢(t) from having a characteristic...
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