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

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Unformatted text preview: n general. However, in t his case, b1 = 0, a nd t here is o nly o ne feedforward connection w ith coefficient bo = 5 f rom t he o utput o f t he i ntegrator t o t he o utput s ummer. O bserve t hat b ecause t here is o nly o ne s ignal t o b e s ummed a t t he o utput s ummer, a s ummer is n ot n eeded. For t his r eason, t he o utput s ummer is o mitted i n Fig. 6.23. . -L 8 +7 o ne. sig~al ~o ~e s ummed a t t he o utput s ummer, t he o utput s ummer is o mitted. T he realizatIOn I II Fig. 6.25a is r edrawn i n a more convenient form, as i llustrated i n Fig. 6.25b. ( d) H (s) = 4 s + 28 s2 + 6s + 5 T his is a second-order s ystem w ith bo = 28, b l = 4, b2 = 0, ao = 5, a l = 6. F igure 6.26 shows a realization w ith t wo feedback connections a nd two feedforward • connections. = s +5 s +7 T he r ealization a ppears in Fig. 6.24. Here H (s) is a first order t ransfer f unction w ith ao = 7 a nd bo = 5 , bl = 1. T here is a single feedback connection ( with coefficient 7) from t he i ntegrator o utput t o t he i nput s ummer. T here a re t wo feedforward connections from t he o utputs o f t he i ntegrator a nd t he i nput s ummer t o t he o utput s ummer.t F ig. 6 .24 R ealization of m F ig. 6 .26 R ealization o f .i~t;!5 6 ( c) H (s) = _ s_ 8 +7 E xercise E 6.9 Realize the transfer function H(s) = T his f irst-order t ransfer f unction is similar t o t hat i n ( b), e xcept t hat bo = 0. T herefore, t he r ealization i s s imilar t o t hat in Fig. 6.24 w ith t he feedforward connection from t he o utput o f t he i ntegrator missing, as d epicted i n Fig. 6.25a. Also, because t here is only tWhen m = n (as in this case), H (s) can also be realized in another way by recognizing t hat H(s) = 1 - _2_ 8 +7 We now realize H ( s) as a parallel combination of two transfer functions, as indicated by t he above equation. 6 .6-2 82 28 + 6 . + 25 Cascade and Parallel Realizations A n n th-order t ransfer f unction H (s) c an b e e xpressed a s a p roduct o r a s um o f n fi~st-order t ransfer f unctions. A ccordingly, w e c an a lso r ealize H (s) a s a c ascade ~senes) o r p arallel f orm o f t hese n f irst-order t ransfer f unctions. I nstance, t he t ransfer f unction i n p art ( d) o f t he l ast e xample: C onsider, f or 420 6 C ontinuous-Time System Analysis Using t he Laplace Transform 6.6 T he complex poles in H (s) s hould be realized as a second-order (quadratic) factor because we c annot implement multiplication by complex numbers. Consider,· for example, We can express H (s) as = 4s + 28 (s + l )(s + 5) (48 + 28) ( 1 ) = ~ 8+ 5 H (s) _ lOs + 50 - (s + 3)(s2 + 4s + 13) (6.76a) ' -v-"---"""-H1(8) lOs + 50 (s + 3)(s + 2 - j3)(s + 2 + j 3) H 2(s) We can also express H (8) as a sum of p artial fractions as 48 + 28 l )(s + 5) 2 1 + j2 1 - j2 s + 2 - j3 8 + 2 + j3 We cannot realize first-order transfer functions individually with t he poles - 2 ± j 3 because they require multiplication by complex numbers in t he feedback a nd t he feed forward path...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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