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

63b where t he two node e quations 663a a nd 663b in

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Unformatted text preview: . . W hen t he o utput is fed back t o t he i nput, as shown I II Fig. 6.18d, t he overall transfer function Y (s) / F (s) c an be computed as follows. T he i nputs t o ~he s ummer are F (s) a nd - H(s)Y(s). Therefore, E (s), t he o utput of t he s ummer, IS The feedback transfer function, when K = 7, 16, and SO, can be obtained by the following MATLAB commands: K =7;c62 num/den 7 K=16,c62 num/den = 16 E (s) = F (s) - H(s)Y(s) B ut y es) = G (s)E(s) = G (s)[F(s) - H(s)Y(s)] T herefore y es) [1 + G(s)H(s)] = G (s)F(s) K =80;c62 num/den = 80 s "2 + 8 s + 8 0 o 414 6 6 .6 Continuous-Time System Analysis Using t he Laplace Transform 6.6 System Realization 415 S ystem realization x (t) (a) We now develop a systematic method for realization (or simulation) of a n arbitrary n th-order t ransfer function. Since realization is basically a synthesis problem, t here is no u nique way of realizing a system. A given transfer function can b e realized in many different ways. We p resent here three different ways o f realization: canonical, cascade a nd parallel realization. T he second form of canonical realization is d iscussed in Appendix 6.1 a t t he e nd of this chapter. A transfer function H (s) c an b e realized by using integrators or differentiators along with summers a nd multipliers. For practical reasons we avoid t he use of differentiators. A differentiator a ccentuates high-frequency signals, which, by their nature, have large rates of change (large derivative). Signals, in practice, are always corrupted b y noise, which happens t o b e a broad-band signal; t hat is, t he noise contains components of frequencies r anging from low t o very high. In processing desired signals by a differentiator, t he high-frequency components of noise are amplified disproportionately. Such amplified noise may swamp t he desired signal. T he i ntegrator, in contrast, tends t o s uppress a high frequency signal by smoothing i t out. In addition, practical differentiators b uilt w ith op amp circuits tend to be unstable. For these reasons we avoid differentia t ors in practical realizations. Consider a n LTIC s ystem with a transfer function ( b) b msm + b m_ls m - l + ... + b ls + bo s n + a n_ls n - l + ... + a lS + ao (e) Hs=-"'----''---''---,:------'--~ () For large s (s ~ (0) H (s) ' " b ms m - n F ig. 6 .20 Realization of H l(S) = Therefore, for m > n , t he s ystem acts as a n (m - n)th-order differentiator [see Eq. (6.55)]. F or t his reason, we r estrict m ::; n for practical systems. W ith t his restriction, the m ost general case is m = n w ith t he t ransfer function 1 83 + a2 8 2 + UI s +ao For convenience of realization, we shall express this transfer function as a cascade of two transfer functions H I(S) a nd H 2(S), as depicted in Fig. 6.19. H (s) = ( s3+ a 28 (6.70) , 2~ als + ao ) '(b3S3+b2S2+bIS+bo), H.-(s) , (6.72) H ;(s) T he o utput of HI(8) is d enoted by X (s), as illustrated in Fig. 6.19b. Therefore b /,+bZ i1+b l s+bO a nd...
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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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