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

Y t c linear system analysis using t he bilateral

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Unformatted text preview: l by which a signal is expressed as a sum o f everlasting exponentials est. T he relati:ve a mount o f a c omponent est is F (s). Therefore F (s), t he Laplace transform of f (t), r epresents t he s pectrum of exponential components of f (t). Moreover, H (s) is t he s ystem response (or gain) t o a s pectral component est, a nd t he o utput signal s pectrum is t he i nput s pectrum F (s) t imes t he s pectral response (gain) H (s) [yes) = F(s)H(s)J. T he Laplace transform changes integro-differential equations of L TIC systems into algebraic equations. Therefore, solving these integro-differential equations reduces t o solving algebraic equations. T he Laplace transform method cannot generally be used for time-varying p arameter s ystems or for nonlinear systems. T he t ransfer function of a system may also be defined as a ratio of t he Laplace transform of t he o utput t o t he Laplace transform of t he i nput when all initial conditions are zero (system in zero state). I f F (s) is t he Laplace transform of t he i nput f (t) a nd Y (s) is t he Laplace transform of t he corresponding o utput yet) (when all initial conditions are zero), t hen y es) = F (s)H(s), where H (s) is t he system transfer function. T he s ystem transfer function H (s) is t he Laplace transform of the system impulse response h (t). Like t he impulse response h (t), t he t ransfer function H (s) is also a n e xternal description of t he s ystem. Electrical circuit analysis can also be carried o ut by using a transformed circuit method, in which all signals (voltages a nd c urrents) are represented by their Laplace transforms, all elements by their impedances (or admittances), a nd initial conditions by their equivalent sources (initial condition generators). In this method, a network can be analyzed as if it were a resistive circuit. Large systems can b e considered as suitably interconnected subsystems represented by blocks. Each subsystem, being a smaller system, can b e readily analyzed a nd r epresented by its i nput-output relationship, such as its transfer function. Analysis o f large systems can be carried o ut w ith t he knowledge of i nput-output relationships of its subsystems a nd t he n ature o f i nterconnection o f various subsystems. LTIC s ystems can be realized by scalar multipliers, summers, a nd i ntegrators. A given transfer function can be synthesized in many different ways. Canonical, cascade, a nd parallel forms of realization are discussed. I n p ractice, all t he building blocks (scalar multipliers, summers, a nd i ntegrators) can be obtained from operational amplifiers. Feedback systems are closed-loop systems mainly used t o c ounteract t he effects of unpredictable variations in t he s ystem parameters, t he load, a nd t he environment. Such systems are designed for a specified speed a nd t he s teady-state error. For speed, t he useful transient parameters are rise time, peak time, a nd s ettling time. Percent overshoot indicates how smoothly t he system o utput rises to its final value. We can relate t he s tea...
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