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

# T hus a differentiator along with one piece of

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Unformatted text preview: namical behavior of t he system. In this chapter w e shall consider only t he continuous-time systems. (Modeling of discrete-time s ystems is discussed in C hapter 8.) To construct a system model, we m ust s tudy t he relationships between different variables in t he system. In electrical systems, for example, we m ust determine a satisfactory model for the voltage-current relationship of each element, such as Ohm's law for a resistor. In addition, we m ust determine the various constraints on voltages a nd c urrents when several electrical elements are interconnected. These are t he laws of i nterconnection-the well-known Kirchhoff's voltage and current laws (KVL a nd KCL). From all these equations, we eliminate unwanted variables t o o btain equation(s) relating t he desired o utput variable(s) to t he i nput(s). T he following examples d emonstrate t he procedure of deriving input-output relationships for some LTI electrical systems. tThe additional piece of information cannot be just any information. For instance, in the above example, if we are given i(O) = 0, it will not help in determining c, and the system is noninvertible. (1.47) 1t y (r) dT = f (t) (1.48) - 00 Differentiating both sides of this equation obtains 2 System Model: Input-output Description f (t) By using the voltage-current laws of each element (inductor, resistor, and capacitor), we can express this equation as d2 y dt 1.8 89 R =30 L =l H dt E xercise E 1.IS Show that a system described by the equation y ( t ) = P ( t) is noninvertible. 1.8 System Model: I nput-Output Description + 3 dy + 2y(t) = '!f.. dt dt (1.49) This differential equation is the input-output relationship between the output y(t) and the input f (t) . • -ft. I t proves convenient t o use a compact notation D for t he differential operator T hus ~~ (1.50) == D y(t) d2 y - 2 == D 2y(t) dt (1.51) a nd so on. W ith t his notation, Eq. (1.49) can be expressed as (D2 + 3 D + 2) y (t) = D f (t) (1.52) T he differential o perator is t he inverse of t he i ntegral operator, so we c an use t he o perator 1 / D t o r epresent integrationt. t 1 - 00 1 y (r)dT == - y(t) (1.53) D t Use of operator 1/ D for integration generates some subtle mathematical difficulties because the operators D and 1 /D do not commute. For instance, we know that D (l/ D ) = 1 because .!!.[ft yeT) dT] = yet). However, 1&gt;D is not necessarily unity. Use of Cramer's rule in solving d t - 00 • simultaneous integro-differential equations will always result I• II cancellatlOn of operators l /D and . D . This procedure may yield erroneous results in those cases where the factor D occurs I II the 90 I ntroduction t o S ignals a nd S ystems 1 .8 R =lSU 5 + D x(t) = 15x(t) + or F ig. 1 .32 15~~ + 5x(t) = ~ C ircuit for E xample 1.11. +3+ % y(t) ) = f (t) dy x(t) = C dt + 3 D + 2) y (t) = ( 1.54) (3D = D f(t) 1 5Dy (t) S ubstitution o f t his r esult in Eq. (1.59a) yields M ultiplying b oth s ides b y D , t hat is, d ifferentiating E q. ( 1.54), w e o btain (D2 (1.59b)...
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