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Unformatted text preview: namical behavior of t he system.
In this chapter w e shall consider only t he continuoustime systems. (Modeling of
discretetime 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 voltagecurrent 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 nterconnectionthe wellknown 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 inputoutput 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: Inputoutput Description f (t) By using the voltagecurrent 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 nputOutput Description + 3 dy + 2y(t) = '!f..
dt dt (1.49) This differential equation is the inputoutput 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>D is not necessarily unity. Use of Cramer's rule in solving
d t  00
•
simultaneous integrodifferential 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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 Spring '13
 Bayliss
 Signal Processing, The Land

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