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Unformatted text preview: . F or this reason noise c ontains a s ignificant a mount o f r apidly
varying c omponents w ith d erivatives t hat a re, consequently, very large. Therefore,
any s ystem specified b y E q. (2.1) in which m > n will magnify t he h igh-frequency
c omponents o f noise t hrough d ifferentiation. I t is e ntirely possible for noise t o b e
m agnified so much t hat i t s wamps t he d esired s ystem o utput even if t he noise signal a t t he s ystem's i nput is t olerably s mall. For t he r est o f t his t ext we a ssume
implicitly t hat m ::; n . F or t he s ake o f g enerality we shall a ssume m = n in Eq.
I n C hapter 1, we d emonstrated t hat a s ystem d escribed by Eq. (2.1) is linear.t T herefore, i ts r esponse c an b e e xpressed as t he s um o f t wo components: t he
z ero-input c omponent a nd t he z ero-state c omponent ( decomposition property).:I:
T otal r esponse = z ero-input r esponse + z ero-state r esponse (2.3) F or t he p urpose o f a nalysis, we shall consider l inear d ifferential s ystems.
T his is t he class o f L TIC s ystems discussed in C hapter 1, for which t he i nput J (t)
a nd t he o utput y (t) a re r elated b y l inear differential equations o f t he form T he z ero-input c omponent is t he s ystem r esponse w hen t he i nput J (t) = 0 so
t hat i t is t he r esult of i nternal s ystem c onditions (such as energy storages, initial
conditions) alone. I t is i ndependent of t he e xternal i nput J (t). I n c ontrast, t he
z ero-state c omponent is t he s ystem r esponse t o t he e xternal i nput J (t) w hen t he
s ystem is in zero s tate, m eaning t he a bsence o f all i nternal e nergy storages; t hat is,
all initial conditions a re zero. (2.1a) tTo demonstrate that any system described by Eq. (2.1) is linear, let the input f l ( t) to the system
generate the output m et), and another input h (t) generate the output Y2(t). From Eq. (2.1c) it
Q (D)Yl(t) = P (D)fl(t)
Q(D)Y2(t) = P (D)h(t) where all t he c oefficients a i a nd bi a re c onstants. U sing o perational n otation D t o
r epresent d /dt, w e c an express t his e quation as Multiplication of these equations by kJ and k2, respectively and then adding yields
( Dn + a n_ID n - 1 + ... + a ID + a o) y et)
= ( bmDm + b m_ID m - 1 + ... + b ID + bo) J (t) ( 2.1b) or
Q (D)y(t) = P (D)J(t) (2.1c) This equation shows that the input k lfl (t) + k 2h(t) generates the response klYJ (t) + k2Y2(t).
Therefore, the system is linear
:I: We can verify readily that the system described by Eq. (2.1) has the decomposition property.
I f yo(t) is the zero-input response, then, by definition,
Q(D)yo(t) = 0 w here t he p olynomials Q (D) a nd P (D) a re
I f yet) P (D) = n + a n_ID n - 1 + ... + a ID + ao
b mDm + b m_ID m - 1 + ... + b ID + bo Q (D) = D (2.2a)
(2.2b) is the zero-state response, then yet) is the solution of
Q (D)y(t) = P (D)f(t) subject to zero initial conditions (zero-state). The addition of these two equations yields
Q(D) [Yo(t) 104 + y(t)[ = P (D)f(t) Clearly, yo(t) + y(t) is th...
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