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Unformatted text preview: gnals generated within t he s ystem will a ct as initial conditions.
Because o f e xponential g rowth, a s tray signal, no m atter how small, will eventually
cause an u nbounded o utput in a n u nstable system.
Marginally stable systems do have one i mportant a pplication in t he oscillator,
which is a s ystem t hat g enerates a signal on its own without t he a pplication of a n
e xternal i nput. C onsequently, t he oscillator o utput is a zeroinput response. I f such
a response is t o b e a sinusoid of frequency wo, t he system should b e m arginally
stable w ith c haracteristic r oots a t ± jwo. T hus, t o design an oscillator of frequency
wo, we s hould p ick a s ystem w ith t he c haracteristic polynomial (A  jWO)(A + jwo) =
A2 + wo 2. A s ystem d escribed by t he differential equation will do t he job.
t This can be shown as follows. I f Ai =
Therefore " 'i + j{3" then e A = e",'ejl>i' and leAi'1 = e ""'.
" if R.e Ai = " 'i <0 and Eq. (2.65) follows. This conclusion is also valid when the integrand is of the form ItkeAi'lu(t).
:I: However, a B IBO stable system is not necessarily asymptotically stable because B IBO stability
is determined from the system's impulse response, which is an external description of the system,
while asymptotic stability is determined from the internal description of the system obtained
from system equations. In certain systems (e.g., uncontrollable or unobservable systems), the two
descriptions may not be the same. Remember that the external description describes only that
part of the system which is coupled to both the input and the output. Hence, a system may be
internally unstable while appearing stable from the system's external terminals ( BIBO stable)2 2 .7 Intuitive Insights into S ystem Behavior T his s ection a ttempts t o provide a n u nderstanding of w hat d etermines system
behavior. Because of its intuitive n ature, t he following discussion will be more or
less qualitative. We shall now show t hat t he m ost i mportant a ttributes of a system
are its characteristic roots or characteristic modes because they determine n ot only
t he z eroinput response b ut also t he e ntire behavior of the system. 2.71 Dependence o f S ystem Behavior on Characteristic Modes Recall t hat t he z eroinput response of a system consists of t he s ystem's characteristic modes. For a stable system, these characteristic modes decay exponentially
a nd e ventually vanish. T his b ehavior may give t he impression t hat t hese modes
do n ot s ubstantially affect system behavior in general a nd s ystem response in p articular. This impression is t otally wrong! We shall now see t hat t he s ystem's
characteristic modes leave their imprint on every aspect of t he s ystem behavior.
We m ay c ompare t he s ystem's c haracteristic m odes (or roots) to a seed which eventually dissolves in t he ground; however, the plant t hat springs from i t is t otally
d etermined by t he seed. T he imprint o f t he seed exists on...
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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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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