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respective o utputs y(t) a nd x(t) a re related by y(t) = P (D)x(t) (2.83) To prove t his r esult, we o perate o n b oth sides of Eq. (2.80) by P (D) t o o btain Q (D)P(D)x(t) = P (D)f(t)
C omparison of t his e quation with Eq. (2.77a) leads immediately t o Eq. (2.83).
Now if t he i nput f (t) = 8(t), t he o utput of So is Yn(t), a nd t he o utput o f
S, according t o Eq. (2.83), is P(D)Yn(t). T his o utput is h(t), t he u nit impulse
response of S. N ote, however, t hat because i t is a n impulse response of a causal
system So, t he f unction Yn(t) is causal. To incorporate this fact we m ust represent
t his function as Yn(t)u(t). Now i t follows t hat h(t), t he u nit impulse response of
t he s ystem S, is given by h(t) = P(D)[Yn(t)u(t)] (2.84) where Yn (t) is a linear combination of the characteristic modes o fthe s ystem subject
t o i nitial conditions (2.82). This expression is valid when
Eq. (2.84) should be used. 2 .9 m :s; n [the form given in Eq. (2.77b)J. When m > n , Summary T his chapter discusses time-domain analysis of LTIC systems. T he t otal response of a linear system is a s um of the zero-input response and zero-state response.
T he zero-input response is t he system response generated only by the internal conditions (initial conditions) of the system, assuming t hat t he external input is zero;
hence t he t erm "zero-input." T he zero-state response is t he system response generated by t he e xternal input, assuming t hat all initial conditions are zero; t hat is,
when t he s ystem is in zero state.
Every system can sustain certain forms ofresponse on its own with no external
input (zero input). These forms are intrinsic characteristics of t he system; t hat
is, they do not depend on any external input. For this reason they are called
characteristic modes of the system. Needless t o say, the zero-input response is
made up of characteristic modes chosen in a suitable combination so as t o satisfy
the initial conditions of the system. For a n n th-order system, there are n d istinct
T he u nit impulse function is a n idealized mathematical model of a signal t hat
c annot be generated in practice. Nevertheless, introduction of such a signal as an
intermediary is very helpful in analysis of signals a nd systems. The unit impulse
response of a system is a combination of t he characteristic modes of the systemt
because the impulse ott) = 0 for t > O. Therefore, t he system response for t > 0
must necessarily be a zero-input response, which, as seen earlier, is a combination
of characteristic modes.
The zero-state response (response due t o e xternal input) of a linear system can
be obtained by breaking the input into simpler components and then adding the
responses to all the components. In this chapter we represent an arbitrary input
f (t) as a sum of narrow rectangular pulses [staircase approximation of f(t)J. In the
limit as the pulse width ...... 0, t he rectangular pulse components approach impulses.
Knowing the impulse response of the system, we c an find t he system response t o
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