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

# L et us denote this impulse response of so b y ynt

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Unformatted text preview: heir 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 &gt; 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 &quot;zero-input.&quot; 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 modes. 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 &gt; O. Therefore, t he system response for t &gt; 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 all the...
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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.

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