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

Unfortunately there are cases where homogeneity does

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Unformatted text preview: m ultiple-input, m Ultiple-output ( MIMO) systems. A system's o utput for t ? 0 is t he r esult of two independent causes: t he initial conditions of t he system (or t he s ystem s tate) a t t = 0 a nd t he i nput J (t) for t ? O. I f a s ystem is t o b e linear, t he o utput m ust be t he sum of t he two components resulting from t hese two causes: first, t he z ero-input r esponse c omponent t hat results only from t he initial conditions a t t = 0 w ith t he i nput J (t) = 0 for t ? 0, a nd t hen t he z ero-state r esponse c omponent t hat results only from t he i nput J (t) for t ? 0 when t he initial conditions ( at t = 0) a re assumed to be zero. When all t he a ppropriate initial conditions are zero, t he s ystem is said t o b e in z ero s tate. T he s ystem o utput is zero when the i nput is zero only if the system is in zero state. In s ummary, a linear system response can be expressed as t he s um of a zeroinput a nd a z ero-state component: T otal r esponse = z ero-input r esponse + z ero-state r esponse (1.41) This p roperty of linear systems which permits t he s eparation of a n o utput into components resulting from t he initial conditions a nd from t he i nput is called t he d ecomposition p roperty. For t he R C circuit of Fig. 1.26, t he response y (t) was found t o b e [see Eq. (1.36c)] y (t) = v dO) ----- z-i c omponent + RJ(t) , + -1 % -8 it . J(r)dr co, and dY2 d t + 3Y2(t) = h (t) Multiplying the first equation by k l, the second with k2, and adding them yields But this equation is the system equation [Eq. (1.43)J with and yet) Therefore, when the input is k dl(t) + k2!2(t), the system response is k1Yl(t) + k2Y2(t). Consequently, the system is linear. Using this argument, we can readily generalize the result to show t hat a system described by a differential equation of the form (1.44) is a linear system. The coefficients a i and bi in this equation can be constants or functions of time . • 6 E xercise E 1.12 Show that the system described by the following equation is linear: (1.42) c omponent From Eq. (1.42), it is clear t hat if t he i nput J (t) = 0 for t ? 0, t he o utput y (t) = v dO). Hence v c(O) is t he zero-input component of t he response y (t). Similarly, if t he s ystem s tate ( the voltage v c in this case) is zero a t t = 0, t he o utput is given by t he second component on t he r ight-hand side of Eq. (1.42). Clearly t his is t he z ero-state c omponent of t he response y (t). I n addition to t he decomposition property, linearity implies t hat b oth t he zeroinput a nd z ero-state components must obey t he principle of superposition with respect t o each of their respective causes. For example, if we increase t he i nitial condition k-fold, t he zero-input component must also increase k-fold. Similarly, if we increase t he i nput k-fold, t he z ero-state component must also increase k-fold. = k lYl(t) + k2Y2(t) 6 E xercise E 1.13 ~ + t 2 y(t) = (2t + 3 )J(t) dt 'V Show that a system described by the following equation is nonlinear: y(t)~ + 3 y(t) = dt J (t...
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