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

Pb2 9 r epeat p rob b4 1 for t he s ignal d epicted in

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Unformatted text preview: n) (9.2c) T his e quation shows t hat y[k], t he o utput a t t he k th i nstant, is c omputed from 2 n+l pieces of information. These are t he p ast n values of t he o utput: y[k - 1], y[k 2], . .. , y [k-n], t he p ast n values o fthe i nput: f [k-l]' f [k-2]' . .. , f [k-n]' a nd the p resent value o f t he i nput f[k). I f t he i nput is causal, t hen f [-IJ = f [-2] = . .. = f [ - n] = 0, a nd we need only n initial conditions y (-I], y [-2], . .. , y [-n]. T his result allows us to compute iteratively or recursively t he o utput y[O], y[l], y[2], y[3], . .. , a nd so o n.t For instance, to find y[O] we s et k = 0 in Eq. (9.2c). T he l eft-hand s ide is y[O), a nd t he right-hand side contains t erms y [-I], y [-2), . .. , y [-n), a nd t he i nputs frO), f [-I), f [-2), . .. , f [-n]. Therefore, t o begin with, we m ust know t he n initial conditions y [-IJ, y [-2J, . .. , y [-n]. Knowing these conditions and t he i nput f[k), we c an iteratively find t he response y[O), y[IJ, y[2], . .. , a nd so on. The following examples demonstrate this procedure. This method basically reflects the m anner in which a computer would solve a difference equation, given t he i nput and i nitial conditions. t For this reason Eq. (9.2) is called a r ecursive d ifference e quation. However in Eq. (9.2), if a l = a 2 = ... = a n-l = 0, then, according to E q. (9.2c), determination of the present o utput y[kJ does not require the past values y[k - 1], y[k - 2], . .. , etc. For this reason, when a i = 0, (i = 0 ,1, . .. , n - 1), the difference E q. (9.2) is n onrecursive. This classification is i mportant in designing and realizing digital filters. In this chapter, however, this classification is n ot important. The analysis techniques developed here (and in Chap. 12) apply t o general ao = recursive and nonrecursive systems. Observe t hat a nonrecursive s ystem is a special case o f a recursive system with ao = a l = . .. = a n-l = O. Design o f recursive and nonrecursive s ystems is discussed in Chapter 12. +0 = 8 Now, s etting k = 1 i n E q. ( 9.3b) a nd u sing t he value y[O] = 8 ( computed i n t he first s tep) a nd f[IJ = (1)2 = 1, we o btain y[I) = 0.5(8) + (1)2 = 5 Next, s etting k = 2 i n Eq. (9.3b) a nd u sing t he value y[l] = 5 ( computed i n t he p revious s tep) a nd f[2) = (2)2, we o btain y[2] = 0.5(5) + (2)2 = 6.5 C ontinuing in this way iteratively, we o btain + (3)2 = 12.25 0.5(12.25) + (4)2 = 22.125 y(3) = 0.5(6.5) y[4] = T he o utput y[k] is d epicted in Fig. 9.1. • We now present one more example o f i terative s olution-this t ime f?r a.secondorder equation. Iterative method can be applied t o a difference e quatIon I II d elay o r advance operator form. In Example 9.1 we considered t he former. Let us now apply t he i terative method t o t he advance o perator form. • E xample 9 .2 S olve i teratively y[k + 2 ]- y[k + 1] + 0.24y[k] = f[k + 2 ]- 2f[k + IJ (9.4) w ith i nitial conditions y [-I] = 2, y[-2] = 1 a nd a c ausal i nput f[k] = k ( starting a t k = 0). T he s ystem e quation c an b e e xpressed a s 5 76 9 T ime...
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