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

# 8 2 2 r epeat p rob 82 1 for t he e xponentials a

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Unformatted text preview: d (8.29) a re f irst-order difference equations, a nd E q. (8.26) is a second-order difference e quation. All t hese e quations a re l inear, w ith c onstant ( not time-varying) coefficients. Before giving a general form o f a n n th-order difference e quation, we recall t hat a difference e quation c an b e w ritten i n two forms; t he first form uses delay t erms s uch as y[k - 1], y[k - 2], I [k - 1], J[k - 2], . .. , e tc., a nd t he a lternate form uses advance t erms s uch as y [k+1]' y [k+2]' . .. , e tc. B oth forms a re useful. We s tart h ere w ith a g eneral n th-order difference e quation, u sing advance o perator form: y[k + nJ + a n-ly[k + n - 1J + ... + aIY[k + 1J + aoy[kJ = bmJ[k + mJ + b m-d[k + m- 1J + ... + b d[k + 1J + bof[kJ (9.1) O bserve t hat t he coefficient of y[k + nJ c an b e a ssumed t o b e u nity w ithout loss o f g enerality. I f t his coefficient is o ther t han u nity, we c an d ivide t he e quation t hroughout b y t he v alue o f t he coefficient o f y[k + nJ t o n ormalize t he e quation t o t he f orm in (9.1). Causality Condition T he l eft-hand side o fEq. (9.1) consists o f t he o utput a t i nstants k +n, k +n-1, k + n - 2, a nd s o on. T he r ight-hand s ide o f E q. (9.1) consists o f t he i nput a t instants k + m , k + m - 1, k + m - 2, a nd so on. For a causal s ystem t he o utput c annot 573 574 9 Time-Domain Analysis o f Discrete-Time Systems 9.1 575 Discrete-Time S ystem e quations y I k] depend on future input values. This fact shows t hat when t he s ystem equation is in the advance operator form (9.1), t he causality requires m s: n. For a general causal case, m = n , a nd Eq. (9.1) can be expressed as 12.25 .. . 8 6.5 y[k + nJ + a n-lyrk + n - IJ + ... + aly[k + 1] + aoY[k] = b nf[k + n] + b n_d[k + n - 1] + ... + b d[k + IJ + bof[k] 5 (9.2a) where some of the coefficients on b oth sides can be zero. However, t he coefficient of y [k +n] is normalized t o unity. Equation (9.2a) is valid for all values of k . Therefore, t;he e quation is still valid if we replace k by k - n t hroughout t he e quation [see E qs. (8.25a) a nd (8.25b)J. Such replacement yields t he a lternative form (the delay o perator form) o f Eq. (9.2a): y[kJ + a n-ly(k - IJ + .. , + aIY[k - n + IJ + aoy(k - n] = bnf[k] + b n-d[k - 1] + ... + b I![k - n + 1] + bof[k - o 3 2 F ig. 9 .1 I terative s olution o f a difference e quation in E xample 9.1. • E xample 9 .1 Solve i teratively y[kJ - 0.5y[k nJ (9.2b) k _5 _ 4 w ith i nitial condition y [-I] = 16 a nd c ausal i nput f[k] = e quation c an b e e xpressed as W e shall designate Form (9.2a) t he a dvance o perator f orm, a nd F orm (9.2b) t he d elay o perator f orm. ( 9.3a) IJ = flkJ e ( starting a t k = 0 ). T his ylk] = 0.5ylk - 1) + f[k) ( 9.3b) I f we s et k = 0 i n t his e quation, we o btain 9 .1-1 Initial Conditions and Iterative Solution o f Difference Equations y[O] =0.5y[-I] + f[O] E quation (9.2b) can be expressed as = 0.5(16) y[kJ = - an-ly[k - IJ - an-2y[k - 2] - .,. - aoy[k - n] + bnf[kJ + b n-I!(k - 1] + ... + bof!k -...
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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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