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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 -...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online