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

# T he direct a nd inverse z transforms can b e

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Unformatted text preview: screte- Time S ystems A nalysis U sing t he Z - Transform 11.1 677 T he Z -Transform M ethod o f F irst-Order F actors w here k= F[z] 2 (3z + 17) _ 2 (3z + 17) - z- = (z - 1)(z2 - 6z + 25) - (z - l)(z - 3 - j4)(z - 3 + j 4) 2Z2 - lIz + 121 - -3 (z - 2)3 %=, - W e find t he p artial f raction o f F [z]/z u sing t he Heaviside "cover-up" m ethod: ao = 2 z2 - llz + 121 -1 = -2 T herefore a nd F [z]=2 F[z] z + 1 .6e-J2246 + 2Z2 - lIz + 12 -3 2 a, a2 =------+---+-(z - l)(z - 2)3 z - 1 (z - 2)3 ( z - 2)2 ( z - 2) (11.13) We c an d etermine a , a nd a2 b y clearing fractions or by using t he s hort c uts discussed in Sec. B.5-3. For example, t o d etermine a2, we multiply b oth sides o f E q. (11.13) by z a nd l et z - + 0 0. T his y ields Z z _ 1 + ( 1.6e- j2 246 1 .6e . z - 3 - j4 F[z] = _ 2_ z z- 1 z;;:;2 z - 3 + j4 j 2 2 46) . z z -3-j4 (1 6 j2.246) + .e z z -3+j4 T he inverse t ransform o f t he f irst t erm o n t he r ight-hand s ide is 2u[k]. T he i nverse t ransform o fthe r emaining two t erms ( complex c onjugate poles) c an b e o btained f rom P air 1 2b (Table 11.1) b y i dentifying ~ = 1.6, () = - 2.246 r ad., "I = 3 + j 4 = 5eJO.927, so t hat 1"11 = 5, {3 = 0.927. T herefore I [k] = [2 + 3 .2(5)k cos ( 0.927k - 2.246)] u[k] o = - 3 - 0 + 0 + a2 ==> a2 = 3 T his r esult leaves only o ne u nknown, a" w hich is readily determined by l etting z t ake a ny convenient value, say z = 0, o n b oth s ides o f E q. (11.13). T his s tep yields M ethod o f Q uadratic F actors F[z] _ 2 (3z + 17) = _ 2_ - z- - (z - 1)(z2 - 6z + 25) z- 1 Multiplying b oth s ides by z a nd l etting z M ultiplying b oth s ides by 8 yields - + 0 0, + Az + B z2 - 6z + 25 we find 0 = 2 + A ==> A = - 2 12 = 24 + 2 + 2 a, - 12 ==> a , = - 1 . a nd T herefore -3 2 1 3 -=----------+-z z - 1 ( z - 2)3 (z - 2)2 z- 2 2 (3z + 17) _ _ _+ - 2z+B 2 ( z - 1)(z2 - 6z + 25) - z - 1 z2 - 6z + 25 F[z] a nd F[z] = z _ 3__ _ 2 _ __ z z- 1 _ __ _ z (z - 2)3 T o find B , we let z t ake a ny c onvenient value, say z = O. T his s tep y ields Z- B - 34 z + 3 __ (z - 2)2 -=-2+25 25 2 M ultiplying b oth s ides b y 25 yields Now t he use of T able 11.1, P airs 7 a nd 10, yields - 34 = - 50 + B ==> B = 16 I = k k (k - 1) [ k k k - 3 - 2 --(2) - 2(2) + 3(2) 8 = - [3 + ~(k2 + k k] u[k] Therefore F[z] _ _ _ 2 z - z- 1 - 12)2k]u[k] a nd 2z F[z] = z _ 1 ( c) C omplex P oles + - 2z + 16 z2 - 6z + 25 z ( - 2z + 16) 6z + 25 + z2 - W e now use P air 12c where we identify A = - 2, B = 16, 1"11 = 5, a = - 3. T herefore F[z] = 2 z(3z + 17) ( z - l)(z2 - 6z + 25) 2 z(3z + 17) ( z - 1)(z - 3 - j4)(z - 3 + j 4) P oles o f F[z] a re 1, 3 + j 4, a nd 3 - j4. W henever t here a re complex c onjugate poles, t he p roblem c an b e w orked o ut i n two ways. I n t he first m ethod we e xpand F[z] i nto (modified) f irst-order p artial f ractions. I n t he s econd m ethod, r ather t han o btaining o ne factor c orresponding t o e ach complex c onjugate pole, we o btain q uadratic f actors corresponding t o e ach p air of complex c onjugate poles. T his p rocedu...
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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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