Question

37z-2 (a) Show using the definition of the Z-Transform that Z({3+4Uk-3} ) 2. Z 3 (b) Using operational theorems and the table of Fourier-Transforms, determine the following: i. F(6e-5te-4lt); ii. F124juw sin (11w) -7 iii. F-1 4w2- 12w + 12 (c) The Fibonacci sequence {fk), is generated via the following second order difference equation fk+2 Z-Transform technique, show that for k 2 1 fk+1 f, for k 0, with fo = 0 and fi = 1. Using the k V5 V5 1+ V5 5 1 1 2 2

Answer 1

2 Hoanstort r R+4 (a) R-3 2-3 -n. We knoco that Gven n+4 k+4 u(k-3) n(n)= uin-3) 3 X(z)= 4 3え n-3 Ihe hndeud fesics con/ have beera odstoactmg addlung and n+3-3 (z)= 3 n-3 3 n-3 k, pd n-3 LL k=O then (k+3) k=の UL 3 Z (32 -3. -k 3.7. 3 . 3. KO 3え 1-32 8NN

(b) Fl 41t1 6 e -alt) We lanoco that houoeo toangben ct 20 e -Alt1 2(4) 4+co 6+0 -jat x10-cOo nlt) e 5jt -Alt 16+ axlf) ntt -4lH -5t 6e 48 anzwer Ajo 2 e Sm (110 71. Sin F Sat Cf)=Sinc gect (t) Sen (2 rect (t) 2 Sm

gect (72s 2*22x Sn 22 2x L0 91e c ( 2 Sm (9 Sen l2) nec 2 4juD 2 e x(o) 2nIt+to) n(t+4x2 Angioen t+4 2 x rect 2 1 22 t44 22 ot Ans 91ect 40-1200+12 Bw+3 -(-3)+3-12 3+ 2 2 A CLd-(3.-39) --30) w-13-21 o-(3-) 1-i) A = 3 12 12 x2i 12.

2 (-(-) () 12 w- cal ult) a+ fiw t . 12 12 e X12 e $12 t Sin ex 2 12 Cc). Aret + the De get toangbon apply 2 + f(z) to F(z) Fl) - 7 +7+ Fl2) 2 F()= 2 Roots d -(-1) 1 44 150 2 2.

F(2)= 2 2-1 2 Noco Coe knoco that I.R.T a utn)-P u z-a(z-b (0F a-b b ucn ucn cohene 041=k, hen ar uthi) b ulk-1) a-b We oo1 get For a-b= 2x àTei-ci 2 2