Question

Question 4. (a) Determine the difference equations that relate the input and output variables for the following block diagram: (b) Determine if the following systems are equivalent? Explain. 2 Delay Delay Delay -1 Delay -1 (c) Choose the coefficients for the block diagram on the right so that the two systems (both left and right) become equivalent. In this question, you are asked to find the values of po, p1, A, and B that will make the system equivalent to the one on the right. (R is the shift-right or the delay operator.) X

Answer 1

43 (a) 2- considering the block diagram in 2- domain. 20-20 10-s you) e Replace, (delay) (2-1) x(2) at point → 2X(z) PS X(z) + x(z)2-= x(z) [1 + 2-'] 3 – 42XC2) [1+z='] 4- 427X(z) [1 +2+] + X(z) = Y (2) Y(z) = X(2) [42+ (1+2+)+1] | 12) = (1+ 4z4+42+2) X(z) b.) Let us findout the transfer function system and then compare them they are equivalent or not. of both to findout X(2) 20 Dip's Y(2) 4 si X3) T TD2TB -> Y(2)

2-37 at hat. yy*(2) 2-1 oft X(2)(-1) 2-1.2-! → -*(2) 2-2 3+ X(2) [21-2-2] 4-61 (2-1) X(2) [24-2-23 = - X(z) [ z so QX (2) [21-2-2] 6- QX(2) [24-2-27 - X(2)[2-22-33 = X(2] [22-7-22-2-2-* + 2-37 . Y62) = X(z) [Qz-+-32-42-3] LO) = (@e;"–33+2+3+37 for figlio): at point 1 + x(2) 6-1)21 - - X(2)2-1 24 x(2) — X(2) Z-' => (2)[1-2"]. 3+ (-1) zt. 2+ x(2) [1-2-'] =) - X(2) 2-2 (1-2"] = -X(z)(2-2-2-"). 43 dz t X (2) (1-2-1] 5 => + XC2) [2-2-2-3) + 22-1862261-2-'] = X(z) [-250+2+3+271-22-] Y(z) = x(2) (dz. *_3252 +2 °] 40) = (23'-332+ 2 =9 - conclution : - Both systems are equivalent.

e) [ Consider delay unit = P = z'] X(2) > 3 yr Yz) x2 B D fénye) At point 17 YC2) 2-1 27 22 24 (2) 3.-) X(z) +402)2+1 +22-242) .: Y(z) = X(2)+ Y(212'+ 22 PY(2) Y(2)[1-2-4-22-?] = X (2) at point x =) M1 po2+ + XC2) 20,= Kipoz t +X(2) at point na ) na pozi+X (2) 12 = xy piz' + X(2) at poin 1 => a, Atxa B=Y (2) => X(2), A + X(2) 8 - (1-port) . (1-8,2") [Y(2) = A + 1 1 1 1 1 1 2 X(z) (1-Poz-1)' (1-242") L(1) (1+2-1 (1-22) Now comparing If I, we get. f = -1 , P. = 21