Question

Two filters described by the formulas below are cascaded.

Yı[n] = x[n] – x[n- 1] y2[n] = x[n] + x[n-1]

What will be the transfer function h(z) for the total system? from these four choices

Answer 1

* Two filters are described by formeelas as 9.(n) = f(n) - X(n-1) 42(n) = x (h) + n(n-1) System / filter 1 :- (n) - (n) - (n-1) y (n) - Fleter X(2) 4.(2) Hi(2) * y, (n) = x(n) - n(n-1) → Applying z-transform, we get x(n) + Zit:, XC7) x(n-no ( Z.T. *(z) z-no | Y, (z) = x(z) - x(2) Z-' | 4,(Z) = x(z) [-2-3 * Transfer function of filtera is given as ratio of output z-transform to input z-transform as H, CZ) = 4(2) - X (2) 1-Z Filter 2 :- x (n) Filter H₂(z) - H X(Z) Y₂(m) 4 (2) > Ba(n) = a(n) + x (n-1) * Applying z-transform, we get | 42(Z) = X(7) + X(2) 7-11

42(2) = x(z) [1 +2 -] * Transfer function H₂(z) of filter 2 is ratio of output z- transforn 43(z) and input 7-transfour Xtz). H2(Z) = 42 (7) X(z) - 1+2 -1 * Now, two filters are cascaded as below ain) - 4.H2(z) -> 4cm) x17) 4(2) * The Transfer function of cascaded System H(Z) geven as the tip multiplication of individual transfer functions. ( H(Z) = H,(z) * H2() 0 * Substituting and ② in ③ equation H(7) = (1-z-1) (1+2-1) HCZ) = 1-2-!+2-12-2 as H (Z) = 1-Z - 2 correct option is 6 H(z) = 1- 2-2