Question

Compare the responses of the following LPF algorithms to a unit step input: a) y(n) = 0.1x(n) +0.9v(n-1) y(n)=0.5x(n) +0.5y(n-1) y(n)=0.05x(n) +0.95y(n-1) y(n) = 0.01x(n) +0.99 y(n-1)

Answer 1

ayon = 0.52 (+0.94" (9+0.9 gon-1). 6) HOL) = 05 7 Apply Z hans form Ylz)[ 0.92'] = 0-1X(z) 10.52 YEZ) = D.LT X(z) I-0.92' hin) = 05 (0.5) huen) Yon) = 0.5 10.50€ k=0 = 0.5 11-(0.59+) (Z) = 0 ) 1-0.9² -0.5 = 0.5 [ 1-0.500.559] 0-5 * Apply Inverse z bransform, hede 0.1 Co.qnJuin ) xc9) = u(n) -→ unot step%% yonda xom * fon) = un) v 0.1 (0.ghuin)) Yons famous [un) * 0.9"um] yan) to nzo, yen) 2001 (6.9k You = [(- (0.5) (0.53") uch) 9) # (2)= 0.05 1-0.9521 20 hin) = 0.0310.95," uch) ges) = 0.05 Ž 20.959² ko = 0.1 x r10.99+) -0.9 = 0.1*1- 0.910.99" 20.05 1-(0.95)'* -0.95 = 0.05 ( 1-60-9590+1) 0.1 — yen) = [- 0.910.9997 uond by step Reuponce 0105 yn = [[_009510.95)] am) ob yon) = [-0.99-0.99)"] uls)

The four step responses are plotted by using MATLAB.

Script;

n=0:1:50;

y1=1-(0.9.^(n+1));figure;subplot(221);stem(n,y1),'r';xlabel('n');ylabel('y1(n)');title('a)step response ')

y2=1-(0.5.^(n+1));subplot(222);stem(n,y2,'m');xlabel('n');ylabel('y2(n)');title('b)step response ')

y3=1-(0.95.^(n+1));subplot(223);stem(n,y3,'b');xlabel('n');ylabel('y3(n)');title('c)step response ')

y4=1-(0.99.^(n+1));subplot(224);stem(n,y4,'g');xlabel('n');ylabel('y4(n)');title('d)step response ')

a) step response b) step response yl(n) y2(n) 0000000 o 20 40 60 0 20 40 60 c)step response d) step response mwomam TADT y3(n) y 4(n) S 20 O o 20 40 60 0 20 40 60