Question

calculate the unit step response of a system with the difference equation

y[n]+1.5y[n-2]=x[n]-x[n-1], y[-1]=2 ,y[-2]=1

Answer 1

E The difference equation given is as follows :- y [n] + 1-5 y[n-2] = x[n]-x [n-1] => z fy [n]+ 1-5 y[n-2)} = 24 x [n] – x[n-1]} => z{y[n]3+1:524 y[n-234 =74x[n]) == {x{n-1}}

>> Y(3) +1.54 z? Y(z)+ z 'yE] + y(-3]} = X(z) – +2X(z) => (1+1.52-2) Y(z) +(1.5z+1)(2) +(1:5)(1) = X(3) (1-21) Since, we are given y[-1]=2&y[-2] = 1 → Y(z) (1+1:577) = X(z) (1-2+4) -327-15 Now, unit step function (u[n]) = {1, 120 1o, nco According to the question, unit step function is our input signal, thus x [n] = u[n] X(z) = z fuin]] to both sides. (taking z transform on )

nEO » X(z) = Eur] zen >> X(z) = 1+ 2++32+..... » X(z) - t (Applying geometric (1-z-1) progression to infinite We took common ratio (8)= x², thus series) X(21= 5021 - Art Now putting the value of X(z) in equation 0.we Y(z) (17 1.522) = 1 (1-21) – 37'-1.5 (1-) => Y(z) (4+1.52+2) = 1-33-1-15 Y(z) = -0.5-324 1+1.522 => Y(z) = -0.5 ( ) -177522 We know that, ztca" cosw.nju[n]} = t-az-cosw, 1- Qaz cosw. + a?z? Z{(asinw.n) u[n]} = az-sinw. 1 I-2az cosw.+a?z? for both cases, 171>1 ali Now comparing equation ® with standard results we get, q² = 1.5, when w.=t/ radians Sie or, a = 11:5 = 1*2247

putting the value and rearranging eq@, 2247 Ye3 = cos(s) or ( = Y(z) = -0.51 t 1- 171522) 449 1 1.2247 1+1.522 taking inverse z transform on both sides, >> Ž +4Y(z)} =z={-0.5( 171571) -2.497 to » y[n] = (-0.5)(1:2207)" cos(an) u[n] +(-2.499) (1.2247)ì sin ) ucn] This is the required step response of the given difference equation. WAS VENT