A filter is described by the difference equation y[n]= x[n] - 0.5x [n -1] - 0.25

Question

A filter is described by the difference equation    y[n]= x[n] - 0.5x [n -1] - 0.25x [n - 2] -0.125x [n -3] Find the first 10 samples of it’s step response A filter is described by the difference equation    y[n]= x[n] - 0.5x [n -1] - 0.25x [n - 2] -0.125x [n -3] Find the first 10 samples of it’s step response    y[n]= x[n] - 0.5x [n -1] - 0.25x [n - 2] -0.125x [n -3] Find the first 10 samples of it’s step response

Explanation / Answer

   y[n]= x[n] - 0.5x [n -1] - 0.25x [n -2] - 0.125x [n -3] --(1) We want to find the step response of the system. i.e. to find y[n]for x[n] = u[n] = 1, for n=0, 1, 2, 3,4......                                                                                                                        = 0, otherwise. i.e. x[0] =x[1=x[2] =x[3]=x[4]=.........=x[n]= 1 for all values ofn=0,1,2,3,4.... y[0] =x[0]-0.5x[-1]-0.25x[-2]-0.125x[-3] =1-0-0-0=1; y[1] = x[1]-0.5 x[0]-0.25 x[-1] - 0.125 x[-2] =1-0.5=0.5; y[2] = x[2]-0.5 x[1]-0.25x[0] - 0.125 x[-1] =1-0.5-0.25 =0.25; y[3] = x[3]-0.5 x[2]-0.25 x[1] - 0.125 x[0]=1-0.5-0.25-0.125 =0.125; since for n >= 3, x[n-3] =x[n-2]=x[n-1] =x[n]=1 y[n] =1-0.5 -0.25-0.125 = 0.125, for alln=3,4,5,6,7,8,9,........... -------------------------------------------------- Note : This problem can also be solved by using Z transformtechniques 1. First apply z transform on both sides 2. Substitute X(z) = 1/(1-z-1) [this is the z transformof x[n] = u[n]] 3. expand 1/(1-z-1) =1+z-1+z-2+z-3+... 4. Simplify and compare the coefficients of Y(Z) with thestandard expression Y(z) = y[n] z-n to obtain values of y[n] for n

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