Assume first entry in all depicted sequences correspond to n = 0. given the disc

Question

Assume first entry in all depicted sequences correspond to n = 0. given the discrete time linear time (shift) invariant system: Using any method find the filter input x[n], given the output to this input, y[n], and given the filter unit sample response h[n] = [3 1 2]

Explanation / Answer

let the length of x be m, length of h be n then the length of y is m+n-1 => length of input here = 6+1-3 =4 let x[n]= {a b c d 0 0 ..0} = a del[n] + b del[n-1]+ c del[n-2] + d del[n-3] h[n]= [3 1 2]=3 del[n]+ del[n-1]+ 2 del[n-2] y[n] =[9 9 11 -1 0 -4] del[n-a] * del[n-b] = del[n-a-b] using this, x[n]* h[n]= a h[n]+ b h[n-1] + c h[n-2] + d h[n-3] y[n] = 3a del[n]+ del[n-1] (a + 3b) + del[n-2] (3c+ b+ 2a) + del[n-3] (3d+ 2b+ c) + del[n-4](d+2c) + del[n-5] (2d) y[n]=[9 9 11 -1 0 4] comparing, we have 3a =9 => a=3 a+3b=9 => b=2 3c+b+2a=11 => c=1 3d+2b+c = -1 => d=-2 so, x[n]={3 2 1 -2}

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