The response of a discrete time system y[n] obeys the following relation: 𝑦[𝑛]=5/6 𝑦[𝑛−1]−1/6 𝑦[𝑛−2]+𝑥[𝑛] ...

 

Q. The response of a discrete time system y[n] obeys the following relation: 𝑦[𝑛]=5/6 𝑦[𝑛−1]−1/6 𝑦[𝑛−2]+𝑥[𝑛]. The input to the system is 𝑥[𝑛]=𝛿[𝑛]−1 3 𝛿[𝑛−1]. Which of the following options is TRUE for 𝑦[𝑛]?   

Answer: 

𝑦[𝑛]=5/6 𝑦[𝑛−1]−1/6 𝑦[𝑛−2]+𝑥[𝑛]

Y[z]=5/6.z^-1.Y[z] - 1/6.z^-2.Y[z] + X[z]

Or, Y[z] (1 - 5/6.z^-1 + 1/6.z^-2) = X[z]

The input to the system is 𝑥[𝑛]=𝛿[𝑛]−1/3 𝛿[𝑛−1]

Or, X(z)= 1 - 1/3.z^-1

So, Y[z] (1 - 5/6.z^-1 + 1/6.z^-2) = 1 - 1/3.z^-1

Or, Y[z]  = (1 - 1/3.z^-1) / (1 - 5/6.z^-1 + 1/6.z^-2)

Or, Y[z]  = (1 - 1/3.z^-1) / (1 - 1/2.z^-1).(1 - 1/3.z^-1)

Or, Y[z]  = 1 / (1 - 1/2.z^-1)

Or, y[n] = (1/2)^n.u[n]

Hence, y[n] is stable and causal

Answer: Option A