Final answer:
To determine the output when the input to the LTI system is x₂[n]=2u[n]−4u[n−2], we can use the linearity property and calculate the output y₂[n]=4δ[n]−6δ[n−1]−6δ[n−2]+12δ[n−3]−4δ[n−4].
Step-by-step explanation:
To determine the output when the input to the system is x₂[n]=2u[n]−4u[n−2], we can use the linearity property of LTI systems. We can decompose x₂[n] into two scaled unit step signals as x₂[n]=2u[n]−4u[n−2]=2(x₁[n]−u[n−2]).
Using the linearity property, we can calculate the output as y₂[n]=2y₁[n]−4y₁[n−2]. Substituting the values of y₁[n] given in the question, we have y₂[n]=2(2δ[n]−3δ[n−1]+δ[n−2])−4(2δ[n−2]−3δ[n−3]+δ[n−4]).
Simplifying further, y₂[n]=4δ[n]−6δ[n−1]+2δ[n−2]−8δ[n−2]+12δ[n−3]−4δ[n−4]. Combining like terms, we get y₂[n]=4δ[n]−6δ[n−1]−6δ[n−2]+12δ[n−3]−4δ[n−4], which is the output of the system.