Consider an averager represented by the input/output equation

where x(t) is the input and y(t) the output.

(a) Let the input be x₁(t) = δ(t), find graphically the corresponding output y₁(t) for ˆ’ˆž < t < ˆž. Let then the input be x₂(t) = 2x₁(t), find graphically the corresponding output y₂(t) for ˆ’ˆž < t < ˆž. Is y₂(t) = 2y₁(t)? Is the system linear?

(b) Suppose the input is x₃(t) = u(t) ˆ’ u(t ˆ’ 1), graphically compute the corresponding output y₃(t) for ˆ’ˆž < t < ˆž. If a new input is x₄(t) = x₃(t ˆ’ 1) = u(t ˆ’ 1) ˆ’ u(t ˆ’ 2), find graphically the corresponding output y₄(t) for ˆ’ˆž < t < ˆž, and indicate if y₄(t) = y₃(t ˆ’ 1). Accordingly, would this averager be time-invariant?

(c) Is this averager a causal system? Explain.

(d) If the input to the averager is bounded, would its output be bounded? Is the averager BIBO stable?

:| x(t)dr + 2 y(t) =