An echo system could be modeled using or not using feedback.

(a) Feedback systems are of great interest in control and in modeling of many systems. An echo is created as the sum of one or more delayed and attenuated output signals that are fed back into the present signal. A possible model for an echo system is

y(t) = x(t) + α₁y(t − τ) + ··· +α_Ny(t − Nτ)

where x(t) is the present input signal and y(t) is the present output and y(t − kτ) the previous delayed outputs, the ∣α_k∣ < 1 values are attenuation factors. Carefully draw a block diagram for this system.

(b) Consider the echo model for N = 1 and parameters τ = 1 and α₁ = 0.1, is the resulting echo system LTI? Explain.

(c) Another possible model is given by a non-recursive, or without feedback, system

z(t) = x(t) + β₁x(t − τ) + ··· + β_Mx(t − Mτ)

where several present and past inputs, are delayed and attenuated, and added up to form the output. The parameters ∣β_k∣ < 1 are attenuation factors and τ a delay. Carefully draw a block diagram for the echo system characterized by the above equation. Does the above equation represent a LTI system? Explain.