An analog averager can be represented by the differential equation

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

(a) If the input-output equation of the averager is

Show how to obtain the above differential equation and that y(t)is the solution of the differential equation.

(b) If x(t) = cos(Ï€t) u(t), choose the value of Tin the averager so that the output y(t) = 0 in the steady state. Graphically show how this is possible for your choice of T. Is there a unique value for T that makes this possible? How does it relate to the frequency Ω₀ = Ï€ of the sinusoid?

(c) Use the impulse response h(t) of the averager, to show using Laplace that the steady state is zero when x(t) = cos(Ï€t) u(t) and T is the above chosen value. Use MATLAB to solve the differential equation and to plot the response for the value of T you chose.

dy(t) dt Нко -в [x(t) – x(t – T)] У() T 1-т гр(1)x