Consider the system shown in Figure 7.19 as a means of approximately measuring R_x(Ï„) where x(t) is stationary.

(a) Show that E[y] = R_x(Ï„).

(b) Find an expression for σ²_y if x(t) is Gaussian and has zero mean. If x₁, x₂, x₃, and x₄ are Gaussian with zero mean, it can be shown that 

E[x₁x₂x₃x₄] = E[x₁x₂]E[x₃x₄] + E[x₁x₃]E(x₂x₄] +E[x₁ x₄]E[x₂x₃]

Figure 7.19

x(t) to +T ()dt to Delay z (variable)