Problem 1RP

In Problems 1 and 2, use the definition of the Laplace transform to determine

Auto-correlation (Wiener-Khinchin’s theorem) The auto-correlation, R(x), of a function, f(x), is de▯ned as ∫ ∞ R(x) ▯ f(x + y)f(y)dy: (1) −∞ Recalling that the convolution between f(x) and g(x) is de▯ned as ∫ ∞ f(x) ▯ g(x) ▯−∞ f(x ▯ y)g(y)dy (2) it can be shown, by setting g(x) ▯ f(▯x), that R(x) = f(x) ▯ f(▯x): (3) According to the Fourier convolution theorem, F(f(x) ▯ g(x)) = F(!)G(!): (4) ∫ ∞ G(!) = g(x)