Let f★g denote the cross-correlation of functions f(t) and g(t). Then f★g | = | integral_(-∞)^∞ f^_(τ) g(t + τ) d τ | = | integral_(-∞)^∞[ integral_(-∞)^∞ F^_(ν) e^(2π i ντ) d ν integral_(-∞)^∞ G(ν') e^(-2 π i ν'(t + τ)) d ν'] d τ | = | integral_(-∞)^∞ integral_(-∞)^∞ integral_(-∞)^∞ F^_(ν) G(ν') e^(-2 π i τ(ν' - ν)) e^(-2 π i ν' t) d τ d ν d ν' | = | integral_(-∞)^∞ integral_(-∞)^∞ F^_(ν) G(ν') e^(-2 π i ν' t) [ integral_(-∞)^∞ e^(-2 π i τ(ν' - ν)) d τ] d ν d ν' | = | integral_(-∞)^∞ integral_(-∞)^∞ F^_(ν) G(ν') e^(-2 π i ν' t) δ(ν' - ν) d ν' d ν | = | integral_(-∞)^∞ F^_(ν) G(ν) e^(-2 π i ν t) d ν | = | ℱ[F^_(ν) G(ν)], where ℱ denotes the Fourier transform, z^_ is the complex conjugate, and

Fourier transform | Wiener-Khinchin theorem