The cross-correlation of two complex functions f(t) and g(t) of a real variable t, denoted f★g is defined by f★g congruent f^_(-t)*g(t), where * denotes convolution and f^_(t) is the complex conjugate of f(t). Since convolution is defined by f*g = integral_(-∞)^∞ f(τ) g(t - τ) d τ, it follows that [f★g](t) congruent integral_(-∞)^∞ f^_(-τ) g(t - τ) d τ.

autocorrelation | convolution | cross-correlation theorem | Fourier transform