신호의 교차상관함수

Definition¹

Analog Signal
For an Energy Signal $f \in L^{2}(\mathbb{R})$, $f \star g$ defined as follows is called the cross-correlation function defined by $f$ and $g$.
$$ (f \star g)(\tau) = R_{fg}(\tau) := \int_{-\infty}^{\infty} \overline{f(t)} g(t + \tau) dt $$
Here, $\overline{f(t)}$ is the conjugate complex number of $f(t)$.
Digital Signal
The cross-correlation function of energy signal $\left\{ x_{n} \right\} \in \ell^{2}$ is defined as follows.
$$ (x\star y)[n] = R_{xy}(m) := \sum\limits_{n \in \mathbb{N}} \overline{x_{n}}y_{n+m} $$

According to the definition, the autocorrelation function $R_{f}(\tau)$ is a special case of the cross-correlation function when $g=f$.

The Fourier transform of the cross-correlation function is called the cross (energy) spectrum.

$$ S_{fg}(\omega) := \int_{-\infty}^{\infty} R_{fg}(\tau)e^{-i\tau \omega} d\tau = \hat{R}_{fg}(\omega) $$