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신호의 교차상관함수

신호의 교차상관함수

Definition1

  • Analog Signal

    For an Energy Signal fL2(R)f \in L^{2}(\mathbb{R}), fgf \star g defined as follows is called the cross-correlation function defined by ff and gg.

    (fg)(τ)=Rfg(τ):=f(t)g(t+τ)dt (f \star g)(\tau) = R_{fg}(\tau) := \int_{-\infty}^{\infty} \overline{f(t)} g(t + \tau) dt

    Here, f(t)\overline{f(t)} is the conjugate complex number of f(t)f(t).

  • Digital Signal

    The cross-correlation function of energy signal {xn}2\left\{ x_{n} \right\} \in \ell^{2} is defined as follows.

    (xy)[n]=Rxy(m):=nNxnyn+m (x\star y)[n] = R_{xy}(m) := \sum\limits_{n \in \mathbb{N}} \overline{x_{n}}y_{n+m}

Explanation

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

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

Sfg(ω):=Rfg(τ)eiτωdτ=R^fg(ω) S_{fg}(\omega) := \int_{-\infty}^{\infty} R_{fg}(\tau)e^{-i\tau \omega} d\tau = \hat{R}_{fg}(\omega)

See Also

Stochastic Process


  1. 최병선, Wavelet 해석 (2001) p24-26 ↩︎