logo

신호의 자기상관함수

신호의 자기상관함수

Definition1

  • Analog Signal

    For an energy signal fL2(R)f \in L^{2}(\mathbb{R}), RfR_{f} defined as follows is called the auto-correlation function.

    Rf(τ):=f(t)f(t+τ)dt R_{f}(\tau) := \int_{-\infty}^{\infty} \overline{f(t)} f(t + \tau) dt

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

  • Digital Signal

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

    Rx(m):=nNxnxn+m R_{x}(m) := \sum\limits_{n \in \mathbb{N}} \overline{x_{n}}x_{n+m}

Explanation

Expressed in terms of inner product, it can be written as follows. We simplify ff’s translation as fτ=Tτff_{-\tau} = T_{-\tau}f,

Rf(τ)=f,Tτf=f,fτ R_{f}(\tau) = \braket{f, T_{-\tau}f} = \braket{f, f_{-\tau}}

The auto-correlation function is a measure of the similarity between signal ff and fτf_{-\tau}. If we define the distance in space L2L^{2} as d(f,g)=fg2=fg,fgd(f,g) = \left\| f - g \right\|_{2} = \sqrt{\braket{f-g, f-g}},

d(f,fτ)2=ffτ,ffτ=f,ff,fτfτ,f+fτ,fτ=f2f,fτf,fτ+fτ=2f22f,fτ=2f22(Rf(τ)) \begin{align*} d(f, f_{-\tau}) ^{2} &= \braket{f-f_{-\tau}, f-f_{-\tau}} \\ &= \braket{f, f} - \braket{f, f_{-\tau}} - \braket{f_{-\tau}, f} + \braket{f_{-\tau}, f_{-\tau}} \\ &= \left\| f \right\|_{2} - \braket{f, f_{-\tau}} - \overline{\braket{f, f_{-\tau}}} + \left\| f_{-\tau} \right\| \\ &= 2\left\| f \right\|_{2} - 2 \Re \braket{f, f_{-\tau}} \\ &= 2\left\| f \right\|_{2} - 2 \Re \left( R_{f}(\tau) \right) \end{align*}

Therefore, when the value of Rf(τ)R_{f}(\tau) increases, the disparity between ff and fτf_{-\tau} decreases, and when the value decreases, the disparity increases.

Theorem

For an analog signal fL2(R)f \in L^{2}(\mathbb{R}), the following holds:

Sf(ω)=Rf(τ)eiωτdτ=Rf^(ω) S_{f}(\omega) = \int_{-\infty}^{\infty} R_{f}(\tau) e^{-i\omega \tau} d\tau = \hat{R_{f}}(\omega)

Here, Sf(ω)S_{f}(\omega) is the energy spectrum density of ff, and Rf^\hat{R_{f}} is the Fourier transform of RfR_{f}.

Proof

By appropriately substituting variables, this can be easily shown. Rf^(ω)=Rf(τ)eiωτdτ=f(t)f(t+τ)dteiωτdτ=f(t)f(t+τ)eiωτdτdt=f(t)f(τ)eiω(τt)dτdt(τ=t+τ)=f(t)f(τ)eiωτdτeiωtdt=f(t)f^(ω)eiωtdt=f^(ω)f(t)eiωtdt=f^(ω)f(t)eiωtdt=f^(ω)f^(ω)=f^(ω)2=Sf(ω) \begin{align*} \hat{R_{f}}(\omega) &= \int_{-\infty}^{\infty} R_{f}(\tau) e^{-i\omega \tau} d\tau \\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \overline{f(t)} f(t + \tau) dt e^{-i\omega \tau} d\tau \\ &= \int_{-\infty}^{\infty} \overline{f(t)} \int_{-\infty}^{\infty} f(t + \tau) e^{-i\omega \tau} d\tau dt \\ &= \int_{-\infty}^{\infty} \overline{f(t)} \int_{-\infty}^{\infty} f(\tau^{\prime}) e^{-i\omega (\tau^{\prime} - t)} d\tau^{\prime} dt \qquad (\tau^{\prime} = t + \tau) \\ &= \int_{-\infty}^{\infty} \overline{f(t)} \int_{-\infty}^{\infty} f(\tau^{\prime}) e^{-i\omega \tau^{\prime}} d\tau^{\prime} e^{i\omega t}dt \\ &= \int_{-\infty}^{\infty} \overline{f(t)} \hat{f}(\omega) e^{i\omega t}dt \\ &= \hat{f}(\omega) \int_{-\infty}^{\infty} \overline{f(t)} e^{i\omega t}dt \\ &= \hat{f}(\omega) \overline{\int_{-\infty}^{\infty} f(t) e^{-i\omega t}dt} \\ &= \hat{f}(\omega) \overline{\hat{f}(\omega)} \\ &= | \hat{f}(\omega) |^{2} \\ &= S_{f}(\omega) \end{align*}

See Also

Stochastic Process


  1. 최병선, Wavelet 해석 (2001) p23 ↩︎