Autocorrelation Function of a Signal

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Autocorrelation Function

The autocorrelation function defines the measure of similarity or coherence between a signal and its time delayed version. The autocorrelation function of a real energy signal $\mathrm{x(t)}$ is given by,

$$\mathrm{R(\tau) \:=\: \int_{-\infty}^{\infty}\:x(t)\:x(t\:-\:\tau)\:dt}$$

Energy Spectral Density (ESD) Function

The distribution of the energy of a signal in the frequency domain is called the energy spectral density. The ESD function of a signal is given by,

$$\mathrm{\psi(\omega)\: =\: |X(\omega)|^2 \:=\: X(\omega)\: X(-\omega)}$$

Autocorrelation Theorem

Statement − The autocorrelation theorem states that the autocorrelation function $\mathrm{R(\tau)}$ and the ESD (Energy Spectral Density) function $\mathrm{\psi(\omega)}$ of an energy signal $\mathrm{x(t)}$ form a Fourier transform pair, i.e.,

$$\mathrm{R(\tau) \:\leftrightarrow \:\psi(\omega)}$$

In other words, the autocorrelation theorem states that the Fourier transform of autocorrelation function $\mathrm{R(\tau)}$ results the energy density function of an energy signal $\mathrm{x(t)}$, i.e.,

$$\mathrm{F [R(\tau)]\: =\: \lvert X (\omega)\lvert^2 \:=\: \psi (\omega)}$$

Proof

From the definition of the Fourier transform, we have,

$$\mathrm{F[x(t)] \:= \:X(\omega) \:=\: \int_{-\infty}^{\infty}x(t)e^{-j\omega t}\: dt}$$

Therefore, the Fourier transform of the autocorrelation function $\mathrm{R(\tau)}$ is given by,

$$\mathrm{F[R(\tau)] \:=\: \int_{-\infty}^{\infty} R(\tau)e^{-j\omega\tau} \: d\tau}$$

The autocorrelation function of a real energy signal $\mathrm{x(t)}$ is defined as,

$$\mathrm{R(\tau) \:= \:\int_{-\infty}^{\infty}x(t)x(t-\tau)\:dt}$$

$$\mathrm{\therefore\:F[R(\tau)]\:=\: \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}\:x(t)\:x(t\:-\:\tau)\: dt\right] e^{-j\omega\tau} d\tau}$$

By rearranging the order of integrations, we get,

$$\mathrm{F[R(\tau)] \:=\:\int_{-\infty}^{\infty}\:x(t)e^{-j\omega\tau}\:dt\int_{-\infty}^{\infty}\: x(t\:-\:\tau) e^{j\omega(t\:-\:\tau)}\:d\tau}$$

$$\mathrm{\Rightarrow\:F[R(\tau)]\:=\: X(\omega)\int_{-\infty}^{\infty}\:x(t\:-\:\tau)e^{j\omega(t\:-\:\tau)}\:d\tau}$$

By substituting $\mathrm{(t\:-\:\tau)\:=\: p}$ and $\mathrm{d\tau\:=\: dp}$ in the above integral, we get,

$$\mathrm{F[R(\tau)] \:=\:X(\omega)\int_{-\infty}^{\infty}\:x(p)e^{j\omega p}\:dp \:=\: X(\omega)X(-\omega)}$$

$$\mathrm{\Rightarrow\:F [R(\tau)]\:=\:|X(\omega)|^2\:=\:\psi (\omega)}$$

Also,

$$\mathrm{R(\tau)\:\leftrightarrow\: \psi (\omega)}$$

Thus, it proves the autocorrelation theorem.