Autocorrelation Function of a Signal

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 $\mathit{x}\mathrm{(\mathit{t})}$ is given by,

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

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,

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

Autocorrelation Theorem

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

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

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

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

Proof

From the definition of the Fourier transform, we have,

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

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

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

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

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

By rearranging the order of integrations, we get,

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

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

$$\mathit{F}\mathrm{[\mathit{R}\mathrm(\mathit{\tau})]} \:\mathrm{=}\:\mathit{X}\mathrm{(\omega)}\int_{-\infty}^{\infty}\mathit{x}\mathrm{(p)}\mathit{e^{j\omega p}}\:\mathit{dp} \:\mathrm{=}\: \mathit{X}\mathrm{(\omega)}\mathit{X}\mathrm{(-\omega)}$$ $$\Rightarrow\mathit{F}\mathrm{[\mathit{R}\mathrm{(\tau)}]}\:\mathrm{=}\:|\mathit{X}\mathrm{(\mathit{\omega})}|^\mathrm{2}\:\mathrm{=}\:\mathit{\psi}\mathrm{(\mathit{\omega})}$$

Also,

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

Thus, it proves the autocorrelation theorem.