# Autocorrelation Function and its Properties

Signals and SystemsElectronics & ElectricalDigital Electronics

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## What is Autocorrelation?

The autocorrelation function of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. Thus, the autocorrelation is the correlation of a signal with itself.

The autocorrelation function is defined separately for energy or aperiodic signals and power or periodic signals.

## Autocorrelation Function for Energy Signals

The autocorrelation function of an energy signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is defined as −

$$\mathrm{\mathit{R_{\mathrm{11}}\left ( \tau \right )\mathrm{=}R\left ( \tau \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )x^{\ast }\left ( t-\tau \right )dt\mathrm{=}\int_{-\infty }^{\infty }x\left ( t\mathrm{+ }\tau \right )x^{\ast }\left ( t \right )dt}}$$

Where, the variable $\mathrm{\mathit{\tau}}$ is called the delay parameter.

## Properties of Autocorrelation Function of Energy Signals

The properties of autocorrelation function for energy signals are given as follows −

### Property 1

The autocorrelation function of energy signals exhibits complex conjugate symmetry, which means the real part of autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ is an even function of delay parameter ( $\mathrm{\mathit{\tau}}$) and the imaginary part of $\mathrm{\mathit{R\left ( \tau \right )}}$ is an odd function of the parameter $\mathrm{\mathit{\tau}}$. Thus,

$$\mathrm{\mathit{R\left ( \tau \right )\mathrm{=}R^{\ast }\left ( -\tau \right )}}$$

### Property 2

When the delay parameter $\mathrm{\mathit{\tau}}$ is increased in either direction, the autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ of an energy signal reduces. Hence, when the parameter $\mathrm{\mathit{\tau}}$ reduces, the autocorrelation $\mathrm{\mathit{R\left ( \tau \right )}}$ increases and it is maximum at $\mathrm{\mathit{\tau \: \mathrm{=}}}$ 0 (or at origin). Therefore,

$$\mathrm{\mathit{\left| R\left ( \tau \right )\right|\leq R\left ( \mathrm{0} \right );\; \; \mathrm{for\: all\: }\tau }}$$

### Property 3

The value of autocorrelation function of an energy signal at the origin (i.e., at $\mathrm{\mathit{\tau\:\mathrm{=}}}$ 0) is equal to total energy of that signal, i.e.,

$$\mathrm{\mathit{ R\left ( \tau \right )|_{\tau \mathrm{=}\mathrm{0}}\mathrm{=}E\mathrm{=}\int_{-\infty }^{\infty }\left|x\left ( t \right ) \right|^{\mathrm{2}}dt}}$$

### Property 4

The autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ and the ESD (Energy Spectral Density) function 𝜓(𝜔) of an energy signal form a Fourier transform pair, i.e.,

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

## Autocorrelation Function for Power Signals

The autocorrelation function of a power or periodic signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ with any time period T is defined as −

$$\mathrm{\mathit{R\left ( \tau \right )\mathrm{=}\displaystyle \lim_{T \to \infty }\frac{\mathrm{1}}{T}\int_{-T/\mathrm{2} }^{T/\mathrm{2}}x\left ( t \right )x^{\ast }\left ( t-\tau \right )dt}}$$

## Properties of Autocorrelation Function for Power Signals

The properties of autocorrelation function for power signals are given as follows −

### Property 1

The autocorrelation function of power or periodic signals exhibits complex conjugate symmetry property, that is,

$$\mathrm{\mathit{R\left ( \tau \right )\mathrm{=}R^{\ast }\left ( -\tau \right )}}$$

### Property 2

The value of the autocorrelation function for a power signal at origin (i.e., at $\mathrm{\mathit{\tau\:\mathrm{=}}}$ 0) is equal to the average power (P) of that signal, i.e.,

$$\mathrm{\mathit{R\left ( \mathrm{0} \right )\mathrm{=}P}}$$

### Property 3

When the delay parameter $\mathrm{\mathit{\tau}}$ reduces, the autocorrelation $\mathrm{\mathit{R\left ( \tau \right )}}$ of the power signal increases and it is maximum at the origin, i.e.,

$$\mathrm{\mathit{\left| R\left ( \tau \right )\right|\leq R\left ( \mathrm{0} \right )}}$$

### Property 4

The autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ be periodic with the same time period as the power (or periodic) signal itself, i.e.,

$$\mathrm{\mathit{R\left ( \tau \right )\mathrm{=}R\left ( \tau\pm nT \right );\; \; \; \; \mathrm{where,\:\mathit{n}\mathrm{=}1,\, 2,\, 3,\, \cdot \cdot \cdot } }}$$

### Property 5

The autocorrelation function $\mathrm{\mathit{R\left ( \tau \right )}}$ and the PSD (Power Spectral Density) function $\mathrm{\mathit{S\left ( \omega \right )}}$ of a power signal form a Fourier transform pair, i.e.,

$$\mathrm{\mathit{R\left ( \tau \right )\leftrightarrow S\left ( \omega \right )}}$$

Updated on 07-Jan-2022 07:24:45