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Autocorrelation Function and its Properties
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.,
$$\mathrm{\mathit{R\left ( \tau \right )\leftrightarrow S\left ( \omega \right )}}$$
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