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Power Spectral Density (PSD) and Autocorrelation Function
Power Spectral Density
The distribution of average power of a signal in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The power spectral density is denoted by $\mathit{S\left (\omega \right )}$ and is given by,
$$\mathrm{\mathit{S\left (\omega \right )\mathrm{=}\lim_{\tau \rightarrow \infty }\frac{\left | X\left (\omega \right ) \right |^{\mathrm{2}}}{\tau }}}$$
Autocorrelation
The autocorrelation function gives the measure of similarity between a signal and its time-delayed version. The autocorrelation function of power (or periodic) signal $\mathit{x\left ( t \right ) }$ with any time period T is given by,
$$\mathrm{\mathit{R\left(\tau \right)=\lim_{T\rightarrow \infty }\mathrm{\frac{1}{\mathit{T}}}\int_{-\left(T/\mathrm{2}\right)}^{T/\mathrm{2}}x\left(t\right)\:x^{*}\left(t-\tau \right)\:dt}}$$
Where, $\tau$ is called the delayed parameter.
Relation between PSD and Autocorrelation Function
The power spectral density function $\mathit{S\left(\omega\right )}$ and the autocorrelation function $\mathit{R\left(\tau \right)}$of a power signal form a Fourier transform pair, i.e.,
$$\mathrm{\mathit{R\left(\tau \right)\overset{FT}{\leftrightarrow}S\left(\omega\right)}}$$
Proof - The autocorrelation function of a power signal $\mathit{x\left ( t \right ) }$ in terms of exponential Fourier series coefficients is given by,
$$\mathrm{\mathit{R\left(\tau \right)=\sum_{n=-\infty }^{\infty } C_{n}\:C_{-n}\:e^{jn\omega _{\mathrm{0}}\tau }}\:\:\:\:\:\:...(1)}$$
Where,$\mathit{C_{n}}$ and $\mathit{C_{-n}}$ are the exponential Fourier series coefficients.
$$\mathrm{\mathit{\because C_{n}\:C_{-n}=\left | C_{n} \right |^{\mathrm{2}}}}$$
Therefore, Eqn.(1) can be written as,
$$\mathrm{\mathit{R\left(\tau\right)=\sum_{n=-\infty }^{\infty }\left | C_{n} \right |^{\mathrm{2}}\:e^{jn\omega _{\mathrm{0}}\tau }}\:\:\:\:\:\:...(2)}$$
By taking the Fourier transform on both sides of eq. (2), we get,
$$\mathrm{\mathit{F\left [ R\left ( \tau \right ) \right ]=F\left [\sum_{n=-\infty }^{\infty }\left | C_{n} \right |^{\mathrm{2}}e^{jn\omega _{\mathrm{0}}\tau } \right ]=\int_{-\infty }^{\infty }\left [ \sum_{n=-\infty }^{\infty }\left | C_{n} \right |^{\mathrm{2}}e^{jn\omega _{\mathrm{0}}\tau } \right ]e^{-j\omega \tau }\:d\tau}}$$
By interchanging the order of integration and summation on RHS of the above expression, we have,
$$\mathrm{\mathit{F[R(\tau )] =\sum_{n=-\infty }^{\infty }\left|C_{n}\right| ^{\mathrm{2}}\int_{-\infty}^{\infty} e^{jn\omega _{0}\tau} e^{-j\omega \tau } \:d\tau = \sum_{n=-\infty }^{\infty }\left|C_{n}\right| ^{\mathrm{2}}\int_{-\infty}^{\infty} e^{-j\tau (\omega -n\omega _{0}) } \:d\tau }}$$
$$\mathrm{\mathit{\because \int_{-\infty }^{\infty}e^{-j \tau(\omega -n\omega _{0}) }\:d\tau=\mathrm{2}\pi \delta (\omega -n\omega _{0})}}$$
$$\mathrm{\mathit{\therefore F\left [ R\left ( \tau \right ) \right ]=\mathrm{2}\pi \sum_{n=-\infty }^{\infty }\left | C_{n} \right |^{\mathrm{2}}\delta (\omega -n\omega _{\mathrm{0}})}\:\:\:\:\:\:...(3)}$$
The RHS of Eqn. (3) is the power spectral density (PSD) of the power function $\mathit{x\left(t\right)}$. Therefore,
$$\mathrm{\mathit{F\left [ R\left(\tau \right)\right ]=S\left(\omega\right)}}$$
Or, it can also be represented as,
$$\mathrm{\mathit{R\left(\tau\right)\overset{FT}{\leftrightarrow}S\left(\omega\right)}}$$
Hence, it proves that the autocorrelation function $\mathit{R\left(\tau\right )}$ and PSD function $\mathit{S\left (\omega \right )}$ of a power signal form the Fourier transform pair.
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