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 $\mathrm{S(\omega)}$ and is given by,

$$\mathrm{S(\omega) \:=\: \lim_{\tau \rightarrow \infty}\:\frac{| X (\omega) |^{2}}{\tau}}$$

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 x(t) with any time period T is given by,

$$\mathrm{R(\tau) \:=\:\lim_{T\rightarrow \infty}\:\frac{1}{T}\:\int_{-(T/{2})}^{T/{2}}\:x(t)\:x^{*}(t\:-\:\tau)\:dt}$$

Where, τ is called the delayed parameter.

Relation between PSD and Autocorrelation Function

The power spectral density function $\mathrm{S(\omega)}$ and the autocorrelation function $\mathrm{R(\tau)}$ of a power signal form a Fourier transform pair, i.e.,

$$\mathrm{R(\tau)\overset{FT}{\leftrightarrow}S(\omega)}$$

Proof - The autocorrelation function of a power signal $\mathrm{x(t)}$ in terms of exponential Fourier series coefficients is given by,

$$\mathrm{R(\tau)\:=\:\sum_{n=-\infty}^{\infty} C_{n}\:C_{-n}\:e^{jn\omega_{0}\tau}\:\:\dotso\:(1)}$$

Where,$\mathrm{C_{n}}$ and $\mathrm{C_{-n}}$ are the exponential Fourier series coefficients.

$$\mathrm{\because\: C_{n}\:C_{-n}\:=\: | C_{n} |^{2}}$$

Therefore, Eqn.(1) can be written as,

$$\mathrm{R(\tau)\:=\:\sum_{n=-\infty}^{\infty} | C_{n} |^{2}\:e^{jn\omega_{0}\tau}\:\:\dotso\:(2)}$$

By taking the Fourier transform on both sides of eq. (2), we get,

$$\mathrm{F[R(\tau)]\:=\:F\left[\sum_{n=-\infty}^{\infty} |C_{n}|^{2}e^{jn\omega_{0}\tau} \right] \:=\: \int_{-\infty }^{\infty}\left[\sum_{n=-\infty}^{\infty} |C_{n}|^{2}e^{jn\omega_{0}\tau} \right]e^{-j\omega \tau }\:d\tau}$$

By interchanging the order of integration and summation on RHS of the above expression, we have,

$$\mathrm{F[R(\tau)] \:=\: \sum_{n=-\infty}^{\infty}|C_{n}|^{2}\int_{-\infty}^{\infty} e^{jn\omega_{0}\tau} e^{-j\omega \tau}\:d\tau\:=\:\sum_{n=-\infty}^{\infty}|C_{n}|^{2}\int_{-\infty}^{\infty}e^{-j\tau(\omega\:-\:n\omega_{0})}\:d\tau}$$

$$\mathrm{\because\: \int_{-\infty}^{\infty}e^{-j \tau(\omega \:-\: n\omega_{0})}\:d\tau \:=\: 2\pi \delta (\omega \:-\: n\omega_{0})}$$

$$\mathrm{\therefore\: F[R(\tau)] \:=\: 2\pi \sum_{n=-\infty}^{\infty} |C_{n}|^{2}\delta (\omega \:-\:n\omega _{\mathrm{0}})\:\:\dotso\:(3)}$$

The RHS of Eqn. (3) is the power spectral density (PSD) of the power function $\mathrm{x(t)}$. Therefore,

$$\mathrm{F[R(\tau)] \:=\: S(\omega)}$$

Or, it can also be represented as,

$$\mathrm{R(\tau)\overset{FT}{\leftrightarrow}S(\omega)}$$

Hence, it proves that the autocorrelation function $\mathrm{R(\tau)}$ and PSD function $\mathrm{S(\omega)}$ of a power signal form the Fourier transform pair.