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What is Power Spectral Density?
Power Spectral Density
The distribution of average power of a signal $x\mathrm{\left(\mathit{t}\right)}$ in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The PSD function is denoted by $\mathit{S\mathrm{\left({\mathit{\omega }}\right)}}$ and is given by,
$$\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\mathrm{=}\displaystyle\lim_{\tau \to \infty }\frac{\left| \mathit{X\mathrm{\left ( \mathit{\omega}\right)}}\right|^{2}}{\tau}\:\:\:\:\:\:...(1)}$$
Explanation
In order to drive the power spectral density (PSD) function, consider a power signal as a limiting case of an energy signal, i.e., the signal $\mathit{Z\mathrm{\left({\mathit{t }}\right)}}$ is zero outside the interval $\left|\tau /2 \right|$ as shown in the figure.
The signal $\mathit{Z\mathrm{\left({\mathit{t }}\right)}}$ is given by,
$$\mathrm{\mathit{Z\mathrm{\left({\mathit{t }}\right)}}\mathrm{=}\begin{cases} x\mathrm{\left(\mathit{t}\right)}\:\left|t \right|<\left ( \frac{\tau }{2} \right )\ 0 \:\:\: {\mathrm{otherwise} } \end{cases}}$$
Where,$x\mathrm{\left(\mathit{t}\right)}$ is a power signal of same magnitude extending to infinity.
As the signal $\mathit{Z\mathrm{\left({\mathit{t }}\right)}}$ is finite duration signal of duration $\tau$ and thus, it is an energy signal having energy E, that is given by,
$$\mathrm{\mathit{E}\:\mathrm{=}\:\int_{-\infty}^{\infty}\left|\mathit{Z\mathrm{\left ( \mathit{t} \right )}}\right|^{2}\:\mathit{dt}\:\mathrm{=}\:\frac{1}{2\pi}\int_{-\infty }^{\infty}\left| \mathit{Z}\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}\:\mathit{d\omega }}$$
Where,
$$\mathrm{\mathit{Z\mathrm{\left({\mathit{t }}\right)}}\overset{\mathit{FT}}{\leftrightarrow}\mathit{Z\mathrm{\left({\mathit{\omega}}\right)}}}$$
Also,
$$\mathrm{\int_{-\infty}^{\infty}\left|\mathit{Z\mathrm{\left ( \mathit{t} \right )}}\right|^{2}\:\mathit{dt}\:\mathrm{=}\:\int_{-\mathrm{\left(\tau/2\right)}}^{\mathrm{\left(\tau /2 \right)}}\left| \mathit{x}\mathrm{\left ( \mathit{t} \right)}\right|^{2}\:\mathit{dt}}$$
Therefore, we have,
$$\mathrm{\frac{1}{\mathit{\tau}}\int_{-\mathrm{\left(\tau/2\right)}}^{\mathrm{\left(\tau /2 \right )}}\left| \mathit{x}\mathrm{\left ( \mathit{t} \right )}\right|^{2}\:\mathit{dt}\:\mathrm{=}\:\frac{1}{2\pi}\mathrm{\left(\frac{1}{\tau}\right)}\int_{-\infty }^{\infty}\left| \mathit{Z}\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}\:\mathit{d\omega}}$$
Hence, when $\tau\to \infty $, then the LHS of the above equation gives the average power (P) of the signal $x\mathrm{\left(\mathit{t}\right)}$ , i.e.,
$$\mathrm{\mathit{P}\:\mathrm{=}\:\frac{1}{2\pi}\int_{-\infty}^{\infty}\displaystyle \lim_{\tau \to\infty}\left(\frac{\left| \mathit{Z}\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}}{\tau}\right)\:\mathit{d\omega}\:\:\:\:\:\:...(2)}$$
If $\tau\to \infty $, then $\left(\frac{\left| Z\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}}{\tau}\right)$ in equation (2) approaches a finite value. Assume this finite value is represented by $\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}}$, i.e.,
$$\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\displaystyle \lim_{\tau \to\infty}\left(\frac{\left| \mathit{Z}\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}}{\tau}\right) \:\:\:\:\:\:...(3)}$$
The expression in the equation (3) is called the power spectral density (PSD) of the signal $z\mathrm{\left(\mathit{t}\right)}$. Therefore, for the function $x\mathrm{\left(\mathit{t}\right)}$, the PSD function is given by,
$$\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\displaystyle \lim_{\tau \to\infty}\left(\frac{\left| \mathit{X}\mathrm{\left ( \mathit{\omega } \right )}\right|^{2}}{\tau}\right)\:\:\:\:\:\:...(4)}$$
Hence, the average power (P) of the signal $x\mathrm{\left(\mathit{t}\right)}$ is given by,
$$\mathrm{\mathit{P}\mathrm{=}\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\:\mathit{d\omega }\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{S}\mathrm{\left(\mathit{f}\right)}\:\mathit{df}\:\:\:\:\:\:...(5)}$$
Also, the power spectral density (PSD) of a periodic function is given by,
$$\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:2\pi\:\sum_{\mathit{n}=-\infty }^{\infty}\left|\mathit{c_{n}}\right|^{2}\delta \mathrm{\left(\mathit{\omega -n\omega _{\mathrm{0}}} \right )}}\:\:\:\:\:\:...(6)$$
Properties of Power Spectral Density (PSD)
Property 1 - For a power signal, the area under the power spectral density curve is equal to the average power of that signal, i.e.,
$$\mathrm{\mathit{P}\:\mathrm{=}\:\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\:\mathit{d\omega }}$$
Property 2 - If the signal $x\mathrm{\left(\mathit{t}\right)}$ is input to an LTI system with impulse response $h\mathrm{\left(\mathit{t}\right)}$, then the input and output PSD functions of the system are related as,
$$\mathrm{\mathit{S}_{y}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\left|\mathit{H}\left(\mathit{\omega}\right)\right|^{2}\:\mathit{S}_{\mathit{x}}\mathrm{\left(\mathit{\omega}\right)}}$$
Where,$\left|\mathit{H}\left(\mathit{\omega}\right)\right|$ is the magnitude of the system transfer function.
Property 3 - The autocorrelation function $R\mathrm{\left(\mathrm{\tau}\right)}$ and the power spectral density function $\mathit{S}\mathrm{\left(\mathit{\omega}\right)}$of a power signal form a Fourier transform pair, i.e.,
$$\mathrm{\mathit{R\mathrm{\left({\mathit{\tau }}\right)}}\overset{\mathit{FT}}{\leftrightarrow}\mathit{S\mathrm{\left({\mathit{\omega}}\right)}}}$$
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