Properties of Power Spectral Density (PSD)
Properties of Power Spectral Density given below :
- Symmetric Property
- Relation Between PSD and Autocorrelation Function
- Relation Between Input and Output PSD of LTI System
- Relation Between Power and Power Spectral Density (PSD)
Property 1: Symmetric Property
For the real valued signal the Power Spectral Density (PSD) [Tex]S(\omega)[/Tex] is symmetric i.e. even function and mathematically expressed as: [Tex]S(\omega) = S(-\omega)[/Tex].
Property 2: Relation Between PSD and Autocorrelation Function
For wide sense stationary process the Power Spectral Density (PSD) [Tex]S(\omega)[/Tex] and Autocorrelation [Tex]R(\tau)[/Tex] of a signal represents Fourier Transform pair and mathematically it can be expressed with the following expression as follows for the continuous time signal [Tex]x(t)[/Tex]:
[Tex]\therefore R_x(\tau) \longleftrightarrow \mathcal{F}[S_x(\omega)][/Tex]
[Tex]\therefore R_x(\tau) = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_x(\omega) e^{i\omega \tau} d\omega = \int_{-\infty}^{\infty} S_x(f) e^{2\pi if\tau} df [/Tex]
Similarly, for a discrete time signal [Tex]x[n][/Tex] the relation between Power Spectral Density [Tex]S_x(\omega)[/Tex] and autocorrelation function [Tex]R_x[m][/Tex] can be given as:
[Tex]\therefore S_x(e^{i\omega}) = \sum_{m=-\infty}^{\infty} R_x[m]e^{-i\omega m}[/Tex]
Property 3: Relation Between Input and Output PSD of LTI System
If [Tex]x(t)[/Tex] is a signal which is given an input for Linear Time Invariant (LTI) system with an impulse response [Tex]h(t)[/Tex] then the input and the output Power Spectral Density (PSD) [Tex]S(\omega)[/Tex] can be related as follows:
[Tex]\therefore S_y(\omega) = |H(\omega)|^2 * S_x(\omega)[/Tex]
Property 4: Relation Between Power and Power Spectral Density (PSD)
The relation between Power Spectral Density (PSD) [Tex]S(\omega)[/Tex] and the power of the continuous signal [Tex]x(t)[/Tex] can be expressed as follows:
[Tex]\therefore P_x = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_x(\omega) d\omega = \int_{-\infty}^{\infty} S_x(f) df[/Tex]
Now, in case of discrete signal [Tex]x[n][/Tex] the expression is described as below. The expression is similar to that of the continuous one but, here the integral is performed over a range of [Tex]\omega[/Tex] from [Tex]-\pi[/Tex] to [Tex]\pi[/Tex] instead of [Tex]-\infty[/Tex] to [Tex]\infty[/Tex]. Also, here [Tex]\omega[/Tex] is normalized by sampling frequency [Tex](f_s)[/Tex] such that [Tex]\omega = 2\pi \frac{f}{f_s}[/Tex].
[Tex]\therefore P_x = \frac{1}{2\pi}\int_{-\pi}^{\pi} S_x(e^{i\omega}) d\omega = \int_{-\frac{fs}{2}}^{\frac{fs}{2}} S_x(f) df[/Tex]
Power Spectral Density
In terms of electronics, Power is defined as the total amount of energy that is getting transferred or converted per unit measurement of time, or in general terms Power is defined as the strength or the intensity level of the signal. Power is generally measured in watts (W).
In this article, we will be going through Power Spectral Density, First we will start our Article with the Definition of Power Spectral Density with an Example, Then we will go through its derivation Properties and Characteristics, At last, we will conclude our Article with Solved Examples, Applications, and Some FAQs.
Table of Content
- Definition
- Derivation
- Characteristics
- Properties
- Solved Problems
- Applications