Derivation of Power Spectral Density (PSD)
Consider a continuous time power signal [Tex]y(t)[/Tex] which is defined as follows. Here, [Tex]x(t)[/Tex] is a power signal that is defined up to infinity.
[Tex]\therefore y(t) = x(t), \ |t| < |\frac{\tau}{2}| [/Tex]
[Tex]\therefore y(t) = 0, \ elsewhere[/Tex]
As the signal is defined for a finite duration so, the energy of the signal could be defined as follows:
[Tex]\therefore E = \int_{-\infty}^{\infty} |y(t)|^2 dt[/Tex]
Now, defining the energy expression in Fourier Domain.
[Tex]\therefore E = \frac{1}{2\pi} \int_{-\infty}^{\infty} |Y(\omega)|^2 d\omega[/Tex]
As the signal [Tex]y(t) [/Tex]is defined for a range of [Tex]-\frac{\tau}{2}[/Tex] to [Tex]\frac{\tau}{2}[/Tex] So,
[Tex]\therefore \int_{-\infty}^{\infty} |y(t)|^2 dt = \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} |x(t)|^2 dt[/Tex]
Now, defining its equivalent Fourier domain representation.
[Tex]\therefore \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |Y(\omega)|^2 d\omega[/Tex]
As, Total Power [Tex](P_t)[/Tex] is defined as the total energy transferred per unit time interval [Tex](\tau)[/Tex]. Mathematically, it is defined as follows:
[Tex]\therefore P_t = \frac {E}{\tau}[/Tex]
[Tex]\therefore P_t = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{|Y(\omega)|^2}{\tau} d\omega[/Tex]
Now, when [Tex]\tau \rightarrow \infty[/Tex] then the Total Power [Tex](P_t)[/Tex] is termed as Average Power [Tex](P_{avg})[/Tex] which is defined with the following expression:
[Tex]\therefore P_{avg} = \frac{1}{2\pi} \int_{-\infty}^{\infty} lim_{\tau \to \infty} \frac{|Y(\omega)|^2}{\tau} d\omega[/Tex]
when [Tex]\tau \to \infty[/Tex] then the expression [Tex](\frac{|Y(\omega)|}{\tau} )[/Tex] tends to a finite value. Considering that finite term to be [Tex]S(\omega)[/Tex]
[Tex]\therefore S(\omega) = lim_{\tau \to \infty} \frac{|Y(\omega)|}{\tau}[/Tex]
Hence, the expression defined above is termed as Power Spectral Density (PSD) of the continuous time signal [Tex]y(t)[/Tex]. Also, defining in terms of [Tex]X(\omega)[/Tex]
[Tex]\therefore S(\omega) = lim_{\tau \to \infty} \frac{|X(\omega)|}{\tau}[/Tex]
So, the Average Power [Tex](P_{avg})[/Tex] of the generalized continuous time signal [Tex]x(t)[/Tex] in terms of Power Spectral Density [Tex](S(\omega))[/Tex] is expressed as follows
[Tex]\therefore P_{avg} = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_x(\omega) d\omega = \int_{-\infty}^{\infty} 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
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