10.2.1 Power Spectral Density
So far, we have studied random processes in the time domain. It is often very useful to study random processes in the frequency domain as well. To do this, we need to use the Fourier transform. Here, we will assume that you are familiar with the Fourier transform. A brief review of the Fourier transform and its properties is given in the appendix.
Consider a WSS random process with autocorrelation function . We define the Power Spectral Density (PSD) of as the Fourier transform of . We show the PSD of , by . More specifically, we can write
where .
- , for all ;
- , for all .
We conclude that the expected power in can be obtained by integrating the PSD of . This fact helps us to understand why is called the power spectral density. In fact, as we will see shortly, we can find the expected power of in a specific frequency range by integrating the PSD over that specific range.
The expected power in can be obtained as
Example
Consider a WSS random process with
where is a positive real number. Find the PSD of .
- Solution
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We need to find the Fourier transform of . We can do this by looking at a Fourier transform table or by finding the Fourier transform directly as follows.
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We need to find the Fourier transform of . We can do this by looking at a Fourier transform table or by finding the Fourier transform directly as follows.