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 X(t) with autocorrelation function RX(τ). We define the Power Spectral Density (PSD) of X(t) as the Fourier transform of RX(τ). We show the PSD of X(t), by SX(f). More specifically, we can write

SX(f)=F{RX(τ)}=RX(τ)e2jπfτdτ,
where j=1.

Power Spectral Density (PSD)

SX(f)=F{RX(τ)}=RX(τ)e2jπfτdτ,where j=1.
From this definition, we can conclude that RX(τ) can be obtained by the inverse Fourier transform of SX(f). That is
RX(τ)=F1{SX(f)}=SX(f)e2jπfτdf.
As we have seen before, if X(t) is a real-valued random process, then RX(τ) is an even, real-valued function of τ. From the properties of the Fourier transform, we conclude that SX(f) is also real-valued and an even function of f. Also, from what we will discuss later on, we can conclude that SX(f) is non-negative for all f.
  1. SX(f)=SX(f), for all f;
  2. SX(f)0, for all f.
Before going any further, let's try to understand the idea behind the PSD. To do so, let's choose τ=0. We know that expected power in X(t) is given by
E[X(t)2]=RX(0)=SX(f)e2jπf0df=SX(f)df.
We conclude that the expected power in X(t) can be obtained by integrating the PSD of X(t). This fact helps us to understand why SX(f) is called the power spectral density. In fact, as we will see shortly, we can find the expected power of X(t) in a specific frequency range by integrating the PSD over that specific range.
The expected power in X(t) can be obtained as
E[X(t)2]=RX(0)=SX(f)df.


Example
Consider a WSS random process X(t) with
RX(τ)=ea|τ|,
where a is a positive real number. Find the PSD of X(t).
  • Solution
    • We need to find the Fourier transform of RX(τ). We can do this by looking at a Fourier transform table or by finding the Fourier transform directly as follows.
      SX(f)=F{RX(τ)}=ea|τ|e2jπfτdτ=0eaτe2jπfτdτ+0eaτe2jπfτdτ=1aj2πf+1a+j2πf=2aa2+4π2f2.


Cross Spectral Density:

For two jointly WSS random processes X(t) and Y(t), we define the cross spectral density SXY(f) as the Fourier transform of the cross-correlation function RXY(τ),
SXY(f)=F{RXY(τ)}=RXY(τ)e2jπfτdτ.


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