1. Properties of the power spectral density (1/4)
The power spectral density of a WSS random process has several important properties:
Property 1:
Property 2:
Proof: 1
Proof:
5.3 1

2. Properties of the power spectral density (2/4)
Property 3: for all f
Let be passed through an idea very narrow-band filter with bandwidth and centered about the frequency , the mean-square value of the filter output is directly proportional to Since , we have
This statement is correct for any , so we have
for all f
Proof: 1
5.3 2

3. Properties of the power spectral density (3/4)
Property 3a: An important properties for autocorrelation function is that the Fourier transform of a proper auto-correlation function is equal or greater than zero for all f.
For example, for positive T is not a suitable autocorrelation function since its Fourier transform
is not always equal or greater than zero for some f .
However, will be a suitable autocorrelation function since
for all f 1
5.3 3

4. Properties of the power spectral density (4/4)
Property 4: for real random process.
Proof: 1
5.3 4