Properties of the power spectral density (1/4)

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Presentation transcript:

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

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

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

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

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