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23 Feb

In [1, p. 62 exercise 3.13] it is requested to prove that the PSD of a real WSS random process is a real even function of frequency.
**Solution:**
The PSD of a WSS process is given by the Fourier transform of its autocorrelation function:

We note that for a real process , will be also real thus we obtain the relations:

First let us prove that for a real WSS this function is even. Having the previous relations in mind:

Let then

Thus we have shown that the PSD is symmetric . The next step is to show that the PSD is also real. We know that the real part of a complex function is given as one half of the sum of the complex function with its complex conjugate form, that is . Let’s determine the complex conjugate of the PSD:

Again relation (2) was used in order to obtain the last result. The real part of the PSD is thus given by

because of (4). So the real part of the PSD is the PSD itself, or stating it in another way: the PSD of a WSS process is real. Thus together with (3) we have shown that the PSD of a real WSS random process is a real even function of frequency. QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.

(1) |

We note that for a real process , will be also real thus we obtain the relations:

(2) |

First let us prove that for a real WSS this function is even. Having the previous relations in mind:

Let then

(3) |

Thus we have shown that the PSD is symmetric . The next step is to show that the PSD is also real. We know that the real part of a complex function is given as one half of the sum of the complex function with its complex conjugate form, that is . Let’s determine the complex conjugate of the PSD:

(4) |

Again relation (2) was used in order to obtain the last result. The real part of the PSD is thus given by

because of (4). So the real part of the PSD is the PSD itself, or stating it in another way: the PSD of a WSS process is real. Thus together with (3) we have shown that the PSD of a real WSS random process is a real even function of frequency. QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.

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