# What is the relation between the PSDs of filter input and output called? $R_Y = |H|^2R_X$

If a wide-sense stationary signal $X$ is fed to an LTI filter with the transfer function $H$, the power spectral density (PSD) of the output $Y$ can be expressed as:

$$R_Y(f) = \left|H(f)\right|^2R_X(f)$$

where $R_X$ denotes the PSD of $X$.

Does this relation have a common name?

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## 2 Answers

I don't know the name of the relationship, but $\vert H(f)\vert^2$ is called the power transfer function of the LTI system. The output power spectrum is the input power spectrum multiplied by the power transfer function, just as for deterministic signals, the output spectrum is the input spectrum multiplied by the transfer function $H(f)$.

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To be extra pedantic, H(f) is the frequency response function. H(w) is the transfer function. – mtrw Sep 20 '11 at 22:44
@mtrw Do you have a citation to back up your pedantry? Bracewell's classical text The Fourier Transform and Its Applications calls $H(f)$ the transfer function; other texts call $H(\omega)$ or $H(j\omega)$ the transfer function as you do; yet others call $H(s)$ the transfer function. So, please provide a citation that says that calling $H(f)$ the transfer function is wrong as this name is reserved for $H(\omega)$. – Dilip Sarwate Sep 21 '11 at 1:18
First, I have to apologize for a stupid mistake. I should have said H(f) is the FRF, and H(s) is the transfer function. Sadly I don't have my copy of Oppenheim, Schafer & Young's Signals and Systems anymore, which is where I remember learning this. The mnemonic I was taught was that the Fourier transform of the impulse response (either H(f) or H(jw)), since it's evaluated for pure sinusoids, gives the response to frequencies. The Laplace and z transforms (H(s) or H(z)) give transfer functions. – mtrw Sep 21 '11 at 6:23

The relation that you have results from the Wiener-Khinchin theorem (WK). The WK theorem primarily relates the autocorrelation of the input and its power spectral density (PSD) as a Fourier transform pair. I have not heard it referred to by any particular name other than explicitly saying "From the WK theorem, we have blah..." From the article cited:

A corollary [of the WK theorem] is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform of the system impulse response.

While it was written and proven for signals (or functions) that are square integrable, and hence have a Fourier transform, it is commonly used to study WSS random processes (which do not have a Fourier transform) by relating the autocorrelation via expectations rather than integrals.

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It is a fine response, but you really don't answer the question? I get the impression that your answer is Wiener-Khincin theorem, but that is not really true I think. I hope I don't appear grumpy, but the question is really precise so the answer should/could be precise. – niaren Sep 20 '11 at 18:47
I disagree with Wikipedia that the result in question is a corollary of the WK theorem. The WK theorem says that the PSD of a WSS process is the Fourier transform of its autocorrelation function. It is a different result entirely that when a WSS process passes through a linear system, the output autocorrelation function is related to the input autocorrelation as $A_Y = h * \tilde{h} * A_X$. This result requires probabilistic analysis and taking expectations etc. that are related to the calculations used to prove the WK theorem, but the result is not a corollary of the WK theorem – Dilip Sarwate Oct 31 '11 at 11:20
Continuing my previous comment, once the probabilistic analysis has established that $A_Y = h * \tilde{h} * A_X$, we can apply the WK theorem and say $A_X(t) \leftrightarrow R_X(f)$ and $A_Y(t) \leftrightarrow R_Y(f)$ via WK, while $h(t) \leftrightarrow H(f)$ and $\tilde{h}(t) \leftrightarrow H^*(f)$ and so $$R_Y(f) = |H(f)|^2 R_X(f)$$ via the convolution theorem which is what the OP was asking about. But all this is inapplicable unless you first show that $A_Y = h * \tilde{h} * A_X$, and this is not a corollary of the WK theorem. – Dilip Sarwate Oct 31 '11 at 11:27
@Dilip I don't disagree with that, and I never make the claim that the result for WSS is a corollary of WK. The text I quoted just talks about the relationship between the autocorrelation and Fourier transforms for inputs and outputs of an LTI system. It does not talk about WSS. I clarified just beneath it, that while WK was proven for square integrable signals, it is used to study WSS using a probabilistic approach and relating the autocorrelation via expectations. It's pretty much what you've said here, but I didn't go into any detail, because the OP never asked for it. – Lorem Ipsum Oct 31 '11 at 13:51
@yoda Please note that I said I was disagreeing with what Wikipedia claims, not what you said. The Wikipedia article states that the WK theorem holds for WSS processes, and then claims that the result $R_Y(f)=|H(f)|^2R_X(f)$ is a corollary of the WK theorem which is not correct. The result $A_Y=h∗\tilde{h}∗A_X$ can be proved for deterministic (square-integrable) signals very straightforwardly and then $R_Y(f)=|H(f)|^2R_X(f)$ follows via the convolution theorem. For WSS processes, establishing $A_Y=h∗\tilde{h}∗A_X$ requires probabilistic analysis (as you correctly say). More below – Dilip Sarwate Oct 31 '11 at 14:09