Is it necessary that cross power spectrum (CPSD) should always be between power spectrum (PSD) of individual signals, irrespective of the method we compute it? The images are shown below.

psd Px and Py and cpsd Cxy computed using fft

power spectrum Px and Py of signal 'x' and 'y' and cpsd  Cxy using Welch method

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    $\begingroup$ What do you mean with "in between"? What has been your approach so far? It's usually a good idea to show the considerations you came up with so far; potential answerers usually try to avoid explaining stuff that you already know. Also, because things like these usually very much depend on normalization, it's usually also a good idea to explicitly state the formula that you're referring to, in this case the CPSD formula you're using. $\endgroup$ Jun 15 '17 at 6:45
  • $\begingroup$ @MarcusMüller , the term 'in between ' means Px<Cxy< Py or Py<Cxy<Px which is evident in the figure above with title ' using fft'. $\endgroup$
    – Akanksha
    Jun 16 '17 at 8:25
  • $\begingroup$ The Welch method does not hold the relation.I have solved the query, It is because of averaging the spectrum over windowed signal, due to which final spectrum does not conforms to it. @MarcusMuller Thanks for replying $\endgroup$
    – Akanksha
    Jun 16 '17 at 8:36
  • $\begingroup$ Could you post an answer with your solutions, including all the formulas? That way, your question would hold a value for future readers. $\endgroup$ Jun 16 '17 at 15:17
  • $\begingroup$ @Akanksha, Could you please review my answer? If it answers your question, please mark it. $\endgroup$
    – Royi
    Nov 10 at 8:15

Whenever talking about Cross Correlation of signals it is important to well define it.

If you mean Cross Correlation in most common sense (Signal Processing):

$$ \left( x \star y \right) \left[ n \right] = \sum_{i = 0}^{N - m -1 } {x} \left[ m \right] y \left[ n + m \right] $$

Then of course not.
You can have cases where the DFT of the Cross Correlation has higher absolute values then the DFT of the Auto Correlation.

The Schwarz inequality answers your question, Yes.
But don't use pwelch for the cross spectrum. The cross spectrum is complex valued.

The above by @Stanley Pawlukiewicz is wrong.
Simple way to see so is by defining $ y \left[ n \right] = 10 x \left[ n \right] $ which of course mean the cross correlation between those is higher than the cross correlation between $ x $ and itself.

The reason is that Cauchy Schwarz Inequality is about Inner Product Space.
So if we talking about signals as vectors then we need to use the Inner Product which is the cross correlation at $ 0 $.
If we talk about Random Process than we need to use their definition (Which normalize the energy and then everything works).

If you use Cross Spectrum while sticking to the Random Process definition and normalize the Cross Correlation by the variance of the processes than indeed by Cauchy Schwarz Inequality you're guaranteed to see what you got.


The Schwarz inequality answers your question, Yes

But don't use pwelch for the cross spectrum. The cross spectrum is complex valued.

  • $\begingroup$ If you mean Cross Correlation is the flipped Convolution (As usually used in Signal Processing) then of course this is wrong. It only holds in cases one use the normalized cross correlation (Correlation Coefficient per shift). $\endgroup$
    – Royi
    Jul 14 '18 at 16:38

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