Significance of imaginary part of auto-/cross-correlation

Question

In this answer to this question, we get a hint that there is a significance to the imaginary part of the auto-correlation function, and that we should try to calculate the correlation with the real and imaginary parts of the signals. But I would like to get some clarity on how to interpret that imaginary part, either for the cross- or auto-correlation.

Let's start from the definition of the cross-correlation $$ R = \sum_{n} f_{n}\,\overline{g_{n}} = \sum_{n} |f_n\,g_n|\exp[j(\theta_n-\phi_n)] = \sum_{n} (a_{n}\,c_{n}+b_{n}\,d_{n})+j(b_{n}\,c_{n}-a_{n}\,d_{n}) $$ for the two signal sequences $f_{n}=|f_n|\exp(j\theta_n)=a_{n}+jb_{n}$ and $g_{n}=|g_n|\exp(j\phi_n)=c_{n}+jd_{n}$, and where $\overline{g_{n}}$ denotes the complex conjugate of $g_{n}$.

From the complex-phase representation, it seems like the imaginary part of $R$ represents the correlation of the signals $90^{\circ}$ out of phase.

• Is there any other interpretation?
• Is there any other interpretation for the auto-correlation $R(\ell) = \sum_{n} f_{n}\,\overline{f_{n+\ell}}$?

Answer 1

For starters, let's assume that a "complex" is simply a combination of two arbitrary real signals, i.e.

$$x(t) = a(t) + jb(t), y(t) = c(t) + jd(t) \tag{1}$$

For the cross correlation we simply find that

$$r_{xy}(t) = [r_{ac}(t) + r_{bd}(t)] + j \left[ r_{bc}(t) - r_{ad}(t) \right] \tag{2}$$

So it's kind of a "mangled" version of the 4 real cross correlations.

The interpretation of that depends a lot on what exactly you mean by a "complex" signal. There's been some philosophical discussions on this forum around this topic. Complex signals don't exist in the physical world, but you can certainly combine two real signals into a complex one as a mathematical convenience. What happens then will depend on the relationship between the two signals you combine (if any).

A trivial but practical use for this would be to calculate two real cross correlations (say $r_{xy}$ and $r_{xz}$) using a single complex FFT to over $y + jz$ and split the result into real and imaginary part.