# Computable Time-Frequency Distribution without Cross-Terms

Context:

I have a single receiver that is receiving multiple signals. Each signal has a few strong harmonics, but I don't know the fundamental frequency. I would like to display them on a spectrogram like display, but when the number of signals increases beyond three or four, the spectrogram gets very cluttered.

In other words:

$$| X_{1}(f) + X_{2}(f) + \cdots + X_{N}(f) |^2 = |X_{1}|^2(f) + |X_{2}|^2(f) + \cdots + |X_{N}|^2(f) + \text{cross-terms}$$

Question:

Is there another time-frequency distribution that eliminates or greatly reduces these cross-terms, and is also computable on a digital computer in a reasonable time [ideally less than O(N^2)]?

I know other distributions exist, e.g. the Wigner-Ville distribution, but generally, they seem to try and improve the resolution of the spectrogram at the expense of greater cross-terms. I would be happy with the opposite -- less cross-terms but also less resolution.

Synchrosqueezing. Diminishes Wigner-Ville interaction disadvantages as described here.

• I have never heard of this -- I need to figure out how to apply it to my situation, but, it seems promising. Aug 30 '21 at 13:54

All time-frequency representations that can be formulated as part of Cohen's class (bilinear ones) will have cross-terms. There are post-processing / filtering approaches (as mentioned on the Wikipedia page) that allow them to be filtered out.

OverLordGoldDragon's approach seems more sensible than anything in Cohen's class --- including the Wigner-Ville, STFT, or the other more esoteric representations.

• Ah okay, I have heard of Cohen's class before, but I didn't realize ALL T-F representations could be formulated as part of his class. Aug 30 '21 at 13:54
• @TheDude Not all, but certainly most of the ones you've heard about.
– Peter K.
Aug 30 '21 at 15:31