Coherence vs Correlation Functions: Understanding the Difference

[fisico30]

hello forum,

coherence function and correlation functions are the same thing.
If we consider a sinusoidal signal like cos(t), and we calculate the correlation function, we obtain a periodic function. That means that in some cases cos(t) and cos(t+tau) are very similar, for some other time delay tau values they are opposite, for other completely not correlated.
As long as there is a constant in time relationship there should be high correlation: even if the signal cos(t) and its shifted version are not very similar, they are interlocked in their phase behavior...the correlation value for that particular shift tau might not be maximum. but the interlocking is still there...
The correlation function seems to just express the degree of similarity and not highlight the phase interlocking...

If cos(t) is expressed as exp(i*t) and the correlation is calculated, we get that the exp(i*t) and exp(i*t+tau) are correlated for all time delays tau: the magnitude of the correlation function (which is complex) is a constant for all tau...

All this to say that if we consider the real function, the correlation function is a real function whose value is indication only of the similarity between the function and its shifted version. There is not mention of the inter-relationship, the constant in time phase difference (which is the actual measure of the coherence)...
If the real function is expressed as a complex exponential, we get a correlation function with constant magnitude for all tau, to indicate the coherence (constant phase relation).
Looking for similarity instead, we need to look at the instantaneous magnitude of the complex correlation function...
IS this correct?

thanks
fisico30

 

[mmwave]

Dear fisico30,

Thank you for bringing up this interesting topic. I understand your confusion about coherence and correlation functions, as they are often used interchangeably in different fields of science. However, coherence and correlation functions are not exactly the same thing, although they are related to each other.

Coherence is a measure of how well two signals are correlated in terms of their phase or timing relationship. It is a measure of the degree to which two signals are synchronized or "in step" with each other. On the other hand, correlation is a measure of the similarity between two signals. It is a measure of how much the shape or pattern of one signal resembles the other signal.

In your example of the cosine function, the correlation function would indeed show a periodic pattern, as the shape of the function repeats itself. However, the correlation function does not take into account the phase relationship between the two signals. This is where coherence comes in. When we consider the phase relationship between the two signals, we can calculate the coherence function, which would show a constant value for all time delays, indicating a perfect phase relationship between the two signals.

So, to answer your question, it is not entirely correct to say that coherence and correlation functions are the same thing. They are related, but they measure different aspects of the relationship between two signals. It is important to consider both when analyzing signals and their behavior.

I hope this helps clarify the difference between the two functions. Keep exploring and asking questions, and best of luck with your research!