As I switch off the moment I see "signal" or "wave" let alone spinning it around the circle here is my take without any engineering intuitions: -you can repres

As I switch off the moment I see "signal" or "wave" let alone spinning it around the circle here is my take without any engineering intuitions: -you can represent any (nice enough) function as sum of sines and cosines of given frequencies and amplitudes (frequency is how often the function hits the whole period, amplitude is by how much you multiple its value). The functions look this way: Amplitude* sine(x* freq) -So you have your sines and cosines at various frequencies and you wonder what the amplitudes should be for every one o them so that if we add them all up we end up with the original function -The idea to answer this question is to see how values of your sine or cosine correlate with values of original function. If it turns out that the original function is high in value when your sine/cosine is high in value then sine/cosine at this frequency needs to have higher weight than sine/cosine at other frequency where it doesn't correlate well with original function. Intuitively if you take functions which correlate well with original one with higher weight and those which doesn't with lower weight then you will have good approximation of the original function. -Euler formula tells you what sine and cosine at given point are representing it as one complex number; to get value of sine/cosine at frequency different than 1 you need to multiple the argument by k*2PI (as frequency is given in beats/period and period is 2PI it's quite obvious why); So e^(xi) is value of cosx and sinx and e^(xik2PI) is value of those functions at frequencies different than 1. So now let's see what this formula is: -it's an average of things (1/N and a sum of N elements) -those things are multplications of values of sine/cosine at given frequency at given point with value of original function at this point -this average is going to be high if places where sine/cosine at given frequency is high and original function is high overlap (you multiple big number by big number) and low if described correlation is low; that's the concept of correlation -the points are evenly spaced from 0 to n-1 (that's why n/N in the formula) and the sum goes over them. That's it. No circles and spinning. The only trick is to realize that: e^(ix) represent value of sine and cosine at x and e^(ixk2PI) represents value of sine/cosine at frequency k (if k is 1, then it's standard sine/cosine as everything simplifies). One sentence explanation could be: the more sine/cosine at given frequency correlates with original function the more that sine/cosine contribute to the original function.