How to understand negative frequency?

[yangjun1222]

Recently I learned some communication theory, which uses negative frequency, what does negative frequency means? So I checked textbook, found the negative frequency comes from the Fourier transform, which comes from the euler formula, e^-jx=cosx+isinx,
or another form cosx=(e^jx+e^-jx)/2, then any physical quantity can be expressed in negative part and positive part.
Can anyone explain this (negative frequency) further? (In physics, I can not understand this, so this is just a expression).
Some communication theory use this much, like AM, DSB,modulation method.
Does this technical develop after mathematical derivation(like the electromagnetic wave) or mathematical is latter developed after experiments. Are there any good textbook introduce this (communication theory and technology and their history) clearly?

 

[anorlunda]

In the equations you show it the OP, the independent variable is x. +x and-x mean waves moving to the left and right (or whichever direction is the x axis). That is not negative frequency, it is negative direction.

 

[LvW]

Negative frequencies do not exist in reality.
As you have shown, they are invented as a fictive quantity (mathematical tool) to simplify some manipulations/expressions in system or communucation theory only.

EDIT (added): In 1800, it was shown by Fourier how any periodic signal with period T can be expanded to an infinite set of complex exponents with frequencies +-k/T, where k = 0, 1, 2, ... That means: Frequencies can be both positive and negative. If to set T infinite, then we arrive at the Fourier transform.
So, negative frequencies are just mathematical quantities (not physical) similar to the imaginary part of a complex signal. In real world, negative frequencies do not exists and the spectral content on negative frequencies must be added to the spectral content at the positive frequencies.