Of Negative Frequencies

A question is often (well at least once in each DSP course) asked by DSP students. (Quoting from Quora).

What negative frequencies actually mean physically?  Considering the fact that bandwidth is specified by the positive part only, do negative frequencies exist or are they just a mathematical side effect?

Here's my answer, with some edits.

Consider a sine wave. What is its Fourier transform?

Fig. 1  A sine wave and its Fourier transform (Image source)

A negative ordinary frequency of $-10$ does not mean a sine wave oscillating at $-10$ cycles per second. It refers to a complex exponential $e^{-j2 \pi 10t}$ When we add another exponential with positive exponent $+10$ we get a real sinusoid which oscillates at $10$ cps. This also explains your bandwidth confusion ("Considering the fact that bandwidth is specified by the positive part only.."). If by some magic, the formula for addition of complex exponentials were $e^{j\omega t} + e^{-j \omega t} = 2 \cos \sqrt \omega t$  then bandwidth would have been specified by the "square root of the positive part".

For real signals, these "negative frequencies" do not occur alone and always are paired with a corresponding "positive frequency" so that the resultant signal is real. The impulse which you see on the negative X axis in the right hand figure does NOT represent a sinusoid but a complex exponential. This is what one may lose sight of while talking about negative frequency.