Negative frequencies
The Fourier series expansion for a periodic signal x(t) isThis includes Fourier series coefficients for the constant term (when k = 0, exp(jkω _{0}t) = 1), plus the fundamental and harmonics (k ≥ 1). But it also includes terms that seem to correspond to negative frequencies. When k ≤ − 1, the frequency kω _{0} is negative. These negative frequencies balance the positive ones so that the resulting sum is real valued. The following applet demonstrates this:
(Applet by Cory Sharp)
