Multiples of frequencies with high Fourier coefficients

Question

As I understand it, any signal that satisfies certain properties can be represented as an infinite sum of sine waves. And the amplitude of each sine wave is reflected in the Fourier coefficient, so terms with higher Fourier coefficients should contribute more to the signal.

I read in a paper that if I have a signal that spikes every two weeks, the Fourier coefficient corresponding to a two-week period should be high. This makes perfect sense. But they also say that the Fourier coefficient corresponding to multiples of that frequency should be high (like for periods of one week, or one day). This seems counterintuitive, because if the one-week frequency has a high Fourier coefficient, then there should be spikes in between the biweekly spikes, and I don't think there are. How can I reconcile these facts?

Thanks!

Answer 1

Not sure if I understand your question correctly, but these are the facts:

If you have a periodic signal with period $T_0$, you get the frequencies

$$f_0=1/T_0, 2f_0, 3f_0, 4f_0, \ldots$$

A signal with period $T_1=T_0/2$ will contain frequencies

$$f_1=1/T_1, 2f_1,3f_1,4f_1,\ldots =\\ 2f_0,4f_0,6f_0,8f_0,\ldots$$

More specifically, if you were to remove the fundamental frequency of a periodic signal, you will not get a signal with twice the fundamental frequency (because you still retain the odd multiples of the fundamental frequency, which would not be present in a signal with twice the frequency).

Answer 2

The point is that signals are rarely a pure sine wave. If you have a signal that repeats every two weeks but is not a pure sine wave, you need to distort the shape. For example a square wave is approximated by $\sin x + \frac 13\sin 3x + \frac 15 \sin 5x \dots$ The result is below