When you do a Fourrier transform of a signal of finite time, you notice that there's more than the fundamental frequency. Basically, when you cut off a signal,

When you do a Fourrier transform of a signal of finite time, you notice that there's more than the fundamental frequency. Basically, when you cut off a signal, then resume it, over and over, then that signal ceases to be of only one frequency. That's why your method doesn't work.