Comment by physicsguy

That wouldn't really work well because the sparsity of the periodic spikes wouldn't fit the assumption that the signal has a frequency component 'everywhere', even though it's periodic. You can see that mathematically - if you take the Fourier transform of an impulse signal you get a smeared result in frequency space.

You'd probably want to use a tool like calculating the cepstrum rather than fourier transform. Cepstral methods are commonly used in mechanical analysis to detect periodic impacts like where a gear tooth gets damaged.