Fourier transform of a noisy signal
Usually, there are two types of noise that you can eliminate by using the spectrum. 

One type is a noise that is in a different frequency band than the signal (it can be a high-frequency noise). In such a case, there is a clear separation between the signal content and the noise in the spectrum and you have to do nothing. 

The second case is when the noise is in the same frequency band as the signal and then you can perform Welch's method of averaging the spectrums to reduce the noise component. 

In case you have a dynamic model of the signal generator, you can reduce the noise using some kind of adaptive filter such as Kalman-filter prior to the evaluation of the spectrum.