Filtering noise from FFT, where noise is known to be shifted delta function
High-pass filter the input with a zero-phase filter to eliminate the signal of interest, leaving only the high-passed noise. If the signal is periodic, you can do this in the frequency domain and convert back to time domain by IFFT. Find the time of the peak and deduce the magnitude of the unfiltered delta function from the magnitude of the filtered delta function (the peak value or root mean square). Knowing the delay and magnitude, subtract the noise from the original input signal. Do FFT.