# Noise in the Fourier transform

I have a smooth but rather complex curve, sampled with a good frequency. I apply the discrete Fourier transform to it using the fast Fourier transform (FFT) algorithm and get its Fourier image. I need to find peaks on the resulting Fourier transform curve, but the image I get contains a substantial amount of noise, which is a real problem because peaks can't be clearly seen. So, my question is as follows: What are the sources of noise in the Fourier transform, and how it can be reduced?

Typical plots look like this:

Initial function Initial function, narrow range Fourier transform Fourier transform, narrow range

Before transformation a constant was subtracted from the the function so that it goes to zero.

• Could you please define your initialisms, or avoid using initialisms? – onestop Feb 16 '11 at 17:10
• Can you post an example plot? -- this seems like a computational error, but it is hard to tell from the description only. – user88 Feb 16 '11 at 18:05
• FT=Fourier transform, FFT = fast Fourier transform, DFT = discrete Fourier transform – Roman Feb 16 '11 at 18:05
• @Roman, I second @mbq's remarks. A figure would be helpful, especially in terms of understanding what you mean by "noise". Just because the time domain signal is smooth does not mean the Fourier transform will be. Just think of a sinusoid to convince yourself. in terms of plotting, the first thing that comes to mind is: are you plotting the magnitudes and not, say, accidentally, just the real part. Also, do you know about windowing? – cardinal Feb 16 '11 at 19:21
• Some idea of the source of the sampled curve could be helpful too. – onestop Feb 16 '11 at 21:05

This seems like $e^{-ax}\sin(bx)$ function -- FT of such are two Dirac deltas, so it is not surprising at all that they appear as a noisy peaks after DFT (this is a variation of ultraviolet crisis). So, well, don't worry -- you can do nothing wise about it, at least smooth the transform (for instance with moving mean) to find peak locations easier (but better do not report smoothed curve).

On the other hand, if you are interested in the later signal rather than this initial "bang", it is better just to cut it off -- this will clear those major peaks and show the more subtle details.