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I have a signal: $f_i(t_i=i\Delta t)$, where $i = 0\ldots n-1$.

The signal seems to vary quickly around a slower varying "trend". I am assuming that the quickly varying part is noise and the slowly varying part is the real signal.

How do I estimate the signal-to-noise ratio (SNR) of the signal?

I guess that if I could decide on a treshold frequency: $\omega_t$ I could use the following expression:

$$S/N=\frac{\displaystyle\int_0^{\omega_t}|F(\omega)|^2}{\displaystyle\int_{\omega_t}^{\infty}|F(\omega)|^2}$$ where $F$ denotes the Fourier transform of $f(t)$.

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    $\begingroup$ Could you post a periodogram? $\endgroup$ – user42 Dec 19 '11 at 15:46
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    $\begingroup$ What are the characteristics of the noise? White/colored? Known distribution? Zero-mean? $\endgroup$ – Jason R Dec 19 '11 at 16:48
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If the spectrum of the signal and the spectrum of the noise do not overlap, you might be able to integrate or sum the energy in each of the two frequency ranges, and take the ratio.

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