![]() This introduces statistical fluctuations into each bin, and also smears all features over a frequency width $\Delta \omega \sim 1 / \tau$ by the uncertainty principle. A spectrum analyzer produces an estimate of this power spectral density by integrating over only a finite time. what the mathematicians use) is defined by taking an ensemble average, or equivalently by integrating over an infinite time. Roughly speaking, a spectrum analyzer looks at the output over a finite time window $\tau$ (which you can set by pushing appropriate buttons, see your user manual), and calculates the Fourier transform of that finite time series.Ī "true" power spectral density (i.e. The Fourier transform does not change over time, while the output of a spectrometer certainly changes over time.If you're unsure, consult the user manual for your spectrometer.) (The power spectral density $S(\omega)$ typically only includes positive frequencies, but whether a factor of $2$ appears in the definition to compensate for the lack of negative frequencies is convention dependent. Therefore, whether you say the domain is all frequencies or only positive frequencies is just a detail, with no real implications. The Fourier transform (and hence $S$) are defined on negative frequencies, while spectrometers only output values at positive frequencies.įor any real-valued signal, $f(\omega) = f(-\omega)^*$, which implies that the negative frequencies don't carry any information not already in the positive frequencies.
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