Precision in estimating the frequency separation between spectral lines.

It is common to estimate the frequency separation between peaks in a digitized frequency-domain spectrum by fitting an appropriate function to the experimental spectrum using least-squares procedures. In this paper, we assess from first principles the precision associated with such measurements of frequency separation. In addition to the frequency separation between the peaks, other parameters involved in fitting the spectrum are the peak widths, the lineshape functions (Gaussian, Lorentzian, etc.) for the peaks, and the peak amplitudes. The precision also depends on the signal-to-noise ratio and the spacing between adjacent data points in the digitized spectrum. It is assumed that the residuals considered in the least-squares fitting procedure are the differences between the intensities of corresponding digitized data points in the experimental and fitted spectra. Under these conditions, analytical expressions for the precision in peak separation are derived for the following cases: (i) when the amplitudes of two peaks are known and the two peaks have known equal widths; (ii) when the ratio of the amplitudes of two peaks is known, and the widths of the two peaks are known to be equal, but the actual value of the peak width is not known. In each case, the situation with two Gaussian peaks and the situation with two Lorentzian peaks are considered. In all cases, the absolute precision P(eta) in the estimated frequency separation eta between the two peaks is approximated by an equation of the type P(eta) approximately F(eta/Delta, alpha)SK, where Delta is the peak width, alpha is the ratio A2/A1 of amplitudes of the two peaks, S is the signal-to-noise ratio, and K is the density of data points in the frequency-domain spectrum. The form of the function F(eta/Delta, alpha) depends on the type of lineshape (Gaussian or Lorentzian), and depends on which of the parameters A1, A2, and Delta are known independently of the fitting procedure. Attempts to extend our first-principles approach to assess the precision in least-squares estimates of frequency separation between peaks in more complex situations than those discussed above generally lead to analytical expressions that are formidably complicated. In such cases, numerical approaches based on the theoretical framework developed here may be employed to assess the precision in estimating the frequency separation.