Jupp P E, Harris K D, Aliev A E
School of Mathematical and Computational Sciences, University of St Andrews, North Haugh, St Andrews, KY16 9SS, United Kingdom.
J Magn Reson. 1998 Nov;135(1):23-9. doi: 10.1006/jmre.1998.1538.
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.
通过使用最小二乘法程序将适当的函数拟合到实验光谱来估计数字化频域光谱中峰值之间的频率间隔是很常见的。在本文中，我们从第一原理评估与这种频率间隔测量相关的精度。除了峰值之间的频率间隔外，拟合光谱所涉及的其他参数是峰值宽度、峰值的线形函数（高斯、洛伦兹等）以及峰值幅度。精度还取决于信噪比和数字化光谱中相邻数据点之间的间距。假设在最小二乘法拟合过程中考虑的残差是实验光谱和拟合光谱中相应数字化数据点强度之间的差异。在这些条件下，针对以下情况推导了峰值分离精度的解析表达式：(i) 当两个峰值的幅度已知且两个峰值具有已知相等宽度时；(ii) 当两个峰值幅度的比值已知且两个峰值的宽度已知相等，但峰值宽度的实际值未知时。在每种情况下，都考虑了两个高斯峰值的情况和两个洛伦兹峰值的情况。在所有情况下，两个峰值之间估计频率间隔η的绝对精度P(η) 由类型为P(η) ≈ F(η/Δ, α)SK的方程近似，其中Δ是峰值宽度，α是两个峰值幅度的比值A2/A1，S是信噪比，K是频域光谱中的数据点密度。函数F(η/Δ, α) 的形式取决于线形类型（高斯或洛伦兹），并且取决于参数A1、A2和Δ中哪些是独立于拟合过程已知的。试图将我们的第一原理方法扩展到评估比上述更复杂情况下峰值之间频率间隔的最小二乘估计精度，通常会导致解析表达式极其复杂。在这种情况下，可以采用基于此处开发的理论框架的数值方法来评估估计频率间隔的精度。
