A Frequency Determination Method for Digitized NMR Signals (Communications in Computational Physics, 2014)
We present a high precision frequency determination method for digitizedNMR FID signals. The method employs high precision numerical integration ratherthan simple summation as in many other techniques. With no independent knowledge of the other parameters of a NMR FID signal (phase $ϕ$, amplitude $A$, and transverse relaxation time $T_2$) this method can determine the signal frequency $f_0$ with aprecision of 1/(8$π^2$$f^2_0$$T^2_2$) if the observation time $T$ ≫ $T_2$. The method is especiallyconvenient when the detailed shape of the observed FT NMR spectrum is not welldefined. When $T_2$ is +∞ and the signal becomes pure sinusoidal, the precision of themethod is 3/(2$π^2$$f^2_0$$T_2$) which is one order more precise than the ±1 count error induced precision of a typical frequency counter. Analysis of this method shows that theintegration reduces the noise by bandwidth narrowing as in a lock-in amplifier, andno extra signal filters are needed. For a pure sinusoidal signal we find from numericalsimulations that the noise-induced error in this method reaches the Cramer-Rao LowerBand (CRLB) on frequency determination. For the damped sinusoidal case of most interest, the noise-induced error is found to be within a factor of 2 of CRLB when themeasurement time $T$ is 2 or 3 times larger than $T_2$. We discuss possible improvementsfor the precision of this method.