A frequency determination method for digitized NMR signals

We present a high precision frequency determination method for digitized NMR FID signals. The method employs high precision numerical integration rather than simple summation as in many other techniques. With no independent knowledge of the other parameters of a NMR FID signal (phase phi, amplitude A, and transverse relaxation time T-2) this method can determine the signal frequency f(0) with a precision of 1/(8 pi(2)f(0)(2)T(2)(2)) if the observation time T >> T-2. The method is especially convenient when the detailed shape of the observed FT NMR spectrum is not well defined. When T-2 is +infinity and the signal becomes pure sinusoidal, the precision of the method is 3/(2 pi(2)f(0)(2)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 the integration reduces the noise by bandwidth narrowing as in a lock-in amplifier, and no extra signal filters are needed. For a pure sinusoidal signal we find from numerical simulations that the noise-induced error in this method reaches the Cramer-Rao Lower Band (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 the measurement time T is 2 or 3 times larger than T-2. We discuss possible improvements for the precision of this method.