_{exp}(t), is essentially a perfect FID, FID(t) multiplied by an error function, E(t), resulting from the inhomogeneous field.

FID

_{exp}(t) = FID(t) * E(t)

If we could find the error function and divide the experimental FID, by it, we could produce a perfect (or at least improved) FID, the Fourier transform of which would be a spectrum corrected from the effects of field inhomogeneity. In the reference deconvolution technique, one selects a high signal-to-noise ratio singlet peak as a reference in the experimental spectrum and sets all other points in the spectrum to zero. This spectrum is inverse Fourier transformed to produce an FID of the distorted singlet, FID

^{r}

_{exp}(t). A synthetic FID, FID

^{r}

_{syn}(t) is constructed for what one would expect the time domain signal to be for a perfect reference peak (e.g. a sharp Lorentzian). The error function for the reference peak E

^{r}(t) is then determined by:

E

^{r}(t) = FID

^{r}

_{exp}(t) / FID

^{r}

_{syn}(t)

Since all peaks in the experimental spectrum are affected equally by the inhomogeneity, E(t) = E

^{r}(t) and we can compute a corrected FID for the entire spectrum,

FID(t) = FID

_{exp}(t) / E

^{r}(t)

The corrected FID is Fourier transformed yielding a much improved spectrum. This technique is available in newer NMR software processing packages and is particularly easy to implement in the MestReNova software package available to NMR users at the University of Ottawa. An example is shown in the figure below for a 300 MHz

^{1}H NMR spectrum of a mixture of compounds.

The top two traces show portions of the spectrum obtained in a carefully shimmed magnet. The middle traces show the same portions of the spectrum obtained in a poorly shimmed magnet. The bottom traces were obtained by applying reference deconvolution to the spectrum obtained in the poorly shimmed magnet. Clearly, there is much improvement in the reference deconvoluted spectrum, allowing one to obtain much more information and recover some of the lost beauty. In fact, the corrected spectrum is very similar to the one obtained in the homogeneous field of a carefully shimmed magnet. The penalty paid is a lower signal-to-noise ratio, as the noise from the experimental reference signal is convolved into the error function which in turn gets convolved into the corrected spectrum. The loss in signal-to-noise ratio can be minimized by choosing a reference signal with a higher signal-to-noise ratio.