In Principio...

The literature contains countless articles on Automatic Phase Correction. They all begin stating that no prior method is satisfactory. Reading each article you get the sensation that the problem has found the definitive solution, at last. Unfortunately, after a while a new article appears and... resets the situation. If we continue in this way, we'll have hundreds of methods and no one will work. The lesson so far has been: devising a novel method is easy, evaluating your own method is above man's possibilities. I have found one old document of an external observer attempting to rate all existing methods:
R.E: Hoffmann and G.C. Levy, "Modern Methods os NMR Data Processing and Data Evaluation", Progress in NMR Spectroscopy, Vol. 23. pp. 211-258, 1991.
They only demonstrated their incompetence by affirming:
The DISPA autophasing method has proved to be very reliable and is often more accurate than automatic phasing. [...] The Simplex method is slower than DISPA and curve fitting and sensitive to digital resolution, baseline and in the presence of asymmetric line-shapes.
They at least mentioned the work of Siegel, the one that I consider the seminal paper in the field, and that has been unjustly forgotten in recent years. For example, it has not been cited at all by Garland et al (J. Magn. Reson. 158, 2002, 164–168), despite their method has many similarities with the one of Siegel. Like all authors, they have filled their paper with cryptic equations, while Siegel obtained the best results with no equations at all...
Marshall M. Siegel sent his communication to Analytica Chimica Acta exactly 27 years ago (received 29 April 1980). He had collected his spectra, while working at FMC Corp. in Princeton NJ, on a Varian FT-80A and had written his program on a Dec PDP 11/70 with the RSX-11M operating system. He decided to exploit the modified simplex method, which is a popular and well-tested general purpose algorithm for optimizing a process. The simplex is a very simple concept in itself. If you know what a Cartesian plane is, you can instantly understand what the method is about: a moving triangle. Try this interactive Java game. When you click somewhere, you are fixing a new optimal position. The triangle will crawl and reach the minimum. Be sure to select the "modified" option!
The simplex is however an empty set of rules: you must feed it with a numerical measure of the correctness of the NMR spectrum. Siegel's first choice was the total integral of the spectrum. You know: when the phase is completely wrong, all peaks are negative and the total integral is negative. When phase is OK, the total integral is positive. Siegel was surprised to discover that the integral is maximized when NMR peaks are slightly out of phase. In such circumstance, they have a small negative dipping on one side that is more than compensated by a large positive tail on the other side. Siegel than decided to only integrate the negative part of the spectrum and to minimize it, and it worked. Eventually he found that even better results could be obtained if the integral was limited to the single most negative point in the plot. Like saying: the only thing that matters is the one little point that has gone astray. Now substitute "sheep" for "point". Where else have you already heard this concept?