We describe a technique termed multichannel Fourier fringe analysis and its application to the problem of automatic phase unwrapping in the presence of surface discontinuities. The technique is especially useful for the analysis of fringe projection contour maps in order to measure surface height distributions. Use is made of multiple fringe patterns that are separated in the frequency space of the Fourier transform by means of a set of bandpass filters. We also describe the design of a special fiber-optic interferometer with features particularly important in the case of this technique: easily adjustable fringe spacing and rotation. Fringe production by the interferometer is analyzed, and the relationship between the fringe phase and the height distribution of an illuminated surface is derived. A method for measuring phase in the case of multichannel Fourier fringe analysis is presented. The application of the technique to automatic phase unwrapping is shown. An example of the technique in operation is given, and a discussion of implementation of the technique is included.