A real-time system to perform Fourier spectroscopy is described. The spectral distribution is synthesized by the successive summation of cosine functions with amplitudes and frequencies which are determined by the sample points of the interferogram. Summation of these cosine functions is performed in a digital memory oscilloscope which has 1024 addresses. The design criteria and the principles of operation of the system are discussed in detail. Sources of error are examined and several methods of error reduction are discussed. The limitations of this type of system are described and improvements are recommended. Experimental data are presented and the performance of the real-time system is compared to that of a special purpose analog computer, and a digital computer. (Author).