Linear-Real Codes and Coders

Many communications channels, including telephone channels, contain noise which comes in short bursts, such as noise from impulses. Such noise is particularly deleterious when the channel is used for the transmission of digital data. * Part of the research for this article was performed at Carnegie Institute of Technology under National Science Foundation grants GP-39 and GK-373. Some of the material contained in this paper is taken from the author's convention article,0 IO6.7 1066 T H E BELL SYSTEM TECHNICAL JOURNAL, J U L Y - A U G U S T 19(58 At least as early as 1958 it was discovered that it is sometimes possible to reduce digital errors in such channels without reducing the noise power by using a scheme such as Fig. 1 shows. In some formulations 1-7 the transformation A consisted of a continuous all-pass filter whose Fourier transform magnitude was unity at all frequencies but whose phase characteristic varied with frequency; the inverse linear transformation was the continuous all-pass filter with the conjugate phase characteristic. The linear filter was called the smear operation, and its inverse the desmear operation. Later papers considered linear transformations to be real-number matrices operating upon the data in blocks.8-10 In all schemes to which Fig. 1 applies, a single impulse of noise into the inverse linear filter will be transformed into an output noise which is dispersed in time. With proper design, this dispersed noise will be small enough at all times to not produce errors at the output of the quantizer.