Upper Bound on the Efficiency of dc-Constrained Codes

In digital transmission systems, the transmission channel often does not pass dc. This causes the well-known problem of baseline wander. One way to overcome this difficulty is to restrict the dc content in the signal stream using suitably devised codes.1-3 As a result many codes having a dc-constrained property have been studied.4-0 The coding requirement is represented by the constraint put upon the running digital sum (RDS) of the coded signal stream. We expect that the efficiency of a dc-constrained code is related to the limits of RDS in some definite way. This is the subject to which we address ourselves in this paper. More specifically, we intend to answer the question: What is the best possible efficiency of any dc-constrained code satisfying a given limit on R D S ? Let {a t , a2 , · · · J be the sequence of the transmitted symbols, the R D S of the signal stream at instant k is defined to be the sum a, . Talcing the RDS at any instant as the state of the signal stream at that point, the limits on RDS define a set of allowable states, and each additional signal symbol may be considered as a transition from one state to another. This transition can be represented by a matrixcalled naturally the transition matrix. For a K-ary signal alphabet, the best possible efficiency t] of dc-constrained codes is found to be