On Information-Lossless Discrete-Time Systems

The concept of lossless state or system is generalized by the definition of k-losslessness. If k>N( N-1)/2 for an N-state machine, k-lossless implies lossless. The series connection of a set of discrete time systems {Ai}, where Ai is ki-lossless, results in a system which is min{ki}-lossless. This result is a simple verification of the intuitive notion that a noiseless communication channel is only as lossless as the most lossy system in the channel. By means of input- output pair analysis, systems with the decomposition property (one of the necessary but not sufficient conditions for linearity) are shown to be information-lossless of finite order m, when one state has finite order m. Further, every state must have exactly the same order. For an N-state system with decomposition and zero-state linearity, information-lossless implies information-lossless of finite order m≤N. As a consequence, all information-lossless linear sequential machines used as encoders allow decoding to begin after m received symbols, provided that the encoder´s starting state is known in advance.