The Design and Implementation of Unequal Error-Correcting Coding Systems

The theoretical basis for using the mean-square error criterion in channel coding systems has been established previously. Such a criterion leads to an unequal error-correcting system. This paper describes the design methods for translating that theory into practice. It also presents some new results concerning the performance of these types of systems. The relationship of the mean-square system to the one based on the minimum probability of error is examined. Except for very noisy channels the two decoders produce identical outputs for all correctable error patterns. However, the mean-square error decoder guarantees various levels of correcting capabilities for the more significant positions during overload conditions. The basic design technique relies upon generalized transforms that are closely affiliated with the Walsh functions. The fundamental parameters that are the key to the system design may be computed using a fast algorithm which is developed over a subspace in the generalized frequency domain. The decoder performs error correction with a code containing
k
information symbols by numerically combining the output from (
k+ 1
) finite impulse response (FIR) type of filters. The filter weights are directly related to the transforms of the system's transition probabilities. Simplification methods and precision requirements in the filter mechnizations are discussed.