Systematic Bits are Better and No Buts About It

Abstract

An explanation is given for the paradoxical fact that, at low signal-to-noise ratios, the systematic feedback encoder results in fewer decoding bit errors than does a nonsystematic feedforward encoder for the same convolutional code, which is the opposite of the situation at high signal-to-noise ratios. The analysis identifies a new convolutional code property, the weight density of distance-d detours for large d. For a given distance-d weight density, the decoding bit error probability depends on the number of taps in the realization of the encoder inverse. Among all encoders for a given convolutional code, the systematic one has the inverse with fewest taps and, hence, gives the smallest bit error probability at low signal-to-noise ratios where decoding error bursts typically result from the decoder following a detour at large distance d from the transmitted codeword.