Spectral Characteristics of Convolutionally Coded Digital Signals

The power spectral density of the output symbol sequence of a convolutional encoder is computed for two different input symbol stream source models, namely, an NRZ signaling format and a first-order Markov source. In the former, the two signaling states of the binary waveform are not necessarily assumed to occur with equal probability. The effects of alternate symbol inversion on this spectrum are also considered. The mathematical results are illustrated with many examples corresponding to optimal performance codes. It is demonstrated that only for the case of a purely random input source (e.g., NRZ data with equiprobable symbols) and a particular class of codes is the output spectrum identical to the input spectrum except for a frequency scaling (expansion) by the reciprocal of the code rate. In all other cases, the output spectrum is sufficiently changed relative to the input spectrum that the commonly quoted statement "a convolutional encoder produces a bandwidth expansion by a factor equal to the reciprocal of the code rate" must be exercised with care.