A spectral shaping PCM modem communication system uses a symbol encoder to encode a data bit stream into a PCM symbol stream. A plurality of different frames are formed from the PCM symbol stream. Performance metrics are computed for each possible frame and its inversion so a decision can be made in order to select a most desirable frame and inversion. Delays are injected in the data flow to allow a look-ahead and thus provide for better frame selection and inversion decisions. A channel multiplexer combines decision bits and output frames to form encoded output frames that are fed to a digital channel connected to a telephone network wherein they are converted to analog signal. A corresponding decoder converts analog signals from the telephone network to digital signals. A timing recovery and equalization means corrects sample timing and removes inter-symbol interference effects introduced by telephone networks. The output of the equalizer is sampled at correct timing to recover the received samples and received frame. A channel de-multiplexer separates the frame selection and frame inversion decision bits from the symbol frames received. Inversion and framing decisions are discerned from the demultiplexed frame selection and frame inversion decision bits. Original PCM symbol frames are reconstructed and PCM symbols are decoded into a data bit stream. A non-negative shell mapping may be used as the symbol encoder. A decoder checks to see if a currently received frame is negative. If it is, then all samples in such current frame are inverted. Otherwise, all samples in the current frame are unchanged. The resulting frames are then forwarded to a non-negative shell mapping decoder for eventual reconversion to digital data.