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An adaptive predictive coding with adaptive bit allocation (APC-AB) is presented for speech encoding at low to medium bit rates (6.4kb/s-24kb/s). In this system, a split-band predictive coding scheme and a bit allocation scheme are employed in order to remove the redundancies due to a periodic concentration of the prediction residual energy as well as nonuniform nature of the speech spectrum. Quantization bits are dynamically allocated both over the sub-bands(frequency domain) and over the subintervals(time domain) in accordance with the distribution of the residual energies in the time-frequency domain. Optimum bit allocation is derived based on the mean square error criterion on the speech waveform, and the SNR gain is presented in relation to the prediction gain of the full-band signal. This system is evaluated in terms of the segmental SNR and speech quality. The result shows that the APC-AB system has advantage over the conventional full-band APC system in the segmental SNR and the stability of the prediction loop. It was also shown that this system can provide speech quality subjectively equivalent to 7 bit Log-PCM at 16 kb/s, and to 6 bit Log-PCM at 9.6 kb/s.