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A technique for nonlinear prediction of speech via local linear prediction (LLP) is presented and applied to LD-CELP at 16 kbps. With 18th-order backward adaptive LLP for voiced frames, the hybrid LD-CELP coder gives higher segmental signal-to-noise ratio (SNR) compared to a reference version of the ITU-T G.728 LD-CELP algorithm, which has a 50th-order backward adaptive linear predictor. The computational complexity for LLP analysis is significantly less than that of a conventional one-step recursive LLP, and the LLP method gives better prediction gain and a remarkably "whiter" residual compared to backward adaptive linear predictor. With an appropriate state space neighborhood for local linear analysis, the short-delay predictor is also able to effectively model long-term correlations without requiring pitch estimation.