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The theory for a new class of Analysis by Synthesis LPC coding algorithms, called Backward Excitation Recovery (BER) LPC, is presented in this paper for the general case where an ARMA type filter is employed in the speech synthesis process. These coders differ from conventional Analysis by Synthesis LPC systems in that their excitation signal is defined from past information which is available at both the transmitter and the receiver. As a result only the filter parameters are required at the receiver for the recovery of the speech signal. A plethora of sequential filter optimization algorithms are proposed for both Single-input and Multi-input Synthesis filters. Two excitation adaptation algorithms are also proposed. The excitation signal is updated by using either `Past Decoded SpeechÂ¿ (PDS) samples or `Past Excitation SamplesÂ¿ (PES). The PES sequence is the signal which drives the all-pole part of the ARMA synthesis filter. It is shown that a specific BER-LPC codec, which employs one of the proposed PES algorithms, is equivalent to the recently proposed 4Â·8 kbit/s Self-Excited LPC system (SE-LPC). Computer simulation results highlight the SNR performance of BER-LPC codecs employing the proposed filter optimization and excitation adaptation algorithms, and demonstrate the superiority of several BER-LPC codecs over the SE-LPC system. Furthermore, it is shown that BER-LPC codecs which use the PDS excitation adaptation approach can operate successfully with a very low encoding delay. This property is valuable to low bit rate speech codecs operating in applications such as digital mobile radio.