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A new method of speech coding has been developed for efficient analysis and synthesis of speech signals in a transform domain. This scheme utilises the non-uniform variance distribution of speech spectrum together with the correlation between the coefficients. In this method, the base-band coefficients of speech spectrum are directly transmitted whilst the remaining coefficients are linearly predicted from the base-band by a few prediction coefficients. The developed linear prediction algorithm, referred to as Non-Uniform Linear Prediction (NULP), can simultaneously take advantage of correlation between the coefficients and non-uniform variance distribution. The NULP method has been employed in the Walsh-Hadamard (WH) and Discrete Cosine (DC) domain. Highly intelligible speech has been reproduced at 6.5 and 4 kbits/sec, from a 16 point DCT and 64 point WHT, respectively. The main advantage of this system is its low complexity which is in the range of relatively simple waveform coders.