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This paper discusses speech coding systems based upon transform coding (TC). It compares several transforms and shows that the cosine transform leads to a nearly optimum performance for almost all speech sounds. Various adaptive coding strategies are then investigated, and a coding scheme is proposed that is based on a nonadaptive discrete cosine transform (DCT), on an adaptive bit assignment, and on adaptive quantization. The adaptation is controlled by a short-term basis spectrum that is derived from the transform coefficients prior to coding and transmission and that is transmitted as side information to the receiver. The main result is that this adaptive transform coder performs better than all known nonpitch-tracking coding schemes; it extends the range of speech waveform coding to lower bit rates and closes the gap between vocoders and predictive waveform coders.