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This paper presents a rate-distortion derived transform trellis coding (TTC) scheme with applications to Gaussian AR sources and speech data. The optimal encoder consists of a Karhunen-Loeve transform (KLT) on the source output, followed by a search on a trellis structured random code, where the decoder is a time-variant nonlinear filter. The scheme is implementable and applicable to stationary Gaussian sources with a bounded and continuous power spectrum and the squared error distortion measure. The code construction is based on the power or eigenvalue spectrum of the source with no restriction on the coding rate. The TTC scheme is first applied to encode a Gaussian AR source often used to model speech. Simulations were conducted at several rates, using an optimal KLT and the suboptimal discrete cosine transform (DCT). Results demonstrate that the DCT performs as well as the KLT, and both yield average distortions very close to the distortion-rate function. For speech data, an adaptive version of the DCT TTC scheme is applied to encode two speech sentences at several coding rates. The adaptation is controlled by an estimate of the short-term eigenvalue spectrum which is transmitted as side information to the receiver. The proposed scheme is a very efficient speech waveform coder that provides reconstructed speech with very high signal-to-noise ratio values and very good perceptual quality at low bit rates.