Abstract: A speech coder is operable to compress digital data representing speech using a Waveform Interpolation speech coding method. The coding method is carried out on the residual signal from a Linear Predicative Coding stage. On the basis of a series of overlapping frames of the residual signal, a series of respective spectra are found. The evolution of the spectra is filtered in a multi-stage filtering process, the filtered phase data being replaced with the original phase data at the end of each stage. This is found to result in the decoder being better able to approximate the original speech signal. This is of particular utility in relation to mobile telephony.