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In this paper an orthogonal set of basis functions is constructed by the use of a simplified Gramm Schmidt orthogonalization process, using non-orthogonal triangular waveforms. The orthogonalized functions consist of a linear combination of, at most, four triangular waveforms. A unique sequency is shown to exist. A complex form of the triangular basis functions is defined and used to develop an efficient discrete transform: matrix which contains many null or trivial elements. A fast triangular transform is developed which allows computation speeds that are comparable to those for computing Fast Fourier Transforms. The application of the triangular transforms to signal processing is discussed and applications to specific types of signals is briefly described.