Windowed discrete cosine and sine transforms for shifting data

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When processing a signal or an image using the discrete cosine transform (DCT) or discrete sine transform (DST), a typical approach is to extract a portion of the signal by windowing and then form the DCT or DST of the window contents. By shifting the window point by point over the signal, the entire signal may be processed. Previously, algorithms have been developed to “update” the DCT and DST simultaneously to reflect the modified window contents using less computation than directly evaluating the modified transform via standard fast transform algorithms. These algorithms are limited to a single point step between successive windows. In this paper, these results are extended to handle larger step sizes, i.e. an algorithm is developed to simultaneously update the DCT and DST to reflect inclusion of r, where 1⩽ r⩽ N−1, additional data points and removal of r old points from the signal. Algorithms are derived for use with rectangular, split triangular (trapezoidal), Hanning, Hamming and Blackman windows, and implementations in the C language are given. Examples of applications where this algorithm would be useful include data communication where time constraints may not permit the immediate processing of every incoming data point, adaptive system identification, real-time analysis of financial market data, etc.