Windowed r-point update algorithm for Discrete Fourier Transform

In the real-time signal processing of the infinite data sequence, a portion of the sequence of length N is sampled by windowing and Discrete Fourier Transform (DFT) of that portion is computed. After computation of DFT, new input data points are shifted into the window and old data points are spilled out from the window. After shifting a new DFT has to be calculated for updated data sequence. The algorithm developed in this research is capable of updating the previously calculated DFT after shifting the data sequence over the window and reflects DFT of the updated data sequence, using less computation than directly evaluating the new transform. Previously developed algorithms were limited to a single point step between successive windows. In this research this result is extended to handle larger step sizes, i.e. the algorithm is capable to update a DFT sequence after shifting r additional data points into the window and spilling out of r old points from the window, where 1 ≤ r ≤ log₂N. This Algorithm reduces the computational order by a factor of (1/ r) log₂ N. The algorithm is developed for the data sequence windowed by Rectangular window and Bartlett window.