Efficient computation of the DFT with only a subset of input or output points

Ways of efficiently computing the discrete Fourier transform (DFT) when the number of input and output data points differ are discussed. The two problems of determining whether the length of the input sequence or the length of the output sequence is reduced can be found to be duals of each other, and the same methods can, to a large extent, be used to solve both. The algorithms utilize the redundancy in the input or output to reduce the number of operations below those of the fast Fourier transform (FFT) algorithms. The usual pruning method is discussed, and an efficient algorithm, called transform decomposition, is introduced. It is based on a mixture of a standard FFT algorithm and the Horner polynomial evaluation scheme equivalent to the one in Goertzel's algorithms. It requires fewer operations and is more flexible than pruning. The algorithm works for power-of-two and prime-factor algorithms, as well as for real-input data.>