Fast Computation of Partial DFT for Comb Spectrum Evaluation

A comb spectrum evaluation problem arises in the channel estimation of pilot symbol-assisted modulation (PSAM) systems and also in the (de)modulation of multichannel communication system based on orthogonal frequency division multiplexing (OFDM). In this letter, we decompose an M-component comb spectrum evaluation with transform length N into 1) a fast Fourier transform evaluation of transform length M of a length M vector, 2) which in turn is obtained from the lengthN input vector (by folding and scaling). This decomposition allows for the application of split-radix and radix-4 algorithms for evaluation of M length transform and efficient computation of the length M vector from the length N input vector, which is classically termed as "pruning." When N=2^p, the comb spectrum computation requires only N/8+M/3log₂ M nontrivial complex multiplications, if N/Mges8. In addition, conditions under which comb spectrum evaluation can be done in O(MlogM) complex multiplies are identified