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 paper we decompose a M-component comb spectrum evaluation with transform length N into i) a FFT evaluation of transform length M of a length M vector, ii) which in turn is obtained from the length N 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 = 2p, the comb spectrum computation requires only N/8+M/3log2M nontrivial complex multiplications, if N/M - 8. In addition, conditions under which comb spectrum evaluation can be done in O(M log M) complex multiplies are identified.