where we take all of the vectors to be periodic in "N" ("XN" = "X"0, et cetera). Thus, just as the DFT transforms a convolution into a pointwise multiplication of complex numbers ("pairs" of real and imaginary parts), the DHT transforms a convolution into a simple combination of "pairs" of real frequency components. The inverse DHT then yields the desired vector z. In this way, a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. (This is slightly more expensive than the corresponding procedure for the DFT, not including the costs of the transforms below, because the pairwise operation above requires 8 real-arithmetic operations compared to the 6 of a complex multiplication. This count doesn't include the division by 2, which can be absorbed e.g. into the 1/"N" normalization of the inverse DHT.)