New Area-Time Lower Bounds for the Multidimensional DFT

Area-time lower bounds are derived for the VLSI computation of the (n1×n2× . . .×nd)-point multidimensional DFT (MDDFT) over different types of rings and different types of input/output protocols. First, an AT 2 = Ω ( (N log |R|) ) bound is obtained for any finite ring R, where N = ∏d k=1 nk, for any word-local protocol. The bound was previously known for the special case when R = ZM , the ring of integers modulo M . Second, an AT 2 = Ω ( (Nb) ) word-local bound is derived when R is the complex field, the components of the input are fixed-point numbers of b bits, and the precision of the output components guarantees that the resulting approximate transform is injective. No area-time lower bound was previously known for the DFT over the complex field. Third, an AT 2 = Ω ( (N log |R|) ) bound is derived when R = GF (p) is a finite field of polynomials of degree m with coefficients in Zp, for certain classes of non word-local protocols. This is the first area-time lower bound derived for the DFT with I/O protocols that are not word-local.