Fast Fourier Transforms

29 O(b log(b)) operations (using standard multiplication). As there are O(b= log(b)) primes in total, the running time of this stage of the algorithm is O(b 2 L), even using the \grammar school" method of integer multiplication. At this stage of the algorithm we have obtained a vector of length L whose entries are integral linear combinations of powers of with coeecients bounded by M in absolute value. For each of these entries we multiply precomputed b-bit approximations of the roots of unity with the log(M)-bit coeecients. Since log(M) = O(b), the total running time is O(LM(b)) for the nal conversion. All in all, the overall running time of the algorithm is of order L log(L)M(log(b))b= log(b)+LM(b)+ b 2 L which is L log(L)M(log(b))b= log(b) + b 2 L as M(b) = O(b 2). This proves Theorem 1.2. Remark 12.1. The most realistic value for M(a) with reasonable values of the parameters is M(a) = a 2 ; in theory, one might replace this with the asymptotically fast value M(a) = O(a log(a) log log(a)) 22]. It is also conceivable that special-purpose chips would be constructed, so that M(a) would represent a more general notion of complexity, such as chip area, running time, or dollars. We should note that in practice the original values of the v j might have been quantized from values known to greater than b-bit precision. Random quantization error of this kind tends to produce less problems in FFT algorithms than computational noise ((23]); a standard stochastic analysis says that on average this sort of error contaminates p ` bits of the result 12]. Since this applies equally to the usual xed-point model and to our model (and even to oating point models), and since it gives lower order terms in any complexity analysis, we have ignored this kind of quantization error throughout. References 1] R.C. Agarwal and C.S. Burrus. Fast convolution using Fermat number transforms with applications to digital ltering. 28 Finally, we convert the elements of Z ] to complex numbers by replacing and its powers by b-bit complex approximations. This nishes the description of the algorithm. In order to assess the asymptotic complexity of the procedure, we need to make assumptions about the various parameters. For the sake of clarity and simplicity, we choose to think of the approximation-degree n as xed, with the precision b and the length of the Fourier transform …