General Index Maps n K 1 n 1 K 2 n 2 N n K 3 k 1 K 4 k 2 N n 1 0 1 … N 1 1 k 1 0 1 … N 1 1 n 2 0 1 … N 2 1 k 2 0 1 … N 2 1 The basic ideas is to simply reorder the DFT computation to expose the redundancies in the DFT, and exploit these to reduce computation! Three conditions must be satisfied to make this map serve our purposes Each map must be one-to-one from 0 to N 1 , because we want to do the same computation, just in a different order. The map must be cleverly chosen so that computation is reduced The map should be chosen to make the short-length transforms be DFTs. (Not essential, since fast algorithms for short-length DFT-like computations could be developed, but it makes our work easier.)

Conditions for one-to-oneness of general index map

Case I N 1 , N 2 relatively prime (greatest common denominator 1 ) i.e. N 1 N 2 1 K 1 a N 2 and/or K 2 b N 1 and K 1 N 1 1 , K 2 N 2 1

Case II N 1 , N 2 not relatively prime: N 1 N 2 1 K 1 a N 2 and K 2 b N 1 and a N 1 1 , K 2 N 2 1 or K 1 a N 2 and K 2 b N 1 and K 1 N 1 1 , b N 2 1 where K 1 , K 2 , K 3 , K 4 , N 1 , N 2 , a , b integers

Requires number-theory/abstract-algebra concepts. Reference: C.S. Burrus Conditions of one-to-oneness must apply to both k and n

Conditions for arithmetic savings X k 1 k 2 n 1 N 1 1 0 n 2 N 2 1 0 x n 1 n 2 W N ( K 1 n 1 + K 2 n 2 ) ( K 3 k 1 + K 4 k 2 n 1 N 1 1 0 n 2 N 2 1 0 x n 1 n 2 W N K 1 K 3 n 1 k 1 W N K 1 K 4 n 1 k 2 W N K 2 K 3 n 2 k 1 W N K 2 K 4 n 2 k 2 K 1 K 4 N 0 exclusive or K 2 K 3 N 0 ⇒ Common Factor Algorithm (CFA). Then X k DFT N i twiddle factors DFT N j x n 1 n 2 K 1 K 4 N and K 2 K 3 N 0 ⇒ Prime Factor Algorithm (PFA). X k DFT N i DFT N j No twiddle factors! A PFA exists only and always for relatively prime N 1 , N 2

Conditions for short-length transforms to be DFTs K 1 K 3 N N 2 and K 2 K 4 N N 1 Convenient choice giving a PFA K 1 N 2 , K 2 N 1 , K 3 N 2 N 2 1 N 1 N 1 , K 4 N 1 N 1 1 N 2 N 2 where N 1 1 N 2 is an integer such that N 1 N 1 1 1

N 1 3 , N 2 5 N 15 n 5 n 1 3 n 2 15 k 10 k 1 6 k 2 15 Checking Conditions for one-to-oneness 5 K 1 a N 2 5 a 3 K 2 b N 1 3 b 5 3 1 3 5 1 10 K 3 a N 2 5 a 6 K 4 b N 1 3 b 10 3 1 6 5 1 Checking conditions for reduced computation K 1 K 4 15 5 6 15 0 K 2 K 3 15 3 10 15 0 Checking Conditions for making the short-length transforms be DFTS K 1 K 3 15 5 10 15 5 N 2 K 2 K 4 15 3 6 15 3 N 1 Therefore, this is a prime factor map.

2-D map n 5 n 1 3 n 2 15 and k 10 k 1 6 k 2 15 Operation Counts N 2 length- N 1 DFTs N 1 length- N 2 DFTs N 2 N 1 2 N 1 N 2 2 N N 1 N 2 complex multiplies Suppose N N 1 N 2 N 3 … N M N N 1 N 2 … N M Complex multiplies radix-2, radix-4 eliminate all multiplies in short-length DFTs, but have twiddle factors: PFA eliminates all twiddle factors, but ends up with multiplies in short-length DFTs. Surprisingly, total operation counts end up being very similar for similar lengths.