Prime Numbers

9.5 Large-integer multiplication 483
formulae to recover the correct DFT of the original signal x. An example of
this pure-real signal approach for number-theoretical transforms as applied to
cyclic convolution is embodied in Algorithm 9.5.22, with split-radix symbolic
pseudocode given in [Crandall 1997b] (see Exercise 9.51 for discussion of the
negacyclic scenario).
Incidentally, there are yet lower-complexity FFTs, called split-radix FFTs,
which employ an identity more complicated than the Danielson–Lanczos
formula. And there is even a split-radix, pure-real-signal FFT due to Sorenson
that is quite efficient and in wide use [Crandall 1994b]. The vast “FFT forest”
is replete with specially optimized FFTs, and whole texts have been written
in regard to the structure of FFT algorithms; see, for example, [Van Loan
1992].
Even at the close of the 20th century there continue to be, every year, a
great many published papers on new FFT optimizations. Because our present
theme is the implementation of FFTs for large-integer arithmetic, we close this
section with one more algorithm: a “parallel,” or “piecemeal,” FFT algorithm
that is quite useful in at least two practical settings. First, when signal data
are particularly numerous, the FFT must be performed in limited memory. In
practical terms, a signal might reside on disk memory, and exceed a machine’s
physical random-access memory. The idea is to “spool” pieces of the signal
off the larger memory, process them, and combine in just the right way to
deposit a final FFT in storage. Because computations occur in large part on
separate pieces of the transform, the algorithm can also be used in a parallel
setting, with each separate processor handling a respective piece of the FFT.
The algorithm following has been studied by various investigators [Agarwal
and Cooley 1986], [Swarztrauber 1987], [Ashworth and Lyne 1988], [Bailey
1990], especially with respect to practical memory usage. It is curious that
the essential ideas seem to have originated with [Gentleman and Sande 1966].
Perhaps, in view of the extreme density and proliferation of FFT research,
one might forgive investigators for overlooking these origins for two decades.
The parallel-FFT algorithm stems from the observation that a length-
(D = WH) DFT can be performed by tracking over rows and columns of an
H × W (height times width) matrix. Everything follows from the following
algebraic reduction of the DFT X of x:
⎛
D−1
X = DFT(x) = ⎝ xjg −jk
⎞
⎠
=
=
W −1
H−1
J=0 M=0
W −1
J=0
H−1
M=0
j=0
D−1
k=0
xJ+MWg −(J+MW)(K+NH)
xJ+MWg −MK
H
g −JK g −JN
W
D−1
K+NH=0
D−1
K+NH=0
,