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;;;"fft.scm" Fast Fourier Transform
;Copyright (C) 1999 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warrantee or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
;;;; See:
;;; Introduction to Algorithms (MIT Electrical
;;; Engineering and Computer Science Series)
;;; by Thomas H. Cormen, Charles E. Leiserson (Contributor),
;;; Ronald L. Rivest (Contributor)
;;; MIT Press; ISBN: 0-262-03141-8 (July 1990)
;;; http://www.astro.virginia.edu/~eww6n/math/DiscreteFourierTransform.html
;;; differs in the direction of rotation of the complex unit vectors.
(require 'array)
(define (fft:shuffled&scaled ara n scale)
(define lgn (integer-length (+ -1 n)))
(define new (apply make-array 0 (array-dimensions ara)))
(define bit-reverse (lambda (width in)
(if (zero? width) 0
(+ (bit-reverse (+ -1 width) (quotient in 2))
(ash (modulo in 2) (+ -1 width))))))
(if (not (eqv? n (expt 2 lgn)))
(slib:error 'fft "array length not power of 2" n))
(do ((k 0 (+ 1 k)))
((>= k n) new)
(array-set! new (* (array-ref ara k) scale) (bit-reverse lgn k))))
(define (dft! ara n dir)
(define lgn (integer-length (+ -1 n)))
(define pi2i (* 0+8i (atan 1)))
(do ((s 1 (+ 1 s)))
((> s lgn) ara)
(let* ((m (expt 2 s))
(w_m (exp (* dir (/ pi2i m))))
(m/2-1 (+ (quotient m 2) -1)))
(do ((j 0 (+ 1 j))
(w 1 (* w w_m)))
((> j m/2-1))
(do ((k j (+ m k)))
((>= k n))
(let* ((k+m/2 (+ k m/2-1 1))
(t (* w (array-ref ara k+m/2)))
(u (array-ref ara k)))
(array-set! ara (+ u t) k)
(array-set! ara (- u t) k+m/2)))))))
(define (fft ara)
(define n (car (array-dimensions ara)))
(dft! (fft:shuffled&scaled ara n 1) n 1))
(define (fft-1 ara)
(define n (car (array-dimensions ara)))
(dft! (fft:shuffled&scaled ara n (/ n)) n -1))
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