summaryrefslogtreecommitdiffstats
path: root/dft.scm
blob: 29180d0fb3bba72f2cb2991789d3273cacc52c5d (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
;;;"dft.scm" Discrete Fourier Transform
;Copyright (C) 1999, 2003, 2006 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 warranty 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.

;;;; For one-dimensional power-of-two length 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)

;;; Flipped polarity of exponent to agree with
;;; http://en.wikipedia.org/wiki/Discrete_Fourier_transform

(require 'array)
(require 'logical)
(require 'subarray)

;;@code{(require 'dft)} or
;;@code{(require 'Fourier-transform)}
;;@ftindex dft, Fourier-transform
;;
;;@code{fft} and @code{fft-1} compute the Fast-Fourier-Transforms
;;(O(n*log(n))) of arrays whose dimensions are all powers of 2.
;;
;;@code{sft} and @code{sft-1} compute the Discrete-Fourier-Transforms
;;for all combinations of dimensions (O(n^2)).

(define (dft:sft1d! new ara n dir)
  (define scl (if (negative? dir) (/ 1.0 n) 1))
  (define pi2i/n (/ (* 0-8i (atan 1) dir) n))
  (do ((k (+ -1 n) (+ -1 k)))
      ((negative? k) new)
    (let ((sum 0))
      (do ((j (+ -1 n) (+ -1 j)))
	  ((negative? j) (array-set! new sum k))
	(set! sum (+ sum (* (exp (* pi2i/n j k))
			    (array-ref ara j)
			    scl)))))))

(define (dft:fft1d! new ara n dir)
  (define scl (if (negative? dir) (/ 1.0 n) 1))
  (define lgn (integer-length (+ -1 n)))
  (define pi2i (* 0-8i (atan 1) dir))
  (do ((k 0 (+ 1 k)))
      ((>= k n))
    (array-set! new (* (array-ref ara k) scl) (reverse-bit-field k 0 lgn)))
  (do ((s 1 (+ 1 s))
       (m (expt 2 1) (expt 2 (+ 1 s))))
      ((> s lgn) new)
    (let ((w_m (exp (/ 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+m/2 (+ j m/2-1 1) (+ m k m/2-1 1)))
	    ((>= k n))
	  (let ((t (* w (array-ref new k+m/2)))
		(u (array-ref new k)))
	    (array-set! new (+ u t) k)
	    (array-set! new (- u t) k+m/2)))))))

;;; Row-major order is suboptimal for Scheme.
;;; N are copied into and operated on in place
;;;  A[a, *, c] --> N1[c, a, *]
;;; N1[c, *, b] --> N2[b, c, *]
;;; N2[b, *, a] --> N3[a, b, *]

(define (dft:rotate-indexes idxs)
  (define ridxs (reverse idxs))
  (cons (car ridxs) (reverse (cdr ridxs))))

(define (dft:dft prot ara dir transform-1d)
  (define (ranker ara rdx dims)
    (define ndims (dft:rotate-indexes dims))
    (if (negative? rdx)
	ara
	(let ((new (apply make-array prot ndims))
	      (rdxlen (car (last-pair ndims))))
	  (define x1d
	    (cond (transform-1d)
		  ((eqv? rdxlen (expt 2 (integer-length (+ -1 rdxlen))))
		   dft:fft1d!)
		  (else dft:sft1d!)))
	  (define (ramap rdims inds)
	    (cond ((null? rdims)
		   (x1d (apply subarray new (dft:rotate-indexes inds))
			(apply subarray ara inds)
			rdxlen dir))
		  ((null? inds)
		   (do ((i (+ -1 (car rdims)) (+ -1 i)))
		       ((negative? i))
		     (ramap (cddr rdims)
			    (cons #f (cons i inds)))))
		  (else
		   (do ((i (+ -1 (car rdims)) (+ -1 i)))
		       ((negative? i))
		     (ramap (cdr rdims) (cons i inds))))))
	  (if (= 1 (length dims))
	      (x1d new ara rdxlen dir)
	      (ramap (reverse dims) '()))
	  (ranker new (+ -1 rdx) ndims))))
  (ranker ara (+ -1 (array-rank ara)) (array-dimensions ara)))

;;@args array prot
;;@args array
;;@var{array} is an array of positive rank.  @code{sft} returns an
;;array of type @2 (defaulting to @1) of complex numbers comprising
;;the @dfn{Discrete Fourier Transform} of @var{array}.
(define (sft ara . prot)
  (dft:dft (if (null? prot) ara (car prot)) ara 1 dft:sft1d!))

;;@args array prot
;;@args array
;;@var{array} is an array of positive rank.  @code{sft-1} returns an
;;array of type @2 (defaulting to @1) of complex numbers comprising
;;the inverse Discrete Fourier Transform of @var{array}.
(define (sft-1 ara . prot)
  (dft:dft (if (null? prot) ara (car prot)) ara -1 dft:sft1d!))

(define (dft:check-dimensions ara name)
  (for-each (lambda (n)
	      (if (not (eqv? n (expt 2 (integer-length (+ -1 n)))))
		  (slib:error name "array length not power of 2" n)))
	    (array-dimensions ara)))

;;@args array prot
;;@args array
;;@var{array} is an array of positive rank whose dimensions are all
;;powers of 2.  @code{fft} returns an array of type @2 (defaulting to
;;@1) of complex numbers comprising the Discrete Fourier Transform of
;;@var{array}.
(define (fft ara . prot)
  (dft:check-dimensions ara 'fft)
  (dft:dft (if (null? prot) ara (car prot)) ara 1 dft:fft1d!))

;;@args array prot
;;@args array
;;@var{array} is an array of positive rank whose dimensions are all
;;powers of 2.  @code{fft-1} returns an array of type @2 (defaulting
;;to @1) of complex numbers comprising the inverse Discrete Fourier
;;Transform of @var{array}.
(define (fft-1 ara . prot)
  (dft:check-dimensions ara 'fft-1)
  (dft:dft (if (null? prot) ara (car prot)) ara -1 dft:fft1d!))

;;@code{dft} and @code{dft-1} compute the discrete Fourier transforms
;;using the best method for decimating each dimension.

;;@args array prot
;;@args array
;;@0 returns an array of type @2 (defaulting to @1) of complex
;;numbers comprising the Discrete Fourier Transform of @var{array}.
(define (dft ara . prot)
  (dft:dft (if (null? prot) ara (car prot)) ara 1 #f))

;;@args array prot
;;@args array
;;@0 returns an array of type @2 (defaulting to @1) of
;;complex numbers comprising the inverse Discrete Fourier Transform of
;;@var{array}.
(define (dft-1 ara . prot)
  (dft:dft (if (null? prot) ara (car prot)) ara -1 #f))

;;@noindent
;;@code{(fft-1 (fft @var{array}))} will return an array of values close to
;;@var{array}.
;;
;;@example
;;(fft '#(1 0+i -1 0-i 1 0+i -1 0-i)) @result{}
;;
;;#(0.0 0.0 0.0+628.0783185208527e-18i 0.0
;;  0.0 0.0 8.0-628.0783185208527e-18i 0.0)
;;
;;(fft-1 '#(0 0 0 0 0 0 8 0)) @result{}
;;
;;#(1.0 -61.23031769111886e-18+1.0i -1.0 61.23031769111886e-18-1.0i
;;  1.0 -61.23031769111886e-18+1.0i -1.0 61.23031769111886e-18-1.0i)
;;@end example