furnace/extern/fftw/genfft/conv.ml

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(*
* Copyright (c) 1997-1999 Massachusetts Institute of Technology
* Copyright (c) 2003, 2007-14 Matteo Frigo
* Copyright (c) 2003, 2007-14 Massachusetts Institute of Technology
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*
*)
open Complex
open Util
let polyphase m a ph i = a (m * i + ph)
let rec divmod n i =
if (i < 0) then
let (a, b) = divmod n (i + n)
in (a - 1, b)
else (i / n, i mod n)
let unpolyphase m a i = let (x, y) = divmod m i in a y x
let lift2 f a b i = f (a i) (b i)
(* convolution of signals A and B *)
let rec conv na a nb b =
let rec naive na a nb b i =
sigma 0 na (fun j -> (a j) @* (b (i - j)))
and recur na a nb b =
if (na <= 1 || nb <= 1) then
naive na a nb b
else
let p = polyphase 2 in
let ee = conv (na - na / 2) (p a 0) (nb - nb / 2) (p b 0)
and eo = conv (na - na / 2) (p a 0) (nb / 2) (p b 1)
and oe = conv (na / 2) (p a 1) (nb - nb / 2) (p b 0)
and oo = conv (na / 2) (p a 1) (nb / 2) (p b 1) in
unpolyphase 2 (function
0 -> fun i -> (ee i) @+ (oo (i - 1))
| 1 -> fun i -> (eo i) @+ (oe i)
| _ -> failwith "recur")
(* Karatsuba variant 1: (a+bx)(c+dx) = (ac+bdxx)+((a+b)(c+d)-ac-bd)x *)
and karatsuba1 na a nb b =
let p = polyphase 2 in
let ae = p a 0 and nae = na - na / 2
and ao = p a 1 and nao = na / 2
and be = p b 0 and nbe = nb - nb / 2
and bo = p b 1 and nbo = nb / 2 in
let ae = infinite nae ae and ao = infinite nao ao
and be = infinite nbe be and bo = infinite nbo bo in
let aeo = lift2 (@+) ae ao and naeo = nae
and beo = lift2 (@+) be bo and nbeo = nbe in
let ee = conv nae ae nbe be
and oo = conv nao ao nbo bo
and eoeo = conv naeo aeo nbeo beo in
let q = function
0 -> fun i -> (ee i) @+ (oo (i - 1))
| 1 -> fun i -> (eoeo i) @- ((ee i) @+ (oo i))
| _ -> failwith "karatsuba1" in
unpolyphase 2 q
(* Karatsuba variant 2:
(a+bx)(c+dx) = ((a+b)c-b(c-dxx))+x((a+b)c-a(c-d)) *)
and karatsuba2 na a nb b =
let p = polyphase 2 in
let ae = p a 0 and nae = na - na / 2
and ao = p a 1 and nao = na / 2
and be = p b 0 and nbe = nb - nb / 2
and bo = p b 1 and nbo = nb / 2 in
let ae = infinite nae ae and ao = infinite nao ao
and be = infinite nbe be and bo = infinite nbo bo in
let c1 = conv nae (lift2 (@+) ae ao) nbe be
and c2 = conv nao ao (nbo + 1) (fun i -> be i @- bo (i - 1))
and c3 = conv nae ae nbe (lift2 (@-) be bo) in
let q = function
0 -> lift2 (@-) c1 c2
| 1 -> lift2 (@-) c1 c3
| _ -> failwith "karatsuba2" in
unpolyphase 2 q
and karatsuba na a nb b =
let m = na + nb - 1 in
if (m < !Magic.karatsuba_min) then
recur na a nb b
else
match !Magic.karatsuba_variant with
1 -> karatsuba1 na a nb b
| 2 -> karatsuba2 na a nb b
| _ -> failwith "unknown karatsuba variant"
and via_circular na a nb b =
let m = na + nb - 1 in
if (m < !Magic.circular_min) then
karatsuba na a nb b
else
let rec find_min n = if n >= m then n else find_min (2 * n) in
circular (find_min 1) a b
in
let a = infinite na a and b = infinite nb b in
let res = array (na + nb - 1) (via_circular na a nb b) in
infinite (na + nb - 1) res
and circular n a b =
let via_dft n a b =
let fa = Fft.dft (-1) n a
and fb = Fft.dft (-1) n b
and scale = inverse_int n in
let fab i = ((fa i) @* (fb i)) @* scale in
Fft.dft 1 n fab
in via_dft n a b