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The convolution of two functions ${\mathit {F}}$ and ${\mathit {H}}$ of an integer variable is defined as the function ${\mathit {G}}$ satisfying

G(n)=\sum _{m=-\infty }^{\infty }F(m)H(n-m)

for all integers ${\mathit {n}}$ . Assume $F(n)$ can be non-zero only for $0$ ≤ ${\mathit {n}}$ ≤ $|{\mathit {F}}|$ , where $|{\mathit {F}}|$ is the "length" of ${\mathit {F}}$ , and similarly for ${\mathit {G}}$ and ${\mathit {H}}$ , so that the functions can be modeled as finite sequences by identifying $f_{0},f_{1},f_{2},\dots$ with $F(0),F(1),F(2),\dots$ , etc. Then for example, values of $|{\mathit {F}}|=6$ and $|{\mathit {H}}|=5$ would determine the following value of ${\mathit {g}}$ by definition.

{\begin{array}{lllllllllll}g_{0}&=&f_{0}h_{0}\\g_{1}&=&f_{1}h_{0}&+&f_{0}h_{1}\\g_{2}&=&f_{2}h_{0}&+&f_{1}h_{1}&+&f_{0}h_{2}\\g_{3}&=&f_{3}h_{0}&+&f_{2}h_{1}&+&f_{1}h_{2}&+&f_{0}h_{3}\\g_{4}&=&f_{4}h_{0}&+&f_{3}h_{1}&+&f_{2}h_{2}&+&f_{1}h_{3}&+&f_{0}h_{4}\\g_{5}&=&f_{5}h_{0}&+&f_{4}h_{1}&+&f_{3}h_{2}&+&f_{2}h_{3}&+&f_{1}h_{4}\\g_{6}&=&&&f_{5}h_{1}&+&f_{4}h_{2}&+&f_{3}h_{3}&+&f_{2}h_{4}\\g_{7}&=&&&&&f_{5}h_{2}&+&f_{4}h_{3}&+&f_{3}h_{4}\\g_{8}&=&&&&&&&f_{5}h_{3}&+&f_{4}h_{4}\\g_{9}&=&&&&&&&&&f_{5}h_{4}\end{array}}

We can write this in matrix form as:

\left({\begin{array}{l}g_{0}\\g_{1}\\g_{2}\\g_{3}\\g_{4}\\g_{5}\\g_{6}\\g_{7}\\g_{8}\\g_{9}\\\end{array}}\right)=\left({\begin{array}{lllll}f_{0}\\f_{1}&f_{0}\\f_{2}&f_{1}&f_{0}\\f_{3}&f_{2}&f_{1}&f_{0}\\f_{4}&f_{3}&f_{2}&f_{1}&f_{0}\\f_{5}&f_{4}&f_{3}&f_{2}&f_{1}\\&f_{5}&f_{4}&f_{3}&f_{2}\\&&f_{5}&f_{4}&f_{3}\\&&&f_{5}&f_{4}\\&&&&f_{5}\end{array}}\right)\;\left({\begin{array}{l}h_{0}\\h_{1}\\h_{2}\\h_{3}\\h_{4}\\\end{array}}\right)

or

g=A\;h

For this task, implement a function (or method, procedure, subroutine, etc.) deconv to perform deconvolution (i.e., the inverse of convolution) by constructing and solving such a system of equations represented by the above matrix $A$ for ${\mathit {h}}$ given ${\mathit {f}}$ and ${\mathit {g}}$ .

The function should work for ${\mathit {G}}$ of arbitrary length (i.e., not hard coded or constant) and ${\mathit {F}}$ of any length up to that of ${\mathit {G}}$ . Note that $|{\mathit {H}}|$ will be given by $|{\mathit {G}}|-|{\mathit {F}}|+1$ .
There may be more equations than unknowns. If convenient, use a function from a library that finds the best fitting solution to an overdetermined system of linear equations (as in the Multiple regression task). Otherwise, prune the set of equations as needed and solve as in the Reduced row echelon form task.
Test your solution on the following data. Be sure to verify both that deconv $(g,f)=h$ and deconv $(g,h)=f$ and display the results in a human readable form.

h = [-8,-9,-3,-1,-6,7] f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]

11l

Translation of: D

<lang 11l>F deconv(g, f)

  V result = [0]*(g.len - f.len + 1)
  L(&e) result
     V n = L.index
     e = g[n]
     V lower_bound = I n >= f.len {n - f.len + 1} E 0
     L(i) lower_bound .< n
        e -= result[i] * f[n - i]
     e /= f[0]
  R result

V h = [-8,-9,-3,-1,-6,7] V f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] V g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7] print(deconv(g, f)) print(deconv(g, h))</lang>

Output:

[-8, -9, -3, -1, -6, 7]
[-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

Ada

This is a translation of the D solution. <lang Ada>with Ada.Text_IO; use Ada.Text_IO;

procedure Main is

  package real_io is new Float_IO (Long_Float);
  use real_io;

  type Vector is array (Natural range <>) of Long_Float;

  function deconv (g, f : Vector) return Vector is
     len : Positive :=
       Integer'Max ((g'Length - f'length), (f'length - g'length));
     h     : Vector (0 .. len);
     Lower : Natural := 0;
  begin
     for n in h'range loop
        h (n) := g (n);
        if n >= f'length then
           Lower := n - f'length + 1;
        end if;
        for i in Lower .. n - 1 loop
           h (n) := h (n) - (h (i) * f (n - i));
        end loop;
        h (n) := h (n) / f (0);
     end loop;
     return h;
  end deconv;

  procedure print (v : Vector) is
  begin
     Put ("(");
     for I in v'range loop
        Put (Item => v (I), Fore => 1, Aft => 1, Exp => 0);
        if I < v'Last then
           Put (" ");
        else
           Put_Line (")");
        end if;
     end loop;
  end print;

  h : Vector := (-8.0, -9.0, -3.0, -1.0, -6.0, 7.0);
  f : Vector :=
    (-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0,
     -2.0, -7.0, -1.0);
  g : Vector :=
    (24.0, 75.0, 71.0, -34.0, 3.0, 22.0, -45.0, 23.0, 245.0, 25.0, 52.0, 25.0,
     -67.0, -96.0, 96.0, 31.0, 55.0, 36.0, 29.0, -43.0, -7.0);

begin

  print (h);
  print (deconv (g, f));
  print (f);
  print (deconv (g, h));

end Main; </lang>

Output:

(-8.0 -9.0 -3.0 -1.0 -6.0 7.0)
(-8.0 -9.0 -3.0 -1.0 -6.0 7.0)
(-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0)
(-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0)

BBC BASIC

Works with: BBC BASIC for Windows

As several others, this is a translation of the D solution. <lang bbcbasic> *FLOAT 64

     DIM h(5), f(15), g(20)
     h() = -8,-9,-3,-1,-6,7
     f() = -3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1
     g() = 24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7
     
     PROCdeconv(g(), f(), x())
     PRINT "deconv(g,f) = " FNprintarray(x())
     x() -= h() : IF SUM(x()) <> 0 PRINT "Error!"
     
     PROCdeconv(g(), h(), y())
     PRINT "deconv(g,h) = " FNprintarray(y())
     y() -= f() : IF SUM(y()) <> 0 PRINT "Error!"
     END
     
     DEF PROCdeconv(g(), f(), RETURN h())
     LOCAL f%, g%, i%, l%, n%
     f% = DIM(f(),1) + 1
     g% = DIM(g(),1) + 1
     DIM h(g% - f%)
     FOR n% = 0 TO g% - f%
       h(n%) = g(n%)
       IF n% < f% THEN l% = 0 ELSE l% = n% - f% + 1
       IF n% THEN
         FOR i% = l% TO n% - 1
           h(n%) -= h(i%) * f(n% - i%)
         NEXT
       ENDIF
       h(n%) /= f(0)
     NEXT n%
     ENDPROC
     
     DEF FNprintarray(a())
     LOCAL i%, a$
     FOR i% = 0 TO DIM(a(),1)
       a$ += STR$(a(i%)) + ", "
     NEXT
     = LEFT$(LEFT$(a$))</lang>

Output:

deconv(g,f) = -8, -9, -3, -1, -6, 7
deconv(g,h) = -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1

C

Using FFT: <lang C>#include <stdio.h>

include <stdlib.h>
include <math.h>
include <complex.h>

double PI; typedef double complex cplx;

void _fft(cplx buf[], cplx out[], int n, int step) { if (step < n) { _fft(out, buf, n, step * 2); _fft(out + step, buf + step, n, step * 2);

for (int i = 0; i < n; i += 2 * step) { cplx t = cexp(-I * PI * i / n) * out[i + step]; buf[i / 2] = out[i] + t; buf[(i + n)/2] = out[i] - t; } } }

void fft(cplx buf[], int n) { cplx out[n]; for (int i = 0; i < n; i++) out[i] = buf[i]; _fft(buf, out, n, 1); }

/* pad array length to power of two */ cplx *pad_two(double g[], int len, int *ns) { int n = 1; if (*ns) n = *ns; else while (n < len) n *= 2;

cplx *buf = calloc(sizeof(cplx), n); for (int i = 0; i < len; i++) buf[i] = g[i]; *ns = n; return buf; }

void deconv(double g[], int lg, double f[], int lf, double out[]) { int ns = 0; cplx *g2 = pad_two(g, lg, &ns); cplx *f2 = pad_two(f, lf, &ns);

fft(g2, ns); fft(f2, ns);

cplx h[ns]; for (int i = 0; i < ns; i++) h[i] = g2[i] / f2[i]; fft(h, ns);

for (int i = 0; i >= lf - lg; i--) out[-i] = h[(i + ns) % ns]/32; free(g2); free(f2); }

int main() { PI = atan2(1,1) * 4; double g[] = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7}; double f[] = { -3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1 }; double h[] = { -8,-9,-3,-1,-6,7 };

int lg = sizeof(g)/sizeof(double); int lf = sizeof(f)/sizeof(double); int lh = sizeof(h)/sizeof(double);

double h2[lh]; double f2[lf];

printf("f[] data is : "); for (int i = 0; i < lf; i++) printf(" %g", f[i]); printf("\n");

printf("deconv(g, h): "); deconv(g, lg, h, lh, f2); for (int i = 0; i < lf; i++) printf(" %g", f2[i]); printf("\n");

printf("h[] data is : "); for (int i = 0; i < lh; i++) printf(" %g", h[i]); printf("\n");

printf("deconv(g, f): "); deconv(g, lg, f, lf, h2); for (int i = 0; i < lh; i++) printf(" %g", h2[i]); printf("\n"); }</lang>

Output:

f[] data is :  -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1
deconv(g, h):  -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1
h[] data is :  -8 -9 -3 -1 -6 7

deconv(g, f):  -8 -9 -3 -1 -6 7

Common Lisp

Uses the routine (lsqr A b) from Multiple regression and (mtp A) from Matrix transposition.

<lang lisp>;; Assemble the mxn matrix A from the 2D row vector x. (defun make-conv-matrix (x m n)

 (let ((lx (cadr (array-dimensions x)))
       (A  (make-array `(,m ,n) :initial-element 0)))

   (loop for j from 0 to (- n 1) do
        (loop for i from 0 to (- m 1) do
             (setf (aref A i j)
                   (cond ((or (< i j) (>= i (+ j lx)))
                          0)
                         ((and (>= i j) (< i (+ j lx)))
                          (aref x 0 (- i j)))))))
   A))

Solve the overdetermined system A(f)*h=g by linear least squares.

(defun deconv (g f)

 (let* ((lg (cadr (array-dimensions g)))
        (lf (cadr (array-dimensions f)))
        (lh (+ (- lg lf) 1))
        (A  (make-conv-matrix f lg lh)))

   (lsqr A (mtp g))))</lang>

Example:

<lang lisp>(setf f #2A((-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1))) (setf h #2A((-8 -9 -3 -1 -6 7))) (setf g #2A((24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7)))

(deconv g f)

2A((-8.0)

   (-9.000000000000002)
   (-2.999999999999999)
   (-0.9999999999999997)
   (-6.0)
   (7.000000000000002))

(deconv g h)

2A((-2.999999999999999)

   (-6.000000000000001)
   (-1.0000000000000002)
   (8.0)
   (-5.999999999999999)
   (3.0000000000000004)
   (-1.0000000000000004)
   (-9.000000000000002)
   (-9.0)
   (2.9999999999999996)
   (-1.9999999999999991)
   (5.0)
   (1.9999999999999996)
   (-2.0000000000000004)
   (-7.000000000000001)
   (-0.9999999999999994))</lang>

D

<lang d>T[] deconv(T)(in T[] g, in T[] f) pure nothrow {

   int flen = f.length;
   int glen = g.length;
   auto result = new T[glen - flen + 1];
   foreach (int n, ref e; result) {
       e = g[n];
       immutable lowerBound = (n >= flen) ? n - flen + 1 : 0;
       foreach (i; lowerBound .. n)
               e -= result[i] * f[n - i];
       e /= f[0];
   }
   return result;

}

void main() {

   import std.stdio;
   immutable h = [-8,-9,-3,-1,-6,7];
   immutable f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1];
   immutable g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,
                  -96,96,31,55,36,29,-43,-7];
   writeln(deconv(g, f) == h, " ", deconv(g, f));
   writeln(deconv(g, h) == f, " ", deconv(g, h));

}</lang>

Output:

true [-8, -9, -3, -1, -6, 7]
true [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

Fortran

This solution uses the LAPACK95 library. <lang fortran> ! Build ! Windows: ifort /I "%IFORT_COMPILER11%\mkl\include\ia32" deconv1d.f90 "%IFORT_COMPILER11%\mkl\ia32\lib\*.lib" ! Linux:

program deconv

 ! Use gelsd from LAPACK95.
 use mkl95_lapack, only : gelsd

 implicit none
 real(8), allocatable :: g(:), href(:), A(:,:), f(:)
 real(8), pointer     :: h(:), r(:)
 integer              :: N
 character(len=16)    :: cbuff
 integer              :: i
 intrinsic            :: nint

 ! Allocate data arrays
 allocate(g(21),f(16))
 g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
 f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]

 ! Calculate deconvolution
 h => deco(f, g)

 ! Check result against reference
 N = size(h)
 allocate(href(N))
 href = [-8,-9,-3,-1,-6,7]
 cbuff = ' '
 write(cbuff,'(a,i0,a)') '(a,',N,'(i0,a),i0)'
 if (any(abs(h-href) > 1.0d-4)) then
    write(*,'(a)') 'deconv(f, g) - FAILED'
 else
    write(*,cbuff) 'deconv(f, g) = ',(nint(h(i)),', ',i=1,N-1),nint(h(N))
 end if

 ! Calculate deconvolution
 r => deco(h, g)

 cbuff = ' '
 N = size(r)
 write(cbuff,'(a,i0,a)') '(a,',N,'(i0,a),i0)'
 if (any(abs(r-f) > 1.0d-4)) then
    write(*,'(a)') 'deconv(h, g) - FAILED'
 else
    write(*,cbuff) 'deconv(h, g) = ',(nint(r(i)),', ',i=1,N-1),nint(r(N))
 end if

contains

 function deco(p, q)
   real(8), pointer    :: deco(:)
   real(8), intent(in) :: p(:), q(:)

   real(8), allocatable, target :: r(:)
   real(8), allocatable         :: A(:,:)
   integer :: N

   ! Construct derived arrays
   N = size(q) - size(p) + 1
   allocate(A(size(q),N),r(size(q)))
   A = 0.0d0
   do i=1,N
      A(i:i+size(p)-1,i) = p
   end do
   
   ! Invoke the LAPACK routine to do the work
   r = q
   call gelsd(A, r)

   deco => r(1:N)
 end function deco

end program deconv </lang> Results: <lang fortran> deconv(f, g) = -8, -9, -3, -1, -6, 7 deconv(h, g) = -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1 </lang>

Go

Translation of: D

<lang go>package main

import "fmt"

func main() {

   h := []float64{-8, -9, -3, -1, -6, 7}
   f := []float64{-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1}
   g := []float64{24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
       96, 31, 55, 36, 29, -43, -7}
   fmt.Println(h)
   fmt.Println(deconv(g, f))
   fmt.Println(f)
   fmt.Println(deconv(g, h))

}

func deconv(g, f []float64) []float64 {

   h := make([]float64, len(g)-len(f)+1)
   for n := range h {
       h[n] = g[n]
       var lower int
       if n >= len(f) {
           lower = n - len(f) + 1
       }
       for i := lower; i < n; i++ {
           h[n] -= h[i] * f[n-i]
       }
       h[n] /= f[0]
   }
   return h

}</lang>

Output:

[-8 -9 -3 -1 -6 7]
[-8 -9 -3 -1 -6 7]
[-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1]
[-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1]

Translation of: C

<lang go>package main

import (

   "fmt"
   "math"
   "math/cmplx"

)

func main() {

   h := []float64{-8, -9, -3, -1, -6, 7}
   f := []float64{-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1}
   g := []float64{24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
       96, 31, 55, 36, 29, -43, -7}
   fmt.Printf("%.1f\n", h)
   fmt.Printf("%.1f\n", deconv(g, f))
   fmt.Printf("%.1f\n", f)
   fmt.Printf("%.1f\n", deconv(g, h))

}

func deconv(g, f []float64) []float64 {

   n := 1
   for n < len(g) {
       n *= 2
   }
   g2 := make([]complex128, n)
   for i, x := range g {
       g2[i] = complex(x, 0)
   }
   f2 := make([]complex128, n)
   for i, x := range f {
       f2[i] = complex(x, 0)
   }
   gt := fft(g2)
   ft := fft(f2)
   for i := range gt {
       gt[i] /= ft[i]
   }
   ht := fft(gt)
   it := 1 / float64(n)
   out := make([]float64, len(g)-len(f)+1)
   out[0] = real(ht[0]) * it
   for i := 1; i < len(out); i++ {
       out[i] = real(ht[n-i]) * it
   }
   return out

}

func fft(in []complex128) []complex128 {

   out := make([]complex128, len(in))
   ditfft2(in, out, len(in), 1)
   return out

}

func ditfft2(x, y []complex128, n, s int) {

   if n == 1 {
       y[0] = x[0]
       return
   }
   ditfft2(x, y, n/2, 2*s)
   ditfft2(x[s:], y[n/2:], n/2, 2*s)
   for k := 0; k < n/2; k++ {
       tf := cmplx.Rect(1, -2*math.Pi*float64(k)/float64(n)) * y[k+n/2]
       y[k], y[k+n/2] = y[k]+tf, y[k]-tf
   }

}</lang>

Output:

Some results have errors out in the last decimal place or so. Only one decimal place shown here to let results fit in 80 columns.

[-8.0 -9.0 -3.0 -1.0 -6.0 7.0]
[-8.0 -9.0 -3.0 -1.0 -6.0 7.0]
[-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0]
[-3.0 -6.0 -1.0 8.0 -6.0 3.0 -1.0 -9.0 -9.0 3.0 -2.0 5.0 2.0 -2.0 -7.0 -1.0]

Library gonum/mat: <lang go>package main

import (

   "fmt"

   "gonum.org/v1/gonum/mat"

)

var (

   h = []float64{-8, -9, -3, -1, -6, 7}
   f = []float64{-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1}
   g = []float64{24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
       96, 31, 55, 36, 29, -43, -7}

)

func band(g, f []float64) *mat.Dense {

   nh := len(g) - len(f) + 1
   b := mat.NewDense(len(g), nh, nil)
   for j := 0; j < nh; j++ {
       for i, fi := range f {
           b.Set(i+j, j, fi)
       }
   }
   return b

}

func deconv(g, f []float64) mat.Matrix {

   z := mat.NewDense(len(g)-len(f)+1, 1, nil)
   z.Solve(band(g, f), mat.NewVecDense(len(g), g))
   return z

}

func main() {

   fmt.Printf("deconv(g, f) =\n%.1f\n\n", mat.Formatted(deconv(g, f)))
   fmt.Printf("deconv(g, h) =\n%.1f\n", mat.Formatted(deconv(g, h)))

}</lang>

Output:

deconv(g, f) =
⎡-8.0⎤
⎢-9.0⎥
⎢-3.0⎥
⎢-1.0⎥
⎢-6.0⎥
⎣ 7.0⎦

deconv(g, h) =
⎡-3.0⎤
⎢-6.0⎥
⎢-1.0⎥
⎢ 8.0⎥
⎢-6.0⎥
⎢ 3.0⎥
⎢-1.0⎥
⎢-9.0⎥
⎢-9.0⎥
⎢ 3.0⎥
⎢-2.0⎥
⎢ 5.0⎥
⎢ 2.0⎥
⎢-2.0⎥
⎢-7.0⎥
⎣-1.0⎦

Haskell

<lang haskell>deconv1d :: [Double] -> [Double] -> [Double] deconv1d xs ys = takeWhile (/= 0) $ deconv xs ys

 where
   [] `deconv` _ = []
   (0:xs) `deconv` (0:ys) = xs `deconv` ys
   (x:xs) `deconv` (y:ys) =
     let q = x / y
     in q : zipWith (-) xs (scale q ys ++ repeat 0) `deconv` (y : ys)

scale :: Double -> [Double] -> [Double] scale = map . (*)

h, f, g :: [Double] h = [-8, -9, -3, -1, -6, 7]

f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

g =

 [ 24
 , 75
 , 71
 , -34
 , 3
 , 22
 , -45
 , 23
 , 245
 , 25
 , 52
 , 25
 , -67
 , -96
 , 96
 , 31
 , 55
 , 36
 , 29
 , -43
 , -7
 ]

main :: IO () main = print $ (h == deconv1d g f) && (f == deconv1d g h)</lang>

Output:

True

J

This solution borrowed from Formal power series:

<lang J>Ai=: (i.@] =/ i.@[ -/ i.@>:@-)&# divide=: [ +/ .*~ [:%.&.x: ] +/ .* Ai</lang>

Sample data:

Example use: <lang J> g divide f _8 _9 _3 _1 _6 7

  g divide h

_3 _6 _1 8 _6 3 _1 _9 _9 3 _2 5 2 _2 _7 _1</lang>

That said, note that this particular implementation is slow since it uses extended precision intermediate results. It will run quite a bit faster for this example with no notable loss of precision if floating point is used. In other words:

<lang J>divide=: [ +/ .*~ [:%. ] +/ .* Ai</lang>

Java

Translation of: Go

<lang java>import java.util.Arrays;

public class Deconvolution1D {

   public static int[] deconv(int[] g, int[] f) {
       int[] h = new int[g.length - f.length + 1];
       for (int n = 0; n < h.length; n++) {
           h[n] = g[n];
           int lower = Math.max(n - f.length + 1, 0);
           for (int i = lower; i < n; i++)
               h[n] -= h[i] * f[n - i];
           h[n] /= f[0];
       }
       return h;
   }

   public static void main(String[] args) {
       int[] h = { -8, -9, -3, -1, -6, 7 };
       int[] f = { -3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1 };
       int[] g = { 24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,
               96, 31, 55, 36, 29, -43, -7 };

       StringBuilder sb = new StringBuilder();
       sb.append("h = " + Arrays.toString(h) + "\n");
       sb.append("deconv(g, f) = " + Arrays.toString(deconv(g, f)) + "\n");
       sb.append("f = " + Arrays.toString(f) + "\n");
       sb.append("deconv(g, h) = " + Arrays.toString(deconv(g, h)) + "\n");
       System.out.println(sb.toString());
   }

}</lang>

Output:

h = [-8, -9, -3, -1, -6, 7]
deconv(g, f) = [-8, -9, -3, -1, -6, 7]
f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
deconv(g, h) = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

Julia

The deconv function for floating point data is built into Julia, though using DSP is required with version 1.0. Integer inputs may need to be converted and copied to floating point to use deconv().

<lang julia>h = [-8, -9, -3, -1, -6, 7] g = [24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7] f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

hanswer = deconv(float.(g), float.(f)) println("The deconvolution deconv(g, f) is $hanswer, which is the same as h = $h\n")

fanswer = deconv(float.(g), float.(h)) println("The deconvolution deconv(g, h) is $fanswer, which is the same as f = $f\n")</lang>

Output:

The deconvolution deconv(g, f) is [-8.0, -9.0, -3.0, -1.0, -6.0, 7.0], 
which is the same as h = [-8, -9, -3, -1, -6, 7]

The deconvolution deconv(g, h) is [-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0], 
which is the same as f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

Kotlin

Translation of: Go

<lang scala>// version 1.1.3

fun deconv(g: DoubleArray, f: DoubleArray): DoubleArray {

   val fs = f.size   
   val h = DoubleArray(g.size - fs + 1)
   for (n in h.indices) {
       h[n] = g[n]
       val lower = if (n >= fs) n - fs + 1 else 0
       for (i in lower until n) h[n] -= h[i] * f[n -i]
       h[n] /= f[0] 
   }
   return h

}

fun main(args: Array<String>) {

   val h = doubleArrayOf(-8.0, -9.0, -3.0, -1.0, -6.0, 7.0)
   val f = doubleArrayOf(-3.0, -6.0, -1.0,  8.0, -6.0,  3.0, -1.0, -9.0, 
                         -9.0,  3.0, -2.0,  5.0,  2.0, -2.0, -7.0, -1.0)
   val g = doubleArrayOf(24.0,  75.0, 71.0, -34.0,  3.0,  22.0, -45.0, 
                         23.0, 245.0, 25.0,  52.0, 25.0, -67.0, -96.0,
                         96.0,  31.0, 55.0,  36.0, 29.0, -43.0,  -7.0)
   println("${h.map { it.toInt() }}")
   println("${deconv(g, f).map { it.toInt() }}")
   println()
   println("${f.map { it.toInt() }}")
   println("${deconv(g, h).map { it.toInt() }}")

}</lang>

Output:

[-8, -9, -3, -1, -6, 7]
[-8, -9, -3, -1, -6, 7]

[-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]
[-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

Lua

Using metatables: <lang lua>function deconvolve(f, g)

  local h = setmetatable({}, {__index = function(self, n)
     if n == 1 then self[1] = g[1] / f[1]
     else
        self[n] = g[n]
        for i = 1, n - 1 do
           self[n] = self[n] - self[i] * (f[n - i + 1] or 0)
        end
        self[n] = self[n] / f[1]
     end
     return self[n]
  end})
  local _ = h[#g - #f + 1]
  return setmetatable(h, nil)

end</lang>

Tests: <lang lua> local f = {-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1} local g = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7} local h = {-8,-9,-3,-1,-6,7} print(unpack(deconvolve(f, g))) --> -8 -9 -3 -1 -6 7 print(unpack(deconvolve(h, g))) --> -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1</lang>

Mathematica / Wolfram Language

This function creates a sparse array for the A matrix and then solves it with a built-in function. It may fail for overdetermined systems, though. Fast approximate methods for deconvolution are also built into Mathematica. See Deconvolution/2D+ <lang Mathematica> deconv[f_List, g_List] :=

Module[{A = 
   SparseArray[
    Table[Band[{n, 1}] -> fn, {n, 1, Length[f]}], {Length[g], Length[f] - 1}]}, 
 Take[LinearSolve[A, g], Length[g] - Length[f] + 1]]

</lang> Usage:

f = {-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1};
g = {24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7};
deconv[f,g]

Output:

{-8, -9, -3, -1, -6, 7}

MATLAB

The deconvolution function is built-in to MATLAB as the "deconv(a,b)" function, where "a" and "b" are vectors storing the convolved function values and the values of one of the deconvoluted vectors of "a". To test that this operates according to the task spec we can test the criteria above: <lang MATLAB>>> h = [-8,-9,-3,-1,-6,7]; >> g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]; >> f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]; >> deconv(g,f)

ans =

  -8.0000   -9.0000   -3.0000   -1.0000   -6.0000    7.0000

>> deconv(g,h)

ans =

   -3    -6    -1     8    -6     3    -1    -9    -9     3    -2     5     2    -2    -7    -1</lang>

Therefore, "deconv(a,b)" behaves as expected.

Nim

<lang Nim>proc deconv(g, f: openArray[float]): seq[float] =

 var h: seq[float] = newSeq[float](len(g) - len(f) + 1)
 for n in 0..<len(h):
   h[n] = g[n]
   var lower: int
   if n >= len(f):
     lower = n - len(f) + 1
   for i in lower..<n:
     h[n] -= h[i] * f[n - i]
   h[n] /= f[0]
 h

let h = [-8'f64, -9, -3, -1, -6, 7] let f = [-3'f64, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] let g = [24'f64, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96,

        96, 31, 55, 36, 29, -43, -7]

echo h echo deconv(g, f) echo f echo deconv(g, h)</lang>

Output:

[-8.0, -9.0, -3.0, -1.0, -6.0, 7.0]
@[-8.0, -9.0, -3.0, -1.0, -6.0, 7.0]
[-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0]
@[-3.0, -6.0, -1.0, 8.0, -6.0, 3.0, -1.0, -9.0, -9.0, 3.0, -2.0, 5.0, 2.0, -2.0, -7.0, -1.0]

Perl

Using rref routine from Reduced row echelon form task.

Translation of: Raku

<lang perl>use Math::Cartesian::Product;

sub deconvolve {

   our @g; local *g = shift;
   our @f; local *f = shift;
   my(@m,@d);

   my $h = 1 + @g - @f;
   push @m, [(0) x $h, $g[$_]] for 0..$#g;
   for my $j (0..$h-1) {
       for my $k (0..$#f) {
           $m[$j + $k][$j] = $f[$k]
       }
   }
   rref(\@m);
   push @d, @{ $m[$_] }[$h] for 0..$h-1;
   @d;

}

sub convolve {

   our @f; local *f = shift;
   our @h; local *h = shift;
   my @i;
   for my $x (cartesian {@_} [0..$#f], [0..$#h]) {
       push @i, @$x[0]+@$x[1];
   }
   my $cnt = 0;
   my @g = (0) x (@f + @h - 1);
   for my $x (cartesian {@_} [@f], [@h]) {
       $g[$i[$cnt++]] += @$x[0]*@$x[1];
   }
   @g;

}

sub rref {

 our @m; local *m = shift;
 @m or return;
 my ($lead, $rows, $cols) = (0, scalar(@m), scalar(@{$m[0]}));

 foreach my $r (0 .. $rows - 1) {
    $lead < $cols or return;
     my $i = $r;

     until ($m[$i][$lead])
        {++$i == $rows or next;
         $i = $r;
         ++$lead == $cols and return;}

     @m[$i, $r] = @m[$r, $i];
     my $lv = $m[$r][$lead];
     $_ /= $lv foreach @{ $m[$r] };

     my @mr = @{ $m[$r] };
     foreach my $i (0 .. $rows - 1)
        {$i == $r and next;
         ($lv, my $n) = ($m[$i][$lead], -1);
         $_ -= $lv * $mr[++$n] foreach @{ $m[$i] };}

     ++$lead;}

}

my @h = qw<-8 -9 -3 -1 -6 7>; my @f = qw<-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1>; print ' conv(f,h) = g = ' . join(' ', my @g = convolve(\@f, \@h)) . "\n"; print 'deconv(g,f) = h = ' . join(' ', deconvolve(\@g, \@f)) . "\n"; print 'deconv(g,h) = f = ' . join(' ', deconvolve(\@g, \@h)) . "\n";</lang>

Output:

  conv(f,h) = g = 24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7
deconv(g,f) = h = -8 -9 -3 -1 -6 7
deconv(g,h) = f = -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1

Phix

Translation of: D

<lang Phix>function deconv(sequence g, sequence f) integer lf = length(f) sequence h = repeat(0,length(g)-lf+1)

   for n = 1 to length(h) do
       atom e = g[n]
       for i=max(n-lf,0) to n-2 do
           e -= h[i+1] * f[n-i]
       end for
       h[n] = e/f[1]
   end for
   return h

end function

constant h = {-8,-9,-3,-1,-6,7} constant f = {-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1} constant g = {24,75,71,-34,3,22,-45,23,245,25,52,25,-67,

                  -96,96,31,55,36,29,-43,-7}

sequence r r = deconv(g, f) ?{r==h,r} r = deconv(g, h) ?{r==f,r}</lang>

Output:

{1,{-8,-9,-3,-1,-6,7}}
{1,{-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1}}

PicoLisp

<lang PicoLisp>(load "@lib/math.l")

(de deconv (G F)

  (let A (pop 'F)
     (make
        (for (N . H) (head (- (length F)) G)
           (for (I . M) (made)
              (dec 'H
                 (*/ M (get F (- N I)) 1.0) ) )
           (link (*/ H 1.0 A)) ) ) ) )</lang>

Test: <lang PicoLisp>(setq

  F (-3. -6. -1. 8. -6. 3. -1. -9. -9. 3. -2. 5. 2. -2. -7. -1.)
  G (24. 75. 71. -34. 3. 22. -45. 23. 245. 25. 52. 25. -67. -96. 96. 31. 55. 36. 29. -43. -7.)
  H (-8. -9. -3. -1. -6. 7.) )

(test H (deconv G F)) (test F (deconv G H))</lang>

Python

Works with: Python version 3.x

Inspired by the TCL solution, and using the ToReducedRowEchelonForm function to reduce to row echelon form from here <lang python>def ToReducedRowEchelonForm( M ):

   if not M: return
   lead = 0
   rowCount = len(M)
   columnCount = len(M[0])
   for r in range(rowCount):
       if lead >= columnCount:
           return
       i = r
       while M[i][lead] == 0:
           i += 1
           if i == rowCount:
               i = r
               lead += 1
               if columnCount == lead:
                   return
       M[i],M[r] = M[r],M[i]
       lv = M[r][lead]
       M[r] = [ mrx / lv for mrx in M[r]]
       for i in range(rowCount):
           if i != r:
               lv = M[i][lead]
               M[i] = [ iv - lv*rv for rv,iv in zip(M[r],M[i])]
       lead += 1
   return M

def pmtx(mtx):

   print ('\n'.join(.join(' %4s' % col for col in row) for row in mtx))

def convolve(f, h):

   g = [0] * (len(f) + len(h) - 1)
   for hindex, hval in enumerate(h):
       for findex, fval in enumerate(f):
           g[hindex + findex] += fval * hval
   return g

def deconvolve(g, f):

   lenh = len(g) - len(f) + 1
   mtx = [[0 for x in range(lenh+1)] for y in g]
   for hindex in range(lenh):
       for findex, fval in enumerate(f):
           gindex = hindex + findex
           mtx[gindex][hindex] = fval
   for gindex, gval in enumerate(g):        
       mtx[gindex][lenh] = gval
   ToReducedRowEchelonForm( mtx )
   return [mtx[i][lenh] for i in range(lenh)]  # h

if __name__ == '__main__':

   h = [-8,-9,-3,-1,-6,7]
   f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1]
   g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
   assert convolve(f,h) == g
   assert deconvolve(g, f) == h</lang>

Based on the R version.

import numpy

h = [-8,-9,-3,-1,-6,7] f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]

https://stackoverflow.com/questions/14267555/find-the-smallest-power-of-2-greater-than-n-in-python

def shift_bit_length(x):

   return 1<<(x-1).bit_length()

def conv(a, b):

   p = len(a)
   q = len(b)
   n = p + q - 1
   r = shift_bit_length(n)
   y = numpy.fft.ifft(numpy.fft.fft(a,r) * numpy.fft.fft(b,r),r)
   return numpy.trim_zeros(numpy.around(numpy.real(y),decimals=6))

def deconv(a, b):

   p = len(a)
   q = len(b)
   n = p - q + 1
   r = shift_bit_length(max(p, q))
   y = numpy.fft.ifft(numpy.fft.fft(a,r) / numpy.fft.fft(b,r), r)
   return numpy.trim_zeros(numpy.around(numpy.real(y),decimals=6))

should return g

print(conv(h,f))

should return h

print(deconv(g,f))

should return f

print(deconv(g,h))

</lang>

Output

[ 24.  75.  71. -34.   3.  22. -45.  23. 245.  25.  52.  25. -67. -96.
  96.  31.  55.  36.  29. -43.  -7.]
[-8. -9. -3. -1. -6.  7.]
[-3. -6. -1.  8. -6.  3. -1. -9. -9.  3. -2.  5.  2. -2. -7. -1.]

R

Here we won't solve the system but use the FFT instead. The method :

extend vector arguments so that they are the same length, a power of 2 larger than the length of the solution,
solution is ifft(fft(a)*fft(b)), truncated.

<lang R>conv <- function(a, b) { p <- length(a) q <- length(b) n <- p + q - 1 r <- nextn(n, f=2) y <- fft(fft(c(a, rep(0, r-p))) * fft(c(b, rep(0, r-q))), inverse=TRUE)/r y[1:n] }

deconv <- function(a, b) { p <- length(a) q <- length(b) n <- p - q + 1 r <- nextn(max(p, q), f=2) y <- fft(fft(c(a, rep(0, r-p))) / fft(c(b, rep(0, r-q))), inverse=TRUE)/r return(y[1:n]) } </lang>

To check :

<lang R> h <- c(-8,-9,-3,-1,-6,7) f <- c(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1) g <- c(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7)

max(abs(conv(f,h) - g)) max(abs(deconv(g,f) - h)) max(abs(deconv(g,h) - f)) </lang>

This solution often introduces complex numbers, with null or tiny imaginary part. If it hurts in applications, type Re(conv(f,h)) and Re(deconv(g,h)) instead, to return only the real part. It's not hard-coded in the functions, since they may be used for complex arguments as well.

R has also a function convolve, <lang R> conv(a, b) == convolve(a, rev(b), type="open") </lang>

Racket

lang racket

(require math/matrix) (define T matrix-transpose)

(define (convolution-matrix f m n)

 (define l (matrix-num-rows f))
 (for*/matrix m n ([i (in-range 0 m)] [j (in-range 0 n)])
     (cond [(or  (< i j) (>= i (+ j l)))  0]
           [(matrix-ref f (- i j) 0)])))

(define (least-square X y)

 (matrix-solve (matrix* (T X) X) (matrix* (T X) y)))

(define (deconvolve g f)

 (define lg (matrix-num-rows g))
 (define lf (matrix-num-rows f))
 (define lh (+ (- lg lf) 1))
 (least-square (convolution-matrix f lg lh) g))

</lang> Test: <lang racket> (define f (col-matrix [-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1])) (define h (col-matrix [-8 -9 -3 -1 -6 7])) (define g (col-matrix [24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7]))

(deconvolve g f) (deconvolve g h) </lang>

Output:

<array '#(6 1) #[-8 -9 -3 -1 -6 7]>
<array '#(16 1) #[-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1]>

</lang>

Raku

(formerly Perl 6)

Works with: Rakudo version 2018.02

Translation of Python, using a modified version of the Reduced Row Echelon Form subroutine rref() from here.

<lang perl6>sub deconvolve (@g, @f) {

   my $h = 1 + @g - @f;
   my @m;
   @m[^@g;^$h] >>+=>> 0;
   @m[^@g;$h] >>=<< @g;
   for ^$h -> $j { for @f.kv -> $k, $v { @m[$j + $k][$j] = $v } }
   return rref( @m )[^$h;$h];

}

sub convolve (@f, @h) {

   my @g = 0 xx + @f + @h - 1;
   @g[^@f X+ ^@h] >>+=<< (@f X* @h);
   return @g;

}

Reduced Row Echelon Form simultaneous equation solver.
Can handle over-specified systems of equations.
(n unknowns in n + m equations)

sub rref ($m is copy) {

   return unless $m;
   my ($lead, $rows, $cols) = 0, +$m, +$m[0];

   # Trim off over specified rows if they exist, for efficiency
   if $rows >= $cols {
       $m = trim_system($m);
       $rows = +$m;
   }

   for ^$rows -> $r {
       $lead < $cols or return $m;
       my $i = $r;
       until $m[$i][$lead] {
           ++$i == $rows or next;
           $i = $r;
           ++$lead == $cols and return $m;
       }
       $m[$i, $r] = $m[$r, $i] if $r != $i;
       my $lv = $m[$r][$lead];
       $m[$r] >>/=>> $lv;
       for ^$rows -> $n {
           next if $n == $r;
           $m[$n] >>-=>> $m[$r] >>*>> ($m[$n][$lead]//0);
       }
       ++$lead;
   }
   return $m;

   # Reduce a system of equations to n equations with n unknowns.
   # Looks for an equation with a true value for each position.
   # If it can't find one, assumes that it has already taken one
   # and pushes in the first equation it sees. This assumtion
   # will alway be successful except in some cases where an
   # under-specified system has been supplied, in which case,
   # it would not have been able to reduce the system anyway.
   sub trim_system ($m is rw) {
       my ($vars, @t) = +$m[0]-1, ();
       for ^$vars -> $lead {
           for ^$m -> $row {
               @t.push: | $m.splice( $row, 1 ) and last if $m[$row][$lead];
           }
       }
       while (+@t < $vars) and +$m { @t.push: $m.splice( 0, 1 ) };
       return @t;
   }

}

my @h = (-8,-9,-3,-1,-6,7); my @f = (-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1); my @g = (24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7);

.say for ~@g, ~convolve(@f, @h),;

.say for ~@h, ~deconvolve(@g, @f),;

.say for ~@f, ~deconvolve(@g, @h),;</lang>

Output:

24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7
24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7

-8 -9 -3 -1 -6 7
-8 -9 -3 -1 -6 7

-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1
-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1

REXX

<lang rexx>/*REXX pgm performs deconvolution of two arrays: deconv(g,f)=h and deconv(g,h)=f */ call make 'H', "-8 -9 -3 -1 -6 7" call make 'F', "-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1" call make 'G', "24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7" call show 'H' /*display the elements of array H. */ call show 'F' /* " " " " " F. */ call show 'G' /* " " " " " G. */ call deco 'G', "F", 'X' /*deconvolution of G and F ───► X */ call test 'X', "H" /*test: is array H equal to array X?*/ call deco 'G', "H", 'Y' /*deconvolution of G and H ───► Y */ call test 'F', "Y" /*test: is array F equal to array Y?*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ deco: parse arg $1,$2,$r; b= @.$2.# + 1; a= @.$1.# + 1 /*get sizes of array 1&2*/

     @.$r.#= a - b                                             /*size of return array. */
           do n=0  to a-b                                      /*define  return array. */
           @.$r.n= @.$1.n                                      /*define RETURN element.*/
           if n0  then do j=L  to n-1;                _= n-j  /*define elements > 0.  */
                        @.$r.n= @.$r.n - @.$r.j * @.$2._       /*compute   "     " "   */
                        end   /*j*/                            /* [↑] subtract product.*/
           @.$r.n= @.$r.n / @.$2.0                             /*divide array element. */
           end   /*n*/;                     return

/*──────────────────────────────────────────────────────────────────────────────────────*/ make: parse arg $,z; @.$.#= words(z) - 1 /*obtain args; set size.*/
do k=0 to @.$.#; @.$.k= word(z, k + 1) /*define array element. */ end /*k*/; return /*array starts at unity.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ show: parse arg $,z,_; do s=0 to @.$.#; _= strip(_ @.$.s) /*obtain the arguments. */
end /*s*/ /* [↑] build the list. */ say 'array' $": " _; return /*show the list; return*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ test: parse arg $1,$2; do t=0 to max(@.$1.#, @.$2.#) /*obtain the arguments. */
if @.$1.t= @.$2.t then iterate /*create array list. */ say "***error*** arrays" $1 ' and ' $2 "aren't equal." end /*t*/; return /* [↑] build the list. */</lang>

output when using the default internal inputs:
array H: -8 -9 -3 -1 -6 7 array F: -3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1 array G: 24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7
Scala

Output:

Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).

<lang Scala>object Deconvolution1D extends App {
val (h, f) = (Array(-8, -9, -3, -1, -6, 7), Array(-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1)) val g = Array(24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7) val sb = new StringBuilder
private def deconv(g: Array[Int], f: Array[Int]) = { val h = Array.ofDim[Int](g.length - f.length + 1)
for (n <- h.indices) { h(n) = g(n) for (i <- math.max(n - f.length + 1, 0) until n) h(n) -= h(i) * f(n - i) h(n) /= f(0) } h }
sb.append(s"h = ${h.mkString("[", ", ", "]")}\n") .append(s"deconv(g, f) = ${deconv(g, f).mkString("[", ", ", "]")}\n") .append(s"f = ${f.mkString("[", ", ", "]")}\n") .append(s"deconv(g, h) = ${deconv(g, h).mkString("[", ", ", "]")}") println(sb.result())
}</lang>
Tcl
Works with: Tcl version 8.5
This builds the a command, 1D, with two subcommands (convolve and deconvolve) for performing convolution and deconvolution of these kinds of arrays. The deconvolution code is based on a reduction to reduced row echelon form. <lang tcl>package require Tcl 8.5 namespace eval 1D {
namespace ensemble create; # Will be same name as namespace namespace export convolve deconvolve # Access core language math utility commands namespace path {::tcl::mathfunc ::tcl::mathop}
# Utility for converting a matrix to Reduced Row Echelon Form # From http://rosettacode.org/wiki/Reduced_row_echelon_form#Tcl proc toRREF {m} {
set lead 0 set rows [llength $m] set cols [llength [lindex $m 0]] for {set r 0} {$r < $rows} {incr r} { if {$cols <= $lead} { break } set i $r while {[lindex $m $i $lead] == 0} { incr i if {$rows == $i} { set i $r incr lead if {$cols == $lead} { # Tcl can't break out of nested loops return $m } } } # swap rows i and r foreach j [list $i $r] row [list [lindex $m $r] [lindex $m $i]] { lset m $j $row } # divide row r by m(r,lead) set val [lindex $m $r $lead] for {set j 0} {$j < $cols} {incr j} { lset m $r $j [/ [double [lindex $m $r $j]] $val] }
for {set i 0} {$i < $rows} {incr i} { if {$i != $r} { # subtract m(i,lead) multiplied by row r from row i set val [lindex $m $i $lead] for {set j 0} {$j < $cols} {incr j} { lset m $i $j \ [- [lindex $m $i $j] [* $val [lindex $m $r $j]]] } } } incr lead } return $m
}
# How to apply a 1D convolution of two "functions" proc convolve {f h} {
set g [lrepeat [+ [llength $f] [llength $h] -1] 0] set fi -1 foreach fv $f { incr fi set hi -1 foreach hv $h { set gi [+ $fi [incr hi]] lset g $gi [+ [lindex $g $gi] [* $fv $hv]] } } return $g
}
# How to apply a 1D deconvolution of two "functions" proc deconvolve {g f} {
# Compute the length of the result vector set hlen [- [llength $g] [llength $f] -1]
# Build a matrix of equations to solve set matrix {} set i -1 foreach gv $g { lappend matrix [list {*}[lrepeat $hlen 0] $gv] set j [incr i] foreach fv $f { if {$j < 0} { break } elseif {$j < $hlen} { lset matrix $i $j $fv } incr j -1 } }
# Convert to RREF, solving the system of simultaneous equations set reduced [toRREF $matrix]
# Extract the deconvolution from the last column of the reduced matrix for {set i 0} {$i<$hlen} {incr i} { lappend result [lindex $reduced $i end] } return $result
}
}</lang> To use the above code, a simple demonstration driver (which solves the specific task): <lang tcl># Simple pretty-printer proc pp {name nlist} {
set sep "" puts -nonewline "$name = \[" foreach n $nlist {
puts -nonewline [format %s%g $sep $n] set sep ,
} puts "\]"
}
set h {-8 -9 -3 -1 -6 7} set f {-3 -6 -1 8 -6 3 -1 -9 -9 3 -2 5 2 -2 -7 -1} set g {24 75 71 -34 3 22 -45 23 245 25 52 25 -67 -96 96 31 55 36 29 -43 -7}
pp "deconv(g,f) = h" [1D deconvolve $g $f] pp "deconv(g,h) = f" [1D deconvolve $g $h] pp " conv(f,h) = g" [1D convolve $f $h]</lang>

Output:
deconv(g,f) = h = [-8,-9,-3,-1,-6,7] deconv(g,h) = f = [-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1] conv(f,h) = g = [24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7]
Ursala
The user defined function band constructs the required matrix as a list of lists given the pair of sequences to be deconvolved, and the lapack..dgelsd function solves the system. Some other library functions used are zipt (zipping two unequal length lists by truncating the longer one) zipp0 (zipping unequal length lists by padding the shorter with zeros) and pad0 (making a list of lists all the same length by appending zeros to the short ones).
<lang Ursala>#import std
import nat
band = pad0+ ~&rSS+ zipt^*D(~&r,^lrrSPT/~&ltK33tx zipt^/~&r ~&lSNyCK33+ zipp0)^/~&rx ~&B->NlNSPC ~&bt
deconv = lapack..dgelsd^\~&l ~&||0.!**+ band </lang> test program: <lang Ursala>h = <-8.,-9.,-3.,-1.,-6.,7.> f = <-3.,-6.,-1.,8.,-6.,3.,-1.,-9.,-9.,3.,-2.,5.,2.,-2.,-7.,-1.> g = <24.,75.,71.,-34.,3.,22.,-45.,23.,245.,25.,52.,25.,-67.,-96.,96.,31.,55.,36.,29.,-43.,-7.>
cast %eLm
test =
<
'h': deconv(g,f), 'f': deconv(g,h)>
</lang>

Output:
< 'h': < -8.000000e+00, -9.000000e+00, -3.000000e+00, -1.000000e+00, -6.000000e+00, 7.000000e+00>, 'f': < -3.000000e+00, -6.000000e+00, -1.000000e+00, 8.000000e+00, -6.000000e+00, 3.000000e+00, -1.000000e+00, -9.000000e+00, -9.000000e+00, 3.000000e+00, -2.000000e+00, 5.000000e+00, 2.000000e+00, -2.000000e+00, -7.000000e+00, -1.000000e+00>>
Wren
Translation of: Go
<lang ecmascript>var deconv = Fn.new { |g, f|
var h = List.filled(g.count - f.count + 1, 0) for (n in 0...h.count) { h[n] = g[n] var lower = (n >= f.count) ? n - f.count + 1 : 0 var i = lower while (i < n) { h[n] = h[n] - h[i]*f[n-i] i = i + 1 } h[n] = h[n] / f[0] } return h
}
var h = [-8, -9, -3, -1, -6, 7] var f = [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] var g = [24, 75, 71, -34, 3, 22, -45, 23, 245, 25, 52, 25, -67, -96, 96, 31, 55, 36, 29, -43, -7] System.print(h) System.print(deconv.call(g, f)) System.print(f) System.print(deconv.call(g, h))</lang>

Output:
[-8, -9, -3, -1, -6, 7] [-8, -9, -3, -1, -6, 7] [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1] [-3, -6, -1, 8, -6, 3, -1, -9, -9, 3, -2, 5, 2, -2, -7, -1]

zkl
Using GNU Scientific Library: <lang zkl>var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library) fcn dconv1D(f,g){
fsz,hsz:=f.len(), g.len() - fsz +1; A:=GSL.Matrix(g.len(),hsz); foreach n,fn in ([0..].zip(f)){ foreach rc in (hsz){ A[rc+n,rc]=fn } } h:=A.AxEQb(g); h
}</lang> <lang zkl>f:=GSL.VectorFromData(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1); g:=GSL.VectorFromData(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,-96,96,31,55,36,29,-43,-7); h:=dconv1D(f,g); h.format().println();
f:=dconv1D(h,g); f.format().println();</lang>

Output:
-8.00,-9.00,-3.00,-1.00,-6.00,7.00 -3.00,-6.00,-1.00,8.00,-6.00,3.00,-1.00,-9.00,-9.00,3.00,-2.00,5.00,2.00,-2.00,-7.00,-1.00
Or, using lists:
Translation of: D
<lang zkl>fcn deconv(g,f){
flen, glen, delta:=f.len(), g.len(), glen - flen + 1; result:=List.createLong(delta); // allocate list with space for items foreach n in (delta){ e:=g[n]; lowerBound:=(if (n>=flen) n - flen + 1 else 0); foreach i in ([lowerBound .. n-1]){ e-=result[i]*f[n - i]; } result.append(e/f[0]); } result;
}</lang> <lang zkl>h:=T(-8,-9,-3,-1,-6,7); f:=T(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1); g:=T(24,75,71,-34,3,22,-45,23,245,25,52,25,-67,
-96,96,31,55,36,29,-43,-7);
println(deconv(g, f) == h, " ", deconv(g, f)); println(deconv(g, h) == f, " ", deconv(g, h));</lang>

Output:
True L(-8,-9,-3,-1,-6,7) True L(-3,-6,-1,8,-6,3,-1,-9,-9,3,-2,5,2,-2,-7,-1)