Problem Statement

The convolution of two functions

F

{\displaystyle {\mathit {F}}}

and

H

{\displaystyle {\mathit {H}}}

of an integer variable is defined as the function

G

{\displaystyle {\mathit {G}}}

satisfying for all integers

n

{\displaystyle {\mathit {n}}}

. Assume

F ( n )

{\displaystyle F(n)}

can be non-zero only for

0

{\displaystyle 0}

≤

n

{\displaystyle {\mathit {n}}}

≤

|

F

|

{\displaystyle |{\mathit {F}}|}

, where

|

F

|

{\displaystyle |{\mathit {F}}|}

is the "length" of

F

{\displaystyle {\mathit {F}}}

, and similarly for

G

{\displaystyle {\mathit {G}}}

and

H

{\displaystyle {\mathit {H}}}

, so that the functions can be modeled as finite sequences by identifying

f

0

,

f

1

,

f

2

, …

{\displaystyle f_{0},f_{1},f_{2},\dots }

with

F ( 0 ) , F ( 1 ) , F ( 2 ) , …

{\displaystyle F(0),F(1),F(2),\dots }

, etc. Then for example, values of

|

F

|

= 6

{\displaystyle |{\mathit {F}}|=6}

and

|

H

|

= 5

{\displaystyle |{\mathit {H}}|=5}

would determine the following value of

g

{\displaystyle {\mathit {g}}}

by definition. We can write this in matrix form as: or 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

{\displaystyle A}

for

h

{\displaystyle {\mathit {h}}}

given

f

{\displaystyle {\mathit {f}}}

and

g

{\displaystyle {\mathit {g}}}

.

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]

Let's start with the solution:

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