1. Offtopic - sampling problem

Hi All,

I am trying to solve a sampling problem and I have simplified it as 
below.
Imagine this set contains precisely accurate real-world data that we 
wish to record into
the computer (could be temperature or anything)

1,   4,    2,    6,    5,    7,   3,   9,   4

Now, suppose we tried to capture each iteration with a sampling system 
that was not
focussed narrowly enough and so each sampled value acquired contained 
some overlap
from each adjacent value, assuming a spread of 25%, 50%, 25% we would 
get:

1.5, 2.75, 3.5, 4.75, 5.75, 5.5, 5.5, 6.25, 4.25

Each captured value tends to get "smeared" toward the average value

My Question is this: Does anyone know of or could point to a way to 
focus
the 2nd sample set back to the original set?

Any help on this is really appreciated.

Yours truly
Mike
vulcan at win.co.nz

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2. Re: Offtopic - sampling problem

Some sound-processing systems use algorithms to "soften" sound and to
"sharpen" it. I think this is what you need. However:
1) I don't know the algorithms.
2) It seems, according to the results obtained, that "soften" algorithms are
OK, but "sharpen" ones do not work very well (in order to restore the
original signal), due to the fact that some information is lost forever.
----- Original Message -----
From: "Mike" <vulcan at win.co.nz>
To: "EUforum" <EUforum at topica.com>
Subject: Offtopic - sampling problem


>
> Hi All,
>
> I am trying to solve a sampling problem and I have simplified it as
> below.
> Imagine this set contains precisely accurate real-world data that we
> wish to record into
> the computer (could be temperature or anything)
>
> 1,   4,    2,    6,    5,    7,   3,   9,   4
>
> Now, suppose we tried to capture each iteration with a sampling system
> that was not
> focussed narrowly enough and so each sampled value acquired contained
> some overlap
> from each adjacent value, assuming a spread of 25%, 50%, 25% we would
> get:
>
> 1.5, 2.75, 3.5, 4.75, 5.75, 5.5, 5.5, 6.25, 4.25
>
> Each captured value tends to get "smeared" toward the average value
>
> My Question is this: Does anyone know of or could point to a way to
> focus
> the 2nd sample set back to the original set?
>
> Any help on this is really appreciated.
>
> Yours truly
> Mike
> vulcan at win.co.nz
>
>
>
>

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3. Re: Offtopic - sampling problem

Mike,

Here's the best I can observe:

let:  a b c d e f g h i        be the actual original values;
let:  A B C D E F G H I   be the perceived values;

then:

A = .25x0 + .5a + .25b
B = .25a  + .5b + .25c
C = .25b  + .5c + .25d
D = .25c  + .5d + .25e
E = .25d  + .5e + .25f
F = .25e  + .5f + .25g
G = .25f  + .5g + .25h
H = .25g  + .5h + .25i
I = .25h  + .5i + .25x0

This is a set of 9 equations in 9 unknowns, which should(?) be solvable.

I suppose you might use matrices to solve them (or just substitution), but
longer sample lengths would probably(?) be prohibitive.  If you do have
longer samples, I suppose you could arbitrarily select groups of data of
some size and "throw-away" the values before the beginning & after the end
of each group (call them whatever the average is?), and "solve" each group
separately?  Dunno if that's a help or not.

Dan Moyer

----- Original Message -----
From: "Mike" <vulcan at win.co.nz>
To: "EUforum" <EUforum at topica.com>
Sent: Saturday, January 26, 2002 1:09 PM
Subject: Offtopic - sampling problem


>
> Hi All,
>
> I am trying to solve a sampling problem and I have simplified it as
> below.
> Imagine this set contains precisely accurate real-world data that we
> wish to record into
> the computer (could be temperature or anything)
>
> 1,   4,    2,    6,    5,    7,   3,   9,   4
>
> Now, suppose we tried to capture each iteration with a sampling system
> that was not
> focussed narrowly enough and so each sampled value acquired contained
> some overlap
> from each adjacent value, assuming a spread of 25%, 50%, 25% we would
> get:
>
> 1.5, 2.75, 3.5, 4.75, 5.75, 5.5, 5.5, 6.25, 4.25
>
> Each captured value tends to get "smeared" toward the average value
>
> My Question is this: Does anyone know of or could point to a way to
> focus
> the 2nd sample set back to the original set?
>
> Any help on this is really appreciated.
>
> Yours truly
> Mike
> vulcan at win.co.nz
>
>
>
>

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4. Re: Offtopic - sampling problem

Mike,

Go to archives, get the simultaneous equations solver by Laurence Draper,
"simul.e"; change the main function "Solve_Simultaneous" to a global
function; then run this modification of his example that has just your
perceived data in it, and it will give you the original actual data:


-- tested code follows:
include simul.e

sequence A --Matrix of coefficients
sequence b --Column vector of results
sequence x --Column vector of solutions
A = {{.5,.25,0,0,0,0,0,0,0},
     {.25,.5,.25,0,0,0,0,0,0},
     {0,.25,.5,.25,0,0,0,0,0},
     {0,0,.25,.5,.25,0,0,0,0},
     {0,0,0,.25,.5,.25,0,0,0},
     {0,0,0,0,.25,.5,.25,0,0},
     {0,0,0,0,0,.25,.5,.25,0},
     {0,0,0,0,0,0,.25,.5,.25},
     {0,0,0,0,0,0,0,.25,.5}}
b = {1.5, 2.75, 3.5, 4.75, 5.75, 5.5, 5.5, 6.25, 4.25}
if length(A[1]) = length(b) then
   dimension = length(A[1])
   x = Solve_Simultaneous(A,b)
   ? x
else
   puts(1, "Matrices are incorrect size!!!!")
end if

-- code ends

This will handle your *simplified* example.

Dan Moyer

----- Original Message -----
From: "Mike" <vulcan at win.co.nz>
To: "EUforum" <EUforum at topica.com>
Sent: Saturday, January 26, 2002 1:09 PM
Subject: Offtopic - sampling problem


>
> Hi All,
>
> I am trying to solve a sampling problem and I have simplified it as
> below.
> Imagine this set contains precisely accurate real-world data that we
> wish to record into
> the computer (could be temperature or anything)
>
> 1,   4,    2,    6,    5,    7,   3,   9,   4
>
> Now, suppose we tried to capture each iteration with a sampling system
> that was not
> focussed narrowly enough and so each sampled value acquired contained
> some overlap
> from each adjacent value, assuming a spread of 25%, 50%, 25% we would
> get:
>
> 1.5, 2.75, 3.5, 4.75, 5.75, 5.5, 5.5, 6.25, 4.25
>
> Each captured value tends to get "smeared" toward the average value
>
> My Question is this: Does anyone know of or could point to a way to
> focus
> the 2nd sample set back to the original set?
>
> Any help on this is really appreciated.
>
> Yours truly
> Mike
> vulcan at win.co.nz
>
>
>
>

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