Invert single vector dimension using only addition and inversion
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I'm trying to perform an audio task, to try and isolate audio from mixed tracks. Specifically, I have two tracks with the same music but different sounds. Both are stereo. The only operations I can apply to them are inverting and mixing them. Doing so gets rid of the common parts and leaves on the difference.
There is a similar question:
Isolating audio tracks through mixing
that solves a similar-looking problem through a system of linear
equations, however the conditions are different and I tried applying
the answer given as well as the solution the OP applied but I can't
get a meaningful answer out of it (Wolfram
Alpha).
I'm not sure if I've interpreted that solution correctly.
I've boiled the problem down to this:
- Track 1:
i + j
- Track 2:
i + k
i represents the common audio, j and k represent the audio unique to each track
I need to isolate either i, j or k. To do this, I believe it is sufficient to create an asymmetrically signed pair, or more explicitly one of these:
-i + ji - j-i + ki - k
Since I can only invert a whole stereo track and mix tracks together, the only operations permitted are inverting the original vector pairs and adding them together (so it's allowed to have -i -j and -i -k but you can't directly go to -i +j etc.). Any new vectors created can also be inverted and added of course.
I'm thinking this is sort of a 3-dimensional geometric problem of adding and inverting vectors only, sort of an "only with a compass and straightedge" problem. In that representation, it would mean that under those constraints I have to produce a vector with exactly one dimension or one with exactly one negative and one positive dimension (such as -i + j). Either that or it's a system of linear equations like the other question but represented that way it seems there's no solution.
So far I've been unable to prove whether there is a solution (a series of steps to arrive from the original 2 to one of the needed 4) or find one by trial and error.
Is this possible given the constraints? Is there a way to prove whether this is possible at all?
linear-algebra vectors systems-of-equations
asked Nov 22 at 13:29
mechalynx
1063
add a comment |
up vote
1
down vote
favorite
I'm trying to perform an audio task, to try and isolate audio from mixed tracks. Specifically, I have two tracks with the same music but different sounds. Both are stereo. The only operations I can apply to them are inverting and mixing them. Doing so gets rid of the common parts and leaves on the difference.
There is a similar question:
Isolating audio tracks through mixing
that solves a similar-looking problem through a system of linear
equations, however the conditions are different and I tried applying
the answer given as well as the solution the OP applied but I can't
get a meaningful answer out of it (Wolfram
Alpha).
I'm not sure if I've interpreted that solution correctly.
I've boiled the problem down to this:
- Track 1:
i + j
- Track 2:
i + k
i represents the common audio, j and k represent the audio unique to each track
I need to isolate either i, j or k. To do this, I believe it is sufficient to create an asymmetrically signed pair, or more explicitly one of these:
-i + ji - j-i + ki - k
Since I can only invert a whole stereo track and mix tracks together, the only operations permitted are inverting the original vector pairs and adding them together (so it's allowed to have -i -j and -i -k but you can't directly go to -i +j etc.). Any new vectors created can also be inverted and added of course.
I'm thinking this is sort of a 3-dimensional geometric problem of adding and inverting vectors only, sort of an "only with a compass and straightedge" problem. In that representation, it would mean that under those constraints I have to produce a vector with exactly one dimension or one with exactly one negative and one positive dimension (such as -i + j). Either that or it's a system of linear equations like the other question but represented that way it seems there's no solution.
So far I've been unable to prove whether there is a solution (a series of steps to arrive from the original 2 to one of the needed 4) or find one by trial and error.
Is this possible given the constraints? Is there a way to prove whether this is possible at all?
linear-algebra vectors systems-of-equations
asked Nov 22 at 13:29
mechalynx
1063
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm trying to perform an audio task, to try and isolate audio from mixed tracks. Specifically, I have two tracks with the same music but different sounds. Both are stereo. The only operations I can apply to them are inverting and mixing them. Doing so gets rid of the common parts and leaves on the difference.
There is a similar question:
Isolating audio tracks through mixing
that solves a similar-looking problem through a system of linear
equations, however the conditions are different and I tried applying
the answer given as well as the solution the OP applied but I can't
get a meaningful answer out of it (Wolfram
Alpha).
I'm not sure if I've interpreted that solution correctly.
I've boiled the problem down to this:
- Track 1:
i + j
- Track 2:
i + k
i represents the common audio, j and k represent the audio unique to each track
I need to isolate either i, j or k. To do this, I believe it is sufficient to create an asymmetrically signed pair, or more explicitly one of these:
-i + ji - j-i + ki - k
Since I can only invert a whole stereo track and mix tracks together, the only operations permitted are inverting the original vector pairs and adding them together (so it's allowed to have -i -j and -i -k but you can't directly go to -i +j etc.). Any new vectors created can also be inverted and added of course.
I'm thinking this is sort of a 3-dimensional geometric problem of adding and inverting vectors only, sort of an "only with a compass and straightedge" problem. In that representation, it would mean that under those constraints I have to produce a vector with exactly one dimension or one with exactly one negative and one positive dimension (such as -i + j). Either that or it's a system of linear equations like the other question but represented that way it seems there's no solution.
So far I've been unable to prove whether there is a solution (a series of steps to arrive from the original 2 to one of the needed 4) or find one by trial and error.
Is this possible given the constraints? Is there a way to prove whether this is possible at all?
linear-algebra vectors systems-of-equations
asked Nov 22 at 13:29
mechalynx
1063
I'm trying to perform an audio task, to try and isolate audio from mixed tracks. Specifically, I have two tracks with the same music but different sounds. Both are stereo. The only operations I can apply to them are inverting and mixing them. Doing so gets rid of the common parts and leaves on the difference.
There is a similar question:
Isolating audio tracks through mixing
that solves a similar-looking problem through a system of linear
equations, however the conditions are different and I tried applying
the answer given as well as the solution the OP applied but I can't
get a meaningful answer out of it (Wolfram
Alpha).
I'm not sure if I've interpreted that solution correctly.
I've boiled the problem down to this:
- Track 1:
i + j
- Track 2:
i + k
i represents the common audio, j and k represent the audio unique to each track
I need to isolate either i, j or k. To do this, I believe it is sufficient to create an asymmetrically signed pair, or more explicitly one of these:
-i + ji - j-i + ki - k
Since I can only invert a whole stereo track and mix tracks together, the only operations permitted are inverting the original vector pairs and adding them together (so it's allowed to have -i -j and -i -k but you can't directly go to -i +j etc.). Any new vectors created can also be inverted and added of course.
I'm thinking this is sort of a 3-dimensional geometric problem of adding and inverting vectors only, sort of an "only with a compass and straightedge" problem. In that representation, it would mean that under those constraints I have to produce a vector with exactly one dimension or one with exactly one negative and one positive dimension (such as -i + j). Either that or it's a system of linear equations like the other question but represented that way it seems there's no solution.
So far I've been unable to prove whether there is a solution (a series of steps to arrive from the original 2 to one of the needed 4) or find one by trial and error.
Is this possible given the constraints? Is there a way to prove whether this is possible at all?
linear-algebra vectors systems-of-equations
linear-algebra vectors systems-of-equations
asked Nov 22 at 13:29
mechalynx
1063
asked Nov 22 at 13:29
mechalynx
1063
asked Nov 22 at 13:29
mechalynx
1063
asked Nov 22 at 13:29
mechalynx
1063
asked Nov 22 at 13:29
mechalynx
1063
1063
add a comment |
add a comment |
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