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Math Help - Signal Recovery

  1. #1
    Senior Member Sampras's Avatar
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    Signal Recovery

    Suppose we want to recover an input vector  f \in \mathbb{R}^n from corrupted measurements  y = Af + \epsilon . Note that  A is an  m \times n matrix and  e is a matrix of unknown errors. So then we can recover  f exactly from the data  y by first identifying the error and subtracting it off. Then we need something that annihilates  A so that we can get  f . In other words, we want a function such that:

     f(Af+ \epsilon) = f(Af)+f(\epsilon) = f(\epsilon) .

    In other words, we are looking at the kernel of  Af . So once we identity the error term...can we recover  f exactly? Because given  Af+ \epsilon and  \epsilon , we want to get  f . So ultimately we want to do the following:


    • Identify and subtract error term:  (Af+ \epsilon)-\epsilon
    • Now we are left with  Af . So we want something that annihilates  A .

    Is this possible?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Sampras View Post
    Suppose we want to recover an input vector  f \in \mathbb{R}^n from corrupted measurements  y = Af + \epsilon . Note that  A is an  m \times n matrix and  e is a matrix of unknown errors. So then we can recover  f exactly from the data  y by first identifying the error and subtracting it off. Then we need something that annihilates  A so that we can get  f . In other words, we want a function such that:

     f(Af+ \epsilon) = f(Af)+f(\epsilon) = f(\epsilon) .

    In other words, we are looking at the kernel of  Af . So once we identity the error term...can we recover  f exactly? Because given  Af+ \epsilon and  \epsilon , we want to get  f . So ultimately we want to do the following:


    • Identify and subtract error term:  (Af+ \epsilon)-\epsilon
    • Now we are left with  Af . So we want something that annihilates  A .

    Is this possible?
    Since \epsilon is a unknown error structure this won't work as you are treating \epsilon as known.

    CB
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  3. #3
    Senior Member Sampras's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Since \epsilon is a unknown error structure this won't work as you are treating \epsilon as known.

    CB

    But can you identify the source/syndrome of the error? Not the error itself?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Sampras View Post
    But can you identify the source/syndrome of the error? Not the error itself?
    Well I don't know what that means, but you cannot subtract off an unknown error term. You are probably looking for a least squares estimate of f

    CB
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