# Signal Recovery

• Feb 8th 2010, 12:33 PM
Sampras
Signal Recovery
Suppose we want to recover an input vector $\displaystyle f \in \mathbb{R}^n$ from corrupted measurements $\displaystyle y = Af + \epsilon$. Note that $\displaystyle A$ is an $\displaystyle m \times n$ matrix and $\displaystyle e$ is a matrix of unknown errors. So then we can recover $\displaystyle f$ exactly from the data $\displaystyle y$ by first identifying the error and subtracting it off. Then we need something that annihilates $\displaystyle A$ so that we can get $\displaystyle f$. In other words, we want a function such that:

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

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

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

Is this possible?
• Feb 9th 2010, 09:30 AM
CaptainBlack
Quote:

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

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

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

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

Is this possible?

Since $\displaystyle \epsilon$ is a unknown error structure this won't work as you are treating $\displaystyle \epsilon$ as known.

CB
• Feb 9th 2010, 10:20 AM
Sampras
Quote:

Originally Posted by CaptainBlack
Since $\displaystyle \epsilon$ is a unknown error structure this won't work as you are treating $\displaystyle \epsilon$ as known.

CB

But can you identify the source/syndrome of the error? Not the error itself?
• Feb 9th 2010, 11:10 AM
CaptainBlack
Quote:

Originally Posted by Sampras
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