# Math Help - Signal Recovery

1. ## 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?

2. Originally Posted by Sampras
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

3. Originally Posted by CaptainBlack
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?

4. 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