I'm currently attempting to study up on adaptive digital filters. My book presents the diagram I've included below and I'm having trouble understanding conceptually what it's indicating. The problem deals with noise cancelation. The idea is that someone is driving and makes a phone call. The x(k) is their voice input. There's a reference mic at v(k) which picks up road noise. I know that the ultimate goal is to filter road noise from our transmitted voice signal.
The desired output is obviously:
$$ d(k)=x(k)+v(k) $$
The error in this case is:
$$ e(k)=x(k)+v(k)-y(k) $$
Taking a quote from my book, it states
If the speech x(k) and the additive road noise v(k) are uncorrelated with one another, then the minimum possible value for $e^2(k)$ occurs when y(k) = v(k), which corresponds to the road noise being removed completely from the transmitted speech signal e(k).
I don't understand how e(k) is our output of the system though. It seems to me that if we minimize our error, then it approaches zero. This means that $d(k)-y(k) = e(k)=0$ Consequently if our output is the error and we've minimized it, it seems like we're outputting 0 not a transmitted signal e(k) with the road noise removed!? I guess I'm asking why our desired output d(k) isn't our output....why is the error the output?
Can somebody help me understand this conceptually? Thank you for your help! Please let me know if I need to clarify anything.