Adaptive Digital Filter Block Diagram Question

Question

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.

Answer 1

Judging from the figure, the situation is slightly different from your explanation in the question. The noise $v(k)$ is the actual noise in the signal, not the noise picked up by the reference microphone. So the noisy signal is $d(k)=x(k)+v(k)$. If you knew $v(k)$ you could simply subtract it from $x(k)$ without the need to use an adaptive filter. What you have is another noise signal $r(k)$, which is a filtered version of the noise $v(k)$. This unknown filter is depicted by the "black box" in the figure. It is filtered because the transfer function from the noise source (road, tires, etc.) to the reference microphone is different from the transfer function to the microphone recording the speech. What the adaptive filter is trying to do is estimate the noise in the speech signal from the reference noise, i.e. it tries to invert the unknown filter in the black box. This can be achieved by minimizing the power of the error signal $e(k)$. The reason is that if you assume that speech and noise are uncorrelated, the output of the adaptive filter can only reduce the noise component in $d(k)$, not the speech component. So the power of $e(k)$ becomes a minimum if the output of the adaptive filter $y(k)$ equals $v(k)$. You don't need to worry that the speech signal is removed because $y(k)$ cannot model the speech signal at all, because noise and speech are assumed to be uncorrelated. So ideally the error signal $e(k)$ contains only clean speech. Note that due to noise and speech being uncorrelated, the power of $e(k)$ can never become zero. Neither can the error signal itself become zero (for all $k$).