LPC residual

The answer is actually ``not precisely'' for a finite $N$.

Rather, we determine the cofficients that give the best prediction by minimizing the difference, or error $e(n)$, between the actual sample values of the input waveform $y(n)$ and the waveform re-created using the derived predictors $\hat{y}(n)$.

$$\min_n\{e(n)\} = \min_n\{\hat{y}(n) - y(n)\}.$$

The smaller the average value of the error, also called the residual, the better the set of predictors.

The residual may be used to exactly reconstruct the original signal $y(n)$ by using it as an input to our all-pole filter, that is

$$y(n) = b_0 e(n) + a_1y(n-1) + a_2y(n-2) + ... + a_Ny(n-N)$$

where $b_0$ is a scaling factor that gives the correct amplitude.

When the speech is voiced, the residual is essentially a periodic pulse waveform with the same fundamental frequency as the speech. When unvoiced, the residual is similar to white noise.