The Theory of Pre-whitening

The central idea of pre-whitening is to use two PE filters. Each PE filter minimizes the mean square energy of its output given its input. However, the second filter is given the output of the first filter. Once we have found the second filter, we apply it to the original input (the input that we used to train the first filter). Obviously, the second filter will not be optimal for removing the predictable events of the original input. As a matter of fact, the second filter is blind with regard to the events the first filter removed from the data. The second filter applied to the original input will remove the components it was trained to remove, but it will preserve the components that the first filter is capable of removing.

Imagine that an input time series contains three predictable components. Given an input time series and a given filter length, we find the first optimal PE filter. The data component this first filter removes is the first predictable component.

If the spectrum of the input time series is Y₁, then a PE filter P₁ corresponds to

$$P_1 = \frac{1}{(<\bar{Y_1} Y_1\gt)^{1/2}}$$

where $(<\bar{Y} Y\gt)^{1/2}$ represents a smooth version of Y's spectrum. The output spectrum after such a PE filter step is

$$\begin{eqnarray} P_1 Y_1 &=& \frac{Y_1}{(<\bar{Y_1} Y_1\gt)^{1/2}} \nonumber \\ &=& W_1 .\nonumber\end{eqnarray}$$

where W₁ represents a spectrum that tends to be white. If P₁ were a perfect PE filter, then W₁ would be white. But since our PE filter implementations are limited to a few coefficients, W₁ only tends to be white.

I define Y₂ oas the utput of the first PE filter

$$\begin{eqnarray} Y_2 &=& \frac{1}{(<\bar{Y_1} Y_1\gt)^{1/2}} Y_1 \nonumber \\ &=& W_1 .\end{eqnarray}$$ (1)

Consequently, the corresponding PE filter is

$$P_2 = \frac{1}{(\ll\bar{Y_2} Y_2 \gg)^{1/2}}$$

where $\ll \gg$ represents a smoothing that might be different from the P₁ smoothing <>. Applying P₂ to Y₂ yields

$$\begin{eqnarray} P_2 Y_2 &=& \frac{Y_2}{(\ll\bar{Y_2} Y_2\gg)^{1/2}} \nonumber \\ &=& W_2 \nonumber\end{eqnarray}$$

where W₂ represents a spectrum that again only tends to be white. Substitution of the the expression 1 for Y₂ yields

W₂ = P₂ Y₂ = P₂ W₁

which indicates that W₂ is whiter than W₁.

If we apply the second PE filter, P₂ to the original data spectrum Y₁, we finally get

$$P_2 Y_1 = \frac{Y_1}{(\ll\bar{Y_2} Y_2\gg)^{1/2}}$$

The simple identity

$$Y_1 = \frac{Y_1}{(<\bar{Y_1} Y_1\gt)^{1/2}} (<\bar{Y_1} Y_1\gt)^{1/2}$$

allows us to introduce Y₂ according to expression 1

$$\begin{eqnarray} P_2 Y_1 &=& P_2 \frac{1}{(<\bar{Y_1} Y_1\gt)^{1/2}} Y_1 (<\bar{... ...)^{1/2} \nonumber \\ &=& W_2 (<\bar{Y_1} Y_1\gt)^{1/2} \nonumber \end{eqnarray}$$

Consequently, the resulting spectrum of the two step filter operation is the product of the first predictable spectral component $(<\bar{Y_1} Y_1\gt)^{1/2}$ and the rather white spectrum W₂. The second predictable component $(\ll \bar{Y} Y\gg)^{1/2}$ is removed. In that sense the cascading of PE filters implements a high resolution, data adaptive notch filter. The component removed, however, is not a fixed frequency component but a data dependent band of predictable components.