KLT Completely Decorrelates the Signal

KLT is the optimal orthogonal transform in the following sense:

• KLT completely decorrelates the signal
• KLT optimally compacts the energy (information) contained in the signal.

The first property is simply due to the definition of KLT, and the second property is due to the fact that KLT redistributes the energy among the $N$ components in such a way that most of the energy is contained in a small number of components of $Y=\Phi^T X$.

To see the first property, consider the mean vector $M_Y$ and covariance matrix $\Sigma_Y$ of $Y$:

$$M_Y = E(Y)=E(\Phi^T X)=\Phi^T E(X)=\Phi^T M_X$$

$$\Sigma_Y = E(YY^{T})-M_Y M_Y^T=E[\Phi^{T}X(\Phi^{T}X)^{T}] -\Phi^T M_X (\Phi^T M_X)^T$$

$$= E[\Phi^{T}(XX^{T})\Phi]-\Phi^T M_X M_X^T \Phi$$

$$= \Phi^T [ E(XX^{T}) -M_X M_X^T ] \Phi$$

$$= \Phi^{T}\Sigma_X\Phi=\Lambda=diag[\lambda_0, \lambda_1, \cdots, \lambda_{N-1}]$$

Or in matrix form, we have

$$\Sigma_Y = \Phi^T \Sigma_X \Phi=\Lambda= \left[ \begin{arra... ...s \\ 0 & 0 & \cdots & \sigma^2_{y_{N-1}} \end{array} \right]$$

We see that after KLT, the correlation matrix of the signal is diagonalized, i.e., the correlation $\sigma_{ij}=0$ between any two components $x_i$ and $x_j$ is always zero. In other words, the signal is completely decorrelated.

Also, we have

$$R_Y = \Sigma_Y+M_Y M_Y^T=\Lambda+M_Y M_Y^T$$