A.23  Spatial Whitening Applied to Interference Mitigation

Assume that a communication channel has been partitioned into independent parallel “subchannels” by DMT or SVD (SVD-based partitioning is summarized in Application 6.4). Then, the predictor of Eq. (A.106) can be used as follows.

Assume a single “subchannel”, which would correspond to a single tone $k$ in DMT. Also, as depicted in figlalien_crosstalk, assume that $M$ interferers provoke crosstalk on the $V$ copper lines of interest of the vectored group through a channel matrix $H$ of dimension $V\; \times \; M$ as in Eq. (A.101). For DMT, the matrix $H$ corresponds to the frequency response (eventually complex-valued) at tone $k$. The vectors corresponding to transmission over tone $k$ in a DMT system are [GP06] (Eq. (15)):

where $T_k$ is a diagonal channel matrix and ${\mathbf{\text{N}}}_k$ is the noise corresponding to both thermal (background) noise and alien crosstalk. Hence, the autocorrelation $R_{nn}$ of ${\mathbf{\text{N}}}_k$ is nondiagonal and the goal is to reduce the noise power via decorrelation (whitening).

Using the innovations representation, $R_{nn}$ is factored as $R_{nn}=\; L{D}_y^2{L}^{\ast }$

For simplicity, consider a noise free condition ($\mathbf{\text{N}}=\; 0$ in Eq. (A.101)) and that the interferers are i.i.d. zero-mean Gaussians with autocorrelation $R_{xx}={\sigma }_x^2\mathbf{\text{I}}$. From Eq. (A.102),

The predictor of Eq. (A.106) can be applied to $\mathbf{\text{Y}}$ to decrease...

codlto-be-done performs the following experiment: calculates the prediction gain over frequency for a set measured crosstalk channels.