Many modern communication channels are modeled as
a Gaussian multiple-input multiple-output (MIMO) channel.
Examples include multi-tone digital subscriber line (DSL),
orthogonal frequency division multiplexing (OFDM) and multiple
transmit-receive antenna systems.

Here, we consider Gaussian multiple-input multiple output
(MIMO) channels with discrete input alphabets. 

The MIMO channel can be transformed into a set of parallel 
subchannels using Singular Value Decomposition (SVD). 
In OFDM the FFT operation will generate the parallel 
subchannels, given a suitable cyclic prefix is added.

We propose a non-diagonal precoder based on the pairing  
of subchannels to increase the mutual information. 
The pairings are given by simple 2x2 real rotation matrices.
(parameterized with a single angle). 

This precoding structure enables us to express the total 
mutual information as a sum of the mutual information of all
the pairs. The problem of finding the optimal precoder with the
above structure, which maximizes the total mutual information,
is solved by i) optimizing the rotation angle and the power
allocation within each pair and ii) finding the optimal 
pairing and power allocation among the pairs. 

It is shown that the mutual information achieved with the 
proposed pairing scheme is very close to that achieved with 
the optimal precoder by Cruz et al., and is significantly 
better than Mercury/waterfilling strategy by Lozano et al.
Our approach greatly simplifies both the precoder optimization 
and the detection complexity, making it suitable for
practical applications.