Mutual coupling

Let us consider the presence of mutual coupling among the M transmission lines. In this case the model is again

However, due to the presence of mutual coupling between the M lines, the matrix H is generally full

wherein the off-diagonal terms take into account the mutual coupling. A natural question is: what is the equivalent decoupled parallel of transmission lines? 

In order to solve this problem let us apply the Singular Value Decomposition to the H matrix. The SVD gives three matrices,

wherein U and V are unitary matrices, and Sigma is a diagonal matrix whose upper submatrix s is:

wherein r is the rank of the H matrix.

The columns of U and V are the singular functions and the elements of the upper submatrix are the singular values of the matrix H.

For sake of simplicity, let us suppose that the matrix has full rank, i.e. r=M.

By using the SVD we have

In practice, instead of the vector x, the vector at the input of the transmission lines must be x'=Vx , while at the output we must consider the vector y=U^H y instead of y. In this way we have

and we can restore the orthogonality of the channels. In other words, due to the mutual coupling between the lines the output y_k is a linear combination of all the inputs x_j , j=1,...,M . By preprocessing and postprocessing the data, for example by introducing a microwave circuit or equivalently introducing a data processing at the input and the output of the transmission lines we can compensate the mutual coupling, and the output y'_k is related only to the input y'_k .

Again, the channel capacity of this MIMO channel is the sum of the capacity of M Single Input-Single Output (SISO) equivalent orthogonal channels, each of them having an equivalent channel response sigma_k : 

In practical instances, the importance of each equivalent orthogonal channel depends on the value of sigma_k . If sigma_k is very low with respect to sigma_1 its contribution is negligible. In particular, if the rank of the channel matrix is r<M the sum in ranges from 1 to r. In this case the communication system consisting of M transmission lines is equivalent to a communication system having only r transmission lines from the point of view of the channel capacity.

It is worth noting that in the example discussed above, the transmitter must have information on the channel, and in particular on the V matrix. As previously discussed, if the transmitter has knowledge of the channel (including the Sigma matrix), it is possible to make better use of the transmitted power using the waterpouring algorithm, that distributes the power among the equivalent orthogonal channels taking into account the singular value amplitude associated to each equivalent channel. Consequently. if the singular values are not all equal, the capacity using the waterpouring algorithm is higher than the one evaluated using the above formul. However, the use of the waterpouring algorithm would cause an increase in the complexity of the formulas without changing the conceptual results shown in this paper. For the sake of simplicity, in the following we will evaluate the channel capacity supposing No Channel State Information (NCSI). This is equivalent to considering a system in which the transmitter has knowledge of V but not of Sigma.