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This post is courtesy of Johan and Nathanaël – a big thanks !

Today Olivier Levêque made a very nice presentation about results of

large random matrix theory applied to Multi Input Multi Output (MIMO)

antenna systems and large random networks.

He started with a short introduction to information theory, especially

to capacity of a communication channel, on the blackboard, with style.

Olivier then showed a very powerful result in the field of MIMO

communications, which states that when the number $\latex n$ of antennas

grows we get a capacity or order $\latex n$, or

.

This result is powerful since it means that theoretically the capacity

grows linarly with the number of antennas (which is a significant

improvement of the previously believed limiting order

for multi antennas systems).

Olivier also enphasized that we can achieve this capacity without

increasing the power P, but simply by distributing it in a clever way,

which can be done using water-filling techniques. He concluded the first

part of the lecture by reminding that, although this result is

demonstrated for n goes to infinity, it actually works already for small

values of n, which explains the joke common amongst larse random matrix

people which states : .

The second main part of Oliver’s talk dealed with large self-organized

wireless communication networks. The main question proposed to answer

was: assuming that n users want to communicate in a given area with an

O(1) density, what is the rate they can hope to achieve ?

And in relation to this statement, how does the rate scale with n ?

In a setup where we wish to establish n communication channels, we use a

model where the signals that are received depend on the gain between two

nodes divided by the distance, with an additive white gaussian noise.

Note that the gain is i.i.d. and follows a complex gaussian

distribution. Finally the power of emission per user is limited and has

a given value P. For this model we would like to achieve

Following this model, a very naive scheme is to dispatch the n different

pairwise communication instances into n different time slots (or

equivalently n different frequency bands). In this case we achieve

which is really bad for n large.

The second approach uses multi-hop communication. And in this case,

Olivier explained that due to an overload of the users in the center of

the network the limiting rate is ,

which is better but still not good.

Finally, Oliver presented a scheme coined Distributed MIMO

communications which proceeds in an iterative way. At each stage, a user

broadcasts its message to his neighborhood, which will then send the

message to the neighborhood of the destination user, which receives all

the messages from its own neighbors. We can then recursively use this

scheme for the communication within the neighborhood. In this case,

Oliver showed that the limiting rate for large n with k recursions is

which tends to when

k goes to infinity.

Oliver concluded his lecture by explaining a controversy about this

model, which states that considering gains to be i.i.d. is in fact

unrealistic. Some authors indeed claims that for this setup we can only

hope to have i.i.d. random variables (instead of the

that are needed to prove the limits presented above. And

this would bring down the rate back to .

The final word provided a way to avoid this problem simply by stating

that practical networks tend to have a lower density, and that we can

achieve interesting rates given that the area is large enough.