Degraded and Non-Degraded MIMO wireless channels

physically degraded

Dear All,

Topic for Discussion:

what are degraded and non-degraded MIMO Broadcast channels?

Waiting for replies,

Cheers,

MAK

non-degraded channel

I know something about Degraded relay channel
In typical relay channel, there is a sender, relay and a receiver.
During first phase of transmission the sender broadcasts its data which is received both by relay and receiver, but the receiver is not able to decode it.
In the second phase sender and relay jointly send some resulation information to receiver so that receive can decode the data.

In this situation a relay channel is said to be Degraded relay channel if the relay reciever is better than the ulimate receiver.

non degraded broadcast channel for example

Dear Mkhan,

Waiting for your reply with some nice explaination.

degraded channel

Hi urwelcome,

urwelcome said:

Dear Mkhan,

Waiting for your reply with some nice explaination.

Click to expand...

,

You are right, since many days I posted this topic, I am thinking to write about it, its just a matter of time, that i am a bit busy these days, but hopefully at the end of this week I will try my best to explain the concept of degraded and non-degraded channels, as it is one of the most important concept in MIMO studies.

Once again thank you very much for reminding me my responsibility for posting the explanation.

Regards,

MAK

mimo relay channel

Refereces
Note: These are few important references I am quoting here but there are many other related articles which I acknowledge fully.

[1]: T.M.Cover and J.A.Cover, "Elements of Information Theory", Chapter 15, Wiley Series, 2005.
[2]: Scribe Notes, Lecturer M.Sudan, "6.441 Transmission of Information", Spring 2006, MIT, Dept. of Electrical Engg. and Comp. Science.
[3]: E.Biglieri et al, "MIMO Wireless Communications", Chapter 2, Cambridge University Press, 2007.

MIMO Broadcast Channel
A traditional broadcast channel is defined as where a transmitter is sending same information to multiple receivers. The examples of this broadcast channel are radio and TV communication. But this definition is more generalized in the conext of MIMO communication systems.

In MIMO communication system, when multiple antennas at BS are communicating with multiple receivers, not necessarily sending the same information, this is called DownLink (DL) communication. This DL communication is referred as Broadcast channel in MIMO as well, but a caution, it is different in the sense from traditional BC that not necessarily the same information is being sent to the receivers. Now lets define formally the MIMO broadcast channel which we will refer here as MIMO BC.

Definition A MIMO BC is defined as when multiple antennas at BS are sending information to multiple receivers.

Suppose we have some messages \[W_1, W_2\] and we have an encoder \[E\] to encode these messages in single word \[X\] and we sent this encoded signal over the channel. Now the channel produces two outputs \[Y_1, Y_2\] and there are two decoders, each at one receiver, which produces the same message \[W_1, W_2\].

\[(W_1, W_2) \rightarrow Encoder (E) \rightarrow X (coded word) \rightarrow Channel \rightarrow (Y_1 , Y_2) \rightarrow Decoders (D_1, D_2) \rightarrow (W_1 , W_2)\]

Now lets divert our attention to degraded BCs. Degraded broadcast channels are special cases of broadcast channels. A degraded broadcast channel can be categorised in two types.

Physically degraded BC and Stochastically degraded BC. A physically degraded BC is one which forms a Mrkove chain, now this is confusing words because we are not going into discussion of Markovian theory. So in simple words, it is \[X \rightarrow Y_1 \rightarrow Y_2\], which means that signal transmitted from source degrades to first receiver and then degrades more to the second receiver. But this is different than the relay channel, because in relay channel the first receiver retransmits the signal for the next receiver, but here the first reciver does not transmits any thing. Stochastically degraded channel is one which is statistically equivalent to physically degraded channel which means it has the same mean, variance and finally same marginal distributions. An example of degraed BC is a scalar broadcast channel but a vector BC channel is not degraded completely. The capacity region (achieveable rate pairs) of a degraded BC is completely known and discussed in [1] but the capacity region of vector BC is not known as it is not degraded. The reason is simple that the scalar degraded channels can be ordered completely according to their channel strength while the vector BC channels (MIMO BC) can not be ordered completely as now the channel of a single user is a block of matrix in the channel matrix \[H\]. In order to achieve the achieve the capacity region of the single antenna AWGN BC, the idea of superpostion coding is used, but this simply can not be used for the MIMO BC bcause it losts its beautiful property of degradedness and now its up-graded BC. Hence in order to achieve the capacity region of MIMO BC, which is not degraded fully, a non-linear coding technique is used, known as Dirty Paper Coding (DPC_ first proposed by Costa in 1983. So in MIMO BC superpostion coding or successive decoding is no longer a capacity achieveing scheme.

However, there is another concept, known as Duality between MAC and BC which can be used to achive the capacity region of MIMO BC by knowing the capacity region of MAC channel. There are two groups of researchers from UC Berkley and Stanford University, who reported the results on MAC-BC duality independenlty in early 2002 and 2003. The idea is extended from scalar channel to vector channel.

Henec the conclusion is that a single antenna AWGN BC is degraded and we can order the channel matrix of this degraded channel according to the strength and by using the superpositon coding we can achieve the capacity region of this degraded BC channel. But unfortunately when this single antenna channel is upgraded to multiple antenna (MIMO BC), it loses its degradedness and hence becomes a difficult problem to solve.

In the end, I tried to summarise the results in a very simple and short way. But please mention any mistakes or confusion in this article. All suggestions are welcome. I mention some of the important references at the top as well, but again these references are not limited to this list and there are lot more reference to be included.

Thanks very much to all,

Cheers~

MAK

non-degraded mimo bc

Dear MAK,

Thank you very much for your explaination,it was very nice.
if you have some time than I would request you to add something about the Superpostion coding,Diry Paper coding and name any practical coding schemes which mimic these.

Thanks,

Hi Everyone,

First, I would like to say that this topic was really helpful to me. I just have one question. In the physically degraded BC, you said that X, Y1 and Y2 follow a markov chain which implies a certain probability distribution dependence. When you came to talk about the stochastically degraded, you said "Stochastically degraded channel is one which is statistically equivalent to physically degraded channel which means it has the same mean, variance and finally same marginal distributions" what do you mean by statistical equivalence ? And which "mean, variance and marginal distributions" that are going to be the same ? I just need more elaboration on the Stochastically Degraded meaning.

Thanks,
Mohamed