Beamforming in the Relay Channel is Hard
Suppose you are exploring the vast landscapes of a foreign country with a friend, and you want to ask your friend what time it is. The main problem is that your friend is half a mile away and may not be able to understand you, assuming he can hear you at all. There is, however, a very friendly local man roughly half way between you and your friend who is willing to help you communicate your message. The downside is that the local man does not speak your language and can only roughly repeat the sounds as he heard them.
Beamforming, then, can be thought of as tailoring what you say for the local man (by, say, speaking very clearly but less loudly) or tailoring your words for your friend (by yelling loudly but losing enunciation). Your friend can use what he hears from you and the local man to figure out what it is you were trying to say. The problem is that choosing between tailoring for either route is strictly suboptimal; there is a sweet spot in between that will allow your friend to decipher your message better than in any other case. This sweet spot, unfortunately, appears to be very computationally expensive to find.
What Can Be Done?
A quick solution to the problem might be to yell as loud as you can while still enunciating above a certain threshold. The problem with this in wireless networks is that such a tradeoff is not easily expressed mathematically. What does in between mean? Further, by picking an arbitrary level of “in-between-ness”, we cannot guarantee ourselves a minimum probability that our friend has understood the message, and this is really what engineering is about.
To extend the analogy to MIMO we need multiple people around you helping you yell your message, multiple local men helping to relay the message, and multiple people helping your friend receive the message. If you understand waves, you will know that if all of your helpers are saying the same thing, there is a good chance their voices will cancel each other out (if they all have the same voice and spoke synchronously). Beamforming is now the art of yelling so that your voices add constructively at your receiver; but again there is no guarantee that if they add constructively at the group of local men that they will add constructively at the group helping your friend, and vice versa.
Main results and future areas of research
We have studied the case where one member of each group is chosen to communicate the message. In wireless systems, this is known as antenna selection. We are choosing to transmit on the antenna that maximizes the signal power at the destination, taking into effect what the relay will do. This problem has been thoroughly studied for the case where no relay is present, whereas we have extended it to include a relay that cannot understand the message (hence why the local man cannot speak your language).
This work is interesting because, compared to other proposed relaying strategies, the mathematical analysis is straightforward, the implementation is relatively simple, and we can prove interesting things. In particular, the math becomes easier because, instead of having all the group members’ voices overlapping, we only have one voice at a time. The implementation is relatively simple because the talkers don’t need to know how their voices are interfering with each other. In beamforming, you basically have each person change what they say by a delay and amplitude change. Thus, for each of your helpers, the group of friends needs to communicate back to you exactly what kind of delay and amplitude change each helper needs to make in order for the voices to properly add constructively. With antenna selection, the only thing that needs to be fed back is which person should communicate. This is a relatively easy thing to measure and communicate.
The simplification of analysis has allowed us to prove that antenna selection is optimal in the sense of spatial diversity gain; that is, antenna selection is able to exploit all the independent paths in space the message might possibly take to the destination. In fact, the only disadvantage between optimal beamforming and our antenna selection is there is a slightly larger probability that the friend will not be able to understand the message if we use antenna selection. However, if we can vary how loud the people helping the source of the message can yell, we see that as we vary this yelling power, the ratio between the probability of misunderstanding for antenna selection and optimal beamforming, respectively, is a constant; the gap does not grow.
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