The WindoWSIL

Note 1: You may want to check out my followup post on the diversity-multiplexing tradeoff. I think it’s a bit clearer than this one.

Note 2: You need to click on the figures to view them.

I’d like to share some intuition on the diversity-multiplexing tradeoff (DMT) given to me by Rahul Vaze. This way of looking at it really makes clear what its values and limitations are.

The first step in the intuition is to note the definitions of diversity and multiplexing. Here, multiplexing is not spatial multiplexing. That is, it is not the number of parallel streams being transmitted. This is one place where a lot of people seem to get confused, because strategies such as spatial multiplexing with maximum likelihood detectors can use the maximum available degrees of freedom in the channel while achieving a diversity gain, whereas the DMT shows that maximum “multiplexing” gain will be at the expense of all diversity.

To understand what multiplexing means here, we first define rate R to be the number of spatial streams being transmitted times the number of bits per transmission per stream. Thus, if we are sending streams of M-QAM, bits per transmission. Note that as SNR increases, we can increase M, but we cannot Ns above its maximum level. For this R, and for a given detection strategy, there is a probability of outage curve that resembles the figure below. At high SNR, the log-log curve is a line with slope -d, the diversity order of the strategy.

Now imagine we are at some SNR g1, as shown, with outage probability p1. Suppose the SNR increases to g2. If our constellation size does not increase at all, we are sending at the same rate, and the outage curve remains as shown; we are now operating with outage probability p2. If for every SNR increase, we keep sending the same constellation, and choose to operate at a lower outage probability, at high SNR we will be benefiting from the full diversity gain of the strategy.

We now define multiplexing gain as . Recall our definition of , which does not readily depend on SNR. However, we can choose to make M depend on SNR by using continuous-rate (ideal) adaptive modulation. So M = f(SNR). In the previous example, M = M0, which was independent of SNR, so r = 0. This corresponds to the y-intercept of the DMT curve where the maximum diversity (of the strategy, not necessarily of the channel) is achieved.

It is now natural to ask “what is the fastest achievable rate of change of M with SNR“? Observe that log M is the number of bits per transmission over a SISO link. By Shannon’s formula for the Gaussian channel, . Therefore, . So what happens when we let , i.e., when we allow our constellation to grow linearly with SNR? We see that now . This corresponds to the x-intercept of the DMT curve. But why is diversity zero in this case? The answer to this will give us intuition about all the other points on the DMT curve.

First we observe that, for a fixed transmit/receive strategy, the probability of outage curve is a function of both SNR and R. For simplicity we usually calculate, and plot, the outage curve for fixed R. This is also because, in practice, only integer-rate constellations are used, so an outage plot as a function of R would be discrete in R. However, if we pretend we can send real values of log M (that is, we have constellations of order M, where M is not an integer power of 2), we can plot outage as a continuous function of R. This is still not done, because 3-D plots are difficult to observe on paper, and are generally unnecessary. Instead, on a single 2-D plot, we take cross sections of the curve for fixed values of R.

So in the figure below I’ve drawn some possible outage curves for a fixed strategy as a function of SNR and for different R. Suppose again that we are operating with R = Ns log M0 at SNR g0 with outage p0. This time, however, when g0 increases to g1, we increase M0 to M1 linearly with SNR. Thus, we have kept the gap between our rate R and capacity C constant, and the outage probability will be unchanged.

Adapting the rate with SNR effectively results in a new outage curve that is not a cross section of the general outage curve using a plane parallel to the R = 0 plane, but instead is a cross section using a plane with slope r in the R dimension. I hope the figure below (click for a larger version) explains it better than I can with words.

Now this new outage curve is for a fixed r. For this fixed r, the log-log outage probability curve will be linear with some slope at high SNR, and we call this slope -d(r). This pair (r,d(r)) is one point on the DMT curve. As we vary the R-slope (r) of the cross-section plane in the figure above, we observe different d(r). Obviously, the steepest slope will occur when the plane is perpendicular to the R-plane, or when r = 0. Correspondingly, if this cross-section plane is such that outage is flat, then r is along the line of constant outage (color in the figure above) and d(r) = 0.

Because this tradeoff curve is only relevant asymptotically with SNR (via the diversity gain), it has no application to adaptive signaling strategies. For instance, if at SNR g1 you decide to switch from, say, the Alamouti code to spatial multiplexing, this switch is completely lost in the DMT tradeoff. The tradeoff only tells you how fast you can scale the rate of a fixed strategy relative to how fast capacity is changing at high SNR and still achieve a linear log-log decrease in outage probability.