After my post on the diversity-multiplexing tradeoff, which seemed to create confusion instead of alleviating it, Prof. Heath asked me to clarify the finite-SNR diversity-multiplexing tradeoff proposed by Narasimhan in [1] (pdf). The previous explanation of the Zheng-Tse DMT really simplifies the explanation of the Narasimhan DMT. If you understood the previous post (in the rest of this post I will assume you didn’t), then the finite-SNR result simply defines diversity as the slope of the outage curve at a given SNR. If you have the 3-D plot of outage versus SNR and R, then fix r. Then, for each SNR, the DMT curve is given by (d,r,SNR), where d is the slope of the outage curve at SNR. There is a little bit more to it, so keep reading even if you understood that.

Now, I’m back to assuming you didn’t understand the previous post. This is a safe assumption, as I feel I didn’t explain it very well. We can all agree, I think, that in an ideal world with real-rate constellations, we could express the probability of outage as a function of both SNR and R. Again, R is simply the spectral efficiency of the signaling mode. For instance, if we are sending two streams of 16-QAM, R=8.

So we have this P_out(R,SNR) expression. We’re going to imagine this curve plotted in three dimensions. First, imagine a normal 2-D plot of outage versus SNR for a fixed R. Suppose, for instance, R = 1; that is, we’re sending BPSK using the Alamouti code. Simple, eh? Now, bring in the third dimension of the plot, pointing toward you. Thus, the x-axis is SNR in dB, the y-axis is R, and the z-axis is outage probability.

We can imagine that as R increases, the curve becomes more spread out in the SNR dimension. Intuitively, at any given SNR, a larger rate means a higher probability of outage. It’s important that you can see this before moving on. The 3D figure from the previous post might help.

Again, our three axes are SNR, R, and outage. For a moment, let’s forget about the probability of outage and focus on the relationship between SNR and R. For a fixed number of transmit and receive antennas (Nt and Nr, respectively), MIMO information theory tells us that R is always less than or equal to m*log2(1+p), where p is the per-stream SNR at the receiver, and m=min{Nt,Nr}. The figure below plots this upper bound on R versus 10*log10(SNR) for m=1,2. The red line corresponds to m=2, and the blue line corresponds to m=1. We also plot the lines m*log2(SNR) for m=1,2. At SNR > 15 dB, the m*log(SNR) and m*log(1+SNR) lines virtually overlap for respective m.

Now, remember these are upper bounds on R; in practice, we will not achieve them. However, for a particular signalling strategy, we can plot our R on this plot. For any strategy that does not adapt the rate with SNR, our rate plot will be rather boring. It will be a step function with transition at the SNR below which we cannot reliably communicate at that rate (assuming no channel coding).

Observe that as SNR grows arbitrarily large, the ratio between our fixed rate R0 and the SISO capacity (the blue curve) will become arbitrarily small, regardless of what R0 is. This ratio, at any particular SNR, is the definition of multiplexing gain given in the finite-SNR DMT. Let me repeat this for clarity. For any adaptive signalling strategy, we can plot its rate as a function of SNR. The finite-SNR DMT multiplexing gain at a given SNR is defined as the ratio of our rate to the SISO capacity at that SNR.

We see right away how this is a generalization of the Zheng-Tse DMT. They define multiplexing gain as the limit of the ratio between our R curve and the linear magenta curve as SNR goes to infinity. If we do not adapt with SNR, this ratio becomes arbitrarily small, such that its limit is zero (r = 0). For any finite SNR, however, this ratio is strictly nonzero.

Recall for a second that, in my previous post, I defined the multiplexing gain as the derivative of R with log(SNR). Strictly speaking, this was inaccurate. It is only valid at high SNR and if R is linear with log(SNR). In practical systems, R will be nonlinear in SNR.

Now, back to our figures. We’ve plotted R as a function of SNR, and defined the multiplexing gain as the ratio between our R and the SISO channel capacity. These curves on this 2D plot, however, are surfaces when extended into the outage dimension. This is very important to visualize. Think of the line y = x in two dimensions. Extending this equation to three dimensions results in a plane perpedicular to the y-x plane. We can do the same with our R-SNR plots.

So now, to recap, in our minds we have a 3 dimensional space. The dimensions of this space are x = SNR, y = R, and z = Probability of outage. In this space we have plotted probability of outage for our signalling mode (say, for example, the Alamouti code). Now, we have a surface sticking up out of the x-y plane that represents how we plan to vary R with SNR. The intersection of these two surfaces is a curve. That curve is the probability of outage curve for our signalling mode and adaptation strategy.

Now we define diversity as the negative slope of that curve at any given SNR. Again, this is easily seen as a generalization of Zheng-Tse, since their diversity is the negative slope of that curve as SNR goes to infiinity.

Then, the point on the finite-SNR DMT curve is the triple (d,r,SNR), where again r is the ratio of our R to the SISO capacity at the given SNR.

I hope that made sense, because that’s about it. I’ll now talk about some intuition.

First, since r is the ratio of our R to the SISO capacity, r > 0. The limit r = 0 doesn’t make any sense, but we can approach it arbitrarily closely at any finite SNR. Thus, we can’t really achieve the full diversity gain of the channel at finite SNR.

Further, since our R will be strictly less than capacity, the ratio of R to the SISO capacity will always be less than the ratio of the system capacity to the SISO capacity. Effectively, we won’t achieve the full multiplexing gain of the channel at finite SNR, either.

Finally, we note what this finite-SNR DMT can tell us that the infinite version cannot. By taking the limit of the ratio of R over the SISO capacity as SNR goes to infinity, we’re effectively saying R is a line of slope r. This is why I could fudge it and say r = dR/dlog SNR for the infinite-SNR DMT but not for the finite-SNR DMT. By not taking this limit, the math becomes all but intractable, but we can now make r nonlinear! It can increase (or decrease) however we wish, although it will be upper bounded by the ratio of our capacity to the SISO capacity.

Also notice that, if we don’t adapt with SNR, r is still greater than zero for any finite SNR. For example, again, say R = 1. At SNR=10dB, r will be approximately 1/3, and we’ll say d is some d0. At SNR = 20 dB, r is approximately 1/6, and d will be some d1 > d0 (the slope of any outage curve is monotonic decreasing with SNR). Thus, as SNR increases, on the DMT curve, we move up (d increases) and to the left (r decreases).

So, it seems the major thing Narasimhan’s finite-SNR DMT tells us is how quickly a certain strategy approaches the optimal Zheng-Tse DMT. Two strategies that both achieve the optimal DMT curve will likely have different outage curves, and will thus approach the optimal curve at a different rate. The strategy that approaches it quicker will have a smaller outage probability above some unknown crossing point (which may not even exist; one strategy might always be better than the other in terms of outage).

This motivates our last point. The finite-SNR DMT still doesn’t give crossing points or tell us precisely when to switch from one mode to another. It can give us an estimate of where to switch, however. If two outage curves cross each other, then at some SNR smaller than the crossing-point SNR, the higher curve started decreasing quicker than the lower curve; that is, its diversity became higher than the other’s. Since d is nondecreasing with SNR (for any sensible r), this switch will only happen once.

Take, for instance, Alamouti coding with 16-QAM versus spatial multiplexing with 4-QAM using an ML receiver. Here our model assumes 2 transmit and 2 receive antennas. The probability of vector symbol error (okay, not outage, but the same concept applies) for these two strategies is below.

For the infinite-SNR DMT, r = 0, and the best strategy is always the Alamouti code, which achieves d = 4, while spatial multiplexing with ML receiving achieves d = 2.

For the finite-SNR DMT, r is varying with SNR, though it varies equally for both strategies. Observe that the slope of the error curve of the Alamouti code is flatter at low SNR than the slope of the SM-ML curve. Thus, at low SNR, d(Alamouti) < d(SM-ML).

At around 10 dB, the slope of the Alamouti curve becomes steeper than the slope of the SM-ML curve. Thus, at apprximately 10 dB, d(Alamouti) > d(SM-ML). There is a crossing point in their DMT curves around 10 dB and r approximately 4/3. Note, however, that the error curves don’t cross until approximately 17 dB. Quite a gap. Since, to find the finite-SNR DMT, you must take the derivate of outage term approximations, however, you probably have a decent estimate of the outage terms and can directly calculate crossing points, if that is what you’re interested in.

In summary, the DMT gives us intuition about the outage curves for different signalling strategies. In particular, the DMT curve does not tell us there is a tradeoff in how we use an extra antenna pair given to us, but instead a tradeoff in how we use extra SNR given to us. We can use extra SNR to increase our rate, or decrease our outage probability. This is the fundamental tradeoff.