0:01

Let's now take a look at a numerical example.

Â Now in all these lectures we will be looking at numerical examples that are

Â very small. The reason we look at small example is

Â that we want to spell out almost every step of the computation.

Â Even though it might be just addition or multiplication.

Â Because we want to avoid any symbol-phobia on the minds of some students, okay.

Â There's nothing mysterious about the symbols once you see them in action with

Â numerical values. Then you may feel a lot more comfortable

Â with the symbolic operation. But in order to show the numerical

Â addition, multiplication within a slide or, or a page of the textbook, we have to

Â stick to small dimensions. Now, in many cases, there actually, does

Â not capture a key difficulty in the challenge, that is the challenge of scale.

Â Okay. So we would not be able to demonstrate why

Â the problem is difficult because it has to scale up.

Â Now in this case it turns out that usually the scale for a single cell is not that

Â great, but still we're talking about in this case a four user cell, which is a lot

Â smaller than any typical cellular traffic that you can expect.

Â Now having said that, the advantage of small example is that we can write out

Â every step in the textbook or during the lecture.

Â So here is the GIJ values captured in a four by four tables, and there are four

Â users. And we're going to make up some number

Â that will make our computation very easy. We say the diagonal entries which are the

Â G1 1's, G2 two, G2 33, and G4 four, they are all the same, and which is one.

Â 2:18

All right, now, these are the given parameters, depicting, the channel

Â conditions, of both, the direct channels, and the interference channels.

Â Now, what about, the target, gammas? We are going to say that, gamma one is

Â two, gamma two is 2.5, gamma three is 1.5, and gamma four is two.

Â It turns out this set of gamma SIR's, is indeed achievable.

Â 3:40

Now, we start with an initialization or a time zero.

Â Let's say we initialize all the power levels to be one milliwatt.

Â Okay? Now let's calculate what is the

Â corresponding SIRs. The SIR, for the first transceiver pair at

Â time zero initialization is simply one C. This is one, okay, there's one times 1.0

Â milliwatt, okay. Then.

Â We'll write down the similar expression for the interference.

Â Let's say without interference channels. Well, we got.1 from the second

Â transmitter, .2 from the third, and.3 as the interference channel gain from the

Â fourth user. So we have to multiply .1 by the transmit

Â power for the second user, which is 1.0, and then .2 times.

Â Again 1.0 because everybody's power is initialized to be one milliwatt.

Â And then.3 times 1.0. Plus don't forget a little noise term 0.1.

Â And this turns out to be 1.43. Similarly, we can calculate the SIR for

Â the second user at initialization. Which turns out to be two.

Â For the third one. Which turns out be two as well.

Â And the fourth one, which turns out to be 2.5.

Â Okay. If you look at these numbers, these four

Â numbers, at initialization, and compare with the target SIR's Okay.

Â You see that the first user is not getting to the target SIR.

Â Neither is second user, okay. But the third and the fourth user are

Â actually getting above the target SIRs. Instead of 1.5 you are getting two already

Â and instead of two you are getting 2.5. So what would you expect.

Â Intuitively and mathematically you expect that the next time iteration number one

Â you will see that the first and second user's transmission power should go up and

Â the third and fourth user's transmission power should go down.

Â And that is exactly the kind of negative feedback that we will see.

Â Okay. Well let's look at the kind of power risk,

Â that. We'll be looking at.

Â P1, okay? Now at iteration one, it's Gamma one over

Â the S I R. Just observe, times the power, okay, At,

Â the last iteration. And this equals two, your target over

Â 1.43, which I currently getting, multiply your current power level 1.0, and that is

Â 1., four. Okay?

Â Indeed, now you're going to blast more power than before.

Â Okay? Cuz you were not achieving your target

Â SIR. Similarly, the second user's transmit

Â power now becomes 1.25, instead of one. Whereas the third and the fourth users

Â transmit power. You can easily verify the calculation are

Â actually smaller than the last round. They are point 75 and point 80 milliwatts,

Â respectively. Everything is in milliwatt for the power

Â unit. And that confirms our intuition.

Â 8:02

Again, the right channel gains one. Multiply the new power, which is 1.40,

Â divided by the indifference channel gain 0.1, times the new power 1.25, plus direct

Â indifference channel gain 0.2, times the new power 0.75, times a plus 0.3 the

Â indifference channel gain, multiplied by the new transmit power 0.8.

Â This is interference plus a noise, and you get 2.28.

Â 8:39

Now notice 2.28 here now is both bigger than the last round SIR, which is 1.43 if

Â you remember, as well as bigger than the target SIR two.

Â So not only you enhanced your SIR, you enhanced a little to much.

Â You overshot the first user after this round.

Â So what would you expect to happen in the next round?

Â You expect the power for the first user will now go down.'Kay?

Â So power go up, power go down. And you hope that this oscillation would

Â dampen and eventually converge. Let's finish this calculation.

Â Okay. Second user at iteration one, the sir is

Â two point 34. Which is bigger than the SIR and the last

Â iteration which was two but not big enough yet.

Â The target is 2.5 so expect that next round the power for the second transceiver

Â pair will still go up in the SIR. Three at this iteration is 1.28 SIR four

Â and this iteration equals 1.82. And both of these are, not, not quite

Â their target SIR. So what you see after one iteration is

Â that. The first user actually overshot, okay.

Â The second user didn't overshoot. The third and fourth user actually now

Â they are dipping below the target SIR now. And next round their power should

Â increase. All right so now you can go through this

Â yourself or look at the textbook and I'm going to just show you the cooked product.

Â In the end, we can plot these transmit power values in milliwatts over the

Â iterations. We just went through one iteration.

Â You can keep going and then you see the ups and down quickly saturates, and around

Â iteration fifteen, you pretty much converge.

Â Now we will not have time to rigorously talk about this so called exit condition

Â of the iterative algorithm, okay? When should you converge to have a

Â guaranteed, error abound. We'll just hand wave away, to say that, if

Â the transmit power is, no longer vary a whole lot from one iteration to the next,

Â we call that a convergence. And this induces, of course, a convergence

Â in the target SIR towards the target SIR over these iterations.

Â In fact around iteration ten, for sure. The S I Rs converge and as you can see,

Â they achieve their target S I Rs, 2.5, two, two, and 1.5, respectively.

Â 11:35

This actually is a very fast convergence within basically a few iterations we

Â achieve the target sir. Now everything we have talked about so

Â far, okay, the formulation, the optimization view, the game view, the

Â numerical example. Are all about what people call the inner

Â loop power control, which says you have a target gamma feed into you, and you have,

Â the current SIR measured and that's going to update your transmit power.

Â There's actually another loop, the outer loop, which operates at a slower time

Â scale. Okay.

Â By picking a certain transmit power, through the effect of wireless network

Â interference, you can to observe, the trans, the receiver a certain error rate.

Â 12:26

And this error rate is an artifact of the kind of gamma that you picked.

Â And the outer loop control says, you know what is this too much error?

Â If so, you should increase gamma. If not, maybe you can decrease gamma.

Â So now in the outer loop gamma is an output not input.

Â Gamma is a variable not a constant. Now, we will not have time to talk about

Â it. But in 3G and 4G networks, clearly, for

Â data centric applications, you can't just have a target gamma.

Â Everybody want a bigger gamma. The bigger the gamma, the higher the data

Â rate. Or correspondingly, the lower the error

Â rate. So now you have to balance different

Â user's demand for a bigger gamma. You know, you can't give a big gamma to

Â everyone. Then, what would be an efficient and fair

Â allocation of gamma? That is something that is very

Â interesting, we just are running out of time.

Â But before we close this lecture, I want to highlight in practice, how is transmit

Â power control used. Now we assumed that everybody has the same

Â clock, of course, in reality you have asynchronous system.

Â The clocks at different mobile stations are run slightly different.

Â 14:04

But if you look at an asynchronous and discreet power level version of DPC that

Â is actually implemented in virtually all the 215G and 3G networks.

Â And, depending on the protocol, the frequency of running the inner loop could

Â be, anywhere between 800 to 1500 Hz. So every second, roughly speaking, this

Â calculation of DPC is run 1000 times. Now you can count how many 3G and 2.5G

Â devices are out there and you see that this algorithm is used with incredible

Â number of times every single day out there.

Â And indeed to transmit power control together with what's called soft handoff.

Â It's what made all the 3G standards work. So not only is this DPC an elegant

Â mathematical entity. It is also a practically, extremely

Â influential and useful artifact. So now I may wonder, how can I make

Â cellular speed run even faster? We'll come to some of these ideas.

Â From splitting the cells overlaying with smaller cells.

Â To using multiple antennas, and dividing the frequency bands more refinedly later

Â in the course. Every lecture will conclude with a summary

Â slide. And in today's summary slide we want to

Â highlight two things that we learned in this very first lecture of the course.

Â One is that different user's signals interfere with each other in the air.

Â And that happens in all wireless communications.

Â In the cellular world, we use transmit power control to manage this interference.

Â 16:31

We also saw a specific mathematical formula.

Â It's called the interference coordination with distributed power control.

Â A DPC. And the conceptual highlight is that there

Â is a negative feedback. And the feedback is all captured in the

Â current SR value. You just compare that with your target

Â fixed gamma and use that ratio to adjust your power level up or down accordingly in

Â the next time slot. And we saw that this can be viewed as a

Â distributed solution to an optimization problem.

Â Which turns out to be a linear programming.

Â Or we can model that as actions by intelligent agent in the non-corroborative

Â gain that models the competition due to interference.

Â 17:18

Both of the conceptual points of interference, tragid of common, negative

Â externality, as well as the mathematical language.

Â Such as what defines optimization, rather what defines the game, will be so

Â frequently used over and over again in the rest of this course.

Â And indeed, in the next lecture we will shift gear from the word of C D M A over

Â to the word of online. Add auction.

Â