0:01

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

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

very small. The reason we look at small example is

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

Even though it might be just addition or multiplication.

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

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

numerical values. Then you may feel a lot more comfortable

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

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

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

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

Okay. So we would not be able to demonstrate why

the problem is difficult because it has to scale up.

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

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

smaller than any typical cellular traffic that you can expect.

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

every step in the textbook or during the lecture.

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

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

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

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

2:18

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

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

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

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

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

3:40

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

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

Okay? Now let's calculate what is the

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

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

milliwatt, okay. Then.

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

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

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

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

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

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

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

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

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

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

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

Okay. If you look at these numbers, these four

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

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

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

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

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

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

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

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

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

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

that. We'll be looking at.

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

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

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

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

1., four. Okay?

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

Okay? Cuz you were not achieving your target

SIR. Similarly, the second user's transmit

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

transmit power. You can easily verify the calculation are

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

respectively. Everything is in milliwatt for the power

unit. And that confirms our intuition.

8:02

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

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

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

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

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

8:39

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

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

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

You overshot the first user after this round.

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

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

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

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

Okay. Second user at iteration one, the sir is

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

iteration which was two but not big enough yet.

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

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

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

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

that. The first user actually overshot, okay.

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

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

increase. All right so now you can go through this

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

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

iterations. We just went through one iteration.

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

iteration fifteen, you pretty much converge.

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

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

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

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

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

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

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

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

11:35

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

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

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

numerical example. Are all about what people call the inner

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

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

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

scale. Okay.

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

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

12:26

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

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

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

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

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

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

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

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

rate. Or correspondingly, the lower the error

rate. So now you have to balance different

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

everyone. Then, what would be an efficient and fair

allocation of gamma? That is something that is very

interesting, we just are running out of time.

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

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

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

The clocks at different mobile stations are run slightly different.

14:04

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

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

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

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

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

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

number of times every single day out there.

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

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

mathematical entity. It is also a practically, extremely

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

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

From splitting the cells overlaying with smaller cells.

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

in the course. Every lecture will conclude with a summary

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

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

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

And that happens in all wireless communications.

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

16:31

We also saw a specific mathematical formula.

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

A DPC. And the conceptual highlight is that there

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

current SR value. You just compare that with your target

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

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

distributed solution to an optimization problem.

Which turns out to be a linear programming.

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

gain that models the competition due to interference.

17:18

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

externality, as well as the mathematical language.

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

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

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

to the word of online. Add auction.