Now one of the issues with CDMA is clearly that everybody is talking at the same time, the same frequency band. In other words, the frequency reuse vector is one. And how do we deal with the resulting interference? Interference in the air for cellular network is the first example in this course of something called negative externality. Externality, here, roughly speaking, refers to the fact that your happiness also depends on what other people do. Your signal, your cup of tea or coffee is other transmission's poison, and this is what we call a negative externality. Now, we will also later look at positive externalities in social and technological networks. Sometimes people also call this the tragedy of commons. In the later lecture, we will talk about, more tragedy of commons and how to deal with that in social and tecnological networks. Now, one famous special case of interference in the air is so-called near-far problem. If we go back to this picture, you see that A mobile station a is much closer to the base station than mobile station B. Okay? Your neighbor or your friend's, let's say, your friend's iPhone might be much closer than your Android phone to the base station. Maybe your friend is sitting right beneath the cell tower, the base station, and therefore, the distance it traverses is much shorter, and the signal received at the base station is much stronger than yours, and it can drown out your signal, and that is what we call the near-far problem. Okay? The near user will overwhelm the farther away users signal. Now, there was a simple solution provided by Qualcomm in the late 80s that uses the idea of feedback control. Now, later, we'll see much more complicated feedback control in the network. In this case, this is a simple one hop from the base station to different mobile stations. The degree of freedom here is exactly transmit power, and, it says that, well, adjust your transmit power based on where you are. It's going to estimate first the channel condition, for example, how much channel loss does your signal suffer? And then it would reverse that, so if your channel of loss is a factor of two, okay, and the other is a, a factor of 0.9 for example, okay? Then, I will tell you, look, you should multiply your signal strength by a factor of two, and you should multiply by a factor of one over 0.9, which is much smaller than two. Then, this power adjustment would cancel out the effect of the channel attenuation, and then I would have equalized received signal power from both of you and the base station. So this is a simple algorithm. There's no iteration and it is through a central command by the base station and it suffices to. achieve equalization of received signal power. And without this feedback control to take off the near-far problem, you would never be able to implement CDMA, because, the amount of interference would be just too much. Now, what if you need to achieve a target signal quality which might be different for different mobile stations? Not just simply equalize all the received power and that was the challenge in the early 90s. Now in order to make sense of that question, we'll have to provide a little bit of symbol. ,, . Okay. Now, we're going to look at a very small cell. There are only two mobile stations. Okay? Two iPhones, one and two, and one base station. Even though there's one physical receiver, we're going to divide it into two logical receivers. Okay? Receiver one and receiver two. And we will look at four different channels, two of them are the so-called direct channels., these are the desired intended communication path. Of course, in the air, there's no real physical pipe of a channel. So, by channel, we really mean a logical link. What actually happens is just energy propagating in the electromagnetic field and being picked up by antenna. We don't have pipes as if we are, you know, drawing pipes, but actually don't. so these are logical channels. And we say that there is a transceiver pair from the transmitter. one t o the logical receiver one, and then, from transmitter to the logical receiver two. So happens these, two logical receivers are physically co-located at the same spot and then, what about interference? When Transmitter 1 talks, the energy propagates, okay? And part of a energy will be picked up by this logical Receiver 2. This is a dot line to represent an interference channel and there is the other interference channel and we will use the following symbol Gij to denote the channel loss on the corresponding direct or interference channel. Now somehow the community does touch with the term channel gain even though this G is numbers between zero and one. So, it should be cloud channel loss, let's say that called channel gains. Now, this is a little tricky. So, we go a little slow here, it says Gij represents the channel gain from the transmitter of logical tranceiver pair to the receiver of logical transceiver pair i. So, we're going to use the terms user, transmitter receiver pair, or logical transceiver pair interchangeably. For example, Gu11 is the channel from the transmitter of logical pair one to the receiver of logical pair one. Let's say, well, that's a good channel indeed. It is the direct channel for desired communication and so is in general, G2 too. So in general, Gii are the direct channels. But when j is not equal, equal to i, for example, G21 that says, this is from the transmitter of pair one over here, to the receiver of pair two. And therefore, this is the channel gain of an interference channel. G12 says this is from the transmitter of pair two to the receiver of pair one. And, in general, G12 is not equal to G21, necessarily. Now, ideally, of course, we want Gii's to be bigger than Gij's, where j is not equal to i. And, indeed, with the help of those spreading codes, this ones minus ones we do have usually, a Gii that's bigger than Gij's. However, there could be many j's not equal to i. Okay, there could be tens of them. And, even though each one provides just a l ittle bit of interference to your transmission, they could add up. Okay. That is the cocktail party you should log in. So, how do we capture the notion of received signal quality? If the quality's high, you'll be able to talk more efficiently. For example, you can talk faster and the receiver can still understand what you are talking about. But sometimes, in cocktail party. because it's so noisy around, so much interference, you have to slow down your conversation. Okay? You have to talk slower. That is, fewer bits per second. So that the, the receiver can hear you correctly. And the way we capture. this ratio, is through the so called SIR, Signal-to-interference ratio plus noise ratio. Now, what is the signal? I'm not talking about a signal power at the transmitter side. I'm talking about the signal power at the receiver side. So that is the transmit power which we denote by P sub I'm going to say i, P sub i multiplied by the channel, that's the direct channel, and getting to the receiver of pair i. That's the received power. Divided by, because it's a ratio, the interference collected at the receiver, that is P sub j times Gij. Okay? From. J transmitter, okay, to the same common ith receiver and there are many of such j is not equal to i. Plus, at the ith receiver, there's usually some noise term, n sub i. This is what we call the SIR associated with the ith user. For example, in our previous case, however just two users and therefore SIR1 is simply P1 G11 over P2 G12 plus n1 and also SIR2 is just P2 times G2 over interference, which is P1 times G21 plus n2. Okay? But in general, you have more than one term here in the denominator. Okay. So now that we have the notation. Let's look at what kind of a function this SIR is, SIR for each transceiver pair i, okay, the received signal differential ratio for pair i is really a function of the entire vector with this notation on top of power vector P. Okay? not just the function of your own power, but also a function of other's po wer. That is the mathematical representation of the very physical fact of interference in wireless communication. If there's no other users easy to increase SIR for user one, I just increased P1, but of course there are other users. A larger P1 helps SIR1, but it will also show up as a larger denominator for the other SIRs that would hurt the other SIRs. Okay? So First observation is that SIR is a function of the entire vector P.. Now, second, what are the constants and what are your degrees of freedom. Because we're talking about transmit power control, so obviously, the degree of freedom is the vector p and that's what we want to design and optimize over. We are given all the Gij's that is determined by the propagation environment. Okay, is this indoors, is this outdoor, what kind of terrain is it, it is given by the location of the transcievers. and that's not up to us to optimize. And we are given all these receiver electronics and therefore their noise. So the Gs and n's are given constants, the Ps are what we'll be calling the optimization variables in transmit power control. So now our goal is to say all difference transceiver pairs index by i will like to get certain SIR achieved. Okay? So, the question is for a given set of Gs and n's, can we find such a vector P? Is there such a vector P at all to achieve certain target SIR values. And if there's more than 1, which one shall we pick? In the next module of the video lecture, we will see the answers to these questions in a distributed mechanism to coordinate interference in wireless cellular networks and that mechanism is now in use in many wireless standards.