In this video we'll discuss about the area throughput of cellular networks. We'll see how to increase area throughput and then we'll discuss about the capacity of interference limited system. We'll start with cellular network. Cellular network consists of multiple base stations and users, where each user is connected to one of the base stations. If you see here, we have multiple cells scenario where this is a base station represented by blue circle and we have multiple users in each, so we call the cellular network. Now to calculate the area throughput of a cellular network in terms of bits per second per kilometer square, we will take into consideration bandwidth with unit in hertz, average cell density with unit in cells per kilometer square and per cell SE with unit. bits per second per hertz per cell. You can see that any new generation wireless network targets 1000 times improvement in area throughput. So, how to achieve this? We can allocate more bandwidth. Right? Because area throughput directly proportional to bandwidth. So, by allocating more bandwidth area throughput will increase. Second, we can increase the cell density by adding more and more base stations. Third, by improving the spectral efficiency per cell. Now let us look into all these things. What are the advantages or what is the disadvantage when we go for the increase in either more bandwidth, higher cell density, or how can we improve the spectral efficiency per cell. Starting with more spectrum and when we go towards more spectrum, the benefit is it leads to higher bandwidth. In millimetre wave bands which is in between 30 to 300 gigahertz of frequency, we have almost two gigabits of bandwidth instead of 2200 megahertz of bandwidth available in conventional network. That's a 4G, which we're using all over the world. Right? Higher bandwidth always leads to higher capacity. Right? Because we saw capacity was equal to bandwidth times log (1+ SNR). So the capacity is always proportional to bandwidth. But now let us look into a fact that whether when we are increasing bandwidth, is there any saturation that is happening in the capacity or capacity always linearly increases with the bandwidth. Okay, now one important aspect that we saw earlier was that the noise power also depends on the bandwidth. Right? So, as we're increasing the bandwidth, the noise power is also increasing this part. Therefore, there's not always a linear increase. There's also some decrement happening in the SNR when we increase the bandwidth. Right? So when we calculate the capacity as B tends to infinity what we're getting is mod h square P/N_0 log base 2 e. This can be calculated using this standard result. Now what are the issues when we are doing or when we are increasing more and more spectrum and we are going towards higher frequency range? Okay, one important issue that we face is higher propagation loss. Therefore, long range communication is not possible. Second, channel changes very fast in time. Therefore, we can say it is more suitable for indoor or low mobility communication. Third, more transmit power or beamforming gain is required. Okay, so these are the issues that we face when we have higher bandwidth and we can see that it is not always a linear increase in the capacity because of the increase is also the noise power when we are increasing the bandwidth. These are the issues that we face when we are trying to increase spectrum. And the most important issue according to me is the high propagation loss. Because as we are moving towards higher and higher frequency is not possible to cater to the need of the user which is at a far distance from the base station or the place where there is large propagation loss. Okay, for example, due to large buildings, mountains, and all those stuff. Now let us look into the densification. High cell density always results in higher throughput. As we can see in the same area if we have initially three cells with one way station in each cell, if we use a densification concept where, in the same area we are assigned having more number of cells with one way station in each cell. So, therefore, we lead to densification and therefore it will increase the throughput, but what are the issues that we will have when we are using densification? Okay, first of all, it will lead to high deployment costs. This is obvious, right? We are having instead of three cells now we are having let's say 20 cells obviously the deployment cost will increase. Second major issue that we will face is inter cell interference. Okay, because the user here would be interfered by only the users of two neighboring cells. Right? Or I would say the opposite way the base station will be interfered by the signal of the base station of two neighboring cells. Here, the base station will be interfered by the base station of so many cells. Right? So the inter cell interference is increasing. This is a major issue. One important issue that we face is the handover issues. Why? If you see in the case of when we have only three cells, it will happen right? But in case of this side or the green cells, the handover will happen very, very often because now the cell size is reduced. So it is high possibility that the user is moving from one cell to another cell when we have large number of cells. Right? Means the area of a cell is lesser. In that case the handover will increase therefore we will face handover issues. Third, which is very important, spectral efficiency or capacity. First of all let me define spectral efficiency or capacity, we have already discussed in detail but just to recall it. For a SISO system, we have the received signal given by y = hx + n, with x distributed as complex normal (0, P) and noise distributed as complex normal (0, sigma square). Now we know if h is fixed or deterministic, the capacity is given by log base 2 (1 + mod h square P upon sigma square). But if h is a random variable, a realization of random variable in that case Ergodic Channel Capacity is given by expectation of log base 2 (1 + mod h square P upon sigma square, we know that. This is for a fixed h, one h. This is a deterministic quantity because we have fixed h, but now because h is varying, we need to average it over all the channel realization, therefore this e is coming where the expectation is with respect to channel h, realization of channel h I would say. In cellular network, received signal is generally corrupted by interference from the other users. Right? The other users can be in the same cell or other cells. So till now we were discussing a system model where it was just one user, therefore, we were just having hx + n, now with multiple users in a cellular network we will always get some kind of interference from other users in the network or other base stations in the network. So to understand that kind of system, we have taken an example where I'm assuming x is the input, we have some channel then some interference is coming. This interference is coming from the users of the other cells, okay, or the base station of other cells, then we have our obvious noise that is being added at the receiver. Therefore, output y can be written as y = h, this is channel hx + v + n, so therefore now instead of having only hx + n, we're having y = hx + ν + n. Okay, this is the practical channel, actually received signal in the presence of interference coming from the multiple users in the network and multiple base station internet. Remember, interference is not necessarily independent of x and h. Okay. It may happen sometimes that the interference for such specific application that or specific scenario that interference is also dependent on the channel or transmit signal. But for our calculation, we will always use the interference to be independent of the h and x, transmit signal x and the channel h. Now like I said, we have our received single expression now, the updated received expression has y = hx + ν + n. And now let us derive the spectral efficiency expression for different scenarios. Case 1, when the channel is assumed to be fixed. We are assuming input and interferences are uncorrelated. Okay, like I said in a previous slide. Let interference power is given by Pv which is known to me, therefore for fixed or deterministic channel the channel capacity is lower bounded by C greater than equal to log base 2, 1 + P. This is an SNR vehicle. We have already derived this capacity formula, that capacity is bounded by this, log base 2 (1 + SNR) and here what is the SNR, signal power (hx), which will be, x power is P, this Mod h square because it is fixed, divide by this is coming from the interference plus noise, so I am using interference power here which is the power of v, power of v, we said interference power is Pv plus noise power. Noise power is sigma square, therefore we have got this and this equality is achieved, you all know when input is complex normal (0, P), okay. Now Case 2. In case of random channel, Case 1 should be trivial because everyone knows we have already seen this expression earlier also. Right. I'm just updating that based on the updated received expression. One more important thing, now instead of calling it SNR expression, signal to noise ratio, we will call it SINR, signal to interference noise ratio. Because interference is also coming and interference is coming in the denominator only with noise. So in Case 2, I'm assuming the channel is not fixed but it is not random. Okay, now we have channel is random and we know the interference is also random because in this case the channel is random. The interference is coming from other users, right? And the channel between the other user and the base station is also random, so what I'm assuming, channel as well as the interference is random and we are assuming few conditions that the expectation of the interference given channel is equal to 0 and coalition between the signal and the interference given h, v = 0 and third, that power of the or mean square value of the interference is given by Pv. Given h, v. Now based on this, the Ergodic channel capacity is given by C greater than equal to expected. Now this expectation will come because now the channel is random. Expected value of log base 2 (1 + P into h square. Now, we have Pv + sigma square, same. But now Pv, remember, this Pv is this one, Pv is calculated by expected value of mod v square given h, v. Okay, so here the expectation is over h and v. I'm defining v. Some realization of the interference. In this video, we saw that the cellular network consists of multiple base stations and multiple users. High spectrum leads to higher area throughput. However, with increased propagation loss, higher cell density leads to higher inter cell interference. And cellular network were always experiences interference from same and other cells.