In this video, we will define massive MIMO and its fundamental properties like favorable propagation, channel hardening, and we’ll also discuss the channel modeling for massive MIMO scenario. What is massive MIMO? Many definitions in different Books-- I have selected two of them which are very much useful for our discussion. According to the first definition that I've selected, it says that massive MIMO is a useful and scalable version of multiuser MIMO with these fundamental distinctions. Only Base Station learns the channel, this is important. Second, M is always very, very greater than K. And simple linear processing is used both on the uplink and on the downlink. When we're talking about simple linear processing, this is nothing but the combiner that we're using and the recorder on the MRT like we used to that at the transmitter, MRC or zero forcing. So it talks about simple linear process. We don't want any complicated signal processing to be used in the transmitter or at the receiver, multi intra transmitter or multi intra receiver. So, according to the second definition, massive MIMO is stated as follows. Massive MIMO is a multi-user MIMO system that serves multiple users through special multiplexing over a channel. With favorable propagation in TDD, time division duplex. And relies on channel reciprocity and uplink pilots to obtain channel state informers. These technologies may be looking alien to you guys, but over time we’ll read about all the things, TDD, tele-reciprocity. Okay. Why uplink pilots are used to obtain channel state information, all these things we’ll be discussing in detail and I hope after the end of this course, you will be having a good idea about these things, okay. So, these are the two definitions that has been generally used and these have been taken from the two references that we're using for our course. One is by the book by Professor MLB Johnson and others and another book is by Professor Indigo and Margaret and others. Okay, so these are two definitions coming from those two books. Favorable propagation. The propagation is said to be favorable when users are mutually orthogonal in practical sense. When I say practical sense, let us take an example, letter h is a vector of size M cross one is the channel vector of one user and g is the channel vector of another user. In ideal case what will happen? The users are said to be orthogonally, simple at charming g is equal to zero. This everyone knows this is nothing but the eternal formula at charming g is equal to 10g are orthogonal. Actually, in real scenario and this is never true, right? This very really it will be happening that the two channels are orthogonal in real scenario, okay. In practical scenario, the users are said to be orthogonal in quantity at hg divide by norm of it into normal g has zero mean and its variance is very, very less than one. Let us look into an example and that's why I plotted a graph where I'm plotting the variance of hg summation, divided by normal g. We see, as I'm increasing the number of antennas, this value variance of this value is decreasing. And around value of 100, 128, what we're getting a variance of very low value of the order of 0.1. So, what we say is number of antennas around 100 or 128. What I'm getting, very low value of variants and at this point I can say we are achieving favorable propagation. So the big question is, how many antennas as a base station need to be massive is it always 100 or 128? No, it depends on the different system model and the processing that we're using. From the above graphs we can see them is approximately 100 or so. Right? For a variance to be less than 0.1. This number may vary depending on different channel conditions. And like I said, it also depends on the system model. Whether we are having correlated channel, we have an uncorrelated channel. Whether we are having LoS scenario or NLoS scenario, whether the maximum is co located or we are using some kind of disputed massive MIMO system and so on. Let us take a scenario of multi cell system and then try to define the formula for favorable propagation. Let’s have at hlkj denoting the channel between the kth user in the elite cell and the BSj. In that case, the pair of channel, hlij and hjkj to BSj provides a synthetically favorable propagation if and only if, this term is to zero. Okay, this is also can be approved using law of large numbers. So what exactly we are saying is the channel directions become orthogonal asymtotically. It's not the channel that is becoming orthogonal, the channel directions are becoming orthogonal. Or we can say the inner product of the normalized channels. This is what? Unit vector. This another unit vector means this is denoting a direction. This is do not take the direction, right? So what is happening? The channel directions are becoming orthogonal and therefore, the inner product of both is going as asymtotically to zero. Okay, understand? So once again I'm repeating the channel is not becoming orthogonal. Channel directions are becoming orthogonal. This is important fact. Remember norms of the channel grow with M. So, like I said, one second I'm repeating, favorable propagation does not imply that the inner product of the two channels hjkj and hlij tends to zero. But actually, the channel directions become orthogonal and not the channel responses. This always useful to remember because this is one main difference that people always ignore. Now, like we discussed earlier also, the variance of the, this term should be equal to one upon M. This was about variable propagation. Now, let us define another important property of massive MIMO, that is channel hardening. A propagation channel at hjkj provides a symptomatic channel hardening if and only if, the term on the left hand side tends to one. We have seen this expression for some trivial scenario earlier, but now we're defining it for multi cell multiuser scenario. Okay, so what we can see that the channel gain is equal to its mean value. All right, this implies that the fading has little impact on communication performance. We have seen all there also that the performance of LoS and NLoS used to be similar a result of the reason that we found out was that the impact of fading was reducing because of the increased diversity, right. One important fact that you should remember is that this does not imply that norm square of hjkj is becoming deterministic, no. The ratio of the norm squared divided by its mean is standing to one, or only the channel. Okay, remember that like for example in favorable propagation, we said channel is not becoming orthogonal. The directions are becoming orthogonal, the inner product of the directions of the two channels were becoming orthogonal. Similarly here also, you should understand this point. And we plot this variance of this quantity that we have. And if we calculate the variance of this quantity, what we get is one up one and this you can derive it. It's very easy to derive and take it as an a homework exercise. So if you plot the variants of this quantity, what we are saying is when we have no correlation scenario, in that case the channel hardening is happening much faster when compared to the scenario when we have some kind of correlation. In that case, we can see that correlation results in lesser hardening in the channel when compared to when there is no correlation in this channel. Now we are discussing about channel modeling, okay. A fading channel, h, especially correlated if the channel gain and channel directions are independent random variables. And we always defined this hlkj, this complex normal zero M comma beta lkjIM but we have not discussed how exactly beta lkj is denoted this beta lkj in practical world is also known as large scale fading coefficient, and this is given as follows. This is coming from a certain path loss model. Sofitel kj is equal to gamma minus 10 alpha log of 10 D is the distance divided by one kilometer plus Flkj. This is nothing but this is coming from a standard. Okay, so where dlkj, I said in one kilometer denotes the distance between the transmitter and receiver letter in our case between the user and base station. Alpha denotes the path loss exponent which determines how fast signal the case with the distance. Gamma here denotes the median channel again at a reference distance of one km. And most importantly this Flkj. denotes the shadow fading which models the blockages in the channel and this is distributed as cosine zero comma sigma sf . With sigma fm denoting that standard division. So this is the way if you want to simulate your channel, let's say anytime taking into picture the distance between the user and the basis station then you need to use this formula, what is there in the screen. So how exactly you do it? You generate your users randomly in a particular coverage area and then say this is my user. And you place your base station at the center of the cell and I said the distance that you're getting from the base station and user you calculate call it as a D, okay? And then you keep on reporting this for different cells and different users. In this video, we learned that the massive MIMO uses simple linear processing both in the uplink And in the downlink, favorable propagation, which is one of the important properties that arises due to massive MINO implies that the users are mutually or terminal. The channel hardening implies that fading has little impact on communication, performance, Five D users, massive MIMO, and the base station.