noc18-ee31-Lec 43 | Applied Optimization | Multi Antenna Channel Estimation | IIT Kanpur

Hello, welcome to another module in this massive open online course. So, we are looking at least squares, we have looked at several aspects including the intuition behind least squares. Let us look at some practical applications of these least squares. In fact, what we have said is, this is something that arises very frequently in signal processing and communication. So, we have the practical application of least squares and what we want to do in this is, let us consider the first problem is related to wireless communication. Of course, one of the very interesting problems is that in a wireless communication system is that of channel estimation. In particular, let us look at the problem of multi-antenna channel. Let us look at the problem of multi-antenna channel estimation. Now, if you look at this, typically what we said is, now let us consider a system with multiple transmit antennas. So, you have a transmitter and this has multiple antennas. So, you have a transmitter, let us say you have L antennas at the transmitter. For simplicity, let us say you have a single receive antenna. Now, you have the channel coefficients, there are L antennas, there are L channel coefficients. So, you have H 1, H 2 up to H L, these are the L channel coefficients. These channel coefficients are unknown, you comprise the channel. We also said this is the channel state information or CSI, which is unknown and has to be estimated. So, this channel vector, if you call this as H bar, this is this channel vector, which basically constitutes the CSI, the channel state information and that has to be estimated. So, this is basically your H bar equals CSI. This is the channel state information and this CSI has to be estimated. Now, to estimate this channel state information, we transmit pilot symbols. Now, this pilot symbols are symbols that are known at the receiver. So, to estimate the CSI implies, we have to transmit pilot symbols. In this case, pilot vectors, you have to transmit pilot vectors. Now, let us say X k is the k th pilot vector or pilot vector at time instant k, pilot vector at time instant k. So, this implies that you are transmitting pilot vector at time instant k. So, what we have is, I can write the received symbol Y of k at time instant k equals the channel coefficients H 1, H 2, H L times X 1 k, X 2 k up to X L k plus the noise sample plus N of k. What is X i of k? Now, remember X i of k, you can think of this X i of k equals pilot symbol on the i th antenna, i th transmit antenna, pilot symbol transmitted on i th antenna, pilot symbol transmitted on i th antenna at time instant k. So, therefore, I can get this as, this implies Y k equals, well, I can write this as the vector. So, this is your H bar transpose, channel vector is H bar times X bar k. So, H bar transpose X bar k plus the noise sample N of k. Therefore, now if you transmit, let us say, M pilot vectors which you can also, now remember, considering real vectors, this can also be written as X bar transpose times H bar plus N k. I can write it as H bar transpose X bar k or X bar k transpose times H bar. Now, considering the transmission of M pilot vectors. So, let us say, you transmit M pilot vectors, I can write the equivalent system as Y 1 equals X bar transpose 1 into H bar plus N 1. Similarly, Y 2 equals X bar transpose 2 into H bar plus N 2, so on. Y m equals X bar transpose m into X bar plus N of, and now what I can do is, I can make a matrix out of this. So, this will become a vector, this will become your M cross N, you can call this as your M cross N output vector. This will call, you can call this as your pilot matrix. In fact, this has M rows each of size L. So, this will be pilot matrix X will be of size M cross L, H bar is a channel vector which is of size L cross 1, and now once you concatenate these noise elements, you will have this will be N bar of size also again M cross 1. So, net what you have is, this is your model for channel estimation. So, what you will have is, you will have Y bar equals X times H bar plus N bar. So, this is the model for channel estimation. In fact, channel estimation model for a, this is the channel estimation model for the multi antenna system, and in fact, this X is the pilot matrix. You can denote it by X or X p, it is the same thing. So, this X equals the pilot matrix, and this is of size M cross L, and this vector H bar, remember this is the CSI, this is unknown, this vector H bar which represents the CSI, this is unknown. And now to estimate this H bar, remember we do not know the of course, there is noise, this is the noisy observation model. So, you have to look at the best vector H bar, which explains the observation observed vector Y bar corresponding to the transmitted pilot symbols in the matrix X. So, therefore, now one can formulate for estimation of H. So, to estimate H, formulate the least squares estimation problem, that is minimize Y bar minus minimize Y bar minus H X bar norm square. So, this is the least square form, and we know the solution now. So, this is your L S problem for channel estimation. In fact, for multi antenna channel estimation, the solution H hat is given by the least square solution X transpose X inverse X transpose Y bar, this is also known as the least squares channel estimate. This is very interesting, this is equal to your L S, the least squares channel estimate. So, this is known as the least squares channel estimate, X is your pilot matrix. So, we have the observation vector Y bar, and the corresponding pilot vectors. The other thing that we are assuming is that, this is an over determined system, which means the number of pilot symbols M is greater than or equal to the number of transmitted. Remember, this works only if the system is over determined. So, when we are doing this, implicitly we are assuming that, M number of pilot symbols is greater than or equal to, typically M is much larger than. So, this implies, this is an over determined system. So, you can also say that, for channel estimation using the least squares technique, the number of pilots, minimum number of pilot symbols required is at least that of the number of transmit antennas. So, this is another interesting result. So, this implies minimum number of pilot symbols required, this is another interesting result that is, you need at least to transmit pilot number of pilot symbols L that is, M has to be greater than equal to L for estimation of this multi antennas. So, this is an interesting application. In fact, one of the most, I would say one of the most popular applications and in fact, one of the most practically viable, practically prevalent applications of the least squares solution, especially in the context of signal processing for a practical wireless communication system. So, we will stop here and look at other applications in the subsequent modules. Thank you.