Hello. In this video, we will learn the objectives of beamforming. We will focus on mathematically formulating the downlink and uplink objective. Subsequently, we will discuss a few of selected algorithms for obtaining the beamforming matrices in terms of analog beamforming and digital beamforming. Lastly, we will summarize this video. In general, the hybrid beamforming design aims to maximize the communication rate by eliminating the interference and reducing the power consumption. It also aims to maximize the sum rate which is equivalent to maximizing the signal to noise ratio at the receiver. The figure here shows, the downlink system model where H_1 and H_2 represent the channel between the base station and the mobile users, y_1 and y_2 represent the receive signal at User 1 and User 2, r_1 and r_2 represent the combined signal at User 1 and User 2. As we aim to maximize the overall sum rate, which is the sum of the individual rates of all the users, where rate of k-th at user is expressed as log Base 2 of mod I plus signal to interference plus noise ratio of the k-th user, where SINR_k is expressed as this. Here, H_k represents the channel between the base station and the user, k, F represents the equivalent beamforming matrix of the base station, W_k represents the equivalent combining matrix of the k-th user, and C_k represents the interference plus the noise at k-th user. Here, I would like to point out that this S is a composite signal of both the users and, therefore, at each user, the signals coming from other users will act as an interference. Similarly, the uplink system model is shown here, where H_1 is the channel between User 1 and the base station, and H_2 is the channel between User 2 and the base station. The signal received at the front end of the base station is represented by y, and the combined signal is represented by r. As we aim to maximize the overall sum rate at the base station, which is expressed as the log base 2 of I plus signal to interference plus noise ratio at the base station, which is a sum of individual SINR of all the users, where SINR of k-th user is expressed as this. This expression is written after assuming that you only receive combining and there is no transmit beamforming. That is F, which is the equivalent beamforming matrix, is considered to be an identity matrix, and W_k represents the equivalent beam combining matrix for k-th user. The objective of the uplink and the downlink system models can be achieved if a suitable beamforming matrix is used. Therefore, now, we will learn sample algorithms for designing the hybrid beamforming matrices, if, specifically, the analog part and the digital part separately. We will begin with an algorithm for the analog part at the receiver, namely, the phase minimization algorithm. In the algorithm, we initialize the phase values, as 2 times of q minus 1 times Pi divided by N capital Q, where N_Q is the total number of phase values or the phase shifters. For example, if you considered the value of N_Q as four, the phase values will be 2 times of 1 minus 1 times Pi upon 4, which is 0, the second phase value would be 2 times of 2 minus 1 times Pi divided by 4, which will be Pi by 2, the third phase values will be, 2 times of 3 minus 1 times Pi divided by 4, which is Pi, and the fourth phase value would be 2 times of 4 minus 1 times Pi divided by 4, which is 3 Pi by 2. So, 0, Pi by 2, Pi, and 3 Pi by 2, will be the four phase values if we consider N_Q as 4. After initializing the phase values, we set a particular user, u, and set the switching matrix at 0. Here, switching matrix represents the connections of switches, 0 represents an open connection, and 1 represents a connected connection. Then we set n as the antenna number and determine the index of the phase values that minimizes the difference with the phase of the channel coefficient of u-th user at n-th antenna. We denote this index at q star, and then in the switching matrix, we make this q-th entry at 1. Which means that the signal of u-th user received at n-th antenna will be combined through the phase value obtained using q star. After determining the switching matrix, the analog beamforming matrix at the receiver or the analog beam combining matrix can be obtained as S_u times p, where p is an N_Q times 1 vector, which is obtained after taking the exponent of the phase values. The same algorithm can be used for the transmitter as well. Now, we will discuss an SVD-based algorithm for finding the digital beamforming or the digital beam combining matrices. As we know that the MIMO system model is expressed as y equals to Hx plus n, where y is the receive signal vector, H is the channel matrix, x is the transmit signal vector, and n is the noise vector. If you consider a singular value decomposition of the channel matrix, which is U times Sigma times V Hermitian. The above system model can be expressed as y equals to u times Sigma V Hermitian times x plus n, where U and V are the unitary matrices, which has the property UU Hermitian, is identity or U Hermitian U is also an identity, where Sigma is a diagonal matrix, where all diagonal terms are non-negative. Multiplying both sides by U Hermitian, we can get the following relation to represent U Hermitian y as y tilde owing to the unitary matrices property, U Hermitian U becomes identity, U Hermitian n we represent as n tilde. If you assume x is V times of x tilde, the above system model can be expressed as y tilde equals to Sigma times x tilde plus n tilde. Here, V is referred as digital beamforming matrix at the transmitter, and U is referred as the digital beam combining matrix at the receiver. In this video, we have mathematically formulated the objective for uplink and downlink system, especially when hybrid beamforming is utilized. We have also learned a sample algorithm to design the analog and the digital part of hybrid beamforming. Thank you.