Channel Estimation using Hybrid Beamforming

For channel estimation in hybrid architecture:

Here I will discuss communication between a BS (base station) and vehicles for mm Wave communication channel. Both N_BS and N_v have linear arrays (ULAs), with half-wavelength spaced antenna elements in a compact area uniformly. Let assume, f_n is the beamforming vector at BS side. BS need to transmit a symbol, s, such that |s|² = 1. On the other hand, vehicle assigns a unit-norm measurement vector w_m. Now, the signal received at MS (user) side is written by

…………………………………………………..(vii)

Where, H = N_v X N_BS matrix

σ²= variance of white Gaussian noise

…………………………………………………………….(viii)

where α = channel gain

a_v(θ)= vehicle’s ULA array response for line of sight AOA (θ)

a_BS(φ)= For Line of sight AOD (φ), BS’s ULA steering vector response

represented as

……………………………(ix)

(Garcia, 2016)

Equation (ix) showing phase difference in succeeding antenna elements of MIMO at vehicle and BS side, respectively. .

…………………………………………….(x)

Where s_p denotes the transmitted symbol on the beamforming vector f_p, and E[s_ps_p^H]=P denotes the average power used for transmission in a hybrid design for the beamforming/measurement vectors f_p and w_q, respectively. If the MS vectors w_q,q=1,2,…,M_MS make those measurements at M_MS intervals to determine the signal transmitted over the beamforming vector f_p, the resultant vector is

……………………………(xi)

Here W=[w₁, w₂, …, w_MS] represent measurement matrix (size of N_MS X M_MS).

The resultant matrix can be written by concatenating the MBS processed vectors f_p,p=1,2,…,MBS, at MBS successive time slots, and the MS uses the same measurement matrix W to combine the obtained signal. Y_p, p=1,2,…,M_BS

……………………………………………………………….(xii)

(Alkhateeb, 2014)

Where, beamforming matrix (size of N_BS X M_BS), F=[f₁, f₂,….,fM_BS] is used by the BS, and Q is noise matrix (size of M_MS X M_BS ) given by concatenating the M_BS noise vectors.